Handbook of Mechanics of Materials 9789811068836, 9789811068843

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Chun-Hway Hsueh Editor-in-Chief Siegfried Schmauder · Chuin-Shan Chen Krishan K. Chawla · Nikhilesh Chawla Weiqiu Chen · Yutaka Kagawa  Editors

Handbook of Mechanics of Materials

Handbook of Mechanics of Materials

Chun-Hway Hsueh Editor-in-Chief

Siegfried Schmauder • Chuin-Shan Chen Krishan K. Chawla • Nikhilesh Chawla Weiqiu Chen • Yutaka Kagawa Editors

Handbook of Mechanics of Materials With 1369 Figures and 129 Tables

Editor-in-Chief Chun-Hway Hsueh Department of Materials Science and Engineering National Taiwan University Taipei, Taiwan Editors Siegfried Schmauder Institute for Materials Testing, Materials Science and Strength of Materials University of Stuttgart Stuttgart, Germany

Chuin-Shan Chen Department of Civil Engineering National Taiwan University Taipei, Taiwan

Krishan K. Chawla UAB Mail BEC 254 Birmingham, AL, USA

Nikhilesh Chawla Arizona State University Fulton School of Engineering Tempe, AZ, USA

Weiqiu Chen Zhejiang University Engineering Mechanics Hangzhou, China

Yutaka Kagawa Tokyo University of Technology Katayanagi Advanced Research Laboratories Tokyo, Japan

ISBN 978-981-10-6883-6 ISBN 978-981-10-6884-3 (eBook) ISBN 978-981-10-6885-0 (print and electronic bundle) https://doi.org/10.1007/978-981-10-6884-3 Library of Congress Control Number: 2018964085 # Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Advances in micromanufacturing and nanotechnology have offered unique opportunities of tailoring the mechanical, electronic, magnetic, optical, and chemical properties of materials from the molecular or the atomic scale. Miniaturization of structural components and functional devices has led to the development of new technologies, such as microelectromechanical systems, micro-engines, smart structures, lab-on-a-chip, spintronics devices, and biomedical sensing devices. However, the abovementioned systems and devices have one issue in common. In order to perform their functionalities, they must maintain their structural integrity and reliability during their service life. Thus, the mechanical properties are major concerns for design engineers and device manufacturers. Characterization of the mechanical properties requires both the tools and the theory to obtain and analyze information over multiple length and time scales. Springer’s Major Reference Work Handbook of Mechanics of Materials provides a comprehensive reference for the studies of mechanical properties of materials over multiple length and time scales. To undertake this monumental work, Professor Hsueh was invited by Springer as the Editor-in-Chief to organize the editorial team. Six renowned leading researchers on mechanics of materials were invited to join the editorial team to solicit handbook chapters from internationally recognized contributors to provide in-depth analysis, overviews, and authoritative guidance on the basic science as well as cutting-edge technologies across a broad spectrum of mechanics of materials including nanomechanics, micromechanics, macromechanics, and measurements/applications. The content of each chapter was reviewed by the editorial team to ensure the high quality of the handbook and a coherent approach throughout the completion of the work. This published handbook consists of 71 state-of-the-art review chapters written by more than 178 worldleading experts from more than 18 countries. With more than 4155 references and over 1,400 figures and tables, this handbook provides a valuable reference resource for everyone who is involved in mechanics of materials. To track the current advances in mechanics of materials, the online edition of this handbook will be updated constantly at https://doi.org/10.1007/978-981-10-6855-3, and your comments concerning the publication of this handbook are greatly appreciated. This handbook is intended for audiences of students, faculty members, researchers, and professionals in the fields of Materials Science, Mechanical v

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Preface

Engineering, Civil Engineering, Engineering Mechanics, Aerospace Engineering, Biomedical Engineering, Electronics, Physics, etc. This handbook covers all types of materials that the researchers and engineers may encounter including metals, ceramics, polymers, composites, biomaterials, shape-memory alloys, metallic glasses, nanomaterials, etc. This 2431 page handbook is the end product of the superb cooperation of many distinguished colleagues who devoted their expertise, valuable time, and efforts to write remarkable state-of-the-art review chapters. We greatly appreciate the tremendous efforts of all contributing authors. National Taiwan University, Taiwan

University of Stuttgart, Germany National Taiwan University, Taiwan University of Alabama at Birmingham, USA Arizona State University, USA Zhejiang University, China Tokyo University of Technology, Japan January 2019

Chun-Hway Hsueh Editor-in-Chief Siegfried Schmauder Chuin-Shan Chen Krishan K. Chawla Nikhilesh Chawla Weiqiu Chen Yutaka Kagawa Editors

Contents

Volume 1 Part I

Nanomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Dislocation Nucleation Mediated Plasticity of FCC Nanowires . . . Seunghwa Ryu, Jaemin Kim, and Sangryun Lee

3

2

Indentation Behavior of Metallic Glass Via Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chun-Yi Wu and Yun-Che Wang

3

Surface/Interface Stress and Thin Film Stress . . . . . . . . . . . . . . . . Chun-Wei Pao

4

Characterizing Mechanical Properties of Polymeric Material: A Bottom-Up Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lik-ho Tam and Denvid Lau

5

Fracture Nanomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yabin Yan, Takashi Sumigawa, Licheng Guo, and Takayuki Kitamura

6

In Situ Transmission Electron Microscopy Investigation of Dislocation Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Josh Kacher, Ben P. Eftink, and Ian M. Robertson

19 33

57 93

131

7

Multiscale Modeling of Radiation Hardening . . . . . . . . . . . . . . . . . Ghiath Monnet and Ludovic Vincent

167

8

Atomistic Simulations of Metal–Al2O3 Interfaces . . . . . . . . . . . . . . Stephen Hocker, Alexander Bakulin, Hansjörg Lipp, Siegfried Schmauder, and Svetlana Kulkova

199

9

Multiscale Simulation of Precipitation in Copper-Alloyed Pipeline Steels and in Cu-Ni-Si Alloys . . . . . . . . . . . . . . . . . . . . . . Dennis Rapp, Seyedsaeid Sajadi, David Molnar, Peter Binkele, Ulrich Weber, Stephen Hocker, Alejandro Mora, Joerg Seeger, and Siegfried Schmauder

241

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10

Contents

Atomistic Simulations of Hydrogen Effects on Lattice Defects in Alpha Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shinya Taketomi and Ryosuke Matsumoto

283

11

Molecular Dynamics Simulations of Nanopolycrystals . . . . . . . . . . Christian Brandl

12

Modeling Dislocation in Binary Magnesium-Based Alloys Using Atomistic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sébastien Groh and Mohammad K. Nahhas

331

Atomistic Simulation Techniques to Model Hydrogen Segregation and Hydrogen Embrittlement in Metallic Materials . . . . . . . . . . . Douglas E. Spearot, Rémi Dingreville, and Christopher J. O’Brien

357

13

301

14

Modeling and Simulation of Bio-inspired Nanoarmors . . . . . . . . . Stefano Signetti and Nicola M. Pugno

391

15

Thermal Vibration of Carbon Nanostructures . . . . . . . . . . . . . . . . Lifeng Wang, Haiyan Hu, and Rumeng Liu

421

16

Mechanics of Carbon Nanotubes and Their Composites . . . . . . . . Jian Wu, Chenxi Zhang, Jizhou Song, and Keh-Chih Hwang

483

17

Flexoelectric Effect at the Nanoscale . . . . . . . . . . . . . . . . . . . . . . . Lele L. Ma, Weijin J. Chen, and Yue Zheng

549

18

Mechanical Properties of Nanostructured Metals: Molecular Dynamics Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Haofei Zhou and Shaoxing Qu

591

Processes in Nano-Length-Scale Copper Crystal Under Dynamic Loads: A Molecular Dynamics Study . . . . . . . . . . . . . . . I. F. Golovnev and E. I. Golovneva

615

Understanding Fracture and Fatigue at the Chemical Bond Scale: Potential of Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . Philippe Colomban

655

19

20

21

Atomistic Modeling of Radiation Damage in Metallic Alloys . . . . . Charlotte S. Becquart, Andrée De Backer, and Christophe Domain

673

22

Monte Carlo Simulations of Precipitation Under Irradiation Charlotte S. Becquart and Frédéric Soisson

....

703

23

Mechanics of Auxetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyeonho Cho, Dongsik Seo, and Do-Nyun Kim

733

24

Nanoindentation and Indentation Size Effects: Continuum Model and Atomistic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . Chi-Hua Yu, Kuan-Po Lin, and Chuin-Shan Chen

759

Contents

25

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Continuum Theory for Deformable Interfaces/Surfaces with Multi-field Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Wu and W. Q. Chen

795

Volume 2 Part II 26

27

Micromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Interaction Between Stress and Diffusion in Lithium-Ion Batteries: Analysis of Diffusion-Induced Buckling of Nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Q. Yang, Yan Li, B. L. Zheng, and K. Zhang Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials: Magnesium Single Crystal, Polycrystalline Magnesium, and Magnesium Alloy . . . . . . . . . . . . . . . . . . . . . . . . . Qizhen Li

823

825

845

28

Micropillar Mechanics of Sn-Based Intermetallic Compounds . . . J. J. Yu, J. Y. Wu, L. J. Yu, and C. R. Kao

873

29

Micro-mechanics in Electrochemical Systems Giovanna Bucci and W. Craig Carter

................

901

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics-Based Approach . . . . . . . . . . . . . . . . . . . . . . . . . Jacques Lamon

955

Micromechanics of Polymeric Materials in Aggressive Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaohong Chen

987

31

32

Crack Paths in Graded and Layered Structures . . . . . . . . . . . . . . 1013 Ivar Reimanis

33

Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 Chi-Hua Yu, Chang-Wei Huang, Chuin-Shan Chen, and Chun-Hway Hsueh

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 Chun-Hway Hsueh

35

Micromechanics of Dual-Phase Steels: Deformation, Damage, and Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127 Behnam Anbarlooie, Javad Kadkhodapour, Hossein Hosseini Toudeshky, and Siegfried Schmauder

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36

Defect Accumulation in Nanoporous Wear-Resistant Coatings Under Collective Recrystallization: Simulation by Hybrid Cellular Automaton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157 Dmitry D. Moiseenko, Pavel V. Maksimov, Sergey V. Panin, Dmitriy S. Babich, and Victor E. Panin

37

Multiscale Fatigue Crack Growth Modeling for Welded Stiffened Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1191 Ž. Božić, Siegfried Schmauder, M. Mlikota, and M. Hummel

38

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213 Tetsuya Ohashi

39

Competing Grain Boundary and Interior Deformation Mechanisms with Varying Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239 Wei Zhang, Yanfei Gao, and Tai-Gang Nieh

40

Multiscale Translation-Rotation Plastic Flow in Polycrystals Victor E. Panin, Valerii E. Egorushkin, Tamara F. Elsukova, Natalya S. Surikova, Yurii I. Pochivalov, and Alexey V. Panin

41

Micromechanics of Hierarchical Materials: Modeling and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293 Leon Mishnaevsky Jr.

42

Modelling the Behavior of Complex Media by Jointly Using Discrete and Continuum Approaches . . . . . . . . . . . . . . . . . . . . . . . 1311 Sergey G. Psakhie, Alexey Yu. Smolin, Evgeny V. Shilko, and Andrey V. Dimaki

43

Spectral Solvers for Crystal Plasticity and Multi-physics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347 Pratheek Shanthraj, Martin Diehl, Philip Eisenlohr, Franz Roters, and Dierk Raabe

44

Interface Delamination Analysis of Dissimilar Materials: Application to Thermal Barrier Coatings . . . . . . . . . . . . . . . . . . . . 1373 Yutaka Kagawa, Makoto Tanaka, and Makoto Hasegawa

. . . . 1255

Volume 3 Part III 45

Macromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1413

Smoothed Particle Hydrodynamics for Ductile Solid Continua . . . 1415 Peter Eberhard and Fabian Spreng

Contents

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46

Simulation of Crack Propagation Under Mixed-Mode Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465 Martin Bäker, Stefanie Reese, and Vadim V. Silberschmidt

47

Relaxation Element Method in Mechanics of Deformable Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503 Ye. Ye. Deryugin, G. V. Lasko, and Siegfried Schmauder

48

Damping Characteristics of Shape Memory Alloys on Their Inherent and Intrinsic Internal Friction . . . . . . . . . . . . . . . . . . . . . 1565 Shih-Hang Chang and Shyi-Kaan Wu

49

Application of Homogenization of Material Properties . . . . . . . . . 1595 Ming Dong and Siegfried Schmauder

50

Fatigue Behavior of 9–12% Cr Ferritic-Martensitic Steel . . . . . . . 1629 Zhen Zhang, Zhengfei Hu, and Siegfried Schmauder

51

Coupling of Discrete and Continuum Approaches in Modeling the Behavior of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675 Alexey Yu. Smolin, Igor Yu. Smolin, Evgeny V. Shilko, Yuri P. Stefanov, and Sergey G. Psakhie

52

Numerical Simulation of Material Separation Using Cohesive Zone Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715 Ingo Scheider

53

Current Applications of Finite Element Methods in Dentistry . . . . 1757 Noriyuki Wakabayashi, Natsuko Murakami, and Atsushi Takaichi

54

Elastic-Plastic and Quasi-Brittle Fracture . . . . . . . . . . . . . . . . . . . 1785 Xiaozhi Hu and Li Liang

55

Coupling Models of New Material Synthesis in Modern Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817 Anna Knyazeva, Olga Kryukova, Svetlana Sorokova, and Sergey Shanin

56

Simulation of Fracture Behavior of Weldments . . . . . . . . . . . . . . . 1859 Haoyun Tu, Siegfried Schmauder, and Yan Li

Part IV

Measurement and Applications

.....................

1877

57

Very High Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1879 Martina Zimmermann

58

High Temperature Mechanical Testing of Metals . . . . . . . . . . . . . . 1917 Birgit Skrotzki, Jürgen Olbricht, and Hans-Joachim Kühn

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Contents

59

Microelectromechanical Systems (MEMS)-Based Testing of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955 Jagannathan Rajagopalan

60

Nanoindentation for Testing Material Properties . . . . . . . . . . . . . . 1981 Yu-Lin Shen

61

3D/4D X-Ray Microtomography: Probing the Mechanical Behavior of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2013 Sudhanshu S. Singh and Nikhilesh Chawla

62

Mechanical Testing of Single Fibers . . . . . . . . . . . . . . . . . . . . . . . . 2035 Krishan K. Chawla

63

Stress Measurement in Thin Films Using Wafer Curvature: Principles and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2051 Eric Chason

64

Testing of Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2083 Nikhil Gupta, Steven Eric Zeltmann, Dung D. Luong, and Mrityunjay Doddamani

65

Crack-Dislocation Interactions Ahead of a Crack Tip . . . . . . . . . . 2123 R. Goswami and C. S. Pande

66

In-Situ Nanomechanical Testing in Electron Microscopes . . . . . . . 2143 Shou-Yi Chang

67

Deformation Measurement for Multiscale and Multifield Problems Using the Digital Image Correlation Method . . . . . . . . . 2189 Chien-Ching Ma and Ching-Yuan Chang

68

High Temperature Nanomechanical Testing . . . . . . . . . . . . . . . . . . 2219 Miguel A. Monclús and Jon M. Molina-Aldareguia

69

The Sliding Wear Response of High-Performance Cermets . . . . . . 2249 Kevin P. Plucknett, C. Jin, C. C. Onuoha, T. L. Stewart, and Z. Memarrashidi

70

Electromechanical Coupling of Botanic Cells: Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2291 C. C. Chen and W. P. Shih

71

Additive Manufacturing of Multidirectional Preforms and Composites: Microstructural Design, Fabrication, and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2353 Zhenzhen Quan and Tsu-Wei Chou

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2407

About the Editor-in-Chief

Professor Chun-Hway Hsueh received his Ph.D. degree from the Department of Materials Science and Engineering at the University of California, Berkeley, in 1981. Before joining National Taiwan University in 2010 as a Distinguished Professor, he was a Distinguished R&D Staff at Oak Ridge National Laboratory. Professor Hsueh’s formal training is analytical modeling. He has developed analytical models and derived closed-form solutions for many complex problems to identify key parameters in controlling properties/performance of materials and to provide guidelines in the material design. His work has been extensively cited, and he was listed as ISI highly cited researcher in Materials Science in 2002. Since joining National Taiwan University, his work has been extended to applied research. His current research work includes metallic glasses, shape-memory alloys, high-entropy alloys, nanoindentation, surface-enhanced Raman scattering, plasmonics nanodevices, etc. He has authored or co-authored more than 260 scientific journal papers. Professor Hsueh is a Fellow of the American Society for Metals (ASM), the American Ceramic Society (ACerS), and the World Innovation Foundation (WIF). Currently, he serves as Associate Editor of seven international journals.

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About the Editors

Professor Dr. rer. nat. Dr. h. c. Siegfried Schmauder received the degree Dipl.-Math. (Information Technology) from the Mathematics Department, University of Stuttgart, in 1981, and in 1988, he received the Doctor of Science degree in Materials Science at the same university. Since 1994 he is a Professor of Strength of Materials and Materials Sciences at the University of Stuttgart. During 2011–2014 he was the Acting Leader of the Institute for Materials Testing, Materials Science and Strength of Materials (IMWF) and during 2011–2013 of the Institute of Thermal Turbomachinery and Machinery Laboratory (ITSM). His research interests comprise computational methods in applied sciences and engineering; finite elements and analytical methods; materials modeling on nano-, meso-, micro-, and macro-mechanical scales; experiments to investigate microstructure/property relationships in heterogeneous materials; and multiscale materials modeling for different materials. Professor Schmauder started his career as Scientific Assistant at Max Planck Institute for Metals Research in Stuttgart in 1982. From 1988 to 1989, he was Representative Leader of the Electron Microscopy Group at Max Planck Institute for Metals Research in Stuttgart. From 1991 to 1994, he was Leader of the group Structural Mechanics at Max Planck Institute for Metals Research. During the period 2004–2008, he was Elected Member of the Council of the Collaborative Research Center 381 “Characterization of Damage in FiberReinforced Composites Using Non-destructive Methods” and during the period 2008–2018 of the Collaborative Research Center 761 “Dynamic Simulations of Systems with Large Numbers of Particles.” xv

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About the Editors

Professor Schmauder was a Guest Professor at the University of Tokyo, Japan, in 1996; a DAAD Guest Professor at the Academy of Sciences in Tomsk, Russia, in 1997; and a Guest Professor at the Science University of Tokyo in 1999 and at the University of Kyoto, Japan, in 2004. He was awarded the JSPS Research Fellowship, University of Tokyo, Japan (1989–1990), and received the Medal of the University of Tokyo in 1989. He obtained a DFG Research Fellowship from the University of California, USA (1990–1991). Professorships were offered to him by the German University in Cairo (GUC) in 2014; by the University of Bochum, Germany, in 2007; and by the University of Rostock, Germany, in 2005. Professor Schmauder was Co-Editor of the International Journal of Computational Materials Science and Surface Engineering from 2003 to 2013, and he is Editorial Board Member of the Journal of Physical Mesomechanics since 1998. He has held distinguished community and university services as a member of numerous committees and was organizer and chairman of 28 international workshops on Computational Mechanics of Materials and chairman, organizer, or member of the organizing committee of more than 10 conferences. Professor Schmauder supervised more than 30 dissertations, and he is External Examiner of the universities Oxford; Lille; Ecole Polytechnique, Saclay; Université Paris 13 – Paris Nord; Karlsruhe; Dresden; Jena; Bochum; Hamburg; and Freiberg. He gave more than 180 invited oral presentations at conferences and workshops and has authored or co-authored over 500 papers, 4 book chapters, and 4 books. Professor Chuin-Shan (David) Chen is Professor in the Department of Civil Engineering and the Deputy Director for iNSIGHT Center (Center of Innovation and Synergy for Intelligent Home and Living Technology), National Taiwan University. He is also the Division Head of Information Technology, National Center for Research on Earthquake Engineering. He is currently the Fellow of the International Association for Computational Mechanics (IACM), a General Council Member of the IACM, an Executive Council

About the Editors

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Member of the Asian-Pacific Association for Computational Mechanics (APACM), a Secretary General and an Executive Council Member of the Association of Computational Mechanics Taiwan (ACMT), a Vice President and an Executive Council Member of International Chinese Association for Computational Mechanics (ICACM), and a Member of EASEC International Steering Committee. He was the Editor-inChief of the international journals Interaction and Multiscale Mechanics (IMM) and Multiscale and Multiphysics Mechanics (MMM) and is now the Editor-in-Chief of Coupled Systems Mechanics (CSM). Professor Chen’s research interests are associated with mechanics and physics of materials at the nanometer and micrometer scales. He has made significant contributions on the area of multiscale computational methods and their applications to nanomechanics, materials modeling, and biosensor simulation. Among his major achievements are the development of physical partition cluster summation and atom-continuum transition for the quasicontinuum method, closed-form real-space absorbing boundary condition for molecular dynamics, multiscale method for microcantilever biosensors, atomistic and continuum modeling of ceramics and piezomaterials, phasefield model for freeze casting and crack growth simulation, p-version finite element and its applications to fracture mechanics, novel contact detection algorithms for polyhedral blocks, discrete elements coupling with fluid dynamics, and subgrid-enriched immersed boundary method for solid-fluid interaction, among others. He has received numerous awards, including International Association for Computational Mechanics (IACM) Fellow Award, International Chinese Association for Computational Mechanics (ICACM) Computational Mechanics Award, Lifelong National Taiwan University Distinguished Teaching Award, National Science Council Distinguished Young Investigator Award, and Distinguished Young Scholar from Society of Theoretical and Applied Mechanics, among others.

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About the Editors

Professor Krishan K. Chawla obtained his B.S. from Banaras Hindu University and his M.S. and Ph.D. degrees from the University of Illinois at UrbanaChampaign. He has taught and/or done research at (in alphabetical order) Arizona State University, Tempe, AZ; Ecole Polytechnique Federale de Lausanne (Switzerland); Federal Institute for Materials Research and Testing (BAM), Berlin (Germany); German Aerospace Research Institute (DLR), Cologne (Germany); Instituto Militar de Engenharia (Brazil); Laval University (Canada); Los Alamos National Lab (USA); New Mexico Tech (USA); Northwestern University (USA); University of Alabama at Birmingham (USA); and University of Illinois at Urbana-Champaign (USA). He served as the Chairman of the Department of Materials and Mechanical Engineering, University of Alabama at Birmingham (UAB), during 1999–2001. Currently, he is a Professor Emeritus in the Department of Materials Science and Engineering at the UAB. He has published extensively in the areas of processing, microstructure, and mechanical behavior of materials, in general, and composite materials and fibers, in particular. His current research projects involve processing, microstructure, and properties of metalmatrix and ceramic-matrix composites. Among the areas of research interest are powder processing of composites, thermal management via composites, interfacial engineering in various composites, and highperformance fibers. Professor Chawla is author or co-author of over 200 papers. He is also the author/coauthor of eight textbooks that are used in classrooms all over the world. In addition, he has co-edited four books. Besides being a member of various professional societies, Professor Chawla is Editor of International Materials Reviews (published jointly by ASM International, USA, and the Institute of Materials, London) and a Member of the Editorial Board of Journal of Materials Science (Springer), Materials Characterization (Elsevier), and Metallography, Microstructure, and Analysis (published by Springer for ASM International) and Trans Tech Publications. During 1989–1990, Professor Chawla served as a Program Director for metals and ceramics in the

About the Editors

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Division of Materials Research, National Science Foundation, Washington, DC. Professor Chawla serves as a consultant to the industry, US national laboratories, and various US federal government agencies. In 1990, he was selected an ASM International-Indian Institute of Metals Lecturer. In 1992, he was the recipient of the Eshbach Society Distinguished Visiting Scholar Award from Northwestern University. During the period of June 1994 through June 1995, he held the US Department of Energy Faculty Fellowship at Oak Ridge National Laboratory. In 1996, he was given the Distinguished Researcher Award by the New Mexico Tech. In 1997, he was made a Fellow of ASM International. In 2000, he was awarded the Distinguished Alumnus Award by Banaras Hindu University. He received the President’s Award for Excellence in Teaching, University of Alabama at Birmingham, in 2006. He was awarded the Educator Award by The Minerals, Metals and Materials Society (TMS) in 2011. The citation of this award reads: “For seminal contributions and text books written for the education and training of students in mechanical behavior and composites.” Professor Chawla has organized a number of international conferences and workshops under the auspices of the National Science Foundation, United Engineering Foundation, Engineering Conferences International, and American Composites Manufacturers Association in different parts of the world. He has also given special courses on composite materials for the engineers and researchers from the industry. Nikhilesh Chawla is Director for the Center for 4D Materials Science and is the Fulton Professor of Materials Science and Engineering (MSE) at Arizona State University. He is also a Professor of Mechanical Engineering. Professor Chawla received his Ph.D. in Materials Science and Engineering from the University of Michigan in 1997. He served as Acting Chair of the MSE program at ASU in 2010. Prior to joining Arizona State University in 2000, he was a Postdoctoral Fellow jointly at Ford Motor Company and the University of Michigan and a Senior Development Engineer at Hoeganaes Corporation.

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About the Editors

Professor Chawla’s research interests encompass the deformation behavior of advanced materials at bulk and small length scales, including four-dimensional (4D) materials science, environmentally benign metallic alloys, composite materials, and nanolaminates. He has co-authored over 230 refereed journal publications (Web of Science h-index of 40, Google Scholar h-index of 50) and close to 470 presentations in these areas. He is the author of the textbook Metal Matrix Composites (co-authored with K.K. Chawla), published by Springer. The second edition of this book was published in 2013. Professor Chawla is a Fellow of ASM International and Past Member of The Minerals, Metals and Materials Society (TMS) Board of Directors. He is the recipient of the Acta Materialia Silver Medal for 2017 and New Mexico Tech Distinguished Alumnus Award for 2016. In addition, he was awarded the 2016 Structural Materials Division Distinguished Scientist/Engineering Award and the 2016 Functional Materials Division Distinguished Scientist/Engineering Award, both from TMS; 2013 Brimacombe Medalist Award from TMS; 2011 Distinguished Lectureship given by Tsinghua University, China; 2004 Bradley Stoughton Award for Young Teachers, given by ASM International; and 2006 TMS Young Leaders Tutorial Lecture. He has also won the National Science Foundation Early Career Development Award and the Office of Naval Research Young Investigator Award. Professor Chawla is Editor of Materials Science and Engineering A published by Elsevier (2016, impact factor of 3.1). He also serves on the Editorial Boards of Advanced Engineering Materials, Materials Characterization, and Materials Chemistry and Physics. He has served or is serving on several external advisory boards, including that of Naval Research Laboratory, the Advanced Photon Source at Argonne National Laboratory, and New Mexico Tech. His work has been featured on the show Modern Marvels on the History Channel, R&D News, Fox News, and the Arizona Republic. He serves on ASU President Michael Crow’s Academic Council, which provides input to the president on academic, structural, and strategic matters.

About the Editors

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Weiqiu Chen is a Professor in the Department of Engineering Mechanics at Zhejiang University, Hangzhou, China. He received his B.S. and Ph.D. degrees from Zhejiang University in 1990 and 1996, respectively. He worked as a Postdoctoral Research Associate at the University of Tokyo, Tokyo, Japan, during 1997–1999. He was promoted as an Associate Professor in 1999 and a Full Professor in 2000. Professor Chen has engaged himself in mechanics of smart materials/structures, vibration/waves in structures, and mechanics of soft materials/structures for over 20 years. In particular, he has made a systematic generalization of the potential theory method and derived a series of exact three-dimensional solutions to crack and contact problems of functional materials with multi-field couplings; he has initiated to use the state-space formulations and the layer-wise approximation jointly to analyze mechanical responses of functionally graded material structures in the three-dimensional setting; he has established the mathematical basis of the method of reverberation-ray matrix, which is generally superior to other matrix techniques for structural dynamics due to its unconditionally stable capability; he has coined a convenient method to establish surface theories from the three-dimensional ones which can easily incorporate all complex factors into the final formulations without the need to introduce any new physical concepts. He has co-authored over 350 peer-reviewed journal articles (with an h-index of 42 from WOS) and 3 English books on Three Dimensional Problems of Piezoelasticity (Nova, 2001), Elasticity of Transversely Isotropic Materials (Springer, 2006), and Static Green’s Functions in Anisotropic Media (Cambridge University Press, 2015), respectively. He is now serving as the Editorial Member (or Associate Editor-in-Chief) of 12 academic journals including Mechanics of Advanced Materials and Structures (Taylor & Francis), Composite Structures (Elsevier), Theoretical and Applied Mechanics Letters (Elsevier), Journal of Thermal Stresses (Taylor & Francis), and Applied Mathematics and Mechanics (English Edition) (Springer).

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About the Editors

Dr. Yutaka Kagawa is a Professor at Tokyo University of Technology and the Director of Katayanagi Advanced Research Institute and Director of the Center for Ceramic Matrix Composites (CCMC) at Tokyo University of Technology. He is also Emeritus Professor of the University of Tokyo and Emeritus Fellow of National Institute for Materials Science (NIMS). He received his M.S. and Dr. Eng. degrees from Waseda University, Tokyo, Japan. His current research interests include design and performance of advanced fiberreinforced ceramic composites, environmental barrier coatings, and fracture of solid materials. Emphasis is placed on understanding of mechanical behavior through experimental and theoretical approaches.

Contributors

Behnam Anbarlooie Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran Dmitriy S. Babich Tomsk State University, Tomsk, Russia Martin Bäker Institut für Werkstoffe, Technische Universität Braunschweig, Braunschweig, Germany Alexander Bakulin Institute of Strength Physics and Materials Science of SB RAS, Tomsk, Russia Tomsk State University, Tomsk, Russia Charlotte S. Becquart University Lille, CNRS, INRA, ENSCL, UMR 8207, UMET, Unité Matériaux et Transformations, Lille, France EM2VM, Joint laboratory Study and Modeling of the Microstructure for Ageing of Materials, Lille and Moret sur Loing, France Peter Binkele Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany Ž. Božić Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia Christian Brandl Institute for Applied Materials (IAM-WBM), Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany Giovanna Bucci Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA W. Craig Carter Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA Ching-Yuan Chang Department of Mechanical Engineering, National Taipei University of Technology, Taipei, Taiwan Shih-Hang Chang Department of Chemical and Materials Engineering, National I-Lan University, I-Lan, Taiwan xxiii

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Contributors

Shou-Yi Chang Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan Eric Chason School of Engineering, Brown University, Providence, RI, USA Krishan K. Chawla Department of Materials Science and Engineering, University of Alabama at Birmingham, Birmingham, AL, USA Nikhilesh Chawla Center for 4D Materials Science, Materials Science and Engineering, Arizona State University, Tempe, AZ, USA C. C. Chen Institute of Physics, Academia Sinica, Taipei, Taiwan Chuin-Shan Chen Department of Civil Engineering, National Taiwan University, Taipei, Taiwan W. Q. Chen Department of Engineering Mechanics, Zhejiang University, Hangzhou, P. R. China State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, P. R. China Weijin J. Chen State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou, China Micro&Nano Physics and Mechanics Research Laboratory, School of Physics, Sun Yat-sen University, Guangzhou, China Xiaohong Chen Aerostructures, UTC Aerospace Systems, Chula Vista, CA, USA Hyeonho Cho Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Republic of Korea Tsu-Wei Chou Department of Mechanical Engineering and Center for Composite Materials, University of Delaware, Newark, DE, USA Philippe Colomban De la Molécule aux Nano-Objets: Réactivité, Interactions Spectroscopies, MONARIS, Sorbonne Université, CNRS, Paris, France Andrée De Backer CCFE, Culham Science Centre, Abingdon, Oxon, UK Ye. Ye. Deryugin Institute of Strength Physics and Materials Science, SB RAS, Tomsk, Russia Martin Diehl Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany Andrey V. Dimaki Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia Rémi Dingreville Sandia National Laboratories, Albuquerque, NM, USA Mrityunjay Doddamani Department of Mechanical Engineering, National Institute of Technology Karanataka, Surathkal, Mangalore, Karanataka, India

Contributors

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Christophe Domain EDF-R&D, Département Matériaux et Mécanique des Composants, Les Renardières, Moret sur Loing, France EM2VM, Joint laboratory Study and Modeling of the Microstructure for Ageing of Materials, Lille and Moret sur Loing, France Ming Dong Eberspächer Exhaust Technology GmbH & Co. KG, Esslingen, Germany Peter Eberhard Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany Ben P. Eftink Los Alamos National Laboratory, MST-8 Materials Science in Radiation and Dynamic Extremes, Los Alamos, NM, USA Valerii E. Egorushkin Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia Philip Eisenlohr Chemical Engineering and Materials Science, Michigan State University, East Lansing, MI, USA Tamara F. Elsukova Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia Yanfei Gao Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA I. F. Golovnev Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, Russia E. I. Golovneva Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, Russia R. Goswami Materials Science and Technology Division, Naval Research Laboratory, Washington, DC, USA Sébastien Groh Department of Biomedical Engineering, University of Basel, Allschwil, Switzerland Licheng Guo Department of Astronautical Science and Mechanics, Harbin Institute of Technology, Harbin, China Nikhil Gupta Composite Materials and Mechanics Laboratory, Department of Mechanical and Aerospace Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA Makoto Hasegawa Division of Systems Research, Faculty of Engineering, Yokohama National University, Yokohama, Japan Stephen Hocker Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany

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Contributors

Hossein Hosseini Toudeshky Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran Chun-Hway Hsueh Department of Materials Science and Engineering, National Taiwan University, Taipei, Taiwan Haiyan Hu State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China Xiaozhi Hu School of Mechanical and Chemical Engineering, University of Western Australia, Perth, WA, Australia Zhengfei Hu School of Material Science and Engineering, Tongji University, Shanghai, China Chang-Wei Huang Department of Civil Engineering, Chung Yuan Christian University, Chung-Li, Taiwan M. Hummel Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany Keh-Chih Hwang AML, Department of Engineering Mechanics, Tsinghua University, Beijing, China Center for Mechanics and Materials, Tsinghua University, Beijing, China C. Jin Department of Mechanical Engineering, Dalhousie University, Halifax, NS, Canada Josh Kacher Georgia Institute of Technology, Materials Science and Engineering, Atlanta, GA, USA Javad Kadkhodapour Shahid Rajaee Teacher Training University, Tehran, Iran Yutaka Kagawa Katayanagi Advanced Research Laboratories, Tokyo University of Technology, Tokyo, Japan C. R. Kao Department of Materials Science and Engineering, National Taiwan University, Taipei City, Taiwan Do-Nyun Kim Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Republic of Korea Jaemin Kim Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea Takayuki Kitamura Department of Mechanical Engineering and Science, Kyoto University, Kyoto-daigaku-katsura, Nishikyo-ku, Kyoto, Japan Anna Knyazeva Tomsk Polytechnic University, Tomsk, Russia Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia

Contributors

xxvii

Olga Kryukova Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia Hans-Joachim Kühn Bundesanstalt für Materialforschung und -prüfung (BAM), Berlin, Germany Svetlana Kulkova Institute of Strength Physics and Materials Science of SB RAS, Tomsk, Russia Tomsk State University, Tomsk, Russia Jacques Lamon LMT (CNRS-ENS Cachan-University Paris Saclay), Cachan, France G. V. Lasko Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany Denvid Lau Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong, China Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA Sangryun Lee Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea Qizhen Li School of Mechanical and Materials Engineering, Washington State University, Pullman, WA, USA Yan Li School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, People’s Republic of China Li Liang School College of Resources and Civil Engineering, Northeastern University, Shenyang, People’s Republic of China Kuan-Po Lin Structural Engineering, University of California, San Diego, CA, USA Hansjörg Lipp Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany Rumeng Liu State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Jiangnan University, Wuxi, China Dung D. Luong Composite Materials and Mechanics Laboratory, Department of Mechanical and Aerospace Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA Chien-Ching Ma Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan

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Contributors

Lele L. Ma State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou, China Micro&Nano Physics and Mechanics Research Laboratory, School of Physics, Sun Yat-sen University, Guangzhou, China School of Engineering, Sun Yat-sen University, Guangzhou, China Pavel V. Maksimov Laboratory of Physical Mesomechanics and Non-destructive testing, Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, Russia Ryosuke Matsumoto Department of Mechanical Engineering and Science, Kyoto University, Kyoto, Japan Z. Memarrashidi Department of Mechanical Engineering, Dalhousie University, Halifax, NS, Canada Leon Mishnaevsky Jr. Department of Wind Energy, Technical University of Denmark, Roskilde, Denmark M. Mlikota Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany Dmitry D. Moiseenko Laboratory of Physical Mesomechanics and Non-destructive testing, Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, Russia Jon M. Molina-Aldareguia IMDEA Materials Institute, Getafe (Madrid), Spain David Molnar Institut für Materialprüfung, Werkstoffkunde und Festigkeitslehre (IMWF), University of Stuttgart, Stuttgart, Germany Miguel A. Monclús IMDEA Materials Institute, Getafe (Madrid), Spain Ghiath Monnet EDF – R&D, MMC, Paris, France Alejandro Mora Institut für Materialprüfung, Werkstoffkunde und Festigkeitslehre (IMWF), University of Stuttgart, Stuttgart, Germany Natsuko Murakami Removable Partial Prosthodontics, Graduate School of Medical and Dental Sciences, Tokyo Medical and Dental University (TMDU), Tokyo, Japan Mohammad K. Nahhas Department of Biomedical Engineering, University of Basel, Allschwil, Switzerland Tai-Gang Nieh Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA Christopher J. O’Brien Sandia National Laboratories, Albuquerque, NM, USA Tetsuya Ohashi Kitami Institute of Technology, Kitami, Hokkaido, Japan Jürgen Olbricht Bundesanstalt für Materialforschung und -prüfung (BAM), Berlin, Germany

Contributors

xxix

C. C. Onuoha Department of Mechanical Engineering, Dalhousie University, Halifax, NS, Canada C. S. Pande Materials Science and Technology Division, Naval Research Laboratory, Washington, DC, USA Alexey V. Panin Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia National Research Tomsk Polytechnic University, Tomsk, Russia Sergey V. Panin National Research Tomsk Polytechnic University, Tomsk, Russia Victor E. Panin Laboratory of Physical Mesomechanics and Non-destructive testing, Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, Russia National Research Tomsk Polytechnic University, Tomsk, Russia National Research Tomsk State University, Tomsk, Russia Chun-Wei Pao Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan Kevin P. Plucknett Department of Mechanical Engineering, Dalhousie University, Halifax, NS, Canada Yurii I. Pochivalov Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia Sergey G. Psakhie Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia Tomsk Polytechnic University, Tomsk, Russia Skolkovo Institute of Science and Technology, Skolkovo, Russia Nicola M. Pugno Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy School of Engineering and Materials Sciences, Queen Mary University of London, London, UK Ket-Lab, Edoardo Amaldi Foundation, Italian Space Agency, Rome, Italy Shaoxing Qu Department of Engineering Mechanics, Zhejiang University, Hangzhou, China Zhenzhen Quan College of Textiles, Donghua University, Shanghai, China Dierk Raabe Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany Sergey G. Psakhie: deceased.

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Contributors

Jagannathan Rajagopalan Mechanical and Aerospace Engineering, School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, USA Dennis Rapp Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany Stefanie Reese Institute of Applied Mechanics, RWTH Aachen University, Aachen, Germany Ivar Reimanis Colorado School of Mines, Golden, CO, USA Ian M. Robertson University of Wisconsin-Madison, Materials Science and Engineering, Madison, WI, USA Franz Roters Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany Seunghwa Ryu Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea Seyedsaeid Sajadi Institut für Materialprüfung, Werkstoffkunde Festigkeitslehre (IMWF), University of Stuttgart, Stuttgart, Germany

und

Ingo Scheider Institute of Materials Research, Materials Mechanics, HelmholtzZentrum Geesthacht, Geesthacht, Germany Siegfried Schmauder Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany Joerg Seeger Wieland-Werke AG, Ulm, Germany Dongsik Seo Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Republic of Korea Sergey Shanin Tomsk Polytechnic University, Tomsk, Russia Pratheek Shanthraj Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany The School of Materials, The University of Manchester, Manchester, UK Yu-Lin Shen Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM, USA W. P. Shih Tiny Machine and Mechanics Lab (Timmel), Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan Evgeny V. Shilko Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia Tomsk State University, Tomsk, Russia

Contributors

xxxi

Stefano Signetti Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy Currently at: Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea Vadim V. Silberschmidt Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, UK Sudhanshu S. Singh Department of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India Birgit Skrotzki Bundesanstalt für Materialforschung und -prüfung (BAM), Berlin, Germany Alexey Yu. Smolin Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia Tomsk State University, Tomsk, Russia Igor Yu. Smolin Tomsk State University, Tomsk, Russia Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia Frédéric Soisson DEN-Service de Recherches de Métallurgie Physique, CEA, Université Paris-Saclay, Gif-sur-Yvette, France Jizhou Song Department of Engineering Mechanics, Soft Matter Research Center and Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou, China Svetlana Sorokova Tomsk Polytechnic University, Tomsk, Russia Douglas E. Spearot University of Florida, Gainesville, FL, USA Fabian Spreng Materials Testing Institute University of Stuttgart (MPA Stuttgart, Otto-Graf-Institut (FMPA)), Stuttgart, Germany Yuri P. Stefanov Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia T. L. Stewart Department of Mechanical Engineering, Dalhousie University, Halifax, NS, Canada Takashi Sumigawa Department of Mechanical Engineering and Science, Kyoto University, Kyoto-daigaku-katsura, Nishikyo-ku, Kyoto, Japan Natalya S. Surikova Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia

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Contributors

Atsushi Takaichi Removable Partial Prosthodontics, Graduate School of Medical and Dental Sciences, Tokyo Medical and Dental University (TMDU), Tokyo, Japan Shinya Taketomi Department of Mechanical Engineering, Saga University, Saga, Japan Lik-ho Tam Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong, China Makoto Tanaka Materials Research and Development Laboratory, Japan Fine Ceramics Center, Nagoya, Japan Haoyun Tu School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, People’s Republic of China Ludovic Vincent CEA, DEN, SRMA, Paris, France Noriyuki Wakabayashi Removable Partial Prosthodontics, Graduate School of Medical and Dental Sciences, Tokyo Medical and Dental University (TMDU), Tokyo, Japan Lifeng Wang State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China Yun-Che Wang Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan Ulrich Weber Materials Testing Institute University of Stuttgart (MPA Stuttgart, Otto-Graf-Institut (FMPA)), Stuttgart, Germany B. Wu Department of Engineering Mechanics, Zhejiang University, Hangzhou, P. R. China Chun-Yi Wu Research and Development Department Yan Men LTD, Tainan, Taiwan J. Y. Wu Department of Materials Science and Engineering, National Taiwan University, Taipei City, Taiwan Jian Wu AML, Department of Engineering Mechanics, Tsinghua University, Beijing, China Center for Mechanics and Materials, Tsinghua University, Beijing, China Shyi-Kaan Wu Department of Materials Science and Engineering, National Taiwan University, Taipei, Taiwan Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan Yabin Yan Department of Mechanical Engineering and Science, Kyoto University, Kyoto-daigaku-katsura, Nishikyo-ku, Kyoto, Japan Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, China

Contributors

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F. Q. Yang Department of Chemical and Materials Engineering, University of Kentucky, Lexington, KY, USA Chi-Hua Yu Division of Research and Development, CoreTech Corp. (Moldex3D), Hsinchu, Taiwan J. J. Yu Department of Materials Science and Engineering, National Taiwan University, Taipei City, Taiwan Department of Materials Science and Engineering, University of California at Los Angeles, Los Angeles, CA, USA L. J. Yu Department of Materials Science and Engineering, National Taiwan University, Taipei City, Taiwan Steven Eric Zeltmann Composite Materials and Mechanics Laboratory, Department of Mechanical and Aerospace Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA Chenxi Zhang Department of Engineering Mechanics, Soft Matter Research Center and Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou, China K. Zhang Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Wei Zhang Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA Zhen Zhang School of Material Science and Engineering, Nanjing Institute of Technology, Nanjing, China B. L. Zheng School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China Yue Zheng State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou, China Micro&Nano Physics and Mechanics Research Laboratory, School of Physics, Sun Yat-sen University, Guangzhou, China School of Engineering, Sun Yat-sen University, Guangzhou, China Haofei Zhou School of Engineering, Brown University, Providence, RI, USA Martina Zimmermann Faculty of Mechanical Engineering, Materials Science Institute, Technical University of Dresden, Dresden, Sachsen, Germany

Part I Nanomechanics

1

Dislocation Nucleation Mediated Plasticity of FCC Nanowires Seunghwa Ryu, Jaemin Kim, and Sangryun Lee

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Slip Systems of FCC Crystalline Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Deformation of FCC Nanowires: Slip Versus Twin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Onset of Plasticity: Dislocation Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Deformation of Penta-Twinned Silver Nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exceptional Deformation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 7 9 12 14 16 16

Abstract

Nanowires have gained a significant attention as a key component applicable to various devices and advanced materials. Hence, it is essential to ensure the mechanical stability and reliability of nanowires for the realization of devices based on nanowires. In addition, nanowires provide a unique test bed for studying the fundamental mechanism associated with plastic deformations of materials. When the size of specimen reduces below a critical scale (sub-100 nm), the population of vacancies, dislocations, as well as the grain boundaries decreases significantly, which allows us to study a completely new deformation mechanism that cannot be observed in bulk scale. This chapter involves a brief review on the deformation mechanism of sub-100 nm crystalline nanowires with FCC (facecentered cubic) crystal structure, which is relatively well characterized via molecular dynamics and dislocation dynamics simulations as well as in-situ and ex-situ nanotension experiments.

S. Ryu (*) · J. Kim · S. Lee Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea e-mail: [email protected]; [email protected]; [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_1

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1

S. Ryu et al.

Introduction

Nanowires have gained a significant attention as a key component applicable to various devices and advanced materials. Semiconductors or oxide nanowires have been investigated for the use in solar cells, batteries, transistors, as well as catalyses [1, 2]. Metal nanowires have been used for an element of NEMS (nano-electromechanical-systems) because of their unique electric, magnetic, and optical properties [3]. Also, metal nanowires have been integrated as key components of stretchable and flexible electronics and sensors [4]. Hence, it is essential to ensure the mechanical stability and reliability of nanowires for the realization of devices based on nanowires. In addition, nanowires provide a unique test bed for studying the fundamental mechanism associated with plastic deformations of materials [5]. When the size of specimen reduces below a critical scale (sub-100 nm), the population of vacancies, dislocations, as well as the grain boundaries decreases significantly, which allows us to study a completely new deformation mechanism that cannot be observed in bulk scale. Also, important unit processes in plastic deformation, such as dislocation nucleation, interaction between dislocation and twin boundary, and competition between slip and twinning, can be distinctly investigated in nanowires due to their small volumes. This chapter involves a brief review on the deformation mechanism of sub100 nm crystalline nanowires with FCC (face-centered cubic) crystal structure, which is relatively well characterized via molecular dynamics and dislocation dynamics simulations as well as in-situ and ex-situ nanotension experiments. Since there are few defects in these FCC nanowires, the plastic deformation begins with dislocation nucleation from the surface in most cases. Depending on the stacking fault energy characteristics as well as Schmid factor, the deformation proceeds with one of the three mechanisms: full slip by perfect dislocation, partial slip by partial dislocation, and twinning by the successive partial slip along the adjacent slip planes. Unlike FCC nanowires, the understanding of deformation mechanics of BCC (body-centered cubic), HCP (hexagonal-closed-packed), and DC (diamond cubic) crystalline nanowires is insufficient due to high anisotropy in Peierls stress (BCC), complex slip systems (HCP and DC), and high surface reconstruction effect (DC), which requires a further study. The chapter is written as follows. In Sect. 2, we provide a brief summary of the slip systems of FCC crystalline metals and describe the choice of deformation mechanism based on Schmid factor and the generalized stacking fault energy (GSFE). In Sect. 3, we describe the deformation mechanism of FCC nanowires with various nanowires under tension, compression, and torsion, based on the extensive review of Weinberger and Cai [6]. Section 4 will be devoted to the prediction of the nanocrystal yield stress based on the dislocation nucleation rate obtained from atomistic simulations. Section 5 summarizes the study on the deformation mechanism of silver nanowires with penta-twin structure, which has received much attention in recent years. In Sect. 6, we discuss the deformation mechanisms that cannot be explained by Schmid factor or GSFE, such as surface-energy

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mediated reorientation and global deformation mechanism mediated by elastic instability. In Sect. 7, we will provide a summary and a brief discussion on the future research prospects.

2

Slip Systems of FCC Crystalline Metals

The concepts of slip system, Schmid factor, and generalized stacking fault energy are essential to understand the plastic deformation of FCC metals. In the FCC metals, a slip occurs in one of four independent {111} planes, the closed-packed-planes in FCC crystals, each of which has three equivalent h110i slip directions. Hence, there are 12 independent slip systems in total. Slip along h110i directions does not proceed immediately, but is divided into two partial slips following two different directions of h112i family, as depicted in Fig. 1. In order to understand the slip phenomena at atomic detail, it is very convenient to use the concept of generalized stacking fault (GSF) energy surface (or curve) suggested by Vitek [7]. GSF surface (Fig. 1a) can be obtained by tracing the change of energy, as we fix the atoms below a {111} slip plane and horizontally translate the rigid block of atoms above. The GSF curve in Fig. 1b is obtained from the data along specific directions in GSF surface. From the projection view along 011 direction (Fig. 1c), one can see that atomic layer of {111} planes are repeatedly stacked in . . .ABCABC.  . . manners. As shown in Fig. 1b, when the atoms above C layer moves by a6 211 to form the intrinsic stacking fault (where a is the lattice constant), they experience relatively small energy barrier because there is no atom at the corresponding positions in the bottom layer. However, very high energy barrier must be overcome to move the atoms further along the same direction because of the presence of atoms in the layer. After forming the intrinsic stacking fault  bottom  configuration, the slip by a6 112 become the preferred choice because   of low  energy  a a barrier along the path. By combining two sequential slips of 211 and 6 6 112 , a   slip of a2 101 is formed which corresponds to the Burgers vector of perfect dislocations. The line defect formed at the frontal boundary of the stacking fault is referred to as leading partial dislocation, while the line defect at the rear boundary is called trailing partial dislocation. When the slip plane is swept only by the leading partial dislocation, it forms stacking fault plane. The crystal recovers to the perfect lattice state after the trailing partial is swept. In the presence of the stacking fault, the stacking along h111i directions is disrupted to form . . .ABCA/CABC. . . order (Fig. 1d). If stacking faults tend to repeatedly form at the adjacent plane of another stacking fault, the stacking order becomes . . .ABCA/CBACBA. . . which has a mirror symmetry. Such deformation mechanism is referred to as twinning. Although the intrinsic stacking fault energy, i.e., the local minimum point from GSF curve, is the only data obtainable from experiments, we can obtain the entire GSF surface from first principle calculations based on well-established protocols. Instead of the first principle calculations requiring large computational power, atomistic modelling based on the empirical potential model shows a comparable

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Fig. 1 (a) Generalized stacking fault energy map for a typical FCC crystal. (b) The black solid line and the red dashed line represent the generalized stacking fault energy curves along the samecolored paths in the bottom of a. The blue dashed dotted line represents the stacking fault energy curve   along the path from d to e. (c) Schematics of perfect FCC crystals projected along [111] and 011 directions. (d) Schematic of stacking fault formation. (e) Schematic of twin formation

predictive power for the FCC crystalline materials with much lower computational costs. Hence, extensive data and information on the FCC crystalline slip systems have been accumulated for the last few decades as summarized in the literature [8] and references therein. Since the slip system of FCC crystals is well characterized via theoretical, experimental, and computational studies, the deformation mechanism of nanowire can be predicted by computing the effective stress along the slip direction in slip plane from the applied axial stress. The ratio between the effective stress and the applied stress is called the Schmid factor. As depicted in Fig. 2, when the angle between the loading axis and the normal vector of the slip plane is θ, the angle between the loading axis and slip direction is ϕ, and the applied tensile stress is σ, the

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Fig. 2 Schematic representing the definition of Schmid factor

effective stress acting on the slip system (or resolved shear stress) becomes τRSS = σ cos θ cos ϕ and the Schmid factor is defined as the ratio, S = cos θ cos ϕ. When the resolved shear stress is larger than the critical resolved shear stress (i.e., Peierls stress) of the corresponding slip system, the slip system is activated and the plastic deformation is initiated.

3

Deformation of FCC Nanowires: Slip Versus Twin

In the case of bulk FCC metals, even if the dislocations are divided into two Shockley partial dislocations, the width of the stacking fault between the two partials is very small compared to the size of the specimen scale. Hence, the deformation can be understood by only considering the Schmid factor of the Burgers vector of perfect dislocation, a2 h110i. However, the diameter of sub-100 nm nanowire is not significantly larger than the equilibrium width of stacking fault, we can observe a stacking fault formed across the entire nanowire cross section. Hence, in the case of FCC metal nanowires, three different deformation mechanisms have been observed in both experiments and molecular dynamics simulations: first, a full slip mechanism by perfect dislocation sweep in the direction of a2 h110i ; second, a partial slip mechanism by partial dislocation sweep, which is accompanied by the formation of stacking fault planes at multiple locations in the nanowire; and third, twinning which is occurred by the successive partial slip along the adjacent slip planes. Another feature of nanowire deformation is the asymmetric deformation behaviors under compressive and tensile loadings. If the perfect dislocation is only taken into account, the deformation mechanism should be identical because Schmid factor is identical for tensile and compressive loading for the single crystal specimen. However, in nanowires, we observe asymmetric phenomena such as twinning in compression and slip in tension. This section explains the origin of asymmetric deformation behavior based on Schmid factor and generalized stacking fault energy concepts.

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Table 1 summarizes the Schmid factors for three cases (leading partial, trailing partial, and perfect dislocation) for three different loading directions. Schmid factor for the perfect dislocation is the same for compression and tensile loadings. On the other hand, since the Schmid factors for the leading partial and trailing partial dislocations are different for compressive and tensile loadings, the difference should be taken into account when predicting the deformation mechanism of nanowires. When the Schmid factor for the leading partial dislocation is higher than that of the trailing partial, partial slip or twin deformation occurred. Otherwise, full slip is expected because once the leading partial is nucleated, the trailing partial will be followed immediately. With the Schmid factor, we can predict the occurrence of the full slip or partial slip/twinning. However, it is impossible to distinguish between the partial slip and the twinning because Schmid factors are identical for two mechanisms. To overcome this limitation, we need to analyze the generalized stacking fault energy curve in detail. Figure 1b presents the generalized stacking fault energy curve (dashed dotted blue line) for the case when another stacking fault is formed at the adjacent plane of a preexisting stacking fault (Fig. 1e). We can predict the preference between partial slip and twinning by comparing the energy barriers of forming a new stacking fault out of perfect lattice (EUSF of Fig. 1b) with the energy barrier of forming another stacking fault at the adjacent plane of an existing stacking fault (EUT  EISF of Fig. 1b). After a stacking fault is formed, if the former is lower, partial slip is preferred because it is energetically favorable to form a stacking fault at a completely new location apart from the existing stacking fault, while if the latter is lower, twinning is preferred because it is energetically favorable to form another stacking fault at the adjacent slip plane. In summary, when we define a dimensionless parameter τ as bellow, τ¼

EUSF EUT  EISF

Table 1 Schmid factors in FCC crystalline solid in terms of crystallographic orientation and loading direction. Deformation mechanism is predicted based on the mechanism with largest Schmid factor Orientation

Loading Tension Compression

Leading partial dislocation 0.236 0.471

Trailing partial dislocation 0.471 0.236

Perfect dislocation 0.408 0.408

Tension

0.471

0.236

0.408

Compression Tension Compression

0.236 0.314 0.157

0.471 0.157 0.314

0.408 0.272 0.272

Predicted mechanism Full slip Partial slip or twin Partial slip or twin Full slip Full slip Partial slip or twin

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partial slip is preferred if τ is lower than 1, while twinning is preferred when it is larger than 1. Weinberger and Cai [6] compared the observed deformation behaviors of nanowires in many molecular dynamics studies with the predicted mechanisms from the Schmid factor and generalized stacking fault energy. They found that the occurrence of the full slip can be predicted correctly, while the preference between the partial slip and twinning was relatively difficult to predict. The difficulty arises because another important factor, the number of nucleation sites for partial dislocation and twinning, was not taken into account. There are many nucleation sites for partial slip in the nanowire because a partial dislocation can be nucleated at any location along the nanowire, while the nucleation site for twinning is very limited because it can only occur at the adjacent planes of preexisting stacking faults. Hence, the preference prediction based on the GSF curve overestimate the rate of twinning compared to the rate of partial slip.

4

Onset of Plasticity: Dislocation Nucleation

Sub-100 nm scale nanocrystals are often free of dislocations because of high surfaceto-volume ratio and hence the image stress pulls the dislocations to annihilate at the surface. Hence, the onset of plasticity or the yield of the nanocrystal is initiated by the nucleation of dislocations, and predicting dislocation nucleation rate is essential to understand the mechanical deformations of nanocrystals. The fundamental quantity of interest is the dislocation nucleation rate I as a function of stress σ and temperature T. However, calculating the rate in experimental conditions is very difficult because the experimental time scale is beyond the reach of molecular dynamics simulations. An alternative method is to combine the reaction rate theories with atomistic models. Full details on the method can be found in the literature [9, 10], and we will provide a short summary in this section. Atomistic simulations are employed to compute the activation free energy barrier which is used as an input for the reaction rate theory to predict dislocation nucleation rate. There are several reaction rate theories which predict the nucleation rate as follows: 

Gc ðσ, T Þ I ðσ, T Þ ¼ N s v0 exp  kB T



where Ns is the number of equivalent nucleation sites, v0 is a frequency prefactor, Gc is the activation Gibbs free energy for dislocation nucleation, and kB is the Boltzmann constant. If strain γ is kept constant instead of stress, activation Helmholtz free energy Fc(γ, T ) is concerned instead of Gc(σ, T ). It was proven that Gc(σ, T ) = Fc(γ, T ) when σ = σ(γ) if the specimen size is infinitely larger than the critical dislocation loop size at the saddle point (i.e., in thermodynamic limit), so that the critical dislocation loop size does not depend on the choice of the constant

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stress or constant strain ensemble. However, the activation entropy depends on the choice of the ensemble even in the thermodynamic list, as follows: @Gc ðσ, T Þ @σ Sc ðσ, T Þ ¼ Sc ðγ, T Þ þ @σ T @T γ ðσ , T Þ With the definition of activation volume Ωc ¼ @Gc@σ , the difference in activaT tion entropy ΔSc = Sc(σ, T )  Sc(γ, T ) is given as Ωc @σ . Because Ωc is always @T γ

positive (corresponding to the volume of the dislocation loop at the saddle point @σ configuration) and @T is usually negative due to thermal softening, ΔSc is γ positive in crystal under most conditions. Due to large activation volume of dislocation loop involving many hundreds to thousands of atoms, it is very important to note that activation entropy difference ΔSc can be significantly large in dislocation nucleation. The difference in theories, such as transition state theory and Becker-Döring theory, lies in the expression for the frequency prefactor ν0. Highest estimate for the frequency prefactor is the Debye frequency of the crystal ~1013 s1, while it can be predicted to be 1 – 3 orders of magnitude lower when we consider the re-crossing near the energy barrier. Because the direct calculation of the activation free energy at finite temperature is challenging, the free energy barrier at zero temperature, i.e., activation enthalpy, Gc(σ, T = 0) = Hc(σ) has been computed by nudged elastic band (NEB) method or closely related string method [9]. Gc(σ, T ) is expected to decrease with T, as characterized by the activation entropy Sc through approximated relationship of Gc(σ, T ) = Hc(σ)  TS  c(σ). The activation entropy contributes the overall Sc multiplicative factor exp kB to the nucleation rate prediction, which can be significant (>2  105) when Sc exceeds 10kB. Having reviewed the basic thermodynamics of dislocation nucleation, we now turn our attention to the calculation of the activation barrier and activation entropy. Because the simulation cell in the atomistic simulations cannot be infinitely large, there should be some difference in the activation Gibbs free energy Gc and activation Helmholtz free energy Fc. If large enough stress (or strain) is applied, the dislocation loop size becomes much smaller than the simulation cell, and hence we can assume that Gc  Fc. Figure 3a depicts the schematic of predicting homogeneous dislocation rate. The simulation cell is subjected to a  nucleation  pure shear stress along 112 and the nucleated dislocation  lies on the (111) plane and has the Burgers vector of a Shockley partial, bp ¼ 112 =6. Figure 3b presents the shear stress-strain relationship of the perfect crystal at different temperatures prior to dislocation nucleation, which clearly shows the thermal softening effect. The activation free energy at zero temperature, i.e., Gc(σ, T = 0) = Hc(σ), can be obtained from the NEB or string method. The finite temperature free energy barrier is computed by umbrella sampling which is performed in Monte Carlo simulations using a bias potential that is a function of the number of atoms within the largest dislocation loop in the simulation cell. Figure 3c shows the critical dislocation loop sampled at T = 0 K and T = 300 K. Although the dislocation

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a

b

σxy [111]

3

100 K 200 K 300 K 400 K 500 K 600 K

σxy (GPa)

2

1.5

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0 0

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10–1410–910–4 101 106 1 1 05 10 0–4 1 10 –9

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10–14 10–9 10–4 101 10 6 1 1 10 –4 0 10 –9

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Compressive Nucleation Stress (GPa)

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100 200 300 400 T (K)

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Fig. 3 (a) Schematic of simulation cell designed for computing homogeneous dislocation nucleation. The spheres represent atoms enclosed by the critical nucleus of a Shockley partial dislocation loop. (b) Shear stress-strain curves of the Cu single crystal before dislocation nucleation at different temperatures. (c) Atomistic configurations of dislocation loops at 0 K (up) and 300 K (down). (d) Activation Gibbs free energy Gc for homogeneous nucleation as a function of shear stress σ at different T. (e) Gc as a function of T at σ = 2.0 GPa. (f) Contour lines of homogeneous dislocation nucleation rates per site I as a function of T and σ. The predictions with and without considering the activation entropy Sc(σ) are plotted in thick and thin lines, respectively. The nucleation rate of I~106 per site is accessible in typical MD timescales whereas the nucleation rate I~104 to 109 is accessible in typical experimental timescales, depending on the number of nucleation sites. (g) Schematic of simulation cell designed for computing homogeneous dislocation nucleation in Cu nanorod. (h) Contour lines of heterogeneous dislocation nucleation rates per site I as a function of T and σ in nanorod. (i) Nucleation stress of the nanorod (i.e., yield stress) under a constant compressive strain loading rate ϵ̇ ¼ 103 which is the experimentally accessible loading rate. The solid lines are the prediction based on the activation free energy computed by umbrella sampling. The dashed lines are the nucleation stress prediction by neglecting activation entropy.

loop at 0 K is symmetric, 300 K configuration is distorted because of thermal fluctuation. By collecting the activation free energy data at different temperatures and stresses, we obtain the activation Gibbs free energy curves Gc(σ, T ) shown in Fig. 3d. We note that curves at different temperatures are wide apart in Gc(σ, T ) which

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shows a strong temperature dependence, i.e., large activation entropy. For example, Fig. 3e plots Gc as a function of T at σ = 2.0 GPa, from which we can obtain an averaged activation entropy Sc ðσ Þ ¼ 48kB in the temperature range of 0 – 300 K.  This activation entropy contributes a multiplicative factor of exp Sc ðσ Þ=kB  1020 to the absolute nucleation rate. Hence, without correct computation of activation entropy, the dislocation nucleation rate prediction will be incorrectly estimated by many orders of magnitude. We present the homogeneous nucleation here for the sake of simplicity. The same method has been applied to study the heterogeneous nucleation in nanorod as shown in Fig. 3g and h. Based on the activation free energy and entropy data, we can discuss how the yield stress is predicted in a constant strain increasing rate condition which is often used in experiments. If the crystal contains no preexisting defects, then the yield strength is the stress at which the first dislocation nucleates. The following implicit equation is derived for the yield strength σ Y by considering a nanorod loaded at a constant strain rate ϵ̇ : Gc ðσ, T Þ kB TN s v0 ¼ ln ̇ kB T E ϵ Ωc ð σ Y , T Þ One can apply the same equation to the homogeneous nucleation by replacing E and ϵ̇ by G and γ ̇ . Although the equation is derived with the assumption of the linear stress-strain relationship with Young’s modulus E, we can predict the yield strength numerically based on nonlinear stress-strain curves obtained from simulations. At a compressive strain rate of ϵ̇ ¼ 103 s1 , the compressive yield stress of nanorod becomes 4.7 GPa at 0 K and 1.7 GPa at 300 K. Interestingly, the yield stresses of nanorod with and without activation entropy (Fig. 3i) are not different significantly as presented in Fig. 3i. This is because the activation volume of the heterogeneous dislocation nucleation in the nanorod is much smaller than that of homogeneous nucleation in a perfect crystal. Also, because the activation free energy Gc decreases very quickly with σ, small change in σ gives rise to a very large change in nucleation rate I. Nanorod with less sharper curvature radius (such as circular cross section) will have large activation volume. Still, we note that the activation entropy effect would contributes only a few hundred MPa correction in yield stress of nanowire under most conditions.

5

Deformation of Penta-Twinned Silver Nanowires

In this section, we will discuss the deformation mechanism of the penta-twinned silver nanowire which is currently in the limelight in many research areas. From a widely used solution synthesis process [11], silver nanowires with high aspect ratio can be synthesized which have the characteristic five twin boundaries {111} along the nanowire axis , as shown in Fig. 4a. Silver nanowires are widely applied to transparent electrodes and stretchable sensors that utilize the piezoresistivity effect of nanowire-elastomer composite. Because these devices rely on the mechanical

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Fig. 4 (a) The microstructure of silver nanowire with the unique five {111} twin boundaries. (b) Formation of dislocation and stacking fault under tension, which serves as a basic unit of deformation [12]. (c) Distributed plasticity under tension [12] with the scale bar (left: 120 nm, middle: 10 nm). (d) Disappearance of dislocation under compression [13]. (e) Atomic stress distribution in the presence of a confined partial dislocation loop within one of five grain [14]

reliability under the repeated bending, compression, and tension, numerous studies have been conducted on the deformation mechanism of silver nanowires which provide an electrically conductive path in these devices. The deformation mechanism of silver nanowires can be summarized by two major characteristics. First, even if there exist minor defects such as surface modulations, plastic deformation occurs at various locations of the nanowire without being concentrated solely on the defect, which is referred to as distributed plasticity (Fig. 4c) [12]. Second, it shows recoverable plasticity in which a significant portion of dislocations disappears in compression and tension cycle (Fig. 4d) [13]. These characteristics can be explained in relation to the unique microstructure which invokes uncommon deformation mechanisms. The deformation of the silver nanowire with no extra defect other than the pentatwin is initiated by the partial dislocation nucleation at the center of one side of pentagonal cross section (Fig. 4b). Further plastic deformation proceeds as the nucleated dislocation moves toward the twin boundaries and transmit through them by leaving sessile dislocations, as shown in Fig. 4b. According to the study

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combining HRTEM (high-resolution transmission electron microscopy) experiments and molecular dynamics simulations, although the dislocation is nucleated and plastic deformation is initiated at the point where the surface is undulated (i.e., at stress concentration sites), the local plastic deformation do not progress until the complete failure. Instead, plastic deformation progresses at other stress concentration positions, which results in the distributed plastic deformation behavior [12]. This is in contrast to the case of macroscopic specimens that experience the complete failure at the same position where the necking is initiated. It has been found that the stress field from existing stacking fault is focused at a considerably far from the nucleated dislocation (Fig. 4e) due to the unique penta-twin boundaries, which helps to nucleate a new dislocation far from the initial dislocation nucleation site [14]. On the other hand, recent HRTEM and molecular dynamics studies have found that the dislocations formed during the tensile loading disappear under the following compressive loading [13]. Because the stresses required for the dislocation nucleation and the dislocation annihilation are different, there exist a significant asymmetry during tension and compression cycle, which is referred to as intrinsic Bauschinger effect. Based on the interaction between the dislocation and the twin boundary, it has been shown that dislocation loop confined by twin boundary is relatively easy to be annihilated [13]. The Bauschinger effect has been explained by difference in the energy barrier of forming a new dislocation and annihilating an existing dislocation loop. In summary, due to the unique microstructure of the silver nanowire with five twin boundaries, dislocation loop confined by twin boundaries causes stress to focus considerably far from the dislocation loop, resulting in distributed plastic deformation. Also, the recoverable plasticity characteristic arises from the fact that a confined dislocation loop can be rather easily annihilated by inverse loading. The distributed and recoverable plasticity could provide an explanation on the observed durability and stability of transparent electrodes or piezoresistive sensors based on silver nanowire composite under repeated tension and compression cycles [4].

6

Exceptional Deformation Mechanisms

FCC metal nanowires exhibit two major exceptional deformation mechanisms. First, there is a case where a twin is observed even though partial slip is predicted from the analysis Schmid factor and GSFE. Second, at very low temperature, a global deformation mechanism that shows discontinuous stress-strain curves originated from an abrupt change in the cross-sectional shape of the entire nanowire, which is very different from the conventional local deformation mechanism by dislocation nucleation and emission. The first case is known to occur due to the high surface area ratio of nanowires [15]. The deformation mechanism of h100i nanowire is predicted to be partial slip from Table 1 in Sect. 3, while it deforms by twinning when side walls are made of four {100} planes (Fig. 5a). Because the deformation mechanism prediction based on GSFE often underestimates the partial slip as stated in Sect. 3, it is very unusual to

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Fig. 5 (a) MD simulation of Ni nanowire under compression. {100} side walls are reoriented to {111} surface upon twinning [15]. (b) h100i Au nanowire under tension at 10 K. Before the nucleation of dislocation, global deformation occurs first, accompanied by abrupt change in cross-sectional shape [16]. (c) Change of width and height as a function of strain. Bifurcation is observed beyond a critical strain [16]

observe twinning in this condition. Due to high surface fraction of nanowires, the surface stress (or energy) contribution affects the deformation mechanism significantly. The energy change due to the surface reorientation or the surface step formed by dislocation motions is not negligible compared to the volumetric energy. In terms of surface energy, it is more preferred to have twin instead of partial slip which produces surface steps at multiple locations. In twinning deformation, new surface steps form right next to the existing steps by successive partial slips occurring at adjacent slip places, which costs much less energy than the partial slip case where new surface steps form at completely new locations. In addition, after the formation of twin, the {100} side walls are reoriented to be the {111} surfaces which have much lower surface energy. The second case is a global deformation mode accompanied by the abrupt change of overall cross-sectional shape of nanowire, which is very different from local deformation mechanisms served by dislocation or twinning (Fig. 5b, c) [16]. Defect free nanowires or nanofilms can be stretched without nucleating dislocation to high strain level where the nonlinear elastic softening is observed in stress-strain curves. The tangent stiffness tensors of the materials becomes significantly different at such high strain, compared to the stiffness tensors at zero strain. When the tangent stiffness tensors changes so much that Hill’s instability criterion is satisfied, global deformation occurs that is accompanied by the abrupt change in cross-sectional

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shape. However, at high temperature, even in the absence of any defect, the dislocation nucleation energy barrier can be overcome relatively easily and global deformation mode is not observed.

7

Summary

In this chapter, we show that most deformation mechanisms of FCC metal nanowires can be explained by a combination of Schmid factor and GSFE. Since the yield stress of defect free nanocrystal is initiated by dislocation nucleation, we discuss the thermodynamics and atomistic simulation methods of dislocation nucleation phenomena. Then, we summarize the causes of distributed and recoverable plasticity of penta-twinned silver nanowires. In addition, we explain two unusual deformation mechanisms of surface-driven twin and global deformation by accounting for the surface effect and elastic instability (Hill’s criterion), respectively. While the deformation mechanisms of FCC metal nanowires have been well established by combining HRTEM experiments and molecular dynamics simulations, two remaining challenges remain to be solved [6, 17]. First, in experiments, the Young’s modulus is found to change significantly when the diameter of nanowire reduces below 100 nm, while atomistic calculations find that the modulus changes only below 10 nm. Second, while the minimum diameter of nanowire tested in tension is 40 nm and maximum strain rate is less than 103 s1 in experiments, most atomistic calculations considers nanowires with less than 20 nm diameter at strain rate of 107 s1, which can cause artifacts from length and time scale discrepancies. Only after solving these challenges, a realistic quantitative connection between atomistic simulations and experiments can be made. In addition to the single crystal or pentatwinned nanowires considered in this chapter, one can find the deformation mechanisms of the FCC metal nanocrystals with more complex polycrystals or twinned microstructures in the literature [18–20].

References 1. Yang P, Yan R, Fardy M. Semiconductor nanowire: what’s next? Nano Lett. 2010;10:1529. 2. Comini E, Baratoo C, Faglia G, Ferroni M, Ponzoni A, Zappa D, Sberveglieri G. Metal oxide nanowire chemical and biochemical sensors. J Mater Res. 2013;28:2911. 3. Ozin GA, Manners I, Fournier-Bidoz S, Arsenault A. Dream nanomachines. Adv Mater. 2005;17:3011. 4. Amjadi M, Pichitpajongkit A, Lee S, Ryu S, Park I. Highly stretchable and sensitive strain sensor based on silver nanowire-elastomer nanocomposite. ACS Nano. 2014;8:5154. 5. Uchic MD, Dimiduk DM, Florando JN, Nix WD. Sample dimensions influence strength and crystal plasticity. Science. 2004;305:986. 6. Weinberger CR, Cai W. Plasticity of metal nanowires. J Mater Chem. 2012;22:3277. 7. Vitek V. Intrinsic stacking faults in body-centred cubic crystals. Philos Mag. 1968;18:773. 8. Wu X-Z, Wang R, Wang S-F, Wei Q-Y. Ab initio calculations of generalized-stacking-fault energy surfaces and surface energies for FCC metals. Appl Surf Sci. 2010;256:6345.

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9. Ryu S, Kang K, Cai W. Entropic effect on the rate of dislocation nucleation. Proc Natl Acad Sci. 2011;108:5174. 10. Ryu S, Kang K, Cai W. J Mater Res. 2011;26:2336. 11. Lee JH, Lee P, Lee D, Lee SS, Ko SH. Large-scale synthesis and characterization of very long silver nanowires via successive multistep growth. Cryst Growth Des. 2012;12:5598. 12. Filleter T, Ryu S, Kang K, Yin J, Bernal RA, Sohn K, Li S, Huang J, Cai W, Espinosa HD. Nucleation-controlled distributed plasticity in penta-twinned silver nanowires. Small. 2012;8:2986. 13. Bernal R, Aghaei A, Lee S, Ryu S, Sohn K, Huang J, Cai W, Espinosa H. Intrinsic Bauschinger effect and recoverable plasticity in Penta-twinned silver nanowires tested in tension. Nano Lett. 2015;15:139. 14. Lee S, Ryu S. Molecular dynamics study on the distributed plasticity of penta-twinned silver nanowires. Front Mater. 2015;2:56. 15. Park H, Gall K, Zimmerman J. Deformation of FCC nanowires by twinning and slip. J Mech Phys Solids. 1862, 2006;54 16. Ho DT, Im Y, Kwon S-Y, Kim SY. Sci Rep. 2015;5:11050. 17. Park HS, Cai W, Espinosa HD, Huang H. Mechanics of crystalline nanowires. MRS Bull. 2009;34:178. 18. Jang D, Li X, Gao H, Greer JR. Deformation mechanisms in nanotwinned metal nanopillars. Nat Nanotechnol. 2012;7:594. 19. Wu ZX, Zhang YW, Jhon MH, Greer JR, Srolovitz DJ. Nanostructure and surface effects on yield in cu nanowires. Acta Mater. 1831, 2013;61 20. Gu XW, Loynachan CN, Wu Z, Zhang YW, Srolovitz DJ, Greer JR. Size-dependent deformation of nanocrystalline Pt nanopillars. Nano Lett. 2012;12:6385.

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Indentation Behavior of Metallic Glass Via Molecular Dynamics Simulation Chun-Yi Wu and Yun-Che Wang

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular-Dynamics Models and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Properties from Indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Sputter Deposition Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Indentation Simulation at Room Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Temperature Effects on Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Temperature Effects on Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 21 22 23 23 23 27 30 30 31

Abstract

Metallic glasses, also known as glassy metals, exhibit unique mechanical properties in terms of their strength and ductility due to their noncrystalline microstructures. By performing molecular dynamics simulations, thin-film metallic glasses can be prepared by simulated sputter deposition processes. The deposition simulations were conducted with a tight-binding interatomic potential, and argon working gas was modeled by the pair-wise Moliere potential. The atomic structures of the glasses are verified by the radial distribution functions. After deposition simulation and suitable equilibration, the deposited amorphous films were simulated for their indentation properties by a right-angle conical indenter tip at selected temperatures. The hardness and Young’s modulus of the glasses show

C.-Y. Wu Research and Development Department Yan Men LTD, Tainan, Taiwan Y.-C. Wang (*) Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_2

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strong temperature dependence. The calculated pileup index of the films may be used to indicate the glass transition temperature. Keywords

Metallic glass · Thin film · Molecular dynamics · Indentation

1

Introduction

Many engineering applications require high strength materials with some ductility, and metallic glasses are one of the candidates for such applications [1, 2]. Among many different species of metallic glasses, the Cu-Zr-Al metallic-glass systems have been extensively studied due to their unique mechanical properties, including experimental and simulation methods [3, 4, 5, 6, 7]. The unique properties may be originated from the composite nature in certain composition of the glasses. Simulation studies have been extensively conducted to obtain a better understanding of the mechanical behavior of metallic glasses in relation to their atomic structures [8]. Most work in this area by inversely determining the amorphous microstructure with the MD through experimental data [9]. The medium-range order of the metallic glass has been revealed and identified with MD simulations [10]. Furthermore, consideration of structural anisotropy in analyzing synchrotron experimental data to obtain correct elastic constants has been emphasized to link the macroscopic properties with microstructures [11]. It has been shown that the bulk metallic glass Cu-Zr-Al composites may contain CuZr crystallites, and hence their tensile properties may be improved [12]. In this system, the Zr42.5Cu42.5Al5 bulk metallic glass has been shown to exhibit the deformation-induced martensitic transformation [13]. Experimentally, it has been reported that the glass transformation temperatures of the Cu-Zr-Al bulk metallic glasses may vary from about 670
C to about 705
C for aluminum atomic percentage less than 10% [3]. Furthermore, structural relaxation has been experimentally demonstraed to delineate the behavior of crystal-like clusters embedded in amorphous matrix [14]. In addition, it has been noticed that densification occurs during strain hardening in metallic glass, as an evidence of structural relaxation [15]. For bulk Cu-Zr-Al metallic glasses, the nanoindentation experiments have been used to reveal the unique properties of the materials [16]. In addition to hardness and modulus that are commonly reported from indentation tests, the pileup index, also determined from indentation tests, can be used to reveal the flow behavior of the material [17]. Molecular-dynamics (MD) simulations have been widely used as a general tool to study deformation mechanisms in materials. For example, hydrogen in carbon nanostructures has been studied in [18] for possible energy applications. Using molecular dynamics simulation to study indentation properties of amorphous metals with a Lennard-Jones interatomic potential has been reported for their shear localization [19]. The approach adopted in this work is to follow physical principles in sputter deposition to create the MD models for subsequent mechanical

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characterization of thin-film metallic glasses to compare with experimental data. It is hypothesized that for materials with less frequency-dependent characteristics, MD results, albeit calculated at a short time scale, can be correlated with physical experimental data. In addition to studying the Cu-Zr-Al metallic glasses, similar approaches have been applied to study the indentation behavior of the Zr-base metallic-glass films [20], as well as polycrystalline aluminum under high loading rates [21]. The indentation techniques have been widely adopted to study mechanical properties of materials. As an example, Cu/Cu(O) multilayers have been shown by indentation data to exhibit the high strength and tunable modulus [22]. Here we use (Cu50Zr50)100  xAlx (x = 0, 2, 4, 5, 6, 8, 10, 12) metallic-glass thin films as an example to demonstrate our methodology to characterize their indentation properties by MD simulations.

2

Molecular-Dynamics Models and Simulations

MD simulations were performed to model the sputter-deposition process to create the initial configuration of the Cu-Zr-Al films, referred to as the as-deposited film. The metallic-glass thin films are deposited on the (0001) hexagonal closed pack (hcp) single crystal titanium. During the deposition, the Cu, Zr, or Al atoms may penetrate into the substrate. In the simulations, a thermalcontrol-layer-marching algorithm was adopted to accelerate the deposition calculations [23]. The basic assumption of the thermal-control-layer-marching algorithm is to neglect atomic interactions between newly deposited atoms and the have-been deposited atoms that are relatively far away from the free surface, in order to save computation time. It has been shown that substrate has limited effects when thickness of the film is large enough [24]. Once the film has been deposited to its desired thickness, thermal equilibration simulations were performed to relax the whole system. In the deposition and indentation simulations, interatomic potentials, which are derived from the many-body, tight-binding (TB) second-moment approximation [25], were adopted to simulate the interactions among unlike atoms. The TB parameters were obtained by reproducing the elastic constants of the corresponding pure crystalline materials, as well as fitting with cohesive energy and lattice parameter. Fitting comparisons can be found in the supporting information associated with this paper. The Lorentz-Berthelot rules for the energy potential parameters are conventionally adopted for mixing interactions between unlike atomic species [26]. The geometric means for the other parameters between unlike atoms are adopted. The simulation ensemble was chosen to maintain constant number of atoms (N), constant volume (V), and constant temperature (T), i.e., the NVT ensemble for deposition and indentation. During the deposition, ensemble temperature was set to be 300 K. During annealing at elevated temperatures, the NPT ensemble was adopted, where pressure (P) was kept to be constant. The high temperature annealing was conducted with LAMMPS [27], and

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the embedded-atom-method (EAM) potential was adopted. It has been shown that TB and EAM are formally equivalent [25]. For the interaction between the working argon gas and film atoms, the pair-wise Moliére potential is adopted [20, 28], and the functional form of the potential is as follows: 


 0:3r ij 1:2r ij 6r ij Z 1 Z 2 e2 V ij ðr Þ ¼ 0:35 exp þ 0:55 exp þ 0:1 exp , r ij a a a (1) where a is the Firsov screening length, and calculated by a = 0.4683 (√Z1 + √Z2)2/3, e is the electron charge, and Exp indicates the exponential function. The atomic numbers of the gas ion and depositing atom are Z1 and Z2, respectively. The potential, Vij is in units of electron volts (eV). The screening length is in units of Angstrom (Å). The interatomic potentials are shown below. ( Ei ¼ 

X j


 )12 X
  r ij r ij ξ exp 2q 1 þ Aexp p 1 : r0 r0 j 2

(2)

The symbol x is an effective hopping integer, rij the distance between atoms, i and j, and r0 is the first-neighbor distance. The parameters, A (eV), p, q, and x (eV), are systemspecific parameters, and can be found in Ref. [7].

3

Mechanical Properties from Indentation

The as-deposited films were subject to long MD runs under constant ensemble parameters for equilibration until total system energy becomes invariant to time. As for the indentation simulations, a right-angle conical indenter tip composed of carbon atoms was adopted. The tip is assumed to be rigid, and the interaction among atoms in the film was governed by the tight-binding potential. The substrate was titanium. From this simulated deposition, the interface between the film and substrate is considered to be “naturally” formed, based on the MD principles. In the deposition and indentation simulations, interatomic potentials, which are derived from the many-body, tight-binding second-moment approximation, were adopted to simulate the interactions among the four species of atoms forming the metallic glass. The interaction between the carbon atoms and the film was of a Lennard-Jones (LJ) type [28]. The displacement loading rate of the indenter tip was on the order of 10 m/s. Hardness and modulus of the films are inferred from indentation loaddisplacement curves by the Oliver-Pharr method [29]. For the deposition and indentation simulations, the periodic boundary conditions were adopted in the in-plane dimensions only. No periodic boundary conditions were applied to the thickness direction.

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4

Results and Discussion

4.1

Sputter Deposition Simulations

23

As shown in Fig. 1, a deposited and MD-equilibrated (CuZr)98Al2 molecular dynamics model has a size of 320 Å in length, 320 Å in width, and 120 Å in thickness. The substrate thickness is about 12 Å, and the titanium atoms on the bottom three atomic layers were fully constrained to mimic the fix end. The Cu, Zr, and Al atoms are colored with yellow, light blue, and red, respectively. The titanium crystalline substrate is dark blue. Some schematics of the observed microstructures from our MD models are shown in Fig. 2 to indicate the distorted nature of the icosahedron building blocks. Similar models for other composition of the (Cu50Zr50)100xAlx (x = 0, 2, 4, 5, 6, 8, 10, 12, atomic percent) are also prepared, but not shown. The total pair distribution functions (PDF) of the films are shown in Fig. 3, indicating they all are amorphous. With higher aluminum content, the first peak magnitude decreases. Our calculated PDF is in reasonable agreements with synchrotron experimental data of the Cu50Zr50 glass [7]. The difference in terms of magnitude between calculated PDF and experimental data around the first peak may be due to the local arrangement of the atoms from sputter deposition simulation.

4.2

Indentation Simulation at Room Temperature

In Fig. 4, an indentation snapshot is shown for depth-to-thickness ratio d/h = 0.4 (h = 10 nm). When d/h equals to unity, the indentation depth is the same as the film thickness. The indentation depth is labeled as d, and the film thickness h. The film sample in the figure is (Cu50Zr50)95Al5, and color is designated for different atomic species. Red is for the carbon atom in the indenter, yellow for aluminum, and light blue and green are for copper and zirconia, respectively. It has been observed significant Fig. 1 Molecular dynamics model for (Cu50Zr50)98Al2 prepared from sputter deposition simulation is shown with the color code: dark blue titanium substrate, yellow copper, light blue zirconia, red aluminum [5]

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Fig. 2 Schematics of the atomic arrangements observed in the MD equilibrated Cu-Zr-Al models [7]

pileup around the indenter for large d/h ratio, indicating homogeneous flow behavior [5]. In addition, around the film/substrate interface, titanium atoms may be embedded in the metallic-glass thin film, and vice versa, due to sputtering process. Following the Oliver-Pharr method [29], the hardness values of the film were calculated by dividing the maximum load to the projected indent area from the MD load-displacement data, as shown in Fig. 5, for two different indentation depth ratios. It can be seen that our calculated hardness is smaller than nanoindentation experimental data from bulk Cu-Zr-Al metallic glass [13]. The rationale for this discrepancy may be due to a number of factors, as discussed in Ref. [5]. Figure 6 shows the Young’s modulus of the amorphous films from indentation simulation with the Oliver-Pharr method for various aluminum content. Young’s modulus is calculated from the indentation modulus (Eind) by E = Eind (1ν2) with variations of the Poisson’s ratio, ν, between 0.2 and 0.4, based on the experimental data in Ref. [3]. The open symbols are calculated with the assumption of Poisson’s ratio being 0.3. The indentation modulus is calculated by

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Fig. 3 The pair distribution functions of the x = 0, 2, 4, 5, 6, 8, 10, and 12 metallic-glass thin films are shown and compared with synchrotron experimental data

Fig. 4 For the 5 at.% aluminum case, the penetration-to-thickness ratio d/h = 0.4. The thickness of the (0001) crystalline hcp titanium substrate is about 1.2 nm [5]

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Fig. 5 Hardness versus aluminum percentage for the Cu-Zr-Al metallic glass thin film, calculated from indentation simulation [5]

Fig. 6 Young’s modulus of the Cu-Zr-Al metallic glass thin films for various aluminum content percentage, calculated from indentation simulation [5]

Eind

S ¼ 2

rffiffiffi π , A

(3)

where S is the unloading slope of the indentation load-displacement curve, and A the projected area, as suggested by the Oliver-Pharr method [29]. Qualitatively, our MD data for Young’s modulus are in agreement with experimental data. Figure 7 shows the pileup index for the various Cu-Zr-Al metallic-glass thin films. The pileup index is defined by c¼

hþz , h

(4)

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Fig. 7 Pileup index for various Cu-Zr-Al species with two different d/h ratios [5]

where z is the maximum height of the pileup measured from the undeformed surface of the film, and h is the film thickness. If the pileup index is equal to unity, it indicates no pileup. The pileup index can be used as an indicator for glass phase transformation, since it describes overall changes in the material under the indent. Since all simulation work here is performed at temperature 300 K, the change in pileup index for the films with various aluminum content is due to stress-induced effects. For d/h = 0.4, the 5 at.% aluminum case shows an anomalous increase in its pileup index. For d/h = 1.0, the 8 at.% aluminum case shows a maximum pileup index value, and around 5 at.% the pileup index starts to increase. It is argued that the signatures around 5 at.% aluminum may be associated to its unique composite microstructures and some regions in the film may undergo phase transformation under stress. It is noted that we follow the Oliver-Pharr method for indentation calculation; hence a hold segment is required in the load profile. Due to weak interaction between the indenter tip and film surface, negative load is sensed while pulling the indenter out. During the hold phase, it can be seen that load decreases with time, and an indication of the system is in a relaxation process. The relaxation may be associated with densification [15] or relaxation [9].

4.3

Temperature Effects on Microstructures

When temperature increases, both structures and properties are affected. In this section, the temperature effects on atomic structures are examined first. It is noted that temperature affects structure, and structure affects properties. Hence, properties can be directly affected by temperature, or be indirectly affected through temperature-induced structure changes. Figure 8a shows the clustering of aluminum atoms (yellow and encircled) at 1000 K due to large atomic movement

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Fig. 8 (a) Formation of aluminum clusters in Cu-ZrAl metallic glass [7], and (b) temperature dependence of the size and number of the aluminum clusters

through diffusion. A sharp peak in the radial distribution function of a selected aluminum cluster, in relative large size about 10 Å in radius, has been observed. To analyze the size and number of the aluminum clusters, we follow the classical theory of homogeneous nucleation [30]. The theory predicts that the stability of a newly formed nucleus with its radius r must be greater than a critical radius r*, defined by r ¼

2γ IM , ΔG

(5)

where γ IM is the surface energy between inclusion (I) and matrix (M), and ΔG = ΔGM – ΔGI the Gibbs free energy difference. In the classical nucleation

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theory, the matrix is the liquid and the inclusion is the newly formed solid phase. The normalized critical radius r~ of the inclusion can be expressed by 2 , 1  T~

(6)

r Hf T , and T~ ¼ : Tr 2γ IM

(7)

r~ ¼ where r~ ¼

Here Hf and Tr are the heat of formation and a reference temperature, respectively. The reference temperature is set to be the melting temperature for the classical homogeneous nucleation theory. The normalized number of aluminum clusters n˜ c, defined as the ratio of the number of the clusters nc to total number of atoms in the system, can be estimated by ΔGc

n~c ¼ e kT ,

(8)

 
 Hf 4 3 1 1 4πr 2 πr ln n~c ¼   , Tr T k 3 αT

(9)

or

indicating the aluminum cluster number increases with temperature increasing when ΔG is negative. Here a, in units of 1/meter, is the coefficient relating the formation energy Hf and the surface energy, i.e., Hf = a γ, and the Boltzmann constant is k = 1.3807  1023 J/K. The aluminum cluster sizes and number of aluminum clusters versus temperature have been plotted in Fig. 8b. By choosing the reference temperature Tr = 800 K, it can be shown that a ~ 1011 m1 from Eq. (6). With the data shown in Fig. 8b and Eq. (9), it can be shown that when a ~ 1011 m1, Hf ~ 7.42  109 J/m3. The minus sign comes from strong surface energy term. The negative formation energy indicates that the aluminum cluster may be formed when temperature increases, which is distinguished from the formation of solid phase in liquid under cooling. One may further correlate experimental heats of mixing for Cu-Zr, Cu-Al, and Zr-Al pair interactions are 23, 1, and 44 kJ/mol, respectively [31], with the first-order-approximation estimations to obtain more accurate results. We remark the atomic configurations from molecular statics and molecular dynamics at 300 K are similar, and no aluminum clusters are formed. Noticeable aluminum cluster formation can be observed when temperature is larger than 800 K. The aluminum clustering may be related to the formation of aluminum-rich crystallites [32]. In addition, significant movements of lighter atoms have been observed in the Ni-C system by atomic simulation [33].

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Fig. 9 Temperature dependence of Young’s modulus and hardness calculated by the MD simulation [7]

4.4

Temperature Effects on Physical Properties

The hardness and Young’s modulus of the films calculated from indentation loaddisplacement curves are shown in Fig. 9 for the (Cu50Zr50)95Al5 sample. Since it is required to know the Poisson’s ratio ν of the film for converting indentation modulus Eind and Young’s modulus E, we follow the same procedure as shown in Fig. 6 to present our Young’s modulus data. It can be seen that when temperature is lower than roughly the glass transformation temperatures, the reduction of modulus and hardness is not severe. However, when at higher temperatures, hardness and modulus strongly decrease with temperature. It is noted that at room temperature, our calculated modulus and hardness are qualitatively comparable with experimental data, as discussed in Figs. 5 and 6.

5

Conclusions

The Cu-Zr-Al metallic glass thin films with various compositions are constructed by sputter deposition simulations in accordance with the conventional molecular dynamics simulation principle. Microstructures in the deposited and equilibrated thin films may partially reflect the microstructures found in physical experiment, as evident by our atomic structure analysis. Then, the films are studied at various temperatures for its atomic structural analysis and mechanical characterization by indentation. Calculated hardness and Young’s modulus are qualitatively in agreement with experimental data. At high temperatures, aluminum aggregates are found in the amorphous matrix. Furthermore, the modulus and hardness of the metallic glass are strongly temperature dependent. Around glass transformation temperatures, pileup index shows anomalies, indicating the glass transformation.

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Acknowledgements The authors are grateful for research grants from Taiwan Ministry of Science and Technology. This research received funding from the Headquarters of University Advancement at the National Cheng Kung University, which is sponsored by the Ministry of Education, Taiwan, ROC. We are also grateful to the National Center for High-Performance Computing for computer time and facilities.

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3

Surface/Interface Stress and Thin Film Stress Chun-Wei Pao

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stress Evolution During Island Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Island Morphology and Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Surface/Interface Stresses and Wafer Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Stress Evolution During Continuous Polycrystalline Film Growth . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Film Morphology and Stress Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Adatom Insertion and Thin Film Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34 39 39 41 43 46 46 48 49 53 54

Abstract

Thin film stress is critical for the reliability and electronic/optoelectronic properties of thin film devices. In this chapter, we systematically discussed the effects of surface and interface stresses on the film stress development during growth of polycrystalline films at the initial and final growth stage. We demonstrate that surface stress plays an important role at the initial stage of film growth (island growth stage), and conventional stress analysis technology such as wafer curveture experiments may not be applicable at this stage. At the late stage of film growth, we also show that adatom insertion into the grain boundaries is the primary mechanism of compressive stress development.

C.-W. Pao (*) Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_3

33

34

1

C.-W. Pao

Introduction

Thin films are solid materials for which the length scale in one dimension is much less than in the other two [1]. Thin solid films have been widely employed in many types of engineering systems. For example, thin film technologies are the essential part of microelectronic industries today in the fabrication of high quality and reliable thin films used in miniature transistors for integrated circuits today. Recent advances in quantum confinement devices such as quantum dots also give rise to new challenges in thin film technologies for material synthesis and modeling. Thin film technologies are also used to fabricate modern-day micro-electro-mechanical (MEM) devices, which are widely employed as miniature sensors or actuators. Other than applications for small-scale devices described above, thin film materials are widely adapted as surface coating materials for various engineering purposes in the larger scale: for example, coatings are commonly used to minimize wear and corrosion of structural materials. Surface coatings can also reduce thermal conductivity to increase efficiency (for example, on turbine blades). Three different growth modes have been identified for films grown from the vapor phase. These three growth modes, known as the Frank-van der Merwe (FM) mode, the Stranski-Krastanov (SK) mode, and the Volmer-Weber (VW) mode, are illustrated in Fig. 1. The growth mode is determined primarily by the strength of interatomic bonding between film and substrate atoms. As Fig. 1 shows, in the Frank – van der Merwe (FM) growth mode, the film is grown layer by layer. If the bonding strength between film atoms and substrate atoms is stronger than that between atoms within the film (film atoms prefer to be with substrate atoms), the growth process will be layer by layer. Single-crystal epitaxial semiconductor films are typically grown in the FM mode. In the Volmer-Weber (VW) growth, at the initial stage of film growth small islands nucleate on the substrate, followed by island growth and impingement to form a continuous film; this is shown in Fig. 1. This growth mode prevails if the bonding strength between film atoms is stronger than the bonding between film atoms and substrate atoms. The film atoms would rather stay close to like atoms than substrate atoms and therefore small islands will form at the beginning of growth. As the islands impinge upon each other, grain boundaries form and therefore the film microstructure is polycrystalline (assuming no epitaxial relation with the substrate is established). The VW growth is commonly observed in the growth of polycrystalline metal films. The third growth mode, known as the Stranski-Krastanov (SK) growth mode, is a combination of both the FM (layer-by-layer growth) and the VW growth mode (island nucleation-growth). In the initial stage of SK growth, the film grows in a layer-by-layer mode, but shifts to an island growth mode as the film further thickens, as shown in Fig. 1. The SK growth mode is commonly observed during the growth of heteroepitaxial films with lattice mismatch. Forming islands can relax the increase in free energy due to misfit stress. Thin film microstructure depends on many different material properties and growth conditions. Perhaps the two most important deposition factors are deposition

3

Surface/Interface Stress and Thin Film Stress

Frank - van der Merwe (FM) mode

Stranski - Krastanov (SK) mode

35

Volmer - Weber (VW) mode

Fig. 1 The schematic diagram of the three thin film growth modes

flux and substrate temperature. The deposition flux, which is also related to the partial pressure of the vapor of the materials deposited, determines the quantity of atoms arriving at the growing film surface per unit area of substrate surface per unit time. If the deposition flux is high, surface adatoms may be buried by other incoming adatoms before they reach an equilibrium location. Thus, high deposition flux generally results in rougher growing surface, smaller grain size (for polycrystalline film), and more defects. The substrate temperature affects the mobility of surface adatoms. Higher substrate temperature can greatly promote the mobility of surface adatoms and therefore surface adatoms have more chances to find stable locations. Therefore, high substrate temperature generally leads to surfaces that are closer to equilibrium, larger grain size for polycrystalline film, and film with fewer defects. Generally speaking, for growth processes with low deposition flux and high substrate temperature, the film growth is thermodynamically determined, which means the structure and morphology of the film are determined by the minimization of the free energy because surface adatoms can diffuse to the most stable locations (i.e., locations that give the lowest system free energy). The growth is slower but the film quality is better. Molecular beam epitaxy (MBE) is often much closer to equilibrium growth than most others. On the other hand, for growth processes with large deposition flux and low substrate temperature, the growth processes are kinetically determined, because surface adatoms are not able to find the most stable locations before being buried. Film stresses play an important role in thin film technology. First, the mechanical properties of thin films are very important for thin film reliability and functionality, and one of the most important factors affecting the mechanical properties of thin films is thin film stresses. Large film stresses can lead to film failure, including debonding, peeling, buckling, and cracking. On the other hand, we can also make use of the film stress to manipulate the structures of the materials to design the mechanical, electronic, or optical properties. For example, by properly controlling the lattice misfit stress, self-assembled quantum dots with different sizes and shapes have been fabricated, and thereby tailoring the optoelectronic properties. Thus, once we have a broader knowledge base of film stress generation mechanisms, we will be

36

C.-W. Pao

able to fabricate films with better reliability and functionality, and be better able to control material properties. However, film stress generation mechanisms are not yet fully understood. There remains a lot of questions that must be answered, particularly for films with small thicknesses and polycrystalline films. Wafer curvature experiments are the most common approach used to measure thin film stress in situ. This method, which was first developed by G. Stoney in 1909, was designed to measure the residual tensile stress generated during the deposition of metallic films using electrolysis, which may cause the peeling of the film [2]. The principle behind wafer curvature measurements is very simple: the residual stresses within a thin film on a substrate can bend the substrate/wafer, and we can obtain the changes in film stress by measuring the changes in the curvature of the wafer. The correlation between wafer curvature κ and film stress σ f can be written as Es t2s κ ¼ σ f tf , 6ð1
νÞ

(1)

where Es, ts, ν are the Young’s modulus, thickness, and Poisson’s ratio of the substrate, respectively. This form of the result is commonly known as Stoney’s equation. The right-hand side of Eq. 1, σ f tf, is commonly called the stress–thickness product, or the film force/width, and it is an important physical quantity for the topics throughout this chapter. It is interesting to note from Eq. 1 (from the left-hand side) that to obtain the film stress, we do not need any knowledge of the material properties of the film. All we need is the biaxial modulus of the wafer material, the thickness of the wafer and the wafer curvature. Wafer curvature experiments can be used to measure the film stress in situ since we can easily measure the real-time change in wafer curvature during growth or annealing. Due to these advantages, wafer curvature experiments are commonly adopted in both academic research and industry as the method of choice to measure stresses in thin films. Figure 2 shows a typical example of stress evolution during VW growth [3]. The stress evolution curves are very similar for a wide variety of materials. This figure shows the film force/width (stress–thickness) curves during the VW growth of Fe on oxidized Si (i.e., on an amorphous substrate to eliminate heteroepitaxy) at two different substrate temperatures: 310 K (low Fe atom mobility) and 520 K (high Fe atom mobility) [3]. In both experiments, the growth was interrupted for 15 min after 20 nm of growth. The growth was then resumed and another 15 nm of film was deposited. The stress evolution curves during the initial 20 nm of film growth and an additional 15 min growth interrupt are shown in Fig. 2a (310 K) and c (520 K), while the stress–thickness curves after the growth interrupt are shown in Fig. 2b (310 K) and d (520 K). Figure 2a, b shows that for the 310 K deposition (low mobility) the film stress is compressive at the beginning, but then turns tensile and keeps increasing until the end of deposition. On the other hand, from Fig. 2c, d we find that for 520 K deposition (high mobility) the film stress goes through a compressive-tensilecompressive evolution.

Surface/Interface Stress and Thin Film Stress

Fig. 2 The film force/width (stress–thickness) curves for VW growth of Fe at different temperatures. (a) Deposition at 310 K for 20 nm, followed by a 15-min growth interrupt. (b) After the growth interrupt, growth is resumed at 310 K for another 15 nm. (c) Deposition at 520 K followed by a 15-min growth interrupt. (d) After the growth interrupt, growth is resumed at 520 K for another 15 nm (Data from [3])

30

Film Force [N/m]

3

37

a

20

b + 15 nm Fe

20 nm Fe

10

310 K

0 10

c

d

0

520 K + 15 nm Fe

20 nm Fe

–10 0

10

20

30

40

50

Mean Thickness [nm] or Time [hs]

One interesting thing to note is how the stress evolves during/after growth interrupt. From Fig. 2c, d we see that for 520 K deposition (high mobility materials), the film stress becomes more tensile upon growth interrupt. However, when the growth resumes, the film stress quickly goes back to the stress–thickness it had prior to the growth interrupts. Figure 2a, b shows that, for 310 K deposition (low mobility materials), however, there is no pronounced change in the film stress–thickness product upon growth interrupt. Since the film stress evolution during VW growth shows a generic curve across a wide variety of different materials, this does not depend explicitly on material properties but rather on the complicated interplay between microstructures, surface morphologies, and growth conditions. Several research groups have been trying to understand the underlying mechanisms leading to the stress evolution observed in experiments. The mechanism driving tensile film stress evolution during island impingement is generally well explained by the Hoffmann-Nix model [4, 5] and models motivated by Hoffmann-Nix [6], which propose that island impingement replaces high-energy surfaces with a low-energy grain boundary resulting in the formation of tensile stress. The origin of the compressive stress measured during the early stages of growth, when the substrate is covered with isolated islands, has been explained in terms of the effect of surface stress on the stress state within islands. Cammarata, Trimble, and Srolovitz (CTS) suggested that this compression is a result of the Gibbs–Thompson effect, in which a tensile surface stress compresses the lattice of a small island, compared to that of a bulk material. This leads to an island sizedependent lattice parameter; i.e., the island will try to expand as it grows [7]. However, since the island adheres to the substrate, this expansion is constrained, leading to a compressive stress. This model, however, cannot explain the observed tensile relaxation during growth interrupt. Friesen et al. [8, 9] proposed another model to explain the film compressive stress at both initial and final stage of VW growth.

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C.-W. Pao

They performed a series of experiments in which they interrupted the growth process during the initial stage of film growth while measuring both the film stress and surface roughness [9]. They found that the stress changes that occur during growth interrupt are reversible (i.e., they return to their original value upon restarting growth), at both the initial stage and post-continuous stage of film growth. They also found that stress evolution and surface roughness were correlated. Based upon these observations, they suggest that during growth, the surface has a high surface defect (e.g., adatoms, ledges, vacancies, and islands) density and that these defects are responsible for changing the average surface stress of the growing surface from the common case of tension to compression. They further attributed the change in stress on growth interrupt to the reduction in the surface defect density. While this model is consistent with the experimental observations, the relationship between surface structure (defect density) and surface stress has not been established. In addition, atomistic simulations on the effects of surface defects on surface stress suggest that this effect may be too small to account for the observations [10]. On the other hand, Chason et al. proposed a model for stress evolution at late stage of VW growth based on adatoms diffusing in and out of grain boundaries during growth and interrupt, respectively [11]. In this model, the direction of adatom diffusion is determined by differences in chemical potential μ between the film surface and grain boundaries. During growth, an atomic flux makes the chemical potential of the growing surface higher than grain boundaries high density of adatoms or defects, which is the driving force for surface adatoms to diffuse into grain boundaries. As adatoms diffuse into grain boundaries, they compress the grains on both sides of the boundary, yielding an overall compressive film stress. During growth interrupt, the surfaces heal, leading to a lower chemical potential on the surface than in the grain boundaries which causes atoms to diffuse back out from the grain boundaries onto the film surface, relieving the compressive film stress. However, this model depends upon grain boundary diffusion, which, until recently [12], was considered too slow compared with the timescale for stress relaxation in experiments [13]. Thus, another model assuming negligible grain-boundary diffusion has been proposed [14]. Nevertheless, mechanisms leading to compressive film stress generation during the final stage of VW growth of high-mobility materials are still unclear. In this chapter, the effects of surface and interfaces on polycrystalline thin film growth stresses will be systematically discussed. In particular, we will focus on the compressive film stress development at the initial and final stage, because the mechanisms leading to compressive growth stresses remain elusive until very recent studies combining atomistic simulations and continuum level theoretical analysis. In the next section, the film stress generation mechanisms leading to the compressive film stress at the initial stage of the VW growth (islands nucleation/growth) will be discussed. It will be shown that the interaction of the surface/interface stresses leads to compressive film stress rather than the intrinsic film stress itself using theoretically modeling. The results will demonstrate that the wafer curvature experiments are unreliable in this case. Next, we will discuss recent work on the mechanisms leading to the compressive film stress during the growth of a continuous polycrystalline film

3

Surface/Interface Stress and Thin Film Stress

39

using molecular dynamics simulations together with a theoretical analysis. Pao et al. [15, 16] demonstrate that surface adatoms diffuse into grain boundaries and introduce compressive strains in the neighboring grains, and the film stress–thickness product is simply a linear function of the density of extra atoms incorporated at the grain boundaries regardless of grain sizes and deposition rates. In short, it will be shown that adatoms incorporation at the grain boundaries is the main source leading to the development of compressive growth stresses.

2

Stress Evolution During Island Growth

2.1

Simulation Methods

In this section, we briefly discuss the combined effects of surface/interface stress on film stress during island growth. The growth of an island on the substrate requires surface diffusion and/or other thermally activated processes, and the diffusional atomic hops are rare events on the timescale of atomic vibrations. To overcome the timescale issues from standard MD simulations, Pao and Srolovitz [17, 18] employed a hybrid simulation scheme to model film growth. The hybrid approach, employed here, combines both static relaxation (using the conjugate gradient method) and finite temperature, molecular dynamics simulations. The algorithm is illustrated in Fig. 3, and is briefly described in the following: 1. First, a fine grid of possible deposition locations above the film and substrate is generated (step 1 in Fig. 3). 2. Next, a test atom is released from each of these grid points and allow it to move to minimize the energy of the system (using a conjugate gradient method) with all other atom positions fixed (step 2). 3. Then, an adatom is deposited at the location corresponding to the test atom that has the lowest energy (compared to all of the other test atoms released from the grid, step 3).

Fig. 3 Schematic diagram of the hybrid algorithm used to simulate island growth. The numbers in the figure represent the steps of the hybrid algorithm employed to grow the island, as described in the text. θ is the wetting angle between the island and the substrate (Figure from [18])

1 2

Z Y X

Island

θ

4

3 ts

Substrate Lx

Ly

40

C.-W. Pao

4. Finally, a short molecular dynamics simulation is ran for 10 ps. This 10 ps MD relaxation time is picked because it is long enough to gather sufficient data to calculate thermodynamic averages (step 4). 5. This process is then repeated until the appropriate number of atoms is deposited.

This hybrid MD approach is able to produce compact-shaped surface structures (or, the structure is formed via thermodynamically determined processes), as will be demonstrated in the next section. However, readers must pay attention that this approach cannot produce surface structures formed via kinetic determined processes such as dendrites. The interatomic interactions are described using a simple parameterized potential appropriate for metallic alloys, that is, the Lennard-Jones EAM (LJ-EAM) potential [19–22]. This particular form of an EAM potential was chosen in order to freely adjust the properties of the film and substrate, rather than modeling particular real materials. Since the system was comprised of islands of B atoms on an A atoms substrate, potentials are required for A-A, A-B, and B-B interactions. The functional form of LJ-EAM potential can be expressed as E¼

X i

Ft i ½ ρ i  þ

X i6¼j

  ϕti tj r ij ,

(2)

where F [ρi] is the energy required  to  embed atom i of element ti (= A or B) in an electron gas of density ρi and ϕti tj r ij denotes the pair interaction between atom i of element ti and atom j of type tj. The functional forms of F, ρi, and ϕ are as described elsewhere [19] and the parameters used herein are summarized in Table 1. The parameters, eti tj and r 0ti tj in Table 1 denote the potential well depth and equilibrium separation between neighboring atoms in the reference Lennard-Jones system. We chose the equilibrium B atom (island) interatomic separation to be 1.3 times larger than that between A atoms (substrate) in the respective bulk single crystals. This was done to avoid epitaxy between the islands and substrate. The other parameters for the A and B atoms, At and Bt were chosen to ensure that island and substrate are both face centered cubic [20].

Table 1 Parameters used in the LJ-EAM interatomic interactions (following the notation of Ref. [19]) Pair A-A B-B A-B (strong) A-B (intermediate) A-B (weak) Table from [17]

eti tj ðeVÞ

r 0ti tj ðÅÞ

0.8 0.5 0.55 0.53 0.5

3 3.9 3.6 3.6 3.6

At 0.8 0.8 – – –

βt 6 6 – – –

Z0 12 12 – – –

3

Surface/Interface Stress and Thin Film Stress

41

There is experimental evidence that suggests that the interface bond strength affects the stress evolution during film growth [23]. To examine the effects of interface bond strength on stress evolution, the interactions between the A (substrate) and B (island) atoms had to be tunable without changing the properties of either A or B. To accomplish this, only the pairwise term in the energy, ϕti tj , for ti 6¼ tj were modified. In the LJ-EAM potential, the potential behaves like a Lennard-Jones potential for atoms in the perfect reference structure [20]. As such, it is characterized by bond length and bond energy parameters, r 0ti tj and eti tj. Three different values of eAB were employed, as indicated in the last three rows of Table 1: 0.5 eV (weak interface bonding), 0.53 eV (intermediate interface bonding), and 0.55 eV (strong interface bonding).

2.2

Island Morphology and Stress

Figure 4 shows the structure of typical islands grown with different interface bond strengths. In all cases, the islands formed were face centered cubic (it is not possible to distinguish between this and hexagonal close packed in Fig. 4a since the island is only two layers thick) and the islands are oriented with a direction perpendicular to the substrate. In all cases, the island/substrate interface contains a regular, triangular array of misfit dislocations. There are four close packed rows of atoms in the substrate for each three close packed rows of atoms in the substrate. Since B atoms are 1.3 times larger than A atoms, this implies that the misfit strain is nearly fully relaxed. Comparison of the shapes of the islands formed with different A/B bond strengths (see Fig. 4) shows that as the A/B bond strength increases (A/B interface energy decreases) the islands become thinner and have a larger substrate contact area (i.e., larger radius/height aspect ratio). The islands are 2, 3, and 5 layers thick in the strong, intermediate, and weak A/B bonds cases, respectively. The stress–thickness product – proportional to substrate wafer curvature normally used as a measure of film stress in experiment – is estimated as follows: we imagine cutting our model system along a plane that is orthogonal to the substrate surface and calculate the difference in the normal force across the cut plane before and after island deposition. The stress–thickness product is   σ xx h ¼ Fx
F0x =Ly

(3)

for the X direction, with h the average thickness of the film (whether a continuous film or an array of islands), Ly is the cell length along the Y direction, and Fx and F0x are the normal forces in the X direction after or before the island is deposited, respectively. Fx is given by 1 Fx ¼ Lx

X i

mi vxi vxi þ

X i6¼j

! f xij r xij ,

(4)

42

C.-W. Pao

Fig. 4 The morphologies of the island grown with (a) strong, (b) intermediate, and (c) weak interface bonding. Dark and light gray atoms represent island and substrate atoms, respectively (Figure from [17])

where Lx is the simulation cell length in the X direction, mi is the mass of atom i, and vxi, f xij, and r xij are the X component of the velocity of atom i, the force that atom j exerts on atom i, and the vector that separates atoms i and j, respectively. This force is the same across all possible cut surfaces perpendicular to the substrate surface. In the following, the biaxial mean value of this stress–thickness product – σh = (σ xxh + σ yyh)/2 – will be reported. Figure 5 shows the calculated stress–thickness σh versus number of adatoms N. The data points shown represent the calculated stress–thickness averaged over 40 measurements, each after a single adatom deposition and MD relaxation (to minimize noise). This figure shows that both the stress–thickness product (average film stress) and its derivative with respect to the number of adatoms (the stress in the newly added material) are negative. As the A/B bond strength increases (the A/B interface energy decreases), the magnitudes of the island stresses increase. In other words, the island is under the largest compressive stress when the interface energy is smallest. These results are consistent with previous observations with Ag and Al islands on amorphous SiO2 substrates [23]. This counterintuitive result may

Surface/Interface Stress and Thin Film Stress

Fig. 5 Stress–thickness product as a function of the number of atoms deposited. Each data point represents an average of the data obtained during the deposition of 40 adatoms (to minimize statistical fluctuations) (Figure from [17])

43

–1

Strong interfacial bonding Intermediate interfacial bonding Weak interfacial bonding

–2 –3

sh (GPa – Å)

3

–4 –5 –6

–7 –8 –9 0

50

100

150

200

250

300

N

be understood by consideration of the effect of interface bond strength on island shape, as discussed in the following.

2.3

Surface/Interface Stresses and Wafer Curvature

To isolate the possible sources of measured compressive stress–thickness product, the Stoney equation was rederived, including the effects of surface or interface stresses. The effect of interface stress was considered by Ruud et al. [24] for the special case of continuous, multilayer films. Here the case when the film is not yet continuous is examined. Consider a cylindrical island with radius ρ and height tf on a cylindrical substrate with radius R and thickness ts. The total energy of the system can be written as U tot ¼ U bending þ U island þ U surface ,

(5)

where subscripts refer to energy contributions from bending of the substrate, the island, and the surface or interface. The biaxial strain associated with pure bending is e =
κz, where κ denotes the curvature of the wafer and z is the distance from the neutral surface of bending (κ > 0 refers to the center of the wafer bent downward relative to the edges). The corresponding bending energy for the substrate is U bending ¼

πR2 Es t3s κ 2 , 12ð1
νÞ

(6)

where Es and ν are the Young’s modulus and Poisson ratio of the substrate, respectively [25]. The elastic energy stored within the island has contributions both from the residual stress (i.e., the mean stress in the absence of wafer curvature) and from substrate bending:

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C.-W. Pao

U island

   2 πρ2 tf σ f
Ef κ ts þ tf =2 ¼ , Ef

(7)

where Ef and σ f are the Young’s modulus of the island and mean stress inside the island. Since the surface or interface is biaxially strained, the energy associated with the surfaces or interfaces has contributions from the surface or interface energies themselves and the surface or interface stresses: h   i U surface ¼ πρ2 γ f
f f ts þ 2tf κ þ πρtf ð2γ t
f t ts κÞ   þ πρ2 ðγ i
f i ts κ Þ þ π R2
ρ2 ðγ s
f s ts κÞ

(8)

þ πR2 ðγ s þ f s ts κ Þ, where γ f, γ t, γ i, and γ s ( ff, ft, fi, and fs) are the surface energies (surface stresses) of the island upper surface, island side face, substrate/island interface, and substrate surface, respectively. Minimizing the system energy with respect to substrate curvature κ yields the relationship between substrate curvature, island stress, and surface or interface stresses (assuming a thick substrate ts  tf – this is valid for the simulations since the substrate is effectively semi-infinite, as described above):    t Es t2s f ¼ Θ σ f tf þ f f þ f i þ f t
f s , 6ð 1
ν Þ ρ

(9)

where the surface coverage (i.e., fraction of the substrate surface covered by the islands) is Θ = πρ2/πR2. Equation above is the surface stress-modified Stoney equation. This correction is important if the residual stress within the film is small (e.g., in homoepitaxy) or the film (island) is thin (small). [The corrected Stoney equation in Eq. (2) can be extended to the case of a continuous thin film on a substrate simply by letting tf /ρ go to zero and Θ go to unity.] If we assume the island is in mechanical equilibrium, the stress within the island is balanced by the surface or interface stresses: σ f tf þ f f þ f i þ

tf f ¼ 0: ρ t

(10)

The equation above is simply a form of mechanical equilibrium. In this case, the surface stress-modified Stoney equation reduces to Es t2s ¼
Θf s : 6ð 1
ν Þ

(11)

This implies that the substrate curvature is a linear function of the substrate surface coverage with a slope given by minus the substrate surface stress, fs. Since the surface stress is most commonly found to be tensile, the substrate curvature measured during the island growth process will be negative with a magnitude that

3

Surface/Interface Stress and Thin Film Stress

45

increases as growth proceeds. However, if the island is not in mechanical equilibrium or other sources of island stress (i.e., islands impingement, lattice mismatch) exist, Eq. 11 no longer holds and Eq. 9 should be applied. Pao and Srolovitz reploted the stress–thickness data in Fig. 5 as a function of substrate surface coverage rather than number of atoms deposited. The result is shown in Fig. 6 for the three interfacial bond strength simulations reported above. The stress–thickness products for all three cases are a nearly linear function of surface coverage, and the σh data for all three interfacial bond strength simulation data sets collapse onto the same line with slope 18.8  0.9 GPa
Å. According to Eq. 11, this value should be equal to minus the surface stress of the substrate. The independent measure of the substrate surface stress reported is 18.4 GPa
Å. The slope of the σh versus substrate surface coverage plot (Fig. 6) is in excellent agreement with the theoretical prediction that was based upon the assumption that the island is self-equilibrated (to within 2%). These observations show that the island itself is in mechanical equilibrium, and the compressive stress comes solely from the combined effects of surface or interface stresses together with substrate surface coverage. The measured stress–thickness product does not reflect the stress state inside the island itself. The reason why islands grown with a strong interface bond exhibit larger compressive stresses (as measured by the stress–thickness product) can also be easily explained: Given the same number of adatoms, islands grown with stronger interfacial bonding will have smaller wetting angles (and, therefore, exhibit larger compressive stress). In summary, in this section, the effects of surface and interfacial stresses on film growth stress during island growth are elucidated. It is shown that when the island is self-equilibrated (the stress in the island balances the interface and surface stresses), the stress–thickness product is simply proportional to the substrate coverage and the substrate surface stress. Hence, the wafer curvature experiment – one widely employed measure for thin film stress – is not suitable for in situ characterization of thin film stress at the early stage of VW growth. In the following section, we will

1 Strong interfacial bonding 0 Intermediate interfacial bonding –1 Weak interfacial bonding –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

sk (GPa – Å)

Fig. 6 Stress–thickness product vs. substrate surface coverage. The dashed curve is the best-fit line. The slope of this line is 18.80.9 GPa-Å (Figure from [18])

θ

46

C.-W. Pao

briefly discuss the effects of grain boundaries on thin film growth stress of continuous polycrystalline films.

3

Stress Evolution During Continuous Polycrystalline Film Growth

Compressive stresses are often observed at the late stage of VW growth, namely, after the films are continuous across the substrate. One promising model to explain experimental observation proposes that adatoms reversibly move in and out of grain boundaries (GBs) and, as such, account for both late stage compressive stress evolution (atoms entering GBs) and tensile evolution upon growth interrupt (exiting GBs) [11, 13]. The proposed driving force for an atom to enter a GB during deposition is associated with the high chemical potential of an atom on the surface. However, the model, as proposed, depends upon GB diffusion to account for stress relaxation during growth interrupts, and it has been asserted that the corresponding kinetics would be too slow to account for the experimental observations [13]. An adaptation of this model has been advanced that addresses adatom insertion at GBs in the absence of GB diffusivity [14]. Still, other challenges have been made to adatom insertion as the stress evolution mechanism; these models instead invoke surface defect evolution [8, 9] or locked-in capillarity stress and recrystallization [3] as alternate mechanisms. However, surface defects have little effect on compressive stresses [10], and the locked-in capillarity stress model cannot explain the tensile relaxation during growth interrupt [3]. Thus, the grain boundary insertion mechanism continues to be called upon to explain experimental data on compressive stress evolution in film growth [26].

3.1

Molecular Dynamics Simulations

To elucidate mechanisms leading to compressive film stress development during growth of continuous, polycrystalline film, Pao et al. carried out a series of molecular dynamics (MD) simulations of Ni deposition onto a Ni(111) surface intersected by two GBs to investigate the interplay of growth stress, grain boundaries, and diffusion [15, 16]. To simulate growth of a polycrystalline film, Ni adatoms were deposited at random locations on the surface of a Ni bicrystal film. The surface was intersected by two Σ79 [111] symmetric tilt grain boundaries with misorientation θ = 33.99 (as shown in Fig. 7). Note that the grain-boundary equilibration process is a complex application of Monte Carlo, molecular dynamics, and energy minimization calculations. Periodic boundary conditions were applied along X and Y directions. Note that the system dimension in Z was 18 nm but only part of this is rendered in Fig. 7. Molecular dynamics simulations were performed at constant temperature (T = 0.5Tm of Ni, that is, T = 782.5 K) prior to deposition to relax the system; all subsequent deposition simulations were run under these same conditions but the thermostat algorithm was not applied to the atom currently being deposited. The interaction between the Ni atoms

3

Surface/Interface Stress and Thin Film Stress

47

Fig. 7 Images of the atomic structure of the simulation cell during deposition simulations on L = 11 nm grain systems for varying coverage. Only surface atoms (colored in light gray) and grainboundary atoms (dark gray) are presented; (a) relaxation time Δt = 50 ps (system B), (b) relaxation time Δt = 12 ps (system C). Axes below show the coordinate system of the simulations (Figure from [16]).

was simulated using the Voter-Chen-type embedded atom method potential [27, 28]. Three deposition simulations were performed; these are distinguished from one another by the grain size simulated, L, and the effective deposition rate (the inverse of the time between successive deposition events, Δt). The first simulation with L = 5.5 nm and Δt = 25 ps is system A; the second simulation with L = 11 nm and Δt = 50 ps is system B; the third simulation with L = 11 nm and Δt = 12 ps is system C. Note that the first and third deposition simulations (i.e., systems A and C) had the same deposition flux. For the 5.5 nm grain-size case (system A), the total deposited film thickness tf was three monolayers (ML), while for the 11 nm cases (systems B and C) the total deposited film thickness tf was two monolayers. The stress–thickness product was obtained from simulations via a method described in the previous section. The difference in the average force in X across a series of imaginary cuttingplanes normal along X before and after deposition was computed. That is, Δσ xx h ¼ Fx F0x =Ly , where Δσ xxh is the change in stress–thickness product compared with the initial continuous film (an absolute value of stress–thickness product cannot be given sincewe do not begin with a singular, pristine, clean substrate surface before deposition), Fx F0x is the force in X after (before) deposition, and Ly is the simulation cell width along Y.

48

3.2

C.-W. Pao

Film Morphology and Stress Evolution

Figure 7 shows surface morphology evolution of systems B and C. In Fig. 7, atoms are shaded according to their centrosymmetry parameter [29] and only atoms with centrosymmetry greater than 6.0 are displayed (centrosymmetry equals zero for atoms in an ideal, bulk FCC environment; larger centrosymmetry parameter values indicate more significant deviations from the ideal, bulk environment). Atoms at grain boundaries are colored dark gray while atoms on the surface are light gray. From Fig. 7 it can be observed that growth is layer by layer, which is due to significant atomic mobility on the {111} surface at the high simulation temperature. By comparing Fig. 7a and b it can be seen that the surface for system B is smoother (i.e., smaller step density) than system C because higher deposition rate in the latter system means adatoms have less time to sample the surface before the next adatom arrives. As such, the probability for new two-dimensional (2D) islands and steps to nucleate is higher for increased deposition rate. Examination of Fig. 7a and b also shows that grain boundaries near the surface migrate. As 2D-surface islands grow, island step edges drag grain boundaries near the surface along with advancing step edges. This observation is similar to what was seen experimentally by Ling et al. [30] and it shows a possible mechanism for grain growth during polycrystalline film growth. Figure 8 shows Δσ xxh versus tf. In all cases, compressive film stress evolutions were observed, consistent with experimental observations for high mobility materials in late stage VW growth. While oscillations are observed in the simulation data, they are not in experiment – presumably because the experiments inherently average over many grains. The period of the oscillations is  1 ML – this may be related to the layer-by-layer growth mode. The incremental film stress was calculated (the derivative of the σ xxh curve) σ inc xx , which is approximated by the following expression: σ inc xx

    Δσ xx h tf ¼ 2:0ML
Δσ xx h tf ¼ 1:0ML  : Δtf

Fig. 8 Stress evolution during deposition for (a) system A, (b) system B, and (c) system C (Figure from [15])

(12)

3

Surface/Interface Stress and Thin Film Stress

49

From formula above, the incremental stresses were σ inc xx  3:14 ,  3.23, and  3.11 GPa for systems A–C, respectively. Decreasing deposition rate yields larger compressive stresses (cf. systems B and C), consistent with experiment [11, 26]. σ inc xx in the present study (c.a.
3 GPa) are larger than observed in experiments (c.a.
1 GPa), and this will be discussed in the subsection addressing the mechanism of compressive stress below.

3.3

Adatom Insertion and Thin Film Stress

Figure 9 a shows the number of atoms per layer (of thickness equal to the {111} plane spacing), NL as a function of distance from the surface, D (D = 0 is the free surface), for system B at three values of tf. These results demonstrate that during deposition, the atomic density in the film, near the surface, increases; i.e., extra atoms are incorporated into the bulk of the film, near the surface. By monitoring atomic centrosymmetry, it was confirmed that no self-interstitials exist away from the GBs; rather, extra atoms reside at the GBs. The number of extra atoms can be expressed as   X    N X tf ¼ N L D, tf
N 0 :

(13)

D

Figure 9b displays NX versus tf. Note that N0 = NL(D = 1). The extra atoms penetrate into the film to a depth of at least 20 Å. Adatom trapping in interstitiallike sites near step edges has been proposed as a compressive stress generation mechanism [31] and recent experiments and calculations confirmed related behavior on Au film surfaces [32]. This is not observed in these simulations, which is likely related to the low step density on our near-ideal planar surface. Examination of Fig. 9a and b shows that the number of extra atoms first increases during deposition and then decreases. This means that some of the extra, incorporated atoms move out of the GB and back onto the free surface. In fact, NX is oscillatory, with a period of 1 ML, as for the stress–thickness product. This strongly suggests that Δσ xxh correlates with NX. These results show that adatom incorporation into GBs and their egress from GBs occurs very rapidly and is responsible for increases and decreases in the compressive film stress, respectively, during deposition. The grain boundary insertion model [11, 13] invokes a chemical potential difference Δμ between the growth surface and GB, namely, Δμ = Δμ0 + δμs + σΩ. Δμ0 is the difference between the free surface and GB chemical potentials in the absence of growth, δμs is the increase in surface chemical potential associated with the deposition flux, and σΩ represents the elastic work done by inserting an atom of volume Ω into the grain boundary in the presence of a normal stress σ across the GB. Results displayed in Fig. 9 can be interpreted in terms of this model. The first term is dominated by the difference in the formation energy of an adatom on the surface versus a self-interstitial in the grain boundary. For the polycrystalline film considered here, this difference is in excess of 0.5 eV/atom (surface 1.5 eV, GB ¡1 eV). Slower

50 Fig. 9 (a) Number of atoms in the film bulk NL vs depth from the surface D for system B at tf = 0, 0.64, and 0.89 ML. (b) The number of extra atoms NX incorporated into the grain boundaries vs film thickness tf (Figure from [16])

C.-W. Pao

a

1580

1575

NL

0 ML 0.64 ML 0.89 ML

1570

1565

1560

1555 –180 –160 –140 –120 –100 –80 –60 –40 –20

0

D (Å)

b

180

System A System B System C

160 140 120

NX 100 80 60 40 20 0

0

0.5

1

1.5

2

2.5

3

tf (monolayer)

deposition rates give atoms more time to diffuse into the GB before they are buried by the next layer of atoms. This is consistent with our simulation results shown in Fig. 9b. One of the unique features of the present work is the observation that the number of extra atoms in the grain boundary increases and decreases with a period that is similar to the time to deposit a single monolayer. At the beginning of the deposition of each monolayer, adatoms are either isolated or in small clusters, and hence have very high energies. As the monolayer fills, the average cluster size increases and the energy per adatom drops. Hence, the chemical potential of the surface drops and adatoms that were in the grain boundary start to come back out. This explains the observed cyclic nature of the stress and NX. It must be noted that, unlike other models, this approach does not require any assumptions about tensile surface stresses, island coalescence, etc. [13] and is consistent with the standard understanding of how surfaces evolve during growth. To reveal the correlations between NX and Δσ xxh, a simple model is derived. The film stress due to adatom incorporation can be written approximately as

3

Surface/Interface Stress and Thin Film Stress

Δσ xx

51


Eða
dÞ π ða=2Þ2 Nx  Lþd 2Ly h  3

πEa d 1
Nx, ¼
a 8LLy h

! (14)

where E is the film elastic modulus, L is the grain size, Ly is the simulation cell size in y, a is the atomic diameter, h is the thickness of the film, and d is the GB “width” in x (i.e., the gap between neighboring grains). d can be related to the GB free volume v by v = dLyh and thus depends on the GB type; its value is typically less than an atomic diameter. The term
E(a
d)/(L + d) refers to the compressive stress acting on the grain if the GB with width d is completely filled with inserted atoms with diameter a. Note that the expression for stress in [11] is the special case with d = 0, which may lead to overestimation of stress because there is always free volume associated with GBs. The term NXπ(a/2)2/2Lyh is the fraction of the boundary area occupied by NX inserted atoms (the factor of 2 accounts for the two GBs in our system). Assuming that L  d, we can rewrite the stress–thickness product as

 d πEa 1
a 
Nx 4Lx Ly

 πEa3 ρX d 1
, ¼
a 4 3

Δσ xx h

(15)

where 2 L  Lx and ρX = NX/LxLy is the density of extra atoms per unit film surface area. This demonstrates that Δσ xxh should be a linear function of ρX with slope
(π/ 4)Ea3(1
d/a) regardless of grain size and deposition rate. The geometric parameter α in Chason et al. [11] is manifested in our model as (π/4)(1
d/a). The stress–thickness product is plotted versus ρX in Fig. 10. All data collapse onto a single line, consistent with our prediction. The fitted slope in Fig. 10 is – 1820 GPaÅ3. From separate molecular dynamics simulations [15], E = 191.6 GPa and a = 2.51 Å and the GB free volume is estimated from the difference in volume of a cell with the GB and bulk crystal with the same number of atoms, i.e., d = 0.31 Å. Substituting these data into Eq. 15 yields a slope of
2099 GPaÅ3. The fitted slope is 87% of that of the theoretical prediction and therefore is in very good agreement (considering the simplicity of the stress approximation). This demonstrates that adatom incorporation and, specifically, ρX is an important factor for predicting the stress–thickness (wafer curvature) during polycrystalline thin film growth. Despite the success, it must be noted that the magnitude of the compressive stress found in the simulations is significantly greater than in experiments. Since it is generally accepted (and observed in our simulations) that lower deposition rates increase compressive stresses, this is clearly not an artifact of the high deposition rates employed in the simulations. The molecular dynamics simulations in this study were also performed at a temperature that is much higher than in most growth experiments; this greatly enhances atomic mobility and the likelihood of adatoms

52

C.-W. Pao 5

System A System B System C

0

Dsxxh (GPa –Å)

–5 –10 –15 –20 –25 –30 –35 –40 –45 0

0.005

0.01

0.015

0.02

0.025

rx (#/ Å2) Fig. 10 The film stress–thickness product Δσ xxh vs the density of incorporated extra atoms ρX for three deposition simulations (Figure from [15])

reaching a GB. Atomic mobility is further enhanced by the extreme planarity (low step densities) of the surfaces in our simulations, relative to the much higher step density surfaces seen experimentally during low temperature growth. Steps are effective adatom traps. Pao et al. suspected that it is the enhanced atomic transport on the surface that is responsible for the larger growth stresses observed in simulation as compared with experiment [15, 16]. Figure 9a shows incorporated atoms penetrate 20 Å after tf = 0.64 ML, corresponding to simulation time 45 ns. While T in simulations is high, adatom advancement along the GB is remarkably rapid. Mishin et al. [12, 33] has shown that self-interstitial diffusion rates in GBs in Cu over a very wide range of T. Earlier literature (e.g., [13]) suggested that GB diffusion kinetics may not be sufficiently fast to account for the rapid compressive stress relief upon interrupting growth. However, data in [12, 33, 34] along with the present results demonstrate that GB incorporation and diffusion are quite fast and may explain late stage VW stress evolution for many material systems. Since atoms can readily exit GBs when the surface structure evolves fast transport can explain both the abrupt change in surface stress during growth interrupt and the periodic nature of the growth stress during deposition. This provides a link between surface structure (e.g., see [35]), surface chemical potential, and growth stress. In summary, MD simulations show adatoms incorporate into GBs and this process directly correlates with compressive stress evolution. A theoretical model suggests the density of atoms incorporated ρX is linearly proportional to stress–thickness product, regardless of grain size and deposition rate; simulations support this conclusion. By refining interpretations in [11], a direct connection between GB free volume, surface structure, and the stress evolution during film growth is provided.

3

4

Surface/Interface Stress and Thin Film Stress

53

Summary

Surface/interface stresses are important factors in determining film stresses as measured via wafer curvature experiments, especially during the early stage of Volmer-Weber (VW) growth (islands nucleation/growth). Recent molecular simulations and theoretical analysis results discussed above demonstrate that, when islands are in mechanical equilibrium, the film stress–thickness (i.e., the wafer curvature) is proportional to the substrate surface coverage Θ with slope equal to minus the surface stress of the substrate. This is a new mechanism to explain the initial compressive stress during island growth processes and shows that wafer curvature experiments may not provide an accurate measure of the true stress state inside the film when the film is still thin. An interesting companion study would be to look at wafer curvature for the case of liquid drops on the substrate surface. On the other hand, at the late stage of VW growth, namely, growth of continuous polycrystalline film, adatom incorporation into grain boundaries during the growth of polycrystalline films represents the most likely mechanism for producing the compressive film stresses commonly measured during growth. Once again, molecular dynamics simulation results demonstrate that extra atoms are incorporated into grain boundaries (not in the form of bulk self-interstitials) and the film stress–thickness product is linearly proportional to the surface density of extra atoms incorporated at the grain boundaries ρX regardless of the grain size and deposition rate. This shows that the grain boundary insertion model is a valid model for the generation of compressive film stress. However, due to the timescale limitations in molecular dynamics simulations ( 109 sec), it is impractical (and almost impossible to date) to simulate the relaxation process during growth interrupt. Thus, what’s really happening during growth interrupt remains a mystery. It is possible that the self-interstitial diffusivity at the grain boundaries is fast enough to reach the timescale of relaxation observed in experiments. It is also possible that grain boundary grooving, during which the grain boundaries are replaced by free surfaces, can relax the compressive film stress because extra atoms are driven out of the grain boundaries via fast surface diffusion rather than the relatively slower grain boundaries diffusion. It is also interesting to observe grain boundary migration near the free surface due to the motions of surface steps. This suggests a new mechanism leading to grain growth during polycrystalline film growth processes. In summary, in this chapter the mechanisms leading to film stress development during the initial and final stages of VW growth have been briefly discussed. It is demonstrated that the effects of surface/interface stresses are very important in the early stages of both VW film growth stress. While surface defects such as surface adatoms, steps, and dislocations do modify the surface stress, these changes are not of sufficient magnitude to create the compressive film stresses commonly measured during the initial and final stages of VW growth. However, since surface processes during thin film growth remain largely unclear, there may be other classes of defects that are yet unknown and possibly of some importance. As for the compressive film stress measured during the late stage of VW growth, it is very likely that the grain boundaries insertion model [11] can explain late stage compressive film stress

54

C.-W. Pao

development. However, due to the limitation in the simulation timescale of current MD simulations, the detail mechanism of stress relaxation during growth interrupt remains a challenge to overcome.

References 1. Freund LB, Suresh S. Thin film materials: stress, defect formation and surface evolution. Cambridge: Cambridge University Press; 2003. 2. Stoney GG. The tension of metallic films deposited by electrolysis. Proc R Soc Lond A Math Phys Eng Sci. 1909;82(553) 3. Koch R, Hu D, Das AK. Compressive stress in polycrystalline Volmer-weber films. Phys Rev Lett. 2005;94(14):146101. 4. Hoffman R. Stresses in thin films: the relevance of grain boundaries and impurities. Thin Solid Films. 1976;34(2):185–90. 5. Nix WD, Clemens BM. Crystallite coalescence: a mechanism for intrinsic tensile stresses in thin films. J Mater Res. 1999;14(08):3467–73. 6. Freund LB, Chason E. Model for stress generated upon contact of neighboring islands on the surface of a substrate. J Appl Phys. 2001;89(9):4866. 7. Cammarata RC, Trimble TM, Srolovitz DJ. Surface stress model for intrinsic stresses in thin films. J Mater Res. 2000;15(11):2468–74. 8. Friesen C, Thompson CV. Reversible stress relaxation during Precoalescence interruptions of Volmer-Weber thin film growth. Phys Rev Lett. 2002;89(12):126103. 9. Friesen C, Thompson CV. Correlation of stress and atomic-scale surface roughness evolution during intermittent Homoepitaxial growth of (111)-oriented Ag and Cu. Phys Rev Lett. 2004;93(5):056104. 10. Pao C-W, Srolovitz D, Thompson C. Effects of surface defects on surface stress of Cu(001) and Cu(111). Phys Rev B. 2006;74(15):1–8. 11. Chason E, Sheldon BW, Freund LB, Floro JA, Hearne SJ. Origin of compressive residual stress in polycrystalline thin films. Phys Rev Lett. 2002;88(15):156103. 12. Suzuki A, Mishin Y. Atomistic modeling of point defects and diffusion in copper grain boundaries. Interface Sci. 2003a;11(1):131–48. 13. Guduru P, Chason E, Freund L. Mechanics of compressive stress evolution during thin film growth. J Mech Phys Solids. 2003;51(11):2127–48. 14. Sheldon BW, Ditkowski A, Beresford R, Chason E, Rankin J. Intrinsic compressive stress in polycrystalline films with negligible grain boundary diffusion. J Appl Phys. 2003;94(2):948. 15. Pao C-W, Foiles S, Webb E, Srolovitz D, Floro J. Thin film compressive stresses due to Adatom insertion into grain boundaries. Phys Rev Lett. 2007;99(3):1–4. 16. Pao C-W, Foiles S, Webb E, Srolovitz D, Floro J. Atomistic simulations of stress and microstructure evolution during polycrystalline Ni film growth. Phys Rev B. 2009;79(22):1–9. 17. Pao C-W, Srolovitz D. Stress and morphology evolution during island growth. Phys Rev Lett. 2006a;96(18):1–4. 18. Pao C-W, Srolovitz DJ. Atomistic simulation of stress evolution during island growth. J Mech Phys Solids. 2006b;54(12):2527–43. 19. Baskes M. An atomistic study of solid/liquid interfaces in binary systems. JOM. 2004;56(4):45–8. 20. Baskes MI. Many-body effects in fcc metals: a Lennard-Jones embedded-atom potential. Phys Rev Lett. 1999;83(13):2592–5. 21. Baskes MI, Stan M. An atomistic study of solid/liquid interfaces and phase equilibrium in binary systems. Metall Mater Trans A. 2003;34(3):435–9. 22. Srinivasan SG, Baskes MI. On the Lennard–Jones EAM potential. Proc R Soc Lond A Math Phys Eng Sci. 2004;460(2046):1649–72.

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23. Floro JA, Hearne SJ, Hunter JA, Kotula P, Chason E, Seel SC, Thompson CV. The dynamic competition between stress generation and relaxation mechanisms during coalescence of Volmer–Weber thin films. J Appl Phys. 2001;89(9):4886. 24. Ruud JA, Witvrouw A, Spaepen F. Bulk and interface stresses in silver-nickel multilayered thin films. J Appl Phys. 1993;74(4):2517. 25. Gill SPA, Gao H, Ramaswamy V, Nix WD. Confined capillary stresses during the initial growth of thin films on amorphous substrates. J Appl Mech. 2002;69(4):425. 26. Bhandari A, Sheldon BW, Hearne SJ. Competition between tensile and compressive stress creation during constrained thin film island coalescence. J Appl Phys. 2007;101(3):033528. 27. Daw MS, Foiles SM, Baskes MI. The embedded-atom method: a review of theory and applications. Mater Sci Rep. 1993;9(7):251–310. 28. Foiles SM, Hoyt J. Computation of grain boundary stiffness and mobility from boundary fluctuations. Acta Mater. 2006;54(12):3351–7. 29. Kelchner CL, Plimpton SJ, Hamilton JC. Dislocation nucleation and defect structure during surface indentation. Phys Rev B. 1998;58(17):11085–8. 30. Ling WL, Bartelt NC, McCarty KF, Carter CB. Twin boundaries can be moved by step edges during film growth. Phys Rev Lett. 2005;95(16):166105. 31. Spaepen F. Interfaces and stresses in thin films. Acta Mater. 2000;48(1):31–42. 32. Zandbergen HW, Pao CW, Srolovitz DJ. Dislocation injection, reconstruction, and atomic transport on {001} Au terraces. Phys Rev Lett. 2007;98(3) 33. Sørensen MR, Mishin Y, Voter AF. Diffusion mechanisms in Cu grain boundaries. Phys Rev B. 2000;62(6):3658–73. 34. Suzuki A, Mishin Y. Interaction of point defects with grain boundaries in fcc metals. Interface Sci. 2003b;11(4):425–37. 35. Friesen C, Seel SC, Thompson CV. Reversible stress changes at all stages of Volmer–Weber film growth. J Appl Phys. 2004;95(3):1011.

4

Characterizing Mechanical Properties of Polymeric Material: A Bottom-Up Approach Lik-ho Tam and Denvid Lau

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Atomistic Modeling of Polymeric Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Atomistic Model of Polymeric Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Atomistic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Physical Properties of Polymeric Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Coarse-Grained Modeling of Polymeric Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Coarse-Grained Model of Polymeric Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Coarse-Grained Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effect of Structural Voids on Polymeric Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58 60 60 69 71 77 78 83 84 87 87

Abstract

Polymeric materials have received tremendous attention in both industrial and scientific communities, and can be readily found in applications across a large range of length scales, ranging from the nanoscale structures, such as the photoresist lithography in the micro-electro-mechanical systems, to the macroscale components, such as the adhesive bonding in the aerospace industry and civil infrastructures. The durability of these applications is mainly determined by the mechanical

L.-h. Tam Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong, China D. Lau (*) Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong, China Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_5

57

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L.-h. Tam and D. Lau

reliability of the constituent polymeric materials. In this chapter, a review of the bottom-up approach to investigate the mechanical properties of the polymeric materials is provided. A dynamic algorithm is developed to achieve the crosslinking process of the atomistic network, which possesses the mechanical properties in a good accordance with the experimental measurements. Meanwhile, the moisture effect on the mechanical properties is studied based on the atomistic model, and it is found that the mechanical properties of the solvated models show no significant deterioration. Furthermore, the predicted mechanical properties at the atomistic level are used to develop the cross-linked network at the mesoscale, which enables the investigation of the effect of the structural voids on the polymeric materials. The simulation results demonstrate the strong mechanical reliability of the synthetic polymeric materials during the long-term service life. The multiscale method summarized in this chapter provides a versatile tool to link the nano-level mechanical properties of the polymeric materials to the macro-level material behaviors.

Keywords

Molecular dynamics · Polymeric material · Mechanical properties · Moisture · Structural voids

1

Introduction

Polymeric materials possess the three-dimensional covalent networks formed by the polymerization process, which generally exhibit remarkable physical properties, such as strong mechanical reliability as well as high thermal and chemical resistance. In practice, these materials are frequently used in various engineering applications across a large range of length scales, including micro-electro-mechanical systems (MEMS), aerospace industry, and civil infrastructures. The increasing usage of the polymeric materials is due to the fact that they have the potential to yield cheaper fabrication and cheaper devices, and more fundamentally, their remarkable physical properties can be put into good use in a variety of products. As the polymeric materials are the basis for various engineering applications, a fundamental understanding of the material structure and mechanical properties is essential for predicting the long-term performance of the polymer-based engineering applications and devices, as well as enabling new technologies. After the quantitative property characterization of the polymeric materials, the mechanical reliability of the polymeric materials under the environmental exposures shall also be quantified. Among various environmental factors, the moisture is the most critical factor. As the polymeric materials are sensitive to the humid environment, the moisture can be absorbed into the polymeric material structure and interact with the constituent materials [1]. The interactions are found to influence the mechanical properties of the constituent materials [2–7]. Such moisture-affected difference in the mechanical properties can be a major concern for the durability of the engineering applications involving the polymeric materials, as it can lead to

4

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the structural failure ahead of the expected service life. However, the origin of the moisture-affected mechanical behaviors is not entirely clear at this stage, which can be a limitation on the further application of the polymeric materials. Consequently, a fundamental knowledge of the dependence of the polymer performances on the moisture condition is required. After determining the mechanical properties of the polymeric materials and the dependence on the moisture condition, the relationship between the structural voids and the polymer mechanical properties needs to be clarified, as the voids affect the moisture diffusion into the system, which may initiate the cracks that lead to the structural failure. During the curing process, microscopic structural voids, such as the free volume, can be developed and trapped in the polymerized structure [8, 9]. Meanwhile, during the long-term service life, the voids can also be formed in the polymerized structure even for the initially void-free specimens, leading to the changes in the material structure [10]. Microscopic structural voids generally exist in the polymerized network and have a detrimental influence on the mechanical properties, but the relationship remains elusive, which is hindered by the complex structure of the polymeric materials. Therefore, the investigations of the microscopic structural voids can provide a basic understanding of the mechanical properties of the polymeric materials, and eventually lead to a more comprehensive understanding on the durability of the entire system. In recent years, molecular dynamics (MD) simulation has been demonstrated as a fundamental and versatile tool that enables us to investigate the molecular structure and interaction of the polymeric materials, which can be useful for understanding the mechanical properties of the polymeric materials and the interaction with the ambient environment (i.e., the moisture in particular). The molecular-level understandings enable one to explore the mechanism of the macroscopic behaviors of the polymeric materials, such as the fiber-reinforced polymer bonded concrete system in the civil infrastructures, which are difficult to obtain through the experimental and theoretical approaches. The MD simulations are first applied to the structural characterization of the polymeric materials at the nanoscale, which have been reported since 1990s [11–28]. Whereas these MD studies have made significant progress in the atomistic modeling and property measurement of the polymeric materials, several key obstacles are still present including the low cross-linking degree [12, 13] and relatively low accuracy in the property prediction [15, 19]. Therefore, it can be favorable to develop a computational modeling algorithm with a significant improvement in terms of the time efficiency and the accuracy, as well as a high extensibility to other cross-linked networks. With the developments in the computational simulation of the polymeric materials, the structural behaviors of the synthetic polymer have been studied under the influence of the surrounding environment, particularly in the presence of the moisture [6, 29–44]. From the previous studies, it can be seen that the effect of moisture may be beneficial or detrimental to the polymeric materials, and MD simulations are useful in providing the molecular mechanics behind the experimental observations. The earlier studies mainly focus on the property measurements of the polymeric materials, with lesser consideration of the structural voids that exist in the structure, and their effect on the mechanical properties of the polymeric materials [45, 46]. The experimental studies have reported that the voids dimension in the polymeric materials is normally of several nanometers [8, 9]. It is noted that the polymeric

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materials with a certain number of voids are generally at the submicron scale, which requires excessive computational power to model the cross-linking process of the structure. To overcome the limitation of the length scale, the coarse-grained (CG) MD simulations can be performed by using the mesoscale models [47, 48]. In this work, one representative polymeric material is chosen and investigated. Specifically, the epoxy-based materials are an important family of the polymeric materials, which are widely used for bonding the reinforced materials to the building structures in the structural applications, and have become favorable in the MEMS applications. Typically, SU-8 is a favorable epoxy photoresist, which possesses the common structural characteristics of the epoxy-based materials. A number of preliminary studies have shown that the SU-8 epoxy has a high Young’s modulus [49–51], which is important for achieving a good mechanical stability of the final products. In addition, the SU-8 epoxy has reasonably good adhesion to the commonly used inorganic substrates, which is very crucial to the functionality of the final products. By investigating the SU-8 epoxy, it is expected that the mechanical behavior of the polymeric materials commonly seen in the engineering fields can be interpreted. In this chapter, we present the characterization of the mechanical behaviors of the polymeric materials at the microscale by using a bottom-up approach, with a focus on the structural characterizations and property measurements, so as to develop a mechanically durable polymeric material for the broad engineering applications, including micro-electronics device fabrication, aerospace industry, and civil construction. MD simulation is used to construct the atomistic and CG polymerized networks, which are close to what we can observe from the experiment. The constructed models are used as the basis for studying the mechanical behavior of the polymeric materials at the nano- and mesoscale. It is inferred that this work can yield much significant scientific knowledge and quantitative information upon the mechanical behavior of the polymeric materials, and provide a bottom-up approach for the microscopic investigation of the polymeric material systems.

2

Atomistic Modeling of Polymeric Material

This section shows the atomistic MD investigation on the structure and mechanical properties of the polymeric material. A computational algorithm is developed to construct the cross-linked network, and the mechanical properties are determined by means of dynamic deformations, which are compared with the experimental measurements to validate the constructed model. Furthermore, the moisture effect on the mechanical behaviors of the polymeric material is investigated.

2.1

Atomistic Model of Polymeric Material

SU-8 epoxy possesses the common structural characteristics of the polymeric materials, and it is selected as the representative here. In the MD simulation, it is a great challenge to build up the atomistic model close to the real system. The knowledge of

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the chemical reaction in the material synthesis is required to model the atomistic structure. The SU-8 epoxy is a homogeneous material formed by the cross-linking reaction between the monomers, which does not require the external liquid curing agent. The cross-linking reaction occured in the material preparation can be simulated by using the MD simulation. The detailed atomistic interactions in the SU-8 molecular structure are described by a force field. A force field is developed for the specific material, which can mimic the nature of the interactions in a realistic or appropriate way. As the accuracy of the MD simulations highly depends on the selected force field [15, 18], the investigations of the SU-8 epoxy are carried out by using three different force fields independently in order to obtain a comprehensive picture, which include Consistent Valence Force Field (CVFF) [52, 53], Dreiding force field [54], and Polymer Consistent Force Field (PCFF) [55, 56]. These three force fields are chosen as they are well reported for the simulations of the polymeric materials [13, 19–23]. In addition, the AMBER (Assisted Model Building with Energy Refinement) and COMPASS (Condensedphase Optimized Molecular Potentials for Atomistic Simulation Studies) force fields have also been used in the MD simulations of the polymeric materials [15, 33, 39, 40, 57]. As AMBER and CVFF belong to the same group of the classical force fields, while COMPASS and PCFF belong to the same group of the second-generation force fields, they are not used in this work. The applicability and limitation of the simulation using these force fields are summarized in Table 1. Partial charges assignment method of the atoms varies among these three force fields. In the simulations using CVFF and PCFF, the partial charges are estimated by using a bond increment method [56, 58]. The bond increment δij is described as the partial charge contributed from atom j to atom i. This method assigns δij with equal magnitude and opposite sign to each pair of the bonded atoms i and j. For atom i, the charge is calculated by the summation of δij as given in Eq. (1), qi ¼

X

δij ,

(1)

j

where j runs over all the atoms which are bonded to atom i directly. In addition, the δij for any pair of the same atom type is zero. Meanwhile, partial charges in Dreiding force field are calculated by the charge equilibrium (QEq) method [59]. It is reported that the charge distributions by QEq result in a good agreement with the experimental measurements and ab initio calculations. During the MD modeling and simulation, the van der Waals (vdW) and Coulombic interactions are calculated with a cutoff distance of 10 Å. Atomistic modeling of the SU-8 epoxy is performed in Materials Studio software from Accelrys [60]. After constructing the cross-linked SU-8 network with a clear definition of the molecular structure and interaction between atoms, MD simulations are performed in the open source code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [61]. The msi2lmp tool in LAMMPS is used to generate the input data containing the structural information and force field parameters. Periodic boundary conditions are applied to all three directions of the crosslinked network to get rid of the boundary effect.

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Table 1 The applicability and limitation of the simulation using the specific force field Force field Materials

CVFF CVFF is derived for small organic crystals and gas phase structures, and further optimized for organic, polymeric, and biological materials.

Properties

CVFF is intended for Dreiding force field studying structures and predicts bulk material binding energies, and it properties with a predicts vibrational moderate accuracy, frequencies, including geometries, conformational energies, conformational energies, and mechanical intermolecular binding properties reasonably energies, and crystal well. packing. In the simulations using these force fields, the atomistic interactions are nonreactive, where no charge transfer and chemical reactions occur. Therefore, these force fields are not applicable to the simulations of electronic transitions, electron transport phenomena, and proton transfer, where ab initio calculations are generally used.

Limitations

Dreiding Dreiding force field is developed for the simulation of biological, organic, and some inorganic molecules.

PCFF PCFF is developed for polymers, organic and inorganic materials, including about 20 inorganic metals, as well as for carbohydrates, lipids, and nucleic acids. PCFF is intended for the study of cohesive energies, mechanical properties, compressibilities, heat capacities, and elastic constants.

The cross-linked SU-8 epoxy is polymerized from the monomers. The SU-8 monomer comprises four components of diglycidyl ether of bisphenol A (DGEBA) and the chemical structure of which is shown in Fig. 1a. With eight epoxide groups in each monomer, SU-8 possesses the highest epoxide functionality among the commercially available photoresists, which enables SU-8 epoxy to grow into a highly cross-linked network. The molecular model of the SU-8 monomer is shown in Fig. 1b. Before the cross-linking process, a total of forty SU-8 monomers (7960 atoms) are packed into a three-dimensional periodic simulation cell with a density of 1.07 gcm3, which is the typical value for SU-8. The initial uncross-linked structure of the SU8 epoxy is constructed at the temperature of 300 K, which uses a Monte Carlo packing algorithm according to the rotational isomeric states model [62]. The amorphous SU8 structure is equilibrated for 10 ps in the canonical (NVT) ensemble at 300 K, followed by another 10 ps equilibration in the isothermal and isobaric (NPT) ensemble at 300 K and 1 atm. The corresponding integration time step is 1 fs. During the entire equilibration process, constant temperature and pressure are controlled by the Nosé–Hoover thermostat and the Andersen barostat, respectively [63]. The reason for choosing this short time span (10 ps NVT þ 10 ps NPT) is based on the observation of the potential energy (PE), as shown in Fig. 2. The results are obtained from the initial equilibration at 300 K before the cross-linking process. The results from the CVFF simulation are used here as an illustration. It can be seen from Fig. 2a that the PE oscillates at a narrow range during the equilibration run, with an average of 56,506 kcalmol1. Additionally, there is a 0.5 ps energy minimization process before and after the equilibration run, which also

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Fig. 1 Monomer of the SU8 epoxy: (a) chemical structure and (b) molecular model

minimizes the energy considerably. In comparison, a longer simulation (50 ps NVT þ 50 ps NPT) has been performed based on the same initial structure. The result indicates that the oscillation of PE continues during the 100 ps simulation, as shown in Fig. 2b, and the average is 56,451 kcalmol1. Based on the similar trend of the PE and the small difference between the averages, the short time span of the simulation (10 ps NVT þ 10 ps NPT) is used in this work. It is always good to simulate for a longer time span for a better equilibration, while the current strategy is believed to be sufficient for

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Fig. 2 Evolution of the potential energy of the SU-8 epoxy during (a) a short and (b) a long initial equilibration run at 300 K

obtaining the SU-8 structure with no severe geometric distortion. As the cross-linking reaction of the SU-8 epoxy is usually carried out at an elevated temperature, the same equilibration process is used to equilibrate the structure at the elevated temperature before performing the cross-linking reaction. A temperature of 368 K (95  C) is commonly used for the fabrication process of the SU-8 sample, and it is chosen here. After the above equilibration processes, the SU-8 model is polymerized by using a crosslinking algorithm. Figure 3 shows the flowchart describing the modeling procedure. Before each cross-linking reaction, the distance between the available reactive atoms has been calculated, and the potential reactive atoms located inside the current reaction radius are recognized, as shown in Fig. 4a. The reaction radius during the cross-linking process is set to be 3 Å initially and with an increment of 0.5 Å. The maximum reaction radius is set to be 10 Å, as the cross-linking process with a reaction radius over 10 Å usually requires a long equilibration process for alleviating the geometric distortion in the model. After the determination of the reactive atoms, the epoxide groups comprising those recognized reactive atoms are open, as shown in Fig. 4b. The recognized reactive atoms are then connected by a new bond to form the cross-link (Fig. 4c). After the bond creation (cross-links being formed), the unreacted atoms at the open epoxide groups are saturated with hydrogen atoms (Fig. 4d). It is noticed that the combination of the distance and energy criteria would require a lot of computational power for the iteration process. In addition, the computation using the distance criteria is more straightforward and also requires lesser computational power, as shown from the various force field definitions. From this perspective, we decide to use the distance-based cross-linking approach, which is also well adopted by various researchers [12, 15, 19]. After each cross-linking reaction, the structural information is updated by introducing new bonds, angles, torsional angles, and improper angles into the cross-linked structure. In order to relax the SU-8 structure after each cross-linking reaction, a combined equilibration process consisting of four steps is employed: (1) a 0.5 ps geometry optimization; (2) a 5 ps NVT ensemble equilibration; (3) a 5 ps NPT ensemble equilibration; and (4) a 0.5 ps geometry optimization. During such equilibration process, the lengths of the newly created bonds are relaxed to the equilibrium value,

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Fig. 3 Flowchart of the cross-linking algorithm used in the construction of the SU8 epoxy. Reproduced from Ref. [45] with permission from the Royal Society of Chemistry

which can alleviate the geometric distortion in the newly cross-linked structure. Within each reaction radius, the cross-linking reaction is performed at most three times or it stops if no reactive atoms are identified. The cross-linking process of the SU-8 model is finished when the maximum reaction radius (10 Å) is achieved or all available potential reactive atoms are reacted. Once the cross-linking process is finished, a short equilibration process is used, where the temperature is gradually changed from elevated temperature to room temperature. Specifically, the SU-8 model is first equilibrated in the NVT ensemble for 10 ps, and then in the NPT ensemble for another 10 ps. The equilibration process is performed at the temperature of 368 K, 334 K and 300 K, respectively. Three SU-8 epoxy networks are built by performing the cross-linking process under the chosen force fields separately. Though the cross-linked structure is equilibrated during the cross-linking process, the relatively short time frames used in the above equilibration processes are not sufficient for the epoxy network to reach the fully equilibrated state. In order to obtain a fully relaxed molecular structure, the cross-linked SU-8 model is further equilibrated. Several

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Fig. 4 Procedures of the cross-linking process of the SU-8 epoxy: (a) the reactive atoms located inside the reaction radius are recognized (marked with filled circles); (b) the epoxide groups comprising the recognized reactive atoms are open by deleting bonds; (c) the recognized reactive atoms are connected by a new bond to form the cross-link; (d) the unreacted atoms at the open epoxide groups are saturated with hydrogen atoms. Atoms located between the two epoxide groups are denoted as “R” for better clarity. Reproduced from Ref. [45] with permission from the Royal Society of Chemistry

groups have developed the equilibration scheme with a control of temperature and pressure that can speed up the equilibration process and relieve the residual stress of the polymeric networks [25, 31, 64]. Here, the equilibration scheme containing MD simulations at high pressures is used for equilibrating the cross-linked network, which accelerates the compression of the cross-linked structure by overcoming the large energy barriers effectively. The details of the equilibration scheme are shown in Table 2, which consists of seven equilibration cycles in both NVT and NPT ensembles at 300 K. During the equilibration process, the density of the SU-8 model is adjusted through the pressure control. The pressure applied in the NPT simulation is gradually increased from atmospheric pressure to 50,000 atm in the first three equilibration cycles (as indicated in Table 2) with a longer time frame (50–150 ps each) that allows an adequate relaxation of the structure. The SU-8 epoxy network is compressed efficiently with the large pressure applied on the cross-linked structure. After that, the pressure is steadily reduced to 1 atm in the last four cycles (as indicated in Table 2) with a shorter time frame (5–15 ps each) in order to control the large pressure jumps. Along with the decrease of the applied pressure, the cross-linked structure is decompressed accordingly. These compression and decompression steps only add a short period of simulation time to the equilibration process, but they greatly improve the accuracy of the achieved densities [25, 64]. Finally, a 5 ns

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Table 2 Equilibration scheme incorporating the high pressure control used in the equilibration process of the SU-8 epoxy Cycle 1 2 3 4 5 6 7

Ensemble NVT NPT NVT NPT NVT NPT NVT NPT NVT NPT NVT NPT NVT NPT

Conditions 300 K 300 K, 1000 atm 300 K 300 K, 30,000 atm 300 K 300 K, 50,000 atm 300 K 300 K, 25,000 atm 300 K 300 K, 5000 atm 300 K 300 K, 5000 atm 300 K 300 K, 1 atm

Time frame (ps) 100 50 150 50 150 50 150 5 15 5 15 5 15 5000

relaxation under a constant temperature of 300 K and a constant pressure of 1 atm is carried out such that a fully equilibrated state can be achieved. By examining the rootmean-square displacement (RMSD) of the atoms, which keeps at a constant level before the 5 ns NPT equilibration run is completed, as shown in Fig. 5a, it implies that the fully equilibrated state has been obtained. Meanwhile, the potential energy of the cross-linked SU-8 epoxy networks as a function of the simulation time is shown in Fig. 5b–d. The average potential energy is 28,480  198 kcalmol1 (CVFF), 23,516  387 kcalmol1 (Dreiding), and 11,811  180 kcalmol1 (PCFF), respectively. The variation of potential energy in the cross-linked system is small along the simulation time, which also implies that the cross-linked structure is close to the equilibrium state. After the equilibration process, the SU-8 epoxy network is further relaxed for 200 ps in NVT ensemble in the case of bulk modulus calculation. One representative model of the highly crosslinked SU-8 epoxy network is shown in Fig. 6. In the investigation of the moisture effect on the SU-8 mechanical properties, the atomistic model constructed by using the CVFF potential is chosen as the starting configuration. The equilibrium moisture concentration is found to be 3.3 wt% in SU8 [65], and is less than 4 wt% of the DGEBA-based epoxy [1, 2, 66], where DGEBA is the basic component of the SU-8 monomer. In this work, different amount of water molecules are added to the cross-linked SU-8 epoxy network. The added water molecules are located at the lowest energy sites in the cross-linked structure, which vary from 1, 2, 3, 4 wt% of the model molar mass. The epoxy network solvated with 4 wt% moisture content is used to investigate the SU-8 mechanical properties at the ultimate wet scenario. Water molecules employ the parameters of the TIP3P model with a cutoff distance of 10 Å [67]. During the MD equilibration, the bond lengths and angles of the water molecules are kept constant by using the SHAKE algorithm [68]. The solvated SU-8 epoxy networks experience the same equilibration process as shown in Table 2.

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Fig. 5 Evolution of (a) root-mean-square displacement (RMSD) and (b, c, and d) potential energy of the SU-8 epoxy during the last 5 ns NPT equilibration run under three different force fields Fig. 6 Cross-linked structure of the SU-8 epoxy

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Atomistic Simulations

After the equilibration process of the SU-8 epoxy models, the Young’s modulus (E) and bulk modulus (K ) of the equilibrated models under dry scenario are calculated by the uniaxial tensile deformations and the volumetric deformations, respectively. The shear modulus (G) and Poisson’s ratio (υ) are computed by applying the linear elasticity theory [69]. Due to the timescale limitation of the MD simulations, the strain rate used in the dynamic deformations [20, 21, 23] is much higher than that used in the experiments. Previous MD studies show that the calculated Young’s modulus of the polymeric materials is not sensitive to the change of the strain rate, while the yield stress increases with the strain rate [21, 23]. In the tensile deformation here, the simulation cell along the loading direction X is elongated continuously with a strain rate of 1  108 s1, which is in a range typical for the MD simulations, and the atmospheric pressure is maintained in the two perpendicular directions. The deformation is carried out at 300 K. At each deformation step, the equilibrated models are deformed by 0.1% in strain followed by a 10 ps equilibration process before next deformation step. For all the deformation processes, the equilibrated models are deformed by 3.0% in total. The virial stress tensors are recorded during the entire deformation process, and they are calculated by using Eq. (2), where a and b denote the values X, Y, Z to represent the six components of the symmetric tensor. 1 ð2K ab þ W ab Þ, V N 1X ¼ mi via vib , 2 n¼1

Sab ¼ K ab

of plane torsion þ W bond þ W out W ab ¼ W pairwise ab ab ab þ W ab ¼

Np Nb 1X 1X ðr 1a F1b þ r 2a F2b Þ þ ðr F þ r 2a F2b Þ 2 n¼1 2 n¼1 1a 1b

þ

Na 1X ðr F þ r 2a F2b þ r 3a F3b Þ 3 n¼1 1a 1b

þ

Nt 1X ðr 1a F1b þ r 2a F2b þ r 3a F3b þ r 4a F4b Þ 4 n¼1 N

þ

oop 1X ðr F þ r 2a F2b þ r 3a F3b þ r 4a F4b Þ, 4 n¼1 1a 1b

(2)

where Sab is the virial stress tensor, Kab is the kinetic energy tensor, and Wab is the virial tensor, which is calculated by considering various potential components. Specifically, Wabpairwise is a pairwise energy contribution consisting of vdW and Coulombic interactions, where r1a and r1b are the positions of the two atoms in the pairwise interaction, and F1a and F1b are the forces on the two atoms resulting from

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the pairwise interaction. There are similar terms for the Wabbond bond, Wabangle angle, Wabtorsion torsion, and Wabout-of-plane out-of-plane interactions. After the deformation, E of the model is determined, which is calculated by performing a regression analysis on the relatively linear portion of the stress–strain curve (i.e., the initial slope of the stress-strain curve) and is illustrated in Eq. (3), E¼

σ XX , eXX

(3)

where σ XX and eXX are the stress and strain tensor components along the loading direction X, respectively. Bulk modulus describes the material response under a uniform pressure. Here, it is determined by the volumetric deformation, in which equal axial strains are applied in all three orthogonal directions simultaneously. The calculation of bulk modulus is carried out at a constant temperature of 300 K. The dynamic deformations are applied in the form of dilatation by keeping the strain rate as 1  108 s1. The overall dilation of the model is determined by e = eXX + eYY + eZZ, where eXX, eYY, and eZZ are the infinitesimal strain tensor components with respect to the coordinate directions X, Y, and Z, respectively. The overall stress of the model is calculated by σ = 1/3 (σ XX + σ YY + σ ZZ), where σ XX, σ YY, and σ ZZ are the volume-averaged virial stress tensor components calculated by using Eq. (2). K is then calculated as the initial slope of the curve representing the overall stress σ against the volumetric deformation e as shown in Eq. (4), K¼

σ 1=3ðσ XX þ σ YY þ σ ZZ Þ ¼ : e eXX þ eYY þ eZZ

(4)

Assuming that the material is homogeneous and isotropic, with any two elastic constants available from direct measurements it is considered to be sufficient for a full characterization of the mechanical properties. Therefore, G and υ are computed based on the calculated E and K by applying the linear elasticity theory [69]. Particularly, G is given in Eq. (5), G¼

3KE , 9K  E

(5)

3K  E : 6K

(6)

and υ is determined by using Eq. (6), υ¼

For the solvated models, E is determined and used as an indicator to illustrate the moisture effect on the mechanical properties of the SU-8 epoxy.

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Physical Properties of Polymeric Material

The physical properties of the SU-8 epoxy obtained from the MD simulations under dry scenario are compared with the experimental data in Table 3. The cross-linking degree achieved after the polymerization process is an important parameter for validating the constructed atomistic model. It is defined as the ratio of the cross-linked reactive atoms divided by all the potential reactive atoms before the cross-linking process. The SU-8 atomistic model is obtained by carrying out the cross-linking algorithm without termination. The resulted cross-linking degree of the SU-8 epoxy networks created by using different force fields is shown in Fig. 7. Initially, there are abundant reactive atoms available for the crosslinking reaction, which is indicated by the strong dependence between the crosslinking degree and the reaction radius. As the reaction goes on with the increasing reaction radius, lesser reactive atoms are available for the cross-linking reaction, and the curve becomes steady when the reaction radius is over 6 Å. The final crosslinking degree of all the constructed atomistic models is higher than 80%. Specifically, the SU-8 epoxy network constructed under Dreiding force field shows the maximum cross-linking degree of 88.1%, followed by 82.5% under PCFF and 81.9% under CVFF. The influence of the equilibration time span on the maximum cross-linking degree is investigated by alternating the equilibration scheme before the cross-linking reaction. In consideration of the similar evolution of the crosslinking degree, the final cross-linked structure from the CVFF simulation is chosen as an illustration. The cross-linked structure undergoes a 10 ns equilibration (5 ns NVT þ 5 ns NPT) before the cross-linking process at a larger reaction radius. Another set of the cross-linking process is carried out on the same structure by using the original equilibration scheme (5 ps NVTþ 5 ps NPT). The maximum reaction radius is found to be 11 Å for both cases, and the final cross-linking degree of the model after the 10 ns equilibration is 85%, while the original model is 84%, as shown in Fig. 8. It can be seen that at the extended reaction radius, the new crosslinks are formed slowly for both cross-linking process. Meanwhile, it is noted that a longer equilibration run can enhance the formation of cross-links, but it demands more computational power than the proposed equilibration scheme. Considering the large computational consumption and a limited improvement in the cross-linking degree, current reaction radius range and equilibration process can be considered Table 3 Simulated and experimental results for the SU-8 epoxy at 300 K. Reproduced from Ref. [45] with permission from the Royal Society of Chemistry Force field Cross-linking degree (%) ρ (gcm3) E (GPa) K (GPa) G (GPa) υ

CVFF 81.9 1.04  0.002 4.43  0.23 4.35  0.18 1.66  0.11 0.33  0.02

Dreiding 88.1 1.05  0.001 4.42  0.12 3.72  0.15 1.70  0.06 0.30  0.01

PCFF 82.5 1.05  0.002 2.67  0.16 2.88  0.10 0.99  0.07 0.35  0.02

Expt.  80 1.07–1.20 2.70–4.02 3.20 1.20 0.33

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Fig. 7 Cross-linking degree of the SU-8 epoxy as a function of the reaction radius under three force fields. Reproduced from Ref.[45] with permission from the Royal Society of Chemistry

Fig. 8 Cross-linking degree of the SU-8 epoxy as a function of the reaction radius under CVFF potential: effect of the equilibration time span

reasonable. Overall, the cross-linking algorithm used in this work is capable of constructing the epoxy network with the cross-linking degree close to those epoxy-based materials through various experimental approaches [70, 71]. For the structural characterization of the SU-8 epoxy network, density (ρ) is another critical parameter which should be consistent with the experimental value. In the last 5 ns equilibration run in the NPT ensemble, the SU-8 epoxy network is equilibrated to reach the minimum potential energy. The three orthogonal directions of the simulation cell are adjusted independently under the atmospheric pressure. The density of the cross-linked SU-8 structure is sampled every 10 ps during this 5 ns equilibration process, as shown in Fig. 9. The density at the last 2 ns simulation is relatively stable in the three cases, which indicates that the equilibrium state is reached in the three systems. In order to minimize the statistical error, only the recorded density from the

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Fig. 9 Evolution of density of the SU-8 epoxy at the last 5 ns NPT equilibration run under three different force fields

final 2 ns equilibration run is accounted for the density calculation. The average density of the SU-8 epoxy network is shown in Table 3, with respect to the chosen force fields. In comparison to the available value in the range of 1.07–1.20 gcm3, the slight underestimations of the density are observed from the three equilibrated SU-8 epoxy networks, with the value of 1.04  0.002 gcm3 (CVFF), 1.05  0.001 gcm3 (Dreiding), and 1.05  0.002 gcm3 (PCFF), respectively. The small discrepancies of the SU-8 epoxy networks demonstrate that the equilibration process as shown in Table 2 is effective to improve the accuracy of the densities achieved, and also indicate that the potential function of the chosen force fields are able to provide good mathematical approximations for calculating the potential energy of the cross-linked SU-8 structure. In view of the good agreement of ρ with the experimental measurement, the generated models of the cross-linked SU8 epoxy are regarded as reasonable structures close to those found in the real systems. The stress–strain curves of the cross-linked SU-8 epoxy network under the uniaxial tensile deformation are shown in Fig. 10. The stress along the loading direction shows a linear elastic response to the applied strain during the whole deformation process. The E is determined by performing a regression analysis on the stress–strain data from the 3% deformation. The computed E of the SU-8 epoxy under the chosen force fields is shown in Table 3 with a comparison to the experimental data. The data reported is from a single run, and the standard deviation comes from the linear regression of the stress–strain curve obtained from that simulation run. The E obtained under CVFF and Dreiding potentials are 4.43  0.23 GPa and 4.42  0.12 GPa, respectively, while a smaller value of 2.67  0.16 GPa is yielded under PCFF potential. In view of the experimental tensile test results ranging from 2.70–4.02 GPa [49–51, 72], the simulated E of the three SU-8 epoxy networks provide excellent agreements. Four other strain rates are examined in the tensile deformations under the chosen force fields independently, including 1  107 s1, 5  107 s1, 5  108 s1, and 1  109 s1, respectively. The relatively superposition of the stress–strain curves with different strain rates confirms that the Young’s modulus is not sensitive towards the change of strain rate, which is consistent with the existing simulation works from other similar epoxy systems [21, 23].

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Fig. 10 Stress–strain curves obtained from the tensile deformation of the SU8 epoxy under three force fields. Reproduced from Ref. [45] with permission from the Royal Society of Chemistry

The K of the SU-8 epoxy network obtained under dry scenario is 4.35  0.18 GPa (CVFF), 3.72  0.15 GPa (Dreiding), and 2.88  0.10 GPa (PCFF), respectively. Using the reported E and G from the experimental measurement [50], K is computed to be 3.20 GPa by using the linear elasticity theory [69], which is the closest point for comparison with our data. Close agreements are observed between the three simulated data and the reference value. By applying the linear elasticity theory, G and υ of the SU-8 epoxy network is computed and compared with the experimental measurements at 300 K, as shown in Table 3. The G of the SU-8 epoxy network under CVFF and Dreiding is 1.66  0.11 GPa and 1.70  0.06 GPa, respectively, which are greater than the reported data (1.20 GPa), while a smaller value of 0.99  0.07 GPa is observed in the case of PCFF. In the meantime, a good agreement is found between the computed υ (0.33  0.02) under CVFF and the experimental value (0.33), while the values for other two force fields are 0.30  0.01 (Dreiding) and 0.35  0.02 (PCFF), which are also in a reasonable accordance. By using the three force fields, the simulated cross-linking and physical properties of the constructed SU-8 epoxy network are in a good accordance with various experimental observables, including cross-linking degree, density, Young’s modulus, bulk modulus, shear modulus, and Poisson’s ratio, which demonstrates that the chosen force fields are able to characterize the atomistic interactions in the crosslinked structure in an appropriate way. Therefore, the three force fields can be used to investigate the cross-linking and physical properties of the epoxy-based materials with reasonable accuracy. To determine the effect of NPT equilibration time span on the physical properties of the cross-linked network, a longer equilibration run (8 ns) has been conducted, in comparison with the original 5 ns NPT simulation. Similarly, CVFF case is used as an example. The ρ and E of the SU-8 cross-linked structure are calculated for comparison. The average ρ from the last 2 ns equilibration is shown in Table 4, together with the ρ obtained from the original 5 ns case. The computed ρ between the two separate simulations is identical and the E is

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Table 4 Density and Young’s modulus of the SU-8 epoxy at 300 K: effect of NPT equilibration time span [45] Equilibration (ns) ρ (gcm3) E (GPa)

CVFF 5 1.04  0.002 4.43  0.23

8 1.04  0.002 4.44  0.10

Expt. 1.07–1.20 2.70–4.02

Fig. 11 Cross-linking degree of the original and larger SU8 epoxy as a function of the reaction radius under CVFF potential

relatively the same, which demonstrates that current time span of the 5 ns NPT simulation is sufficiently long for the equilibration process. With the current simulation run, it can be observed that the original predicted properties (e.g., ρ and E) are very close to the results from the longer simulation run, as well as from the experimental observations. Furthermore, in order to investigate the size effect on the physical properties of the SU-8 epoxy, a larger system originated from 15,920 atoms (eighty monomers) is simulated and compared with the original smaller system (forty monomers). The simulation is performed by using CVFF potential as an example. The modeling and simulation conditions are the same as described in previous section. The crosslinking degree of the two SU-8 epoxy networks is shown in Fig. 11. The trend of the two resulted curves is very similar, leading to a cross-linking degree of 78% for the larger system, while the value of the smaller system is 82%. Meanwhile, the computed ρ and E of the larger cross-linked structure are compared with the results of the smaller system, as shown in Table 5. The densities of the two cross-linked structures are nearly the same, and the measured Young’s moduli are also very close between these two systems. In view of the computed cross-linking degree, ρ, and E, a good agreement is observed between the two models with different sizes. Thus, it indicates the size adopted in the original simulation is large enough for evaluating the mechanical properties of the cross-linked SU-8 epoxy.

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Table 5 Density and Young’s modulus of the SU-8 epoxy at 300 K: effect of model size [45] Monomers ρ (gcm3) E (GPa)

CVFF 40 1.04  0.002 4.43  0.23

80 1.05  0.001 4.29  0.47

Expt. 1.07–1.20 2.70–4.02

Table 6 Density and Young’s modulus of the SU-8 epoxy at 300 K: effect of moisture Moisture content (wt%) 0 1 2 ρ (gcm3) 1.044  0.002 1.049  0.001 1.050  0.002 E (GPa) 4.43  0.23 5.03  0.15 4.46  0.09

3 1.053  0.001 4.18  0.23

4 1.056  0.001 4.37  0.19

After characterizing the mechanical properties of the SU-8 epoxy network, the moisture effect on the epoxy mechanical properties can now be quantified. The ρ of the SU-8 epoxy networks is calculated and shown in Table 6, with respect to the moisture content ranging from 0 to 4 wt%. For the solvated SU-8 epoxy networks, the ρ shows a steady increase with the increasing moisture content, and reaches 1.056  0.001 gcm3 at 4 wt% moisture content. For the SU-8 epoxy, no related experimental data are available for a direct comparison with the moisture-affected ρ. It is believed that the monotonic increase of ρ is due to the weight gain from the added water molecules. In view of the small variation of the ρ between dry and wet scenarios, it is inferred that the added water molecules does not make a significant difference to the SU-8 structural properties. Further investigation on the moistureaffected SU-8 mechanical properties can consolidate this finding. During the uniaxial tensile deformation, the stress–strain curves of the cross-linked SU-8 epoxy network are recorded as shown in Fig. 12 with respect to the moisture content. It can be observed that the stress of all the SU-8 epoxy networks along the loading direction shows a linear response to the applied strain, which indicates that the SU-8 epoxy networks are elongated elastically within the 3% deformation. Based on the obtained stress–strain curves, the E is calculated by performing a linear regression analysis on the stress–strain data ranging within the 3% deformation. The calculated E is shown in Table 6, and the data reported is from a single run. Aside from that, two independent simulation runs are performed on each epoxy network. As the computed E shows no significant difference to the reported data, it is not accounted for the reported value. For the solvated SU-8 epoxy networks, the E reaches the peak of 5.03  0.15 GPa at 1 wt% moisture content, and then decreased steadily to 4.18  0.23 GPa at moisture content of 3 wt%, with another small increase to 4.37  0.19 GPa at moisture content of 4 wt%. Our results show that at low moisture content, the absorbed moisture may have a beneficial effect on the SU-8 Young’s modulus, which has been reported in various polymeric materials [73–75]. The explanation of this phenomenon can be made from a viewpoint of the moisture antiplasticization effect. As the SU-8 epoxy network is highly cross-linked after the

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Fig. 12 Stress–strain curves obtained from the tensile deformation of the SU8 epoxy with respect to the moisture content

cross-linking process, the added water molecules start to fill in the free volume of the epoxy structure, and thus enhances the stiffness of the SU-8 epoxy. As the moisture absorption in the SU-8 structure continues to increase during the long-term service life, the absorbed moisture decreases the modulus of epoxy and it becomes softened. This moisture effect is called the plasticization. However, our results indicate that even with large amount of the moisture addition, the largest decline of E is only 5.5% at 3 wt% moisture content. Independently, the SU-8 epoxy network solvated with larger moisture content (i.e., 5, 6 wt%) are simulated to see if any change of the modulus can be observed with the moisture content beyond the limit of 4 wt%. The results show that the downturn in E is not obvious in these epoxy networks with larger moisture content. The discrepancy between the simulation results and the prediction based on plasticization is resulted from the hydrophobic nature of the SU-8 epoxy, as it is only slightly moisture-permeable, and the equilibrium moisture content is reported to be less than 4 wt% [1, 2, 65, 66]. It is noted that a high moisture absorption, e.g., 10 wt%, may cause obvious plasticization of the SU-8 epoxy, but such high moisture absorption may not occur in the SU-8 epoxy. Equipped with the relationship between the SU-8 Young’s modulus and the added moisture content, it can be concluded that the moisture has no severe impact on the SU-8 mechanical properties, which further demonstrates that SU8 is able to provide a strong mechanical reliability for the practical application during the whole life-circle.

3

Coarse-Grained Modeling of Polymeric Material

This section presents the study on the mechanical behaviors of the polymeric material at the mesoscale by using CG MD simulation. The CG model of the polymer is developed by reducing some of the atomistic degrees of freedom, where the polymerized structure is represented by a collection of beads connected by springs, as shown in Fig. 13. The interactions between the beads are characterized by the force

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Fig. 13 Coarse-grained representation of the SU8 epoxy: (a) atomistic, and corresponding (b) mesoscale bead-spring model

field, which describes the resistance to the tensile load, bending and intramolecular interaction of the polymerized structure. Furthermore, the effect of the structural voids on the mechanical behaviors of the investigated material is determined.

3.1

Coarse-Grained Model of Polymeric Material

In this section, a series of mechanical loading cases to determine the force field parameters for the CG epoxy model are described. These studies consist of the following two loading cases: (i) the tensile loading to determine the Young’s modulus of the epoxy; and (ii) the adhesion test of an assembly of two epoxy

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Fig. 14 Mechanical loading cases of the SU-8 epoxy for deriving the force field parameters for the coarse-grained (CG) model: (a) tensile loading and (b) adhesion test

cross-linked molecules to determine their adhesion energy. The different loading cases are shown in Fig. 14. All studies are carried out by using the SU-8 atomistic model, which is constructed based on the method as describe previously. The atomistic simulations are performed by using the classical MD [76]. The detailed molecular interactions of the epoxy structure are described by a force field, which is the core of the classical MD methods. Here the PCFF potential is used to describe the molecular interactions of the epoxy structure, which captures not only the pairwise interactions of the non-bonded atoms but also additional cross-coupling terms from the local geometric configuration of the neighboring atoms [55, 56]. The PCFF potential has been reported to be reliable for the simulations of the polymeric materials [13, 23, 45]. The time step is chosen to be 1 fs, and the vdW and Coulombic interactions are truncated at a cutoff distance of 13.5 Å. The MD simulations are performed by using the open source code LAMMPS [61]. The computational uniaxial tensile deformation of the epoxy structure is carried out by keeping one end of the structure fixed, while slowly stretching the other end in the axial direction of the structure. The loading configuration is shown in Fig. 14a. Three different displacement loading rates are applied in the tensile deformation, including 10, 5, and 2.5 ms1. During the entire tensile deformation, the virial stress tensors are monitored and averaged over the structure volume. The stress tensor component in the loading direction is used to extract the information about the stress as a function of the applied uniaxial strain. The measured stress–strain curve is used to calculate the Young’s modulus of the epoxy network. For the small deformation, the results of all the applied loading rates are similar, indicating the convergence of the elastic properties. Young’s modulus is determined by performing a regression analysis on the stress–strain data, which is estimated to be 3.21  0.69 GPa [77], close to the experimental observation of 2.70–4.02 GPa [45].

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In the epoxy three-dimensional covalent network, there are molecules that are not directly connected, where no direct covalent interactions exist between these epoxy molecules. The primary interaction between these epoxy molecules are the weak dispersive interaction. The adhesion energy of the epoxy molecules is calculated by using the metadynamics method [7, 78–85]. The geometry of the two epoxy molecules is depicted schematically in Fig. 14b. The system is first equilibrated without the application of any external mechanical loading. The equilibrated state of the system is affirmed by examining the RMSD of the epoxy atoms, which becomes stable at the end of the equilibration. After that, the metadynamics simulation is performed by using the PLUMED package in LAMMPS [86, 61]. The free energy is measured as a function of the center-to-center distance between the two epoxy molecules. The metadynamics simulation converges after a 17 ns time span, permitting a full exploration of all the possible states [77]. The equilibrium distance between the two epoxy molecules is found to be ΔD  31.40 Å, which approximates to the thickness h0 of the epoxy molecule. A relationship between ΔD and h0 that depends on the dimension of the epoxy molecules can be arrived, ΔD  1: h0

(7)

For the epoxy molecules with the same size, the equilibrium distance ΔD in the weak dispersive interactions can be approximated as equal to the thickness. The free energy difference between the attached state and the separated state is 102.36 kcalmol1 [77]. With the surface area being 14.00 nm2, the normalized adhesion energy Es of the two epoxy molecules equals 25.40 mJm2. From the atomistic simulation results, it can develop a better understanding of the interaction and force during the deformation of the epoxy at the mesoscale. The information is used to develop a CG model, in which the beads are connected by springs to represent the epoxy covalent network, whereas all parameters are completely derived from the atomistic simulations. With the reducing degrees of freedom in the CG model, it can model the epoxy structure with the length on the order of few tens of nanometers. Therefore, this approach enables one to study the microscopic structural voids. The goal here is to develop a CG model suitable to perform the large-scale simulation of the mechanics of the epoxy-based materials, eventually leading to the understanding of the structural voids effect on the behavior of these materials. The total energy of the system can be expressed as E ¼ ET þ EB þ Eweak ,

(8)

where ET is the energy stored in chemical bonds due to the stretching along the axial direction, EB is the total bending energy, and Eweak constitutes the weak, dispersive interactions between the molecules of the epoxy structure that are not directly connected. Similar techniques have been used effectively to model the behavior of carbon nanotubes, collagen molecules, wood cell walls, and silica nanocomposites [77, 87–92].

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The energy of the axial strain and bending is expressed by the harmonic potential of the form ET ðrÞ ¼

X 1 K T ðr  r 0 Þ2 , 2 bonds

(9)

EB ðθÞ ¼

X 1 K B ðθ  θ0 Þ2 , 2 angles

(10)

where r is the distance between the bonded beads, θ is the angle formed by three consecutive beads, and r0, θ0 refer to the equilibrium distance and angle, respectively. Meanwhile, the weak, dispersive interactions are modeled by a 12: 6 LennardJones (LJ) potential of the form Eweak ðr Þ ¼

X pairs

4e

   σ 12 σ 6  , r r

(11)

with e as the energy at equilibrium and σ as the distance parameter for each pairwise interaction. It is assumed that a pairwise interaction between different particles is sufficient and that there are no multi-body contributions [87, 88]. Based on these assumptions, the interaction between different molecules is modeled by using a LJ 12:6 potential. The mass of each bead can be determined by assuming a homogeneous distribution of the mass in the model, which is an excellent assumption for the homogeneous structure of the epoxy-based materials. The beads are connected through springs to form a network model of a primitive cubic system, which is considered as a representative of the epoxy cross-linked structure. Such threedimensional bead-spring network is analogous to the CG crystalline silica, in which one bead is connected with six neighboring beads by springs [90]. In the epoxy network with structural voids, the beads in the void locations can be removed to create the structural voids. Previous studies have shown that the void volume is 87–111 Å3 for the DGEBA-based epoxy resin, and 97  33 Å3 for the uncured epoxy [8, 9, 93]. In this work, it is designated that one bead has the same mean volume of the voids, i.e., 97 Å3. The equilibrium distance r0 of two beads is calculated to be 4.60 Å. As SU-8 has a density of 1.20 gcm3, it is determined that the mass of one bead with a volume of 97 Å3 is around 70 amu. Therefore, each bead has a weight of 70 amu. The LJ parameters are chosen to reproduce the adhesion energy determined from the full atomistic simulations. The equilibrium bond distance is related to the distance ΔD between the two epoxy molecules in contact by the weak, dispersive interactions, with D ¼ ΔD,

(12)

where ΔD can be approximated to the thickness h0 of the epoxy molecule, i.e., ΔD  4.60 Å. The distance parameter in the LJ potential is then given by

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D ffiffiffi  4:10, σ¼ p 6 2

(13)

where D is the equilibrium bond distance. The LJ potential minimum is at r = D and is given by e. Given that per unit cell of bonds in this setup, the energy per unit area is given by ES ¼

1 ~ þ . . . , ½ϕ ðDÞ þ ϕweak ðDÞ S0 weak

(14)

where ϕweak ðDÞ ¼ e,

(15)

~ þ . . . denote the interaction energy to the second nearest neighbors and so and ϕ(D) on. The numerical value for the adhesion strength of two epoxy molecules is determined from MD simulation as ES = 25.40 mJm2. The parameter e in the CG model is chosen so that the atomistic and CG model feature the same adhesion energy per unit area. For the nearest neighbors only, e ¼ E1 S 0 :

(16)

For more than one nearest neighbors in the case of the larger cutoff radius, e ¼ ES S0 ½ð1 þ π ð2Þ þ π ð3Þ þ . . .Þ1 ,

(17)

~ weak (D). The term (1 þ π(2) + π (3) þ ... þ π (N )) = β(N ) is where π = ϕweak ( D)/ϕ defined and it is found that β(6)  1.0988, which leads to e  0.70 kcalmol1, with a cutoff distance at rcut = 30 Å. The tensile spring constant is determined from previous tensile loading deformation, where the tensile deformation is carried out in the regime of small loads and consequently small displacements. The stress–strain response of the SU-8 epoxy obtained from the full atomistic calculations is used to develop the interaction of the bonded beads. The spring constant kT is then defined as kT ¼

Ac E, r0

(18)

with Ac being the cross-sectional area of the SU-8 epoxy structure with a value of 0.21 nm2, and r0 the equilibrium distance of the two beads. Based on the low-strain rate tensile testing data discussed in previous section, kT is found to be around 2.12 kcalmol1Å2. Using the concept of energy conservation between the atomistic and CG model, the bending stiffness parameter kB is expressed as

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kB ¼

3EI , 2r 0

83

(19)

with r0 denoting the equilibrium bead distance, and EI as the bending stiffness of the epoxy. The cross-sectional moment of inertia for the epoxy is calculated by assuming a square cross section with a length of r0 and using I = r04/12. Based on the Young’s modulus discussed in previous section, kB is calculated to be about 5.58 kcalmol1rad2.

3.2

Coarse-Grained Simulations

The parameters used in the CG simulations of the epoxy are summarized in Table 7. A series of samples are modeled to study the mechanical properties of the polymeric material at the mesoscale. Two cases are investigated: (i) a simple validation to compare the CG simulation results to the experimental data; and (ii) a study of the structural voids effect on the epoxy structure. The initial CG model is set up by distributing the 9216 (36  16  16) beads in the 17  7  7 nm3 simulation box. These beads are equally separated by 0.46 nm, and connected with springs to form a covalent network. A large time step of 10 fs can be used for the integration of the equations of motion, as each bead has a large mass of few tens of atomic mass units [88]. Nonperiodic and shrink-wrapped boundary conditions are used in all three dimensions during the simulations, where the position of the simulation box edge is adjusted in order to include all the beads in that dimension, no matter how far they move. During the simulation, the system is subjected to an equilibration scheme with a control of temperature that can speed up the equilibration process and relieve the residual stress inside the structure. Specifically, the system is first relaxed at absolute zero temperature for 0.5 ns in the microcanonical (NVE) ensemble with constant number of particles, volume, and energy of the system. Subsequently, the system is equilibrated in the NVT ensemble with constant number of particles, volume, and temperature, which is corresponded to the constant temperature experiments. The temperature imposed during the NVT equilibration is gradually ramped up from absolute zero to a maximum of 600 K with an increment of 100 K, and then gradually stepped back down to 300 K, so as to allow the adequate fluctuation and Table 7 Summary of the CG force field parameters derived from the atomistic simulation Parameter Equilibrium bead distance r0 (Å) Tensile stiffness parameter kT (kcalmol1Å2) Equilibrium angle θ0 (o) Bending stiffness parameter kB (kcalmol1rad2) Dispersive parameter e (kcalmol1) Dispersive parameter σ (Å)

Numerical value 4.60 2.12 90, 180 5.58 0.70 4.10

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Table 8 Weight of the CG SU-8 model Weight (amu)

Without voids 645,120

With voids 604,310

Loss of weight 6.3%

relaxation of the structure. At each temperature, the equilibration runs for 0.5 ns. Finally, the two layers of beads in the outermost, left part of the structure are restricted in motion to mimic the fixed boundary condition, and a 2.5 ns NVT equilibration is carried out before the tensile deformation, as similar to the loading case shown in Fig. 14a. The temperature is controlled by Nosé–Hoover thermostat [63]. The equilibrium state of the simulated system is identified by examining the RMSD of the beads in the system, which reaches a constant level at the end of the equilibration. Based on the previous defined CG model, the structural voids are introduced to quantify the effect on the mechanical properties. It is reported that the mean number of structural voids per mass unit is 0.65 nm3 [9]. For the initial CG model, its volume is around 897 nm3, and thus the number of structural voids in the model is 583. In this work, it is designated that one bead has the same mean volume of one void. To create the structural voids, 583 beads are removed from the initial CG model, and the springs connected to the removed beads are deleted accordingly. These 583 beads for structural voids are selected randomly across the CG model. To determine the effect of the void locations, two independent models with structural voids have been constructed. Same as the original model, 583 beads are selected randomly and removed accordingly, but the distributions of the removed beads are different from each other. The volume fraction of the voids in the CG model is calculated as about 6.3%. The weight of the CG models is summarized in Table 8. The CG SU-8 model is equilibrated adequately using the equilibration conditions corresponding to previous case, and then the computational tensile deformation is carried out to investigate the mechanical response of the model with structural voids.

3.3

Effect of Structural Voids on Polymeric Material

In equilibrium state, the epoxy network is relaxed to the potential energy minimum. Due to the incorporation of the angle bending term in the force field, the epoxy model can maintain the primitive cubic system after the equilibration, as shown in Fig. 15a. The evolution of the potential energy is plotted in Fig. 15b, where the potential energy keeps at a stable value during the final 2.5 ns NVT equilibration at 300 K. For the validation of the developed CG model, the density (ρ) is a critical parameter which shall be consistent with the experimental value. During the final 2.5 ns NVT equilibration, the ρ of the epoxy network is sampled every 250 ps. The average ρ of the CG network without the structural voids is shown in Fig. 15c. The error bars shown in the figure are obtained from the standard derivations of the averages. A ρ of 1.26  0.003 gcm3 is obtained for the epoxy network with no structural voids. A slight overestimation is observed in comparison with the experimental value of 1.07–1.20 gcm3. Meanwhile, the Young’s modulus (E) of the CG model is another important material parameter for validation.

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Fig. 15 Equilibration analysis: (a) snapshot of the equilibrium configuration, and (b) plot of the potential energy evolution of the CG network without voids; (c) density of the equilibrated models; and (d) evolution of RMSD of the beads

Fig. 16 Tensile test of the CG network: (a) stress versus strain data from tensile test; (b) histograms showing the Young’s modulus

During the uniaxial tensile deformation, the stress versus strain as obtained by using the CG model is recorded and shown in Fig. 16a. The stress along the loading direction shows a linear response to the applied strain, which indicates that CG epoxy network is elongated elastically within the small strain deformation. The E is calculated by performing a linear regression analysis based on the obtained stress–strain curve, and

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the result is shown in Fig. 16b. The data reported is from a single run. The E of the CG network is 5.99  0.01 GPa, which is larger than the experimental tensile test measurements in the range of 2.70–4.02 GPa [49–51]. Realistically, MD simulation probes the modulus in the athermal limit. Compared with the experiment, the strain rate in MD simulation is several orders of magnitude higher, and the contribution of the thermal motions to the mechanical response of the material is smaller. Both factors can result in the higher moduli as observed. Therefore, MD simulation result can be in a closer agreement with that from the high-strain rate or low-temperature pulling experiment. Meanwhile, the overestimation of ρ and E could be due to the fact that the constructed CG model is free of structural voids, as they normally exist in the experimental samples. With the quantitative mechanical characterization of the pristine epoxy network, the effect of structural voids on the epoxy can now be quantified. During the equilibration process, the ρ of the epoxy network with structural voids is measured and compared with the model with no voids, as shown in Fig. 15c. The computed ρ with voids is 1.13  0.003 gcm3, which is slightly smaller than the model without voids, and agrees very well with the experimental value of 1.07–1.20 gcm3. Meanwhile, the density ρ of the two independent models with different distributions of structural voids is found to be in a good accordance with the reported value. In comparison with the sample without voids, a relatively smaller density is because that the existence of structural voids in the epoxy network causes a loss of weight, and more essentially, it weakens the covalent interactions in the cross-linked network, where the monomers near the structural voids can flow more freely, which can lead to the expansion of the network. The movement of the monomers in the epoxy crosslinked structure, i.e., the beads in the CG model, can be quantified by using the RMSD of the beads during the equilibration run. By comparing the RMSD as shown in Fig. 15d, it is observed that the RMSD of the beads in the model with structural voids keeps at a higher level than that of the model without voids, which demonstrates the larger movements of beads, i.e., the beads flow more freely in the model with voids. Further investigation on the effect of structural voids focuses on the analysis of the Young’s modulus. The stress–strain data of the epoxy network with voids is shown in Fig. 16a. The E is calculated to be 4.32  0.04 GPa, which is closer to the experimental results of 2.70–4.02 GPa when compared to the model without voids. In the meanwhile, the measured E of the two independent models is also close to the reported value. In view of the computed ρ and E of the three models with structural voids, no significant variation is observed from the models with different distributions of the void. Furthermore, our results show that with the existence of structural voids, the epoxy model possesses the properties, i.e., ρ and E closer to experimental data, in comparison with the larger variation of the model without voids. Therefore, the developed CG model of the epoxy with structural voids can be regarded as a reasonable structure close to those found in the real systems, which can be used as a basis in the various investigations involving the cross-linked network at the mesoscale. Equipped with the relationship between the epoxy properties and the structural voids, it can be concluded that the structural voids are intrinsic to the epoxy-based materials, and a certain number of the voids with a volume of 97 Å3 have no severe impact on the epoxy elastic properties.

4

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Conclusion

In this chapter, the atomistic and CG MD simulations have been applied to study the mechanical properties of the polymeric material.The SU-8 epoxy is chosen as the representative polymeric material used in the engineering applications. An effective dynamic cross-linking algorithm has been developed to construct the atomistic crosslinked network, with the cross-linking degree and density close to the experimental observations. Mechanical properties of the constructed epoxy network are measured by means of dynamic deformations, which are in a good accordance with various experimental observables, including Young’s modulus, bulk modulus, shear modulus, and Poisson’s ratio. For the investigation of the moisture effect, the constructed epoxy network is solvated with a successive amount of moisture content from 0 to 4 wt% to mimic the different moisture conditions. The MD simulation results show that the Young’s modulus of the epoxy model is not deteriorated even under excessive moisture absorption, which is close to the experimental observation, and thus it can be concluded that moisture has no severe impact on the SU-8 mechanical properties. Meanwhile, the CG model of the epoxy is developed based on the results of atomistic simulation. After subjected to different mechanical loadings, the elastic and adhesion properties of the epoxy are obtained, which are used to derive the force field parameters of the CG epoxy model. The developed CG epoxy model possesses the density and Young’s modulus larger than the experimental measurements. Further investigations shed light on the property changes due to the influence of the structural voids. The epoxy structure with structural voids possesses mechanical properties closer to the experimental data than that with no structure voids, which leads to a more realistic CG modeling of the epoxy-based material. The present MD simulation work provides a bottom-up approach for the microscopic investigation of the polymeric materials. The developed models enable us to understand the structural and mechanical behavior of the polymeric materials more in-depth, which can be used to investigate the effect of the environment and the voids in the structure. With the availability of the additive manufacturing, the models can also be useful for the design and development of the polymer-based composites for the enhanced mechanical properties. It is expected that this work can provide the foundation for further investigation of the polymeric materials at different hierarchical levels, which can enable us to understand the mechanics in a more systematic way.

References 1. Zhou J, Lucas JP. Hygrothermal effects of epoxy resin. Part I: the nature of water in epoxy. Polymer. 1999;40:5505. 2. Núñez L, Villanueva M, Fraga F, Núñez MR. Influence of water absorption on the mechanical properties of a DGEBA (n=0)/1,2 DCH epoxy system. J Appl Polym Sci. 1999;74:353. 3. Lyons JS, Ahmed MR. Factors affecting the bond between polymer composites and wood. J Reinf Plast Compos. 2005;24:405. 4. Custódio J, Broughton J, Cruz H. A review of factors influencing the durability of structural bonded timber joints. Int J Adhes Adhes. 2009;29:173.

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Fracture Nanomechanics Yabin Yan, Takashi Sumigawa, Licheng Guo, and Takayuki Kitamura

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 In Situ Experiments on Crack Initiation at the Interface Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Experimental Method and Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Criterion for Crack Initiation at the Free Edge of Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Modulation of the Stress Field on Interface in Nanoscale Components . . . . . . . . . . . . . . . . . . . 3.1 Modulation of the Location for Crack Initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modulation of the Mode Mixity for Mixed-Mode Interface Cracking . . . . . . . . . . . . . . 3.3 Simplification of Cracking Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Effect of Microscopic Structure on Interface Cracking in Nanoscale Components . . . . . . . 4.1 Specimen Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Plastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Crack Initiation at Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Creep Cracking of Interface in Nanoscale Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Fatigue in Nanoscale Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Fatigue Fracture of Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 High-Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Fracture of Nanoscale Single Crystalline Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 In Situ Tensile Experiment of Single Crystalline Nanorod . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Fracture of Single Crystalline Si in Nanoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Summary and Further Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Y. Yan Department of Mechanical Engineering and Science, Kyoto University, Kyoto-daigaku-katsura, Nishikyo-ku, Kyoto, Japan Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, China T. Sumigawa · T. Kitamura (*) Department of Mechanical Engineering and Science, Kyoto University, Kyoto-daigaku-katsura, Nishikyo-ku, Kyoto, Japan e-mail: [email protected] L. Guo Department of Astronautical Science and Mechanics, Harbin Institute of Technology, Harbin, China # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_6

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Abstract

This chapter reviews recent advances in experimental studies on fracture mechanics of small materials on nanometer scales. In particular, experimental systems and some testing methods developed by the current authors are introduced for investigating the fracture behavior of interface in nanoscale multilayered components and low-dimensional single crystalline materials, and main experimental results are presented as well. The experimental studies discussed are: (1) crack initiation from the free edge of interface and its mechanical criterion, (2) modulation of the location of crack initiation and the mode mixity for interface cracking with different types of cantilever specimens, (3) evaluation of the effect of microscopic structure on the interface cracking by an inverted-T-shaped cantilever method, (4) creep crack initiation at an interface edge in nanoscale components, (5) fatigue fracture behavior of interface and the environmental effect, (6) novel resonant vibration based high-cycle fatigue method and the fatigue properties of nano-metals, and (7) evaluation of deformation and fracture properties of nanoscale single crystalline materials. From the obtained results, authors pointed out the applicability of conventional fracture mechanics in nanoscale components. Meanwhile, the main challenges and difficulties in experimental studies on the fracture behavior of nanoscale materials are demonstrated, and several future research topics are outlined.

1

Introduction

With the recent development in miniaturization of advanced electronic devices due to the increasing demands for a high-density integration, the size of the constitutive components has been approaching a few nanometers. In addition, the extreme lowdimensional atom-by-atom structure, such as atomic chain [1], carbon naontubes [2], and graphenes [3], has been synthesized for future application. Moreover, microscale advanced electronic devices including small sensors and actuators with multifarious functions, i.e., micro- and nano-electro-mechanical-system (MEMS/NEMS), have already been extensively developed by industry. During the fabrication and application of advanced electronic devices, the nanoscale components suffers to the stress stems from various sources, such as the mechanical loading, residual stress, and thermal/elastic elongation mismatch. Therefore, to insure the long-term reliability of these devices, it is critically important to precisely design these nanoscale components basing on their mechanical properties. However, it should be noted that bulk material properties cannot be directly extrapolated to nanoscale materials due to the size dependence of properties [4, 5]. However, due to experimental difficulties, the mechanical properties of nano-materials have not been clearly investigated yet. For the bulk materials, tension and compression experiments are commonly employed to characterize their mechanical properties, where the stress/strain is uniform in these specimens and various mechanical properties related with the

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deformation and fracture behaviors could be obtained. However, different from traditional experimental methods for macroscale materials, the extremely small volume of nano-materials induces a series of experimental difficulties, such as the grip of specimen, applying load to interfaces inside of specimen, and precisely measurement of minute mechanical quantities. Thus, it is difficult to make an accurate evaluation of the mechanical properties of nano-materials. A generic requirement for materials characterization at the nanoscale is highresolution force and displacement. Recently, with the advancement in MEMS technology, several methods for evaluating the mechanical properties of the nano- and submicron-scaled films have been developed. Typical methods include tensile test [6], bulge test [7], indenting test [8], compression of bar [9], and so on. Until now, by using these methods, the deformation and fracture behaviors of different types of micro/nanoscale thin films have been widely investigated. At nanoscale components with different materials, crack initiation at the free edge of bi-material interface is a prominent problem for the interface delamination in low-dimensional tiny components. In such small materials, the failure at a minute area usually causes the malfunction of entire system. Especially at the free edge of interface, where the interface meets the free surface of component, the concentrated stress field induced by deformation mismatch makes the interfacial crack to be liable to initiate at this region. Thus, it is essential to understand both the mechanism and mechanics of failure characteristics at interface in the nanoscale components. Fracture mechanics theory has been widely used to evaluate the resistance to crack propagation along interfaces in bulk materials and the intensity of singular stress field near the crack tip characterizes the toughness of interface [10]. The concept is extended to investigate the crack initiation at the edge of interface where a singular stress field exists induced by the deformation mismatch [11]. On the other hand, since the stress concentration region is proportionally scaled down for the shrinkage of component size, the stress concentration region at the interface edge is a few nanometers or at most a few tens of nanometers in advanced devices [12]. In such case, it is questionable whether the conventional fracture mechanics is still valid, and thus mechanical factors dominating the interface cracking in nanoscale components become unclear. In recent years, several experimental methods have been developed to evaluate the interface toughness between submicron- or nanometer-thick thin film and between them and the substrate, such as the indentation method [13], scratch test [14], fourpoint bending test [15], compact tension test [16], cantilever test [12], atomic force microscope [17], and so on. However, a universal experiment method has not been proposed and the fracture behavior of interface in nanoscale components has not been clearly clarified yet. This chapter aims to introduce our recent advances to investigate the fracture mechanics of interface in multilayered nanoscale components and low-dimensional single crystalline materials.

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2

In Situ Experiments on Crack Initiation at the Interface Edge

2.1

Experimental Method and Apparatus

Characterizing mechanical properties of nanoscale components is a challenging work due to the following requirements: (1) constructing appropriate tools to manipulate, position, and align specimens, (2) applying and measuring forces with nano-Newton resolution, and (3) measuring local deformation with nanometer resolution. To overcome these difficulties, we have developed a novel experimental methodology by combining a transmission electron microscopy (TEM) and a highprecision minute loading apparatus to in situ observe the fracture behavior of nanoscale components [18]. Figure 1 shows the schematic illustration of the in situ testing system used in this study. The in situ loading experiment of nanoscale samples is conducted in a TEM (JEOL Ltd., JEM-2100) by using a minute mechanical loading apparatus (see Fig. 2) which is integrated into the top of a TEM specimen holder. The observation is carried out by the TEM with the electron beam energy of 200 kV in a vacuum condition of 1.5
105 Pa. The TEM image of the deformation of cantilever beam is continuously recorded through a digital camera (Gatan Inc., ES500) at the frame rate of 60 Hz. The minute mechanical loading apparatus (Nanofactory Instruments AB, SA2000N) is shown in Fig. 2. The apparatus consists of a movable sample stage and a diamond loading tip with a load sensor. The Au wire (φ = 0.25 mm) with a nanoscale specimen mounted on top is attached to the sample stage, which is actuated three-dimensionally by a piezoelectric actuator. The load resolution of the sensor is 0.1 μN and the alignment resolution of the piezoelectric actuator in each direction is approximately 1 nm.

2.2

Criterion for Crack Initiation at the Free Edge of Interface

Figure 3 schematically shows a nano-cantilever bending specimen used to investigate the crack initiation at interface edge in nanoscale components [18]. This specimen is fabricated from a multilayered thin film consisting of silicon/copper/ silicon nitride (Si/Cu/SiN) by using focused ion beam (FIB). Load is applied onto the SiN layer with a diamond loading tip, and the Cu/Si interface was stressed with a bending moment. Although residual stress is inevitably introduced into the thin films, its effect on the crack initiation at interface edge is proved to be negligible [19]. Figure 4 shows the load–time curve (P–t) obtained for a nano-cantilever specimen with a 200-nm-thick Cu layer and corresponding in situ TEM micrographs of the specimen at three specific loading levels (A, B, and C). As shown in Fig. 4(a), after the loading tip hits the SiN layer at point A, the applied load increases monotonically up to a peak magnitude (point B), after which it abruptly decreases

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Load signal

Electron beam

Piezo/Sensor controller

Loading apparatus Amplifier Digital camera PC Digital image

Specimen holder Actuating signal

PC

Sintillator

Fig. 1 Schematic illustration of the in situ TEM testing system for nanoscale specimens

Fig. 2 Minute mechanical loading apparatus for nanoscale samples

to zero (point C). Figure 4(b) clarifies that no crack initiation is recognized before the peak load while the specimen has an interface crack after, which indicates that the crack initiates at point B at the top edge of the Cu/Si interface and instantly propagates along the interface. Figure 4(c) shows the relationship between the applied load and the deflection at the free end of cantilever. With the increase of applied load, the cantilever deflection increases nonlinearly, which is solely induced by the plastic deformation of the Cu layer because the yield stresses of Si substrate and the SiN layer is much higher than that of Cu. The elastoplastic constitutive relationship of the 200nm-thick and 20-nm-thick Cu layers is determined by elongated nano-cantilever specimens by using the inverse analysis, respectively [20, 21].

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Fig. 3 Schematic illustration of a nano-cantilever specimen for crack initiation at the interface edge [18]

Loading tip Interface edge of interest Load h

αSi αCu Si Cu SIN

d

WCu WSin

Electron beam direction

y z

x

Figure 5 shows the distribution of critical normal and shear stresses along the Cu/ Si interface at crack initiation. Although normal stresses away from the edge differ from each other, they show good consistence in the region 40 nm away from the interface edge for specimens with different geometric dimensions. On the other hand, the magnitude of shear stress near the interface edge is much lower than that of normal stress, and thus the concentrated normal stress field with the region of r < 40 nm governs the crack initiation. Therefore, these studies experimentally approve the applicability of stress, a concept defined in the framework of continuum mechanics, for evaluating the interfacial fracture behavior on the order of tens nanometers. Similar experiments have been conducted by using specimens with 20-nm-thick Cu layer, and the singular stress field near the Cu/Si interface edge governs the crack initiation as well independent on the difference in the thickness of Cu layer [19].

3

Modulation of the Stress Field on Interface in Nanoscale Components

3.1

Modulation of the Location for Crack Initiation

In real electronic devices, the material combination, their shape, and loading condition cause complex stress field at the interface. Therefore, the crack might also initiate at the inside of interface or even other interface with relative stronger

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Fig. 4 In situ loading experiment in TEM for a nano-cantilever specimen [18]. (a) Load–time curve of the nano-cantilever specimen. (b) In situ observations of interfacial cracking in nanocantilever specimen. (c) Load–deflection curve of nano-cantilever specimen

bonding strength. For this reason, it is necessary to develop experimental methods which can modulate the location of crack initiation in nanoscale components. Figure 6 shows a two-step nano-cantilever specimen in which a concentrated stress field is introduced to the internal part of interface [19]. By applying the load to the lower step of cantilever, the crack initiation is in situ observed at the internal part of interface at the stress concentrated site. Figure 7 shows the normal stress distribution on the interface for crack initiation. The critical normal stress at the crack initiation site is about 1.1 GPa and the size of this region is about 60 nm, which is consistent with the results of crack initiation from the interface edge in nano-cantilever specimens [19]. Thus, this indicates that the singular stress field on the scale of tens nanometers governs the crack initiation both at the free edge and the internal part of interface, and the criterion has the universal applicability for the evaluation of interface strength.

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Fig. 5 Elastoplastic stress distribution along the Cu/Si interface at crack initiation [18]

Fig. 6 TEM micrograph of a two-step nano-cantilever specimen [19]

On the other hand, the complex internal stress status of electronic devices will also induce the crack initiation at different interface in multilayered nanoscale components. So, it is also important to develop an experimental method to investigate the crack behavior of an arbitrary interface in nanoscale components. Figure 8 shows a scanning electron microscopy (SEM) image of nano-cantilever specimen with a sharp-notch [22]. Since a notch on the nanoscale is

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Fig. 7 Critical stress distribution on the interface for crack initiation in the two-step cantilever specimen [19]

Fig. 8 SEM image of the nano-cantilever specimen with a nanoscale sharp notch [22]

introduced in the SiN layer near Cu/SiN interface, a concentrated stress field exists at this region. In previous nano-cantilever bending specimen with the multilayered material of Si/Cu/SiN, the crack was always initiated at the free edge of Cu/Si interface due to its weaker bonding strength compared with Cu/SiN interface. However, by introducing a nanoscale notch, the crack is initiated at the Cu/SiN interface because of the stress concentration in the region near notch tip, as shown in Fig. 9. Therefore, the concentrated stress field near the notch tip could apply a local load to the Cu/Si interface selectively. This means that the notched nano-cantilever method can modulate the location of stress concentration and evaluate a particular interface strength. In addition, in the notched nanocantilever specimen, the crack stably propagates without rapid rate. Thus, by using this method, the crack propagation behavior on the interface can also be investigated.

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Fig. 9 In situ TEM observation of the crack initiation at interface in the notched nano-cantilever specimen [22]

3.2

Modulation of the Mode Mixity for Mixed-Mode Interface Cracking

Because the state of internal stress is complicated in advanced electronic devices, the interfacial cracking usually appears under a mixed mode induced by the asymmetric loading across interface and the difference in elastic properties [23]. Thus, to ensure the reliability of these devices, it is necessary to investigate the mixed-mode fracture criterion under a nanoscale stress concentration. Figure 10 shows a SEM micrograph of a nano-cantilever torsion specimen which possesses a three-dimensional bent structure [24]. As shown in Fig. 10(a), the Cu/Si interface is located at arm ①. The load is applied to the SiN layer in arm ③ by a diamond loading tip to stress the Cu/Si interface by a torque and bending moment transferred through arm ②. The ratio of torque (shear stress at the interface) to bending moments (normal stress at the interface) can be precisely controlled by changing the loading position (L/l). Figure 11 shows the plot of the normal stress σ mc versus the maximum shear stress τmc for four specimens where the crack is initiated with different mode mixities. By fitting the experimental results with the least-squares method, the criterion of the mixed-mode interface cracking in nanoscale components is represented as the following circular relation: σ mC 2 þ τmC 2 ¼ 11432

(1)

Here, it should be noted that the general applicability of proposed circular criterion for mixed-mode interfacial cracking in nanoscale components is not discussed, and it remains for further experimental and analytical investigations.

3.3

Simplification of Cracking Criterion

The applicability of fracture mechanics for evaluating the interface cracking behavior in nanoscale components has been clarified in previous sections. However, this approach clearly has disadvantages to universally describe the interface cracking behaviors.

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Because the stress singularity at the interface edge is dependent on both the material combination and the corner shape of the interface edge, the corresponding stress intensity factors have different dimensions. In addition, due to the asymmetric loading across the interface and the difference in elastic properties, the local stress state near the interface edge usually becomes mixed mode, even under a simple global loading mode. Therefore, the combination of the stress intensities for normal and shear stresses is required to be considered as well. As a result, to apply the fracture mechanics concept to the design of an interface in industrial components, an enormous number of experiments with different corner sharps of edges and loading modes are needed to obtain a complete understanding of the cracking criterion of an interface. Since it is extremely difficult to conduct experimental studies which could cover all the possibilities, it is necessary to find a reasonable simplification of the

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cracking criterion. On the other hand, although the energy release rate, which is an energy-based concept, might be the possible candidate for evaluating the simplified criterion for interface cracking, it generally converges to zero when the crack length approaches zero. Therefore, the energy release rate could not act as the fracture mechanics parameter for describing the crack initiation at the interface edge. Moreover, the validity of energy-based criterion becomes questionable since energy dissipation exists induced by plastic deformation. An alternative method to interface fracture, which overcomes aforementioned difficulties, is based on the use of cohesive zone model (CZM) [25]. The CZM is a phenomenological mode within the framework of continuum mechanics, and regards the fracture as a gradual phenomenon where the separation takes place between two adjacent virtual surfaces across an interface and is resisted by cohesive tractions. Until now, different types of CZMs such as trapezoid type, bilinear type, and exponential type have been proposed to simulate the interface cracking in bulk materials [25]. However, the applicability of CZM for describing the interface cracking in nanoscale component has not been clarified so far. In this section, a CZM of exponential type [25] is adopted to simulate the crack initiation at the edge of Cu/Si interface in nano-cantilever experiments (including 20nm-thick and 200-nm-thick Cu layers, respectively) and the propagation along the Cu/Si interface in macroscopic four-point bending experiments (including 200-nmthick Cu layer) [26]. Figure 12 shows the normal and shear traction (σ/τ)-separation (δ) relations of exponential CZM. It can be found that with increasing interfacial separation, the tractions across the interface increase to reach a maximum, and then decrease, eventually vanishing with complete decohesion. For numerical analysis, the traction-separation relation of exponential CZM is incorporated with FEM codes with user subroutine, such as UEL [26]. The independent parameters of exponential CZM include the cohesive strength (σ max and τmax), the interface characteristic length parameter (δn and δt), and the

Fig. 12 (a) Normal and (b) shear traction-separation relations of exponential CZM [26]

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coupling parameter r. Because the normal stress is dominant on the interface near the edge in nano-cantilever experiments and the crack tip in the four-point bending experiments, only the parameters for normal traction-separation relation need to be determined. Figure 13 shows procedures of determining CZM parameters for Cu/Si interface by calibrating with the experimental load–deflection curve of a nano-cantilever specimen with 200-nm-thick Cu layer. As shown in Fig. 13(a), the slope of calculated load–deflection curve is sensitive to the magnitude of δmax, and it becomes steep with the decrease of δmax. The simulation of δmax = 1 nm gives the best prediction to the slope of experimental curve. Hence, δmax = 1 nm is adopted in subsequent calculations. On the other hand, Fig. 13(b) shows the effect of cohesive strength, σ max, on the calculated load–deflection curves, where the inserted figure is an enlarged view of the square region. With the increase of σ max, the calculated critical load becomes larger while the slope of calculated curve remains constant. The calculated curve with σ max = 1060 MPa results in the good correspondence to experimental critical load. Therefore, the CZM parameters for Cu/Si interface are determined to be σ max = 1060 MPa and δmax = 1 nm. Figure 14 shows the comparison of all calculated critical loads obtained by CZM simulation and experimental ones from nano-cantilever specimens and four-point bending specimens. It should be noted that the macroscopic four-point bending specimens (P1-P3) possess the length and the width on millimeter scale, which are thousands of times larger than those of nano-cantilever specimens (A1-A3, I1-I4). In addition, the huge difference in the geometric dimension of specimens results in the different of nearly 6 orders in the magnitude of critical load. However, it can be found that the CZM parameters solely determined by a nano-cantilever specimen can predict the toughness along the Cu/Si interface in all specimens. This means that the

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Fig. 13 Determination of CZM parameters for Cu/Si interface by calibrating with a nano-cantilever experiment [26]

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Fig. 14 Comparison of experimental and calculated critical load of all specimens [26]

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CZM criterion could be adopted as the simplified criterion for the interface cracking in nanoscale components, and can predict the interface cracking in all specimens regardless of the specimen geometry, plastic behavior, and residual stress. On the other hand, the CZM method is a phenomenological model, where the rationality of the physical process during cracking of interface is ambiguous. Thus, to elucidate reasonably the fracture criterion on nanoscale where continuum mechanics breaks down, the molecular dynamics (MD) and first-principle calculations are required and these remain for further investigation.

4

Effect of Microscopic Structure on Interface Cracking in Nanoscale Components

4.1

Specimen Design

Because the nanoscale metal is a typical coarse-grained material consisting of a few or a few tens of grains, the microscopic structure could strongly affect the deformation and fracture behaviors of nano-materials. Although TEM possesses a good ability to investigate the microscopic structure in nanoscale materials, the thickness of tested section must be thinned to about 100 nm at least for the in situ observation of the microscopic structure during loading. However, such a thin specimen subjected to compressive load is liable to buckle during bending test. Thus, as shown in Fig. 15, a nano-cantilever specimen with an inverted-T-shaped cross section is developed [27]. In this specimen, the neutral plane is located near the bottom to limit the compressive stress to the wider

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Fig. 15 Inverted-T-shaped nano-cantilever specimen for in situ TEM observation of microscopic structure during loading [27]

portion of specimen. In addition, by fabricating a horizontal through-slit at the center of thinner part, only tensile stress exists in the tested section. Finally, a rigid block is added to the end of cantilever for applying load by the diamond tip and eliminating the out-of-plane deformation.

4.2

Plastic Deformation

In the bright-field image of TEM, the dark region (shadow) appears when the transmitted electrons are diffracted by the distortion of crystal and cannot reach the screen behind the specimen. Since both the elastic and plastic distortions can induce the appearance of the shadow in TEM image, the unloading process is required after the applied load reaching a preset magnitude to remove the effect of elastic distortion. The remained shadow under the unloading indicates the plasticity due to its irreversibility. Figure 16 shows the inverted-T-shaped nano-cantilever specimen consisting of two grains. After the unloading process, the shadow still remains at the region near to the interface edge in Grain A. Therefore, the plastic deformation in Cu layer is clearly identified in such area by using the in situ TEM observation. On the other hand, numerical analysis is conducted to determine the yield stress of Cu layer in this specimen to be about 400–420 MPa, which is about two orders of magnitude larger than that of Cu bulk (0.5–1.0 MPa [28]). Moreover, with the increase of applied load, the plastic zone grows along the top surface and the Cu/Si interface in Grain A, and finally expands to the internal part of Cu layer.

4.3

Crack Initiation at Interface

Figure 17 shows the load–time curve and corresponding TEM images in the last loading before fracture. As shown in Fig. 17(b)-(i), a gradual strain contrast exists in the Si substrate and a plastic zone exists in Grain A near the interface. With the increase of applied load, the gradation in Si substrate smoothly changes and the

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Fig. 16 In situ TEM observation of the plastic deformation in the loaded specimen [27]

plastic region in Cu layer monotonically expands (Fig. 17(b)-(ii)). After the applied load reaches a maximum (Fig. 17(b)-(iii)), two local diffraction contrasts emerge in the Si substrate near the Cu/Si interface, i.e., the intersection between the grain boundary and the Cu/Si interface, and the area a few tens of nanometers below the intersection. Finally, these contrasts are completely removed by unloading process (Fig. 17(b)-(iv)). Because these contrasts reappear at the same place after reloading, it is confirmed that these contrasts are not induced by the plastic deformation of the Cu layer. Since the local diffraction contrast appears near an opening crack tip in the TEM image due to the concentration of elastic strain, the local diffraction contrasts near the Cu/Si interface in Fig. 17(b)-(iii) and (iv) indicate the tips of an opening crack at Cu/Si interface, which is initiated at the intersection. In the reverted-T-shaped nano-cantilever specimen, the crack initiates at the intersection between the grain boundary and the interface instead of the interface edge. In such an ultrathin bi-crystal specimen, the deformation anisotropy of grains strongly affects the stress distribution. Numerical analysis including the effect of crystallographic orientation and shape of each grain shows that the highest stress exists at the intersection between the Cu/Si interface and the grain boundary rather than at the free edge of interface. In addition, it should be noted that a stress concentrated region with a width of 2–4 nm induces the crack initiation at the intersection. Therefore, the interfacial cracking in nanoscale components is governed by the stress field with a width of few nanometers by considering the effect of microscopic structure.

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5

Creep Cracking of Interface in Nanoscale Components

Advanced electronic devices include extensive small scale components consisting of low melting metals and polymers, such as the micro ball grid array. Creep deformation could appear even at low stress levels and relatively low temperature during

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Fig. 18 Nano-cantilever specimen for crack initiation at the interface edge under creep [29]. (a) Schematic illustration of the nanocantilever specimen and the loading method. (b) SEM image of the specimen

fabrication and application. Thus, it is necessary to investigate the mechanics of creep crack initiation at the interface edge in nanoscale components. Figure 18 shows the schematic illustration of a nano-cantilever specimen and its SEM image for studying the creep crack initiation from an interface edge at room temperature [29]. This cantilever specimen is fabricated from a multilayered material consisting of a 400-nm-thick Sn layer, a 450-nm-thick Ta2O5 layer, and a Si substrate by using FIB. The interface of interested in this study is between the Sn layer and Si substrate. Since Sn has a relatively low melting temperature of 505 K, the creep deformation becomes significant even at room temperature while creep hardly ever appears in the Ta2O5 layer and the Si substrate. For conducting the creep cracking experiment, a loading apparatus attached to an atomic force microscope is adopted to apply the load to the top surface of the Ta2O5 layer by a tetrahedral diamond tip with a vertical angle of 60 [17, 29].

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Fig. 19 Experimental results of the creep crack initiation under a constant load [29]. (a) Displacement–time curve during creep experiment. (b) SEM image of the cantilever specimen after the experiment

Figure 19 shows the displacement (δ)–time (t) curve at the loading point during creep experiment and the SEM image after creep experiment. After the initial load increases up to 140 μN, the displacement δ increases from points “a” to “b” with the passage of time under a constant load. As shown in Fig. 19 (a), a transient creep region exists in the early period up to about 200 s and the displacement rate gradually decreases in this region. Subsequently, the creep rate becomes almost constant, which is a steady-state creep region. After the applied load reaches the level of point “b,” the displacement suddenly increases and the interface between the Sn layer and the Si substrate completely delaminates. As shown in Fig. 19(b), the crack clearly goes through the specimen along the Sn/Si interface, and the dent mark due to the tip contact on the Ta2O5 layer confirms that the load is applied to the intended layer. Since creep properties depend on the microscale structures, the properties of nanoscale components are expected to be different from those of the bulk. Basing on experiments and FEM analysis, the creep property (Norton’s power law) of Sn layer under steady-state creep is evaluated as e_ ¼ 1:9
1010 σ 3:0

(2)

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Fig. 20 Transition of Mises stress distribution along the Sn/ Si interface edge during the crack initiation experiment [29]

where σ and e_ are the Mises equivalent stress and strain rate, respectively. After the creep property of Sn layer is determined, FEM analysis is carried out to inquire the stress field near the Sn/Si interface edge. Figure 20 shows the distribution of Mises stress along the Sn/Si interface near the edge. When the loading instant t = 0.1 s, the creep region is very small and the elastic stress intensity λel s ¼ 0:028 of is dominant. With the increase of the passage of time, the creep region apparently expands, and the stress field reaches a steady state at about t = 17 s. Finally, the stress field does not change up to t  1300 s, and the stress singularity in the creep region λel s ¼ 0:23 including the transition period. Figure 21 shows the variation of the stress intensity near the Sn/Si interface edge. Since the creep region is extremely small in the beginning stage for evaluating stress intensity, the magnitude of Mises stress at r = 1 nm is plotted in the figure. It can be found that although the stress intensity increases in the very early region, it is in the steady state in almost all of the period. Therefore, the effect of transient state on the creep crack initiation at the Sn/Si interface edge could be ignored. Figure 22 shows the relation between the intensity of Mises stress at the steady state, K cr s m steady, and the time for crack initiation, tC. Excellent repeatability exists in these experiments, which indicates the stress intensity at the interface edge dominates the crack initiation from the edge. The stress intensity is determined as K cr s m

steady

 t0:14 ¼ 10:6 MPam0:23 s0:14 C

(3)

These facts show that fracture mechanics could be applied to the creep cracking in nanoscale components.

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Fig. 21 Variation of Mises stress at r = 1 nm from the Sn/ Si interface edge [29]

Fig. 22 Relation between creep stress intensity factor at the steady state and the time for crack initiation [29]

6

Fatigue in Nanoscale Components

6.1

Fatigue Fracture of Interface

Distinctive dislocation structures such as ladder-like structures exist in bulk metals during fatigued, and form persistent slip bands and initiate cracks at surfaces because of strain localization [30]. However, since stress concentration regions scale down proportionally with reduction in component size, the strain concentration region is a

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few nanometer or at most a few tens of nanometer in advanced devices [12]. Thus, fatigue substructures typically observed in bulk materials cannot form in nanoscale components. However, due to the difficulty in handling and applying effective loads to nanoscale specimens, the fatigue behavior of nanoscale components has not been clearly clarified yet. Figure 23 shows a nano-cantilever specimen including a 20-nm-thick Cu layer adopted to conduct the cyclic loading experiment by using the minute loading apparatus (see Fig. 2) [31]. During the fatigue experiment, a cyclic load with a constant range of ΔP = Pmax  Pmin and a ratio of Pmin/Pmax = 0 (Pmax: maximum load; Pmin: minimum load) is applied to the SiN layer by a diamond loading tip and the Cu/Si interface is stressed with the applied bending moment. The cantilever deflection at the free end, δ, is continuously recorded from in situ TEM observation during loading. Moreover, the cyclic tensile strain concentrates in the region of about 40 nm in the Cu layer and on the Cu/Si interface from the top surface [31]. Figure 24 shows the load–time curve with ΔP = 18 μN during fatigue experiment and corresponding TEM images in the last cycle of the nano-cantilever specimen. As shown in Fig. 24(a), the specimen is cyclically loaded until the 6th cycle without any damage observed on the interface. In the 7th cycle, the load sharply drops before the peak of the cyclic load and the crack is initiated at the Cu/Si interface (Fig. 24(b)), which is clearly different from the fracture in monotonic loading. This indicates the existence of fatigue in nanoscale metal. The load–deflection curves of nano-cantilever specimen in each cycle are shown in Fig. 25. These curves show apparent nonlinearity and a distinct hysteresis loop exists, which indicates the cyclic plasticity of Cu layer, since only elastic deformation appears in the Si substrate and the SiN layer at this loading level. The cyclic hardening is also found from the change in the hysteresis, which indicates the development of a substructure in the Cu layer during fatigue experiment. In bulk metals, the irreversible cyclic deformation leads to the formation of characteristic dislocation structures including ladder-like structures, veins, and cells. The dislocation structures play a central role to the deformation and fracture behavior of bulk metals. However, as aforementioned, the characteristic size of dislocation structures in bulk metals is usually on the scale of a few micrometers,

Fig. 23 (a) Schematic illustration of the nano-cantilever experiment for fatigue experiments, and (b) a TEM micrograph of the specimen (side view) [31]

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Fig. 24 (a) Load–time relationship under the load range of 18 μN, and (b) corresponding TEM images in the 7th cycle [31]

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Fig. 25 Load–deflection curves under the load range of 18 μN during fatigue [31]

which is much larger than the size of strain concentrated region of Cu layer in nanocantilever specimen. Thus, these typical dislocation structures in bulk metals are difficult to develop in nanoscale components. However, due to the limitation on experimental techniques, the features of dislocation substructures in nanoscale components are totally unknown and remain for the further investigations. Figure 26 shows the relations between the applied stress range, Δσ, and the number of cycles to fracture, Nf (S–N relation) for nano-cantilever specimen including a 20-nm-thick and 200-nm-thick Cu layer, respectively [31, 32]. On the left side of Fig. 26(a) and (b), the facture stress of the Cu/Si interface in the monotonic loading experiment is used for Δσ of Nf = 1. As shown in Fig. 26(a), the fatigue life (i.e., the number of cycles to fracture) clearly exhibits a strong dependence on Δσ, and the magnitude of stress range is much higher than that of bulk Cu (0.05–0.2 GPa [33]). On the other hand, Fig. 26(b) shows the nano-cantilever specimen including 20-nm-thick Cu layer possesses much higher stress range than that of specimen with 200-nm-thick Cu layer both in monotonic and cyclic loading experiments. The

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Fig. 26 Relationship between the stress range, Δσ, and number of cycles to fracture, Nf, of nanocantilever specimens with 20-nm-thick and 200-nm-thick Cu layers [31, 32]

obvious difference is induced by the significant difference in the plastic properties of 20-nm-thick and 200-nm-thick Cu layers. On the other hand, during the actual application of micro-devices, the nanoscale components are usually subjected to the influence of environment factors, such as temperature [34] and humidity [35]. Thus, to assure the long-term durability of these devices, it is important to evaluate the effect of environment factors on fatigue strength of the interface in nanoscale components. Figure 27 shows the comparison of relationships between the applied stress range, Δσ, and the number of cycles to fracture, Nf, for environmentally treated (a temperature of 80  C and relative humidity of 80% for 15 min) and non-treated nano-cantilever specimens [36]. An obvious difference exists between these curves. This means that environment factors including temperature and humidity affect both the toughness of the interface for cracking in the monotonic loading experiments and the fatigue life of interface in fatigue tests.

6.2

High-Cycle Fatigue

6.2.1 Experimental Method Basing on Resonant Vibration The fatigue experiments in previous section are conducted by directly applying a mechanical load to the nanoscale specimen with the contact between load tip and the specimen. Therefore, it is difficult to extend this method to a high-cycle fatigue experiment. In addition, it is extremely difficult to apply a reverse load to nanoscale specimens as well. Thus, a novel methodology by using resonant vibration is proposed to investigate the high-cycle fatigue behavior in nanoscale components [37, 38]. For nanoscale material, due to its extremely small volume, it usually possesses a resonant frequency on the scale of few tens of gigahertz. This high resonant frequency induces extreme difficulties in the precise control of fatigue cycle, because a number of 107 cycles, which is usually defined as the number of cycles of fatigue

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Fig. 27 Relations between the applied stress range, Δσ, and the number of cycles to fracture, Nf, of environmentally treated and non-treated specimens [36]

limit, is attained in less than 1 second. To reduce the resonant frequency of the test section, a weight is attached to the specimen tip. The vertical resonant frequency of the cantilever (the cross section of w (width)
h (height)), f0, with a weight at the free end could be approximately evaluated by next equation: 1 f0 ¼ 2π

! rffiffiffiffi k Ewh3 ... k ¼ 3 , m 4lG

(4)

where k is the spring constant of tested section in the vibration direction, m is the mass of weight, E is the Young’s modulus of cantilever, and lG is the length from the cantilever root to the gravity center of weight. Basing on Eq. (4), the geometric shape and dimension is designed to achieve the targeted f0. Figure 28 schematically shows the experimental system and testing method [37]. This system includes a laminated piezoelectric actuator, an operational amplifier, a function generator, a laser Doppler vibrometer, and a control computer. During the fatigue experiment, the silicon plate with the specimen is mounted on the top face of actuator with an adhesive. The specimen is oscillated at room temperature by an alternating voltage of constant amplitude. The data management system controls the measurement condition and stores the measurement data. The displacement at the specimen root is measured by the laser Doppler vibrometer.

6.2.2 Fatigue Fracture in Nanoscale Components New type of cantilever specimens with a weight attached on the free end are fabricated from a multilayered material, i.e., silicon/titanium/polycrystalline

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Fig. 28 Experimental system for the resonant vibration fatigue experiment [37]

copper/silicon nitride (Si/Ti/Cu/SiN) and adopted to conduct the resonance fatigue experiments [37]. Figure 29 shows the SEM images of the fabricated specimen and the geometric dimensions. In the test part, the surface is flat and there are no steps or defects. As aforementioned, the Cu layer is a polycrystalline material; the crystallographic information on the upper surface of Cu layer in the test part is analyzed by electron backscatter diffraction (EBSD). Figure 30 shows the shape and crystal orientation of grains in the Cu layer by EBSD. The color in the map corresponds to that in a standard stereo-triangle shown in the figure and indicates the crystallographic orientation to the y-axis. Finally, a total of thirteen coarse grains are recognized on the top surface of Cu layer and are numbered 1–13. Figure 31 shows the SEM images of the upper Cu surface after the high-cycle fatigue experiment. A straight line-like trace inclined at an angle of about 30 to the normal stress applied (x-axis) appears on the Cu surface in Grain 3. By analyzing the stereographic projection of Grain 3 (in Fig. 31(a)), these traces are found to be slip bands induced by the activity of slip plane B. A close SEM observation shows that slip bands consist of concavity and convexity with a width of about 30 nm. In addition, as shown in Fig. 31(b), the extrusion grows in the upper-right direction corresponding to the slip direction 4 in Grain 3 with the height of about 40 nm. It should be noted that, although similar extrusions/intrusions have been extensively observed in fatigued bulk materials [39], the width ( 30 nm) in this experiment greatly differs from that of bulk materials ( 1 μm [39]). On the other hand, a crack with a length of about 140 nm is observed along the Cu/Ti interface on the top surface of specimen (Fig. 31(a)). The crack seems to be initiated at the collision point of extrusion with the Cu/Ti interface in which stress concentration exists. FEM model constructed basing on the exact shapes and crystal orientation of grains in the specimen is adopted to inquire the microscopic stress distributions, and the maximum resolved shear stress, τmrss, is obtained. In these 13 grains, Grain 3 possesses the highest τmrss and the slip system is B4, which is consistent with the slip

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Fig. 29 (a) SEM image of fabricated specimen and (b) its dimensions [37]

Fig. 30 Crystallographic orientation in the polycrystalline Cu layer [37]

bands identified in fatigue experiment. This shows that the crystallographic slip induces the appearance of traces and τmrss governs the damage during the high-cycle fatigue in nanoscale metal. Therefore, the generation of extrusion/intrusion with a width of 30 nm is governed by the nanoscopic stress field due to the deformation constraint. The stress defined on the basis of continuum mechanics is applicable for

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Fig. 31 SEM images at Grain 3 after the fatigue experiment [37]

the dominating quantity for slip band formation in fatigue of nanoscale materials with an understructure. By applying the displacement magnitude for the generation of slip bands (Δδ/2 = 1224 nm) to the FEM simulation, the critical resolved shear stress amplitude Δτcrss/2 to form slip bands is identified to be 418 MPa, which is remarkably higher than the Δτcrss/2 of the persistent slip bands formation in Cu bulk single crystal (27 MPa [40]).

6.2.3 Fatigue Mechanism of Nanoscale Components To investigate the fatigue mechanism of nanoscale components, a single crystalline gold (Au) cantilever specimen is adopted to conduct the resonant fatigue experiment [38], and the crystallographic effects on the formation of PSB in a low-dimensional specimen is expected to be clarified. Figure 32(a) shows SEM images of the upper surface and both side surfaces, where the slip bands cross the test section diagonally. Inclined slip bands with angles of 67.5 and 51 to the longitudinal direction of the test section exist at the upper and side surfaces, respectively. By FEM analysis, it is found that the slip bands appear due to the activation of slip system with the maximum resolved shear stress amplitudes in the 12 slip systems based on the crystal orientation, Δτmrss/2. The critical Δτmrss/2 at B4 slip system along the slip bands is about 150 MPa, which is almost six times larger than that of the PSBs in Au bulk (23.4 MPa [41]). On the other hand, a 15-nm-thick layer is removed from the upper surface by using FIB to investigate the cross-sectional morphology of the slip bands. As shown in Fig. 32(b), extrusions/intrusions possess heights of 50–100 nm and widths of about 15 nm exist. Although the extrusions/intrusions show similar features to PSBs in bulk [42], they have an extremely narrow width. Thus, a different nanoscale fatigue structure from that in the bulk is induced by the dimensional constraint. The

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Fig. 32 SEM images of single Au crystal after the resonant fatigue [38] (a) SEM images of top, Side1, and Side 2 surfaces. (b) SEM images of the cross section of extrusion/intrusion near the Side 2 surface

intrusion is expected to be a liable crack initiation site in cyclic deformation, which is similar with that in the PSBs of bulk [43].

7

Fracture of Nanoscale Single Crystalline Materials

7.1

In Situ Tensile Experiment of Single Crystalline Nanorod

Up to now, although bending experiments by using different types of nano-cantilever specimens have been used to examine the deformation and fracture properties of nanoscale components [44], severe strain gradients exist in these bending specimens. For micro/nanoscale pillar compression experiments [4], although the strain gradients are apparently eliminated, the fracture properties could not be investigated at same time. Thus, it is necessary to develop the methodology for tensile experiment where the stress distributes uniformly and the fracture behavior can be investigated at large strains. Figure 33 shows the SEM images of a new in situ tensile specimen for evaluating the deformation and fracture properties of nanoscale components and applying it to a single crystalline gold nanorod with a square section of 189 nm [45]. After carved out from a bulk material by FIB processing, the nanorod is

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Fig. 33 SEM images of the specimen with a single crystalline Au nanorod mounted on the Si frame and the stereographic projection of the Au nanorod [45]

attached to the central part of the lozenge-shaped Si frame. During experiment, a perpendicular downward load is applied to the upper surface of the Si frame by an indenter, and thus the frame expands horizontally and a tensile load is applied to the nanorod. The stereographic projection of nanorod in Fig. 33 indicates that single slip deformation is expected because the magnitude of the slip system B4 is enormously large. Figure 34(a) shows the load–time relationship of the specimen in which the Au nanorod is attached to the Si frame. The applied load, P, firstly increases linearly in the early stage of deformation, and drops rapidly at P = 23.3 μN. Similar drops emerge at the load level of 36.0 μN, 42.6 μN, and 49.0 μN, respectively. The sudden drop of applied load is induced by the deformation of Au nanorod. Figure 34(b) shows TEM images corresponding to Fig. 34(a) (I), (II), (III). Before applying load, the surface of the nanorod is flat and smooth without conspicuous unevenness. With increasing the applied load, the nanorod begins to extend uniformly. At P = 23.3 μN where the first drop of applied load appears, a discontinuous slippage appears on the upper and lower surfaces on the left side of the nanorod (“a” in Fig. 34(b)-II), indicating yielding of the nanorod. Similar slippages appear in subsequent loading process. At P = 42.6 μN, a large slippage is generated on the left side of the nanorod (“b” in Fig. 34(b)-III). FEM analysis is carried out to investigate the constitutive relationship of Au nanorod. By considering the Schmid factor of primary slip system B4, the critical resolved shear stress of the Au nanorod is determined to be 326 MPa, which is about 600 times larger than that of the bulk (0.5 MPa [46]). In addition, no macroscopic work-hardening is recognized after the yielding of Au nanorod. This is similar to that of a whisker [47], which possesses a structure similar as a perfect crystal. Because the loss of dislocations through the surface causes the lower dislocation density in the low-dimensional materials, the nanorod requires a high applied stress and shows similar plastic deformation behavior as the whisker. Figure 35 shows the SEM images of the nanorod after the loading experiment. At the left side of the nanorod where initial slip deformation and final fracture appear, apparent slip traces are recognized near the surface, which correspond to the activity of the primary slip system B4 in the nanorod. In the center of the fracture part, cracking induced by necking is observed. In this region, the specimen deforms isotropically and no crystallographic slip deformation is observed. In addition, at the front tip of the fracture part, the nanorod is thinned down to about 15 nm, which

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Fig. 34 (a) Load–deflection curve of the specimen and (b) corresponding in situ TEM observation of the nanorod during loading experiment [45]

Fig. 35 SEM images of the nanorod after the tensile experiment [45]

means that the nanorod deforms plastically regardless of the crystalline structure in large strain region.

7.2

Fracture of Single Crystalline Si in Nanoscale

The tensile strength (fracture strength) and the fracture toughness are typical parameters for representing mechanical properties of materials. Tensile/fracture strength is well known to be strongly dependent on material size at the micro- and nanoscales [48]. However, the size effect of fracture toughness has not been precisely investigated due to experimental difficulties, such as introducing a pre-crack in a small specimen and specifying of the precise position of the pre-crack tip at nanoscale. Thus, small single crystalline silicon specimens with different pre-crack lengths are adopted to examine the size effect of a singular stress field on the fracture toughness at the nanoscale. Figure 36 shows the schematic illustration of the experimental method and the SEM images of the fabricated specimen by FIB [49]. During the experiment, by using a wedge-shaped diamond indenter, an opening displacement is applied to the notch mouth to introduce a pre-crack at the notch root as shown in Fig. 36(a). After that, the upper part of the notch is cut by FIB to shorten the pre-crack (Fig. 36(b)). In addition, the central region of the specimen is thinned down to 300–600 nm for specifying the pre-crack tip position through in situ TEM observation. Figure 36(c)

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Fig. 36 (a) schematic illustration of experimental method and (b) SEM images of the single crystalline silicon micro-specimen [49]

shows SEM images of a fabricated specimen by FIB processing. There is no unevenness or observed slip lines on the surface of the specimen. Figure 37(a) shows the in situ TEM observation of the crack propagation and corresponding load–displacement curve during loading test. Although deviation exists between the specimen axis and the (110), the crack starts to propagate along a (110) plane at P = 594 μN. Figure 37(b) shows the load–displacement curve. With the increase of applied displacement, the load linearly increases up to a maximum. Therefore, it is the critical load for the unstable crack propagation along (011) plane, and the critical crack opening displacement is identified by in situ TEM observation. Similar cracking behaviors are also observed in other specimens. The stress fields in all specimens near the crack tip at the critical load are calculated by FEM analysis. The fracture toughness for cracking along the (110) plane is evaluated by the singular field intensity. Here, the ɅK is the distance from the crack tip where the stress deviates 5% from the singular field of r1/2. Figure 38 shows the dependence of the fracture toughness, KIC, on ɅK. The fracture toughness of (110) plane cracking in bulk single crystalline Si was evaluated to be 0.75–1.08 MPam1/2 [50], which is also included in the figure. It should be noted that ɅK is larger than a micrometer size for an ordinary fracture toughness specimen. As shown in Fig. 38, the fracture toughness in the nanoscale is equivalent to that of the bulk

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Fig. 37 (a) In situ TEM images of the specimen and (b) load–displacement curve [49]

material and shows no dependence on ɅK. The cracking under the nanoscale ɅK means that the local fracture takes place at that size. The consistence of KIC between the bulk and the low-dimensional specimens clarifies that the fracture of bulk specimen is dominated by the nanoscale local area in carefully controlled homogeneous materials.

8

Summary and Further Directions

Recent advances in the field of fracture mechanics of nanoscale components have been reviewed in this chapter. Different types of nano- and microscale specimens have been designed, fabricated, and employed for characterizations of mechanical properties ranging from basic bending and tensile testing to fatigue to multi-physical testing. In addition, in situ loading experiments inside TEM are adopted and the deformation and fracture behaviors of nanoscale components are directly observed and recorded. Through these studies, the fracture criterions for interface cracking in

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Fig. 38 Relationship between critical stress intensity factor, KIC, at crack propagation and size of stress singular field, ɅK [49]

nanoscale components, the low- and high-cycle fatigue behaviors and the environmental effect of nanoscale metals, the fracture behavior of nanoscale single crystalline metals, and the applicability of continuum fracture mechanics for nanoscale components have been deeply discussed, and corresponding fracture criterion and governing mechanical factors have been obtained. On the other hand, great challenges still remain in the further investigation of fracture mechanics of nanoscale materials and can be depicted as follows: 1. The crack initiation from a free edge of the interface in nanoscale components has been extensively studied by using different types of nano-cantilever specimens. However, due to difficulties in introducing a pre-crack along an interface at the nanometer level and precise measuring the load and deformation during the unstable crack propagation process, new experimental techniques are required and the mechanical criterion for crack propagation along the interface in nanoscale remains for further investigations. 2. In actual applications, crack might initiate and propagate from the microscopic defects inside an ingredient layer in multilayered nanoscale materials and finally interact with the interface. Different interaction behaviors including crack deflecting from the interface, crack propagating along the interface, and crack penetrating through the interface appear dependent on boundary conditions. Therefore, the prediction of interacting behavior between the crack and interface is also critically important for the deep understanding of fracture mechanics at nanometer scale. 3. Through experimental studies in previous sections, the fracture behaviors induced by the highly confined strain concentration of “10–1000 nm” has been successfully investigated. The result indicates that the “stress” within the framework of continuum mechanics is still applicable for the fracture behavior as the dominating parameter while the size approaches the atomic level. On the other hand, it still remains for challenge to establish the fracture mechanics in single-digit nanometer scale. Although some experiments were carried out by using components with an entire size of 1–10 nm, only the fracture behavior under a

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homogeneous load/displacement field is investigated. They critically lack the nature of fracture, because the mechanical failure is a local phenomenon driven by an inhomogeneous deformation field due to component shape, material combination, loading condition, and so on. Therefore, it is necessary to verify the applicability of “stress” on the single-digit nanometer scale attributed to the discreteness of atoms, and to clarify the governing mechanical parameter for fracture in the inhomogeneous field. 4. Graphene as a typical two-dimensional material has recently attracted significant attention from the scientific community because of its extraordinary mechanical and electrical properties. Although extensive theoretical and computational modeling has investigated its intrinsic strength that indicates a uniform rupture of carbon bonds, the useful strength of large-area graphene with engineering relevance is usually evaluated by the fracture toughness which means the ability of a material containing a crack to resist fracture. However, due to difficulties in the mechanical testing of atomically thin membranes, it has been a great challenge to quantitatively measure the mechanical properties of graphene. Thus, further efforts are needed to develop the appropriate experimental methodology for conducting the fracture testing and to systemically investigate the fracture properties of graphene. Acknowledgement This work was supported by JSPS KAKENHI (Grant Numbers 25000012 and 15H02210).

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In Situ Transmission Electron Microscopy Investigation of Dislocation Interactions Josh Kacher, Ben P. Eftink, and Ian M. Robertson

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dislocation Nucleation, Propagation, and Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Dislocation Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dislocation Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Dislocation Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Dislocation Interactions with Defect Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Solid Solution Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Oxide Dispersion Strengthened Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Irradiation-Induced Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Dislocation Interactions with Isolated Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Dislocation Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stacking Fault Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Precipitates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Dislocation–Grain Boundary Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Temperature Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Coherent Twin Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Propagation Limited Systems: Irradiated Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Confined Microstructure: Ultrafine Grained and Nanotwinned Materials . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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J. Kacher (*) Georgia Institute of Technology, Materials Science and Engineering, Atlanta, GA, USA e-mail: [email protected] B. P. Eftink Los Alamos National Laboratory, MST-8 Materials Science in Radiation and Dynamic Extremes, Los Alamos, NM, USA I. M. Robertson University of Wisconsin-Madison, Materials Science and Engineering, Madison, WI, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_7

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Abstract

This chapter provides a broad overview of dislocation interactions investigated via in situ transmission electron microscopy (TEM) deformation experiments. The discussion of these interactions is divided according to the interaction of interest, with the first section exploring the mechanics and energetics governing dislocation nucleation, propagation, and multiplication. The following two sections investigate dislocation interactions with isolated defects and defect fields, including interactions involving irradiation-induced defects, solute atoms, and second-phase particles. The final section discusses dislocation–grain boundary interactions with a focus on understanding how the local grain boundary structure and surrounding microstructure dictate the dislocation transfer process. Two unique advantages of TEM imaging for dislocation interactions will be highlighted throughout this chapter: the ability to capture dislocation interactions at sufficient spatial and temporal resolution to resolve complex interactions, and the ability to resolve salient features of the dislocation interactions using diffraction contrast imaging. This second advantage is used to characterize structural and geometrical factors influencing dislocation interactions, including the dislocation Burgers vector, line direction, and slip plane, crystallographic orientation, and boundary habitat planes. Keywords

Transmission electron microscopy · In situ deformation · Dislocations · Plasticity · Strengthening mechanisms

1

Introduction

The mechanical properties of metals are largely dictated by the activity of dislocations, including their generation, propagation, and interaction with other defects such as precipitates and grain boundaries. Transmission electron microscopy (TEM) is uniquely capable of resolving complex dislocation interactions in real time and at sufficient resolution to capture the unit processes dictating an interaction. In addition, using diffraction-contrast analysis, relevant dislocation parameters such as Burgers vector, slip plane, and line direction can be characterized and applied to understand dislocation processes. This chapter will review in situ TEM-based experiments devoted to understanding the following dislocation interactions: 1. 2. 3. 4.

Dislocation nucleation, propagation, and multiplication Dislocation interactions with defect fields Dislocation interactions with isolated obstacles Dislocation–grain boundary interactions

Although defects play a role in the deformation of ceramics and polymers, this chapter will focus primarily on dislocation interactions in metals.

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In Situ Transmission Electron Microscopy Investigation of Dislocation. . .

2

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One of the original successes of electron microscopy characterization was proving the existence of dislocations [1]. Since that time, many studies have been published seeking to characterize the most fundamental aspects of dislocation behavior. These studies have two broad objectives: understanding the mechanisms by which dislocations operate and calculating the various energies/energy barriers associated with dislocation activity. This section will review studies performed to understand dislocation nucleation, multiplication, and propagation in a variety of systems.

2.1

Dislocation Nucleation

The original impetus for proposing the existence of dislocations was to explain the discrepancy between the theoretical strength of a material, or the strength needed to shear the atomic bonds in a material, and the observed strength. It is now understood that the strength of a material is related to the resolved shear stress needed to propagate dislocations through the matrix. However, a notable exception is in the case of dislocation free materials in which there are no preexisting dislocations for stresses to act on. In these materials, the theoretical strength and measured strength converge. This behavior was first observed in dislocation-free Sn whiskers [2], and, with the prevalence of nanomaterials, is becoming increasingly applicable. This emergence of nanomaterials, as well as advances in sample fabrication and nanoscale testing technologies, has motivated a number of studies into the nucleation mechanisms and energy of dislocations in initially dislocation-free materials. Due to the imposing energy barrier against homogeneous nucleation, the vast majority of dislocation nucleation occurs at preexisting defects, including interfaces. Understanding the energy barriers to dislocation nucleation is important in modelling the strength of nanoscale materials in which the initial defect population is often near zero. Surface dislocation nucleation has been explored under both tension [3–5] and compression [6, 7] using quantitative in situ TEM indentation holders. These holders record the load and displacement while the sample is deformed via direct indentation or in tension using a push-to-pull device. Lu et al. investigated surface dislocation nucleation in ultrathin (7–10 nm diameter) gold wires [4]. Using an atomic force microscopy (AFM) tip inside the TEM, they conducted tensile tests while imaging the sample in high resolution. They found that, upon the onset of plasticity, the stress dropped precipitously. High resolution imaging coupled with fast Fourier transforms of the atomic resolution images showed that necking of the sample occurred along {111} slip bands, suggesting that dislocation propagation was responsible for the observed deformation (Fig. 1). The first instance of dislocation activity in this test, as suggested by morphological changes of the nanowire, is indicated by the arrow in Fig. 1c. With increasing strain, an adjacent dislocation system activated, contributing to the necking of the wire. Dislocation activity and necking continued until final fracture occurred. Although

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Fig. 1 HRTEM images of a short gold nanowire under tensile testing (a)–(h): After the elastic deformation region (a)–(b), initial surface dislocation nucleation is indicated by the arrow in (c). Arrows in (d)–(f) indicate more similar dislocations emitted from both sides of the nanowire surfaces, and a neck (e)–(g) was formed in the middle section of the nanowire until the final fracture occurred as shown in (h). The insert in (b) shows a fast Fourier transform image calculated from the image inside the white square, showing the wire orientation (Image reused with permission of Springer from [4])

the strength of the nanowires was calculated, the authors did not relate this to the dislocation nucleation stress. Chen et al. studied the effects of temperature and strain rate on the surface nucleation of dislocations in dislocation-free Pd nanowhiskers under uniaxial tensile loads using a picoindenter holder combined with a push-to-pull device [3] (Fig. 2a). Multiple samples were tested in situ in the TEM while systematically varying the applied strain rate and test temperature. They found that the samples could experience significant stresses, in the GPa regime, with no dislocation nucleation. That is, the samples underwent true elastic deformation. At a sufficiently high stress level, TEM imaging revealed that the plastic deformation was accommodated by the nucleation of surface dislocations or twins (Fig. 2f–i). Varying the strain rate was not found to influence the dislocation nucleation strength. Temperature variations, in contrast, strongly influenced nucleation strength, with the sample yield strength decreasing from 5.8 to 2.8 GPa with a temperature increase from 93 to 447 K. This temperature correlation is significantly stronger than the strength–temperature correlation found in bulk materials, suggesting that the barrier strength to dislocation nucleation is much more temperature dependent than the barrier to propagation of preexisting dislocations or the nucleation of dislocations from grain boundaries. Chen et al. attributed this temperature dependence to a thermally active surface diffusion mechanism for dislocation nucleation. In contrast to Chen et al., who used indirect measures to calculate the dislocation nucleation stress, Li et al. directly quantified the stress needed to nucleate dislocations using measured atomic displacements during in situ TEM deformation coupled with first principles calculations [6]. They indented a TiN thin film along two different crystallographic directions, the and , to selectively activate

Fig. 2 (a) Image of push-to-pull device with enlarged view showing mounted nanowires. (b–e) Image sequence of nanowire during in situ straining experiment. (f) Stress/strain curve associated with image sequence. (g) Fracture surface of nanowire after test with diffraction pattern showing fracture planes and active dislocation systems. (h–i) Stress strain curve and fracture surface of nanowire test conducted under similar conditions. Both tests were conducted at room temperature. Reprinted by permission from Macmillan Publishers Ltd: Nature Materials (3) copyright 2015

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the {111} and {110} slip systems, respectively. While indenting along a direction, they observed the emission of a partial dislocation from the region beneath the indenter tip. High resolution imaging showed that the dislocation resided on the (1 1 1) plane with Burgers vector a0/6[ 1 12] and line direction parallel to [110]. With continued compression, the trailing partial dislocation was also emitted and recombined with the lead partial dislocation, resulting in the atomic lattice being restored to its original state. The character of the trailing partial dislocation was not reported. In order to quantify the dislocation nucleation stress, the lattice strain was calculated by measuring the atomic displacements from the perfect lattice configuration near the indenter tip 0.3 s prior to the emission of the lead partial dislocation. Density functional theory (DFT) calculations were used to determine the nonlinear elastic behavior at high stresses. The critical nucleation stress was measured as 13.8 GPa. Repeating the experiment with a loading direction parallel to , they observed the emission of a perfect edge dislocation on a {011} plane. The critical stress in this second experiment was calculated as 6.7 GPa. The difference in the measured critical stresses was attributed to dislocations in the {111} systems experiencing a larger Peierls stress.

2.2

Dislocation Propagation

Dislocation propagation occurs via glide, cross-slip, and climb. Dislocation motion and the energetics dictating their motion is central to understanding and modelling the flow stress of materials. In FCC systems, dislocation motion is limited almost exclusively to close packed planes and directions. In lower symmetry crystals such as HCP, the lack of available slip systems often forces the activation of less energetically favorable systems. The behavior of these dislocations and mechanisms that dictate their behavior has been studied extensively by in situ TEM deformation experiments [8–13]. Here we focus on propagation of dislocations in HCP materials. In most HCP materials, dislocation activity is dominated by propagation of a-type dislocations gliding on either the basal plane or the prismatic plane, with the energetically more favorable plane generally determined by the c/a ratio of the crystal lattice. Due to the restricted number of available slip systems, HCP systems tend to be significantly less ductile than FCC materials, with twin activation serving as an additional deformation mechanism. As dislocation glide on the different available slip planes varies energetically, deformation of HCP materials has a strong thermal component. For example, Couret and Caillard investigated the behavior of prismatic dislocation systems in Mg over a temperature range from 50 to 673 K [11, 12]. Crystal directions were carefully aligned with the tensile axis such that dislocation activity on the basal plane, the energetically favorable plane in Mg, was suppressed and prismatic glide was activated. They observed that dislocation motion was strongly anisotropic, with edge components of dislocations gliding easily at low temperatures but screw components requiring significantly higher stress levels to operate. This behavior was strongly dependent on the temperature. At 50 K, no

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Fig. 3 Dislocation loop expansion at 373 K in Mg during in situ TEM deformation showing asymmetric glide of edge and screw components (Image modified with permission from [11])

screw dislocation activity was observed, though limited motion of edge dislocations continued. As the temperature increased, the screw components increased in activity and the anisotropy between the screw and edge dislocation mobilities decreased (see, for example, dislocation glide at 373 K in Fig. 3). At the peak temperature, ~673 K, climb mechanisms activated and the dislocations formed sub-boundaries. At all temperatures, dislocation propagation proceeded via cross-slip between the basal and prismatic planes. Investigations of the tension/compression behavior of certain HCP alloys such as Ti have shown a marked asymmetry in the mechanical properties. This asymmetry has been tied to the dislocation core structure, which can have a strong influence on the dislocation propagation behavior [14]. Clouet et al. investigated this behavior in two different systems, Ti and Zr [10]. Video frames of the dislocation motion taken during in situ TEM deformation and difference images showing the dislocation motion between frames are shown in Fig. 4. Despite their similar electronic structures, dislocation motion in the two materials was seen to proceed in very different manners. In both systems, the dominant slip plane is the prismatic plane. However, while dislocations in Zr propagated smoothly, the motion of dislocations in Ti was jerky and intermittent. Intermittent flow of dislocations in α-Ti was also observed by Kacher and Robertson, who noted that this behavior led to dislocation generation via cross-slip mechanisms [15]. Clouet et al. traced the different behaviors to the core structure of the dislocations. In both systems, the dislocation core spreads on the pyramidal and first order prismatic planes. In Zr, the stable plane is the prismatic plane, which corresponds to the lowest energy glide plane. In Ti, the prismatic plane is the lowest energy glide plane, but core spreading onto the pyramidal plane is more stable. As a result, the dislocations in Zr exhibit easy glide during deformation while dislocations in Ti display frequent cross-slip events onto the pyramidal plane, creating sessile jogs during the deformation process. They hypothesized that this behavior could be influenced by the solute state of the material, suggesting a pathway to understanding the strengthening effects of impurities such as O in Ti.

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Fig. 4 (a–c) Dislocation glide in α-Ti during in situ TEM deformation. Difference image shown in (c). Slope and line direction shown in (a) and plane trace is shown in (b). (d–f) Dislocation glide in Zr during in situ TEM deformation. Difference image shown in (f). Slip plane trace indicated in (d) and Burgers vector in (f) (Reprinted by permission from Macmillan Publishers Ltd.: Nature Materials [10] copyright 2015)

2.3

Dislocation Multiplication

The two primary dislocation-based mechanisms for dislocation multiplication (excluding boundary and interface sources) are the activation of Frank-Read sources and via double cross-slip events. As both of these mechanisms involve the expansion of dislocation loops, their operation can be significantly influenced by the thin film geometry needed for TEM analysis. For example, although Frank-Read sources are expected to be a common mechanism for dislocation multiplication, observations made during in situ TEM deformation are limited almost exclusively to single-arm sources. One way the effects of free surfaces have been partly alleviated in in situ TEM investigations is by using high-voltage electron microscopes (HVEM) in which voltages range from 1 to 3 MV and sample thicknesses can extend into the μm range. Appel et al. investigated dislocation motion under tensile deformation in 1 μm thick MgO films [16]. Video frames of the interaction are shown in Fig. 5. They observed that gliding screw dislocations formed large jogged sections as they propagated through the matrix (Fig. 5a–c). As these jogged sections were sessile, they trailed the propagating dislocation front (Fig. 5d–e). The connecting segments, which were edge in nature, continued to elongate as the screw segment propagated, eventually pinching off and remaining in the matrix as dislocation debris (Fig. 5e–f). Similar experiments have been conducted in a range of materials, though mostly in oxides or ceramics where the electron beam damage effects are less pronounced. Werner et al. investigated dislocation motion in Si during high temperature straining [17]. They observed a mechanism for dislocation source formation where a screw dislocation and mixed-character dislocation with parallel Burgers vectors but propagating on different

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Fig. 5 Image sequence showing an edge dislocation dipole forming behind gliding screw dislocation during in situ TEM deformation of a MgO film (Image used with permission from [16])

glide planes intersect and form a node. Once formed, the node acts as a pinning point for the operation of a single arm dislocation source. Baither et al. investigated high temperature deformation behavior of dislocations in cubic zirconia [18]. They found that the dislocation multiplication mechanisms were dependent on the crystallographic orientation in relation to the tensile axis and were strongly influenced by the free surface. Samples with a (110) foil surface facilitated easy escape of screw dislocations through the sample surface, restricting the number of super jogs forming from cross-slip events. As the super jogs were the primary source for dislocation multiplication, few sources were seen to operate. In contrast, samples with a ( 1 11) free surface, multiple α-shaped

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Fig. 6 Dislocations generated from a single arm dislocation source in an austenite grain during in situ TEM deformation of a duplex steel. Frames (b) and (c) were captured 58 s and 126 s after (a), respectively. Arrow in (a) indicates location of dislocation source (Image used with permission from [19])

dislocation segments were found in the deformed samples. These dislocation configurations operated as dislocation sources, producing shear loops repeatedly during deformation. Messerschmidt and Bartsch summarized many of these findings, focusing their remarks on the influence of the cross-slip behavior of the materials on dislocation multiplication mechanisms [19]. In materials where the cross-slip plane is equivalent to the primary glide plane, such as is found in FCC materials, the barrier to dislocation multiplication via cross-slip is relatively low. In many materials, especially those with lower symmetry, the flow stress for glide on the cross-slip plane is substantially higher than on the primary plane. This significantly decreases the crossslip height, restricting the operation of cross-slip based dislocation multiplication mechanisms. Thus, dislocation generation via a Frank-Read source mechanism was found to be more prevalent in lower symmetry crystals. Messerschmidt and Bartsch also experimentally demonstrated the effects of back-stress in terminating dislocation sources. Figure 6 shows dislocation multiplication occurring at a single-arm dislocation source in an austenite grain in duplex steel. In the middle of the grain, a single-arm source is visible, identifiable by the helical shape indicated by an arrow. As the sample was deformed, dislocations were generated and propagated from the source, piling up at a nearby phase boundary. After the accumulation of several dislocations, the back stress became sufficient to shield the stresses operating on the source and terminate further dislocation multiplication. The source activity restarted once the dislocations transferred across the boundary and away from the source.

3

Dislocation Interactions with Defect Fields

Fields of defects in materials can be introduced to strengthen a material such as in the case of oxide dispersoids or solid solution interstitial and substitutional atoms. Dispersed defects can also be detrimental to the properties of materials such as in

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the case of dislocation loops or stacking fault tetrahedra introduced during irradiation by energetic particles. The impact on mechanical properties, whether beneficial or detrimental, is directly related to how dislocations interact with the defects. In situ TEM characterization techniques are well suited to directly observe dislocation interactions with individual and fields of defects and have the temporal and spatial resolution necessary to determine the mechanisms associated with the interactions. While atomistic simulations provide a higher temporal and spatial resolution than experimental techniques, the simulations are limited in size- and time-scales.

3.1

Solid Solution Atoms

Solid solution strengthening is a common strengthening approach involving additions of interstitial or substitutional atoms to a material. Depending on the solute and concentration, mechanical properties may vary. For example, the addition of carbon increases the strength of Fe whereas the addition of hydrogen to most metals has a detrimental embrittling effect. The mechanisms proposed to account for these changes are attributed to their interaction with dislocations; particularly, how the stress field of dislocations interacts with the solutes. In Fe, solute carbon atoms diffuse to and occupy the octahedral position in the BCC lattice. The resultant matrix strain fields interact with glissile dislocations and add resistance to their movement. This is believed to be the cause of dynamic strain aging and yield stress drop in Fe. The impact of carbon concentration on dislocation slip in Fe has not been directly observed until recently. Fe samples with carbon concentrations ranging from 1 to 230 ppm were subjected to in situ TEM deformation at temperatures ranging from 293 to 473 K [20]. The experiments confirmed the influence of temperature on dynamic strain aging and its effect on the bulk mechanical response of Fe. At room temperature, dislocation glide was smooth for both 1 and 16 ppm carbon samples, with the 230 ppm carbon sample exhibiting serrated dislocation flow. Higher temperatures, 373 K for 16 ppm carbon and 453 K for 1 ppm carbon, induced serrated dislocation flow at lower carbon concentrations. Higher temperatures increase the mobility of carbon and aids carbon diffusion to the dislocations. Carbon at the dislocations also had the effect of changing the character of the dislocations to screw dislocations. In contrast to carbon in Fe, in situ TEM deformation experiments have found that hydrogen additions to most metals increase the dislocation mobility [21]. Shih et al. used an aperture-limited environmental cell TEM in combination with in situ deformation to investigate the influence of a hydrogen-environment on dislocation mobility [22]. By introducing the hydrogen environment in situ during deformation, the dislocation behavior with and without the influence of hydrogen could be directly compared. Shih et al. observed that the dislocation velocity increased from 5
102 μm s1 in vacuum to an average 7.6
102 μm in 13 kPa of hydrogen. They hypothesized that this change in dislocation velocity was due to hydrogen decreasing the stress fields associated with the dislocations such that the interaction energy with other elastic obstacles was decreased – the so-called hydrogen-shielding

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mechanism of hydrogen embrittlement [23]. Comparing the carbon and hydrogen additions to materials shows dislocation motion can be either inhibited or enhanced by solutes. Substitutional atom replacement in Fe has been investigated by alloying controlled concentrations of Ni, Si, Cr, and Al [20]. Alloys were prepared such that the carbon content was constant at 16 ppm. For the case of Ni and Si, minimal effect was observed for dislocation motion, the only notable difference being the more frequent occurrence of superjog formation. This was thought to be due to cross-slip at substitutional atoms. Additions of Cr at 5 and 14 at% and Al at 17 wt% were observed to increase the onset temperature of dynamic strain aging as well as reduce the prevalence of screw character dislocations. This behavior was attributed to an attraction of the carbon atoms to Cr and Al atoms. Clusters of Mg in an Al–Mg alloy have also been observed to interact with and pin dislocations, resulting in the formation of dislocation debris during dislocation propagation [24]. Using electron energy loss spectroscopy (EELS) analysis, the authors found an increased Mg concentration surrounding dislocation loops formed during deformation, suggesting that Mg was preferentially inhabiting the dislocation core. These systematic studies on the influence of solute additions on dislocation behavior can act as guides in the development and application of new alloys.

3.2

Oxide Dispersion Strengthened Materials

Similar to solid solution atoms, oxide dispersions introduced in materials have the effect of inhibiting dislocation motion. Metals with oxide dispersions have been subject to interest for improved high temperature creep strength as well as irradiation tolerance. Studies have used in situ TEM deformation experiments to observe dislocation interactions with fine oxide precipitates at elevated temperatures and, to a limited extent, at room temperature [25–27]. Elevated temperature in situ deformation studies of several oxide dispersion strengthened (ODS) alloys have confirmed the mechanism for precipitate bypass is rate limited by the detachment process at the departure side of the precipitate [25–27]. This mechanism is consistent for several ODS alloys, including INCOLOY MA 956, INCONEL MA 754, NiAl, Ni3Al, and FeAl. At room temperature, particle shearing or Orowan bypass mechanisms are expected to be active. The combination of in situ heating and straining TEM experiments allows the differences in deformation behavior across temperature ranges to be compared directly. When considering ODS alloys, the bypass mechanism of precipitates for dislocations has been found to change with temperature. An Fe–Cr, INCOLOY MA956, ODS alloy has been investigated using in situ TEM heating and deformation experiments to determine the dynamics of this process [25]. At room temperature, strengthening can be partially attributed to the Orowan precipitate bypass mechanism while at elevated temperatures the dislocations are able to climb to bypass the oxide dispersoids. During in situ straining at 700  C, the authors grouped the bypass mechanism into four steps: (i) viscous slip until encountering a precipitate pinning

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point, (ii) encountering a precipitate and quickly overcoming it by climb, (iii) pinning of the dislocation at the departure side of the precipitate as the dislocation bows out on both sides until reaching elastic equilibrium, and (iv) detachment from the precipitate and slip to the next obstacle. From the in situ TEM results, it was confirmed that the rate limiting step of the bypass mechanism was the detachment of the dislocation from the departure side of the precipitate. In other words, the time required for dislocation climb to occur was less than the time required for dislocation bowing and detachment. Häussler et al. also found that in a Ni-base superalloy at elevated temperatures, the rate limiting step of dislocations passing an oxide is the pinning at the interface [26]. The addition of Y2Ti2O7 hard and non-shearable incoherent oxide particles inhibits the motion of dislocations, though how this occurs was seen to depend on the test temperature. At low temperatures, Orowan looping mechanism dominated, while at higher temperatures, dislocation climb was prevalent. Again, dislocation climb was not found to be the rate limiting step during particle bypass. Rather, attraction between the dislocation and the particle interface was found to be the primary strengthening mechanism, imparting creep resistance to the alloy. Häussler et al. were able to directly observe this mechanism during in situ HVEM straining experiments and verify the climb and detachment mechanism at 1020 and 1165 K in Inconel MA 754 Ni superalloy. At elevated temperatures, they found strain-rate dependent dislocation behavior. At low strain-rates, it was found that detachment is the rate-limiting step and is slower than the bowing out of the dislocation. At higher strain-rates, bowing out is the slower step compared to detachment and the deformation process is smoother. Yeh et al. found that dislocations were attracted to the precipitate interfaces in Al with Be precipitates when the samples were heated [28]. In their experiment, in situ TEM heating to 473 and 623 K was used to induce thermal stresses and drive dislocation motion. One key finding in the study was that the stress for detachment was near the Orowan stress, highlighting the attraction of the dislocations to the matrix–particle interface. The authors suggested that the interfaces relieve stress associated with the dislocation and thereby lower the energy. This is dependent on the interface characteristics, which has not been directly explored experimentally. Dislocation–precipitate interactions in ODS intermetallics exhibit a bypass mechanism composed of dislocation climb followed by detachment, again with detachment as the rate limiting step [27]. Using in situ TEM straining experiments in HVEM at 1 MeV and a temperature of 1173 K, the deformation response of NiAl, Ni3Al, and FeAl were investigate. Dislocations were found to split into super partial dislocations and, by measuring the distance between the leading and trailing super partial dislocations, it was seen that the distance decreased near precipitates. This suggests the interface of the precipitate is a low energy position for the dislocations and exerts an attractive force. Figure 7 is a weak-beam dark field micrograph of a super dislocation slipping through the matrix and restricted by several precipitates. Dislocation bowing between oxide particles is apparent in the micrograph, indicating that the oxides are pinning the dislocations during glide. As the dislocation interactions were observed in real time, the time-resolved position data of the lead and trailing partial dislocations could be tracked individually. These data are shown

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Fig. 7 Dissociated lattice dislocation in Ni3Al ODS alloy propagating through a field of precipitates. The dislocation is blocked at several precipitates along its length and is bowed out between them (Image used with permission from [27])

Fig. 8 Position as a function of time for the leading and trailing partial dislocations P1 and P2, black and white shapes, respectively. Two dislocation line segments, the first is in the matrix and second in contact with the particle and labeled as such. The notable feature is between 4 and 8 s, the leading and trailing partial dislocations have a nearly constant position and have a smaller separation (Image used with permission from [27])

graphically in Fig. 8. Notably, the propagation of both the leading and trailing dislocations was inhibited by the particle, resulting in a decrease in their separation distance as they neared the particle interface. Praud et al. investigated the active deformation mechanisms in a Fe–14Cr ODS alloy from 293 to 773 K [29]. During in situ TEM straining at 773 K, they found significant grain boundary source activity as well as grain boundary fracture. Straining at lower temperatures, 293 and 673 K, they found that the importance of

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grain boundary sources was diminished. Dislocation pinning and Orowan bypass mechanisms at the oxides were prevalent. Finely dispersed Cu precipitates in Fe, though not oxides, can be investigated using similar approaches. Nogiwa et al. investigated the impact of precipitate size on strengthening at room temperature by using heat treatments of 20 min and 10 h at 798 K of Fe with Cu additions [30]. Without the heat treatment, the precipitates were predicted to be less than 0.6 nm and with heat treatments, less than 4 nm. This was not confirmed in the experiments. They found that, when no heat treatment was used, dislocation pinning was not observed. In the heat-treated samples, dislocations were pinned at precipitates, with the pinning strength higher in the sample annealed for 10 h. However, the general strength increase due to the precipitates was consistent when measuring the bow-out of dislocations before detachment from the precipitates for each case. Comparing the direct observations of dislocation bowing, the obstacle strength parameter of the precipitates was found using [31]: Δσ c ¼ TΔτc ¼ T ðμb=LÞ cos ðφc =2Þ3=2 ð1  φ0 =5π Þ Where Δσ c is the tensile stress increase, Δτc is the hardening, T is the Taylor factor, μ is the shear stress, b is the dislocation Burgers vector, and φ is the breaking angle between the dislocation and the precipitate. The obstacle strength was in good agreement with the increase in strength measured by tensile testing. Tougou et al. also used this approach to measure the obstacle strength parameter in a V-4Cr-4Ti alloy with 4 nm Ti(OCN) precipitates at room temperature [32]. Though the studies confirmed the applicability of the equation, it is limited by the necessity to conduct in situ TEM experiments for accurate dislocation bowing measurements.

3.3

Irradiation-Induced Defects

Irradiation of materials with energetic particles is known to result in the formation of defects, including perfect and faulted dislocation loops, stacking-fault tetrahedra, and voids. The density of these defects increases with dose, usually measured in displacements per atom (dpa), increasing the strength and decreasing the ductility of the irradiated material. Dislocation interactions with irradiation-induced defects have been found to result in strain localization into dislocation channels, or channels through the material in which the barrier to glide is lower than the surrounding matrix [33–36]. Despite the importance of these channels in modelling deformation behavior, the specific channel formation mechanisms, including channel nucleation and broadening, are not clear from post mortem characterization and remain controversial in the literature. Dislocation channel formation has been observed in in situ TEM experiments in 304 stainless steel [37, 38] and Cu [39] irradiated with ions as well as Mo and Fe–Cr–Ni irradiated with protons [40, 41]. Suzuki et al. investigated dislocation channel formation in 99.99% pure Mo and a Fe–Cr–Ni alloy, covering both BCC and FCC, respectively [40, 42, 43]. Their studies were conducted in a HVEM to

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observe thicker, non-perforated samples. In 1 μm thick Mo specimens, dislocation channel formation corresponded to slip on a specific slip plane by dislocations clearing the irradiation defects. Though the width of the channels was observed to saturate, formation and thickening of channel walls, consisting of dislocation tangles, continued with increasing strain. It was found that the sample thickness changed the behavior such that thinner Mo samples, less than 0.5 μm, did not form dislocation channels but instead a uniform decrease in loop number density occurred. Other studies on FCC materials, including Cu and 304 stainless steel, observe defect free channel formation at lower thicknesses [37, 39]. Studies on the FCC metals conducted after that of Suzuki et al. more closely investigated the channel formation mechanisms. Robach et al. combined in situ straining and ion irradiation of Cu in the TEM [39]. They observed that glissile dislocations present before irradiation were pinned by the irradiation defects and post-irradiation slip of those dislocations was limited. Channel formation was the result of new dislocations generated at grain boundary and crack tip sources. As the dislocations slipped and encountered the irradiation defects, bowing occurred and the breakaway stress was found to be between 15 and 175 MPa with a mode around 40 MPa. Dislocation glide was observed to eliminate the irradiation induced defects, leading to the formation of defect free, or at least partially cleared, channels. During channel formation, it was found that the passage of a single dislocation was insufficient to clear an irradiation-induced defect from the matrix. Instead, the passage of multiple defects was required. Double cross-slip of screw dislocations was observed frequently and found to widen the dislocations channels. Similar findings are available for 304 stainless steel, with the additional observation that dislocation source widening at grain boundaries was also found to contribute to channel widening [37, 38].

4

Dislocation Interactions with Isolated Obstacles

Several studies have used in situ TEM deformation experiments to investigate the interactions of dislocations with isolated defects, determining how characteristics of the defects affect the dislocation interaction mechanisms. TEM experiments have the advantage of allowing the characterization of the crystallographic details of interactions including determining the Burgers vectors of dislocations. Some applications of in situ TEM deformation techniques include dislocation interactions with irradiation produced dislocation loops, stacking fault tetrahedra, and precipitates. This section focuses on the specific interaction mechanisms with isolated defects, as opposed to the previous section which dealt with interactions with defect fields.

4.1

Dislocation Loops

Dislocation loops can be generated during deformation, heating, or irradiation of metals and alloys. These loops can act as obstacles to dislocation motion, traps for solute particles, and nucleation sites for additional dislocation activity.

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Dislocation–loop interactions can modify the dislocation loop character, either through a change in Burgers vector or via an unfaulting reaction. In an irradiated BCC Fe–Cr alloy, in situ TEM deformation experiments showed that loops with Burgers vector ah100i could be transformed to loops with Burgers vector a2 h111i by interacting with a gliding   dislocation [44]. This could be achieved by the dislocation reaction a½100 þ a2 111 ¼ a2 ½111. The importance of this is the ah100i type loops are immobile while a2 h111i are mobile and more energetically favorable. In a FCC Fe–Cr–Ni that loop unfaulting can occur according to   aalloy, Suzuki  demonstrated  a a 110 þ ½ 111  ! 112 [43]. 2 3 6 Drouet et al. demonstrated the ability of glissile dislocations to interact with and incorporate vacancy loops during deformation. They observed dislocation glide during in situ TEM deformation of an ion-irradiated Zircaloy-4 sample at 773 K   [45]. A dislocation with Burgers vector a3 2110 gliding on the pyramidal plane was seen to interact with a vacancy loop with the same Burgers vector (Fig. 9). This interaction was also simulated using dislocation dynamics. The lattice dislocation encountered the vacancy loop in frame (b) which was then incorporated into the dislocation in frames (c) and (d). Incorporation of the vacancy loop resulted in the formation of a helical segment on the dislocation, increasing its resistance to glide. This interaction mechanism would not be observable by post mortem analysis.

4.2

Stacking Fault Tetrahedra

Stacking fault tetrahedra from irradiation or quenching have also been investigated in regards to how dislocations interact with them. Several types of interactions of dislocations with stacking fault tetrahedra have been observed, including shearing and shrinking of tetrahedra and cross-slip facilitated dislocation bypass mechanisms. Matsukawa et al. investigated dislocation interactions with stacking fault tetrahedra in Au and found that dislocations could shrink the tetrahedra [46]. In some cases, the passage of several dislocations was required to clear a stacking fault tetrahedron while in other cases it only required one dislocation. Figure 10 shows an example of dislocations cutting through and collapsing a tetrahedron. The authors suggested that shearing of the tetrahedron results in an unstable atomic configuration, collapsing the base portion of the sheared tetrahedron. The top portion of the tetrahedron and a stable Frank loop, in addition to the glissile dislocation, remained in the matrix after the interaction. The authors acknowledged the mechanism is still unclear for clearing stacking fault tetrahedra as it is known tetrahedra can be completely removed by interactions with lattice dislocations. Different mechanisms were proposed by Robach et al. that include collapsing a tetrahedron to a glissile dislocation loop as well as simple shearing of the tetrahedron [47]. Both mechanisms were based on observations from in situ straining TEM experiments but disagreed with molecular dynamics simulations of the same interaction. At elevated temperatures, 573–773 K, dislocations have been observed to bypass stacking fault tetrahedra via cross-slip or to annihilate, or partially annihilate, the

Fig. 9 Time resolved transmission electron microscope micrographs of in situstraining  experiment in Zr irradiated
Zircaloy-4 in the left column. Dislocation dynamics simulation of the interaction of a dislocation with Burgers vector or a3 2110 on the pyramidal plane 0111 with vacancy loop with Burgers vector of   a 3 2110 which is in good agreement with the interaction observed in the TEM experiment (Image used with permission from [45])

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Fig. 10 Time resolved TEM micrographs of interactions of three lattice dislocations with a stacking fault tetrahedron. The first, second and third dislocation interactions are observed in the left, middle and right columns, respectively. Collapse of the tetrahedron occurs after the third dislocation (Image used with permission from [46])

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tetrahedra upon contact in Au [48]. The observed mechanism was found to depend on the location at which the dislocations intersected a tetrahedron. When encountering the stacking fault tetrahedra at one of the edges, cross-slip to avoid the obstacle occurred. Whereas encountering a tetrahedron at one of the faces resulted in partial or complete annihilation of the tetrahedron. The variety of the interactions demonstrates the complexity of lattice dislocation interactions with stacking fault tetrahedra. It also shows the need to progress the characterization of these interactions to four dimensions, where time and all three spatial dimensions are resolved.

4.3

Precipitates

Interactions between low angle grain boundaries, comprised of multiple dislocations, as well as individual dislocations with precipitates have been observed during in situ TEM deformation. There are several factors that impact how dislocations interact with precipitates, including size and coherency of the precipitate, environment temperature, and intersection location. During in situ straining of an Al-4 Mg-0.3Sc alloy at 733 K, Clark et al. observed individual dislocations as well as mobile low-angle grain boundaries interacting with precipitates [49]. They found that as low-angle grain boundaries intersected large precipitates, the boundary structure was ruptured, resulting in the boundary breaking into smaller segments. Conversely, interactions with small precipitates did not alter the boundary structure. Clark et al. further studied the same alloy at 673 K and found that, when gliding dislocation arrays encountered large precipitates, the individual dislocations encountering the precipitates interacted with the precipitate–matrix interface dislocations [50]. As the dislocation array passed, dislocations remained pinned at the precipitate. This interaction is shown in the time-resolved series of electron micrographs presented in Fig. 11. Breakup of the low-angle grain boundaries was not reported. Single dislocation interactions with the large incoherent Al3Sc precipitates at 673 K were observed to be attractive or repulsive. The difference was attributed to the relationship of the matrix and particle, including differences in: (i) lattice parameter, (ii) shear modulus, (iii) Poisson’s ratio, and (iv) coefficient of thermal expansion. The location of intersection was also found to be an important factor dictating whether the interaction was repulsive or attractive.

5

Dislocation–Grain Boundary Interactions

One of the primary factors dictating the mechanical properties of a material is the distribution of grain boundaries. Grain boundaries have long been known to strengthen materials as well as to influence the corrosion resistance, fatigue damage nucleation, embrittlement, and susceptibility to irradiation damage. Much of this influence can be related to the stress state surrounding grain boundaries and how it evolves during dislocation–grain boundary interactions. Changes in matrix and boundary properties can vary the rate limiting step affecting the transfer of

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Fig. 11 Time-resolved electron micrographs of a dislocation array interacting with an Al3Sc precipitate at 673 K. Corresponding diagram of the interaction below the micrographs (Image used with permission from [50])

dislocations across the boundary between absorption limited, nucleation limited, or propagation limited, with each having a different influence on the stress state around the boundary. These dislocation transfer mechanisms can have profound influences on the material response to external stimuli and environments. In situ TEM deformation remains the only experimental technique capable of resolving unit processes occurring during dislocation–grain boundary interactions and has been instrumental in identifying factors affecting each stage of the interaction, shedding light on material behavior under a variety of ambient or extreme conditions. Specifically, in situ TEM deformation seeks to identify (1) the primary factors dictating the interaction (giving access to predictive capabilities) and (2) the

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rate limiting step in the interaction; see review by Kacher et al. [51] and references therein. The following sections will investigate dislocation–grain boundary interactions in different environments and microstructures and comment on the resultant variations in transfer mechanisms with an emphasis on the local stress state.

5.1

General Characteristics

Dislocation–grain boundary interactions can be separated into four stages of evolution: initial dislocation pileup at the boundary, absorption of the dislocation into the boundary structure, nucleation of a dislocation from the grain boundary, and propagation of the dislocation from the boundary. At every stage of the interaction, the net Burgers vector of the system must be conserved, either through the complete transfer of the dislocation across the grain boundary, incorporation of the Burgers vector into the grain boundary through the creation of an extrinsic grain boundary dislocation or its conversion to intrinsic grain boundary dislocations, back emission of a dislocation into the original grain, or, as is most common, some combination of these mechanisms. For example, depending on geometric and environmental considerations, a dislocation, once absorbed, can glide along the boundary plane, instigate the nucleation of a dislocation into the original grain (back-emission) or into the neighboring grain (forward emission), or retain its character and transfer into the neighboring grain via a cross-slip interaction (direct transfer) [51–53]. Examples of these interactions are shown in the electron micrographs presented in Fig. 12. Grain boundary characteristics, geometric factors, defect state of the matrix, test temperature, and test environment can all influence the relative importance of each stage of the interaction. In situ TEM deformation experiments have been instrumental in establishing broad predictive criteria for understanding dislocation–grain boundary interactions. By fully characterizing the incoming and outgoing dislocation systems involved in an interaction as well as the crystallographic relationship between the two grains, it has been shown that dislocation–grain boundary interactions satisfy the following three criteria [53]: 1. The angle between the traces of the incoming and outgoing slip systems in the grain boundary should be minimized; 2. The magnitude of the Burgers vector of the residual grain boundary dislocation (|bres|) should be minimized; 3. The resolved shear stress acting on the outgoing slip system should be maximized. Of the three, the magnitude of the Burgers vector of the residual grain boundary dislocation has been found to have the largest influence. Criterion 1 has been found to only be important in limited cases where the slip planes of the incoming and outgoing dislocation systems aligned exactly in the boundary, as may be encountered in the transfer of screw dislocations across twin boundaries. The magnitude of

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Fig. 12 Possible dislocation–grain boundary interactions. (a) Dislocation absorption into and nucleation from a random high angle grain boundary in 304 stainless steel. (b) Dislocation absorption into a coherent twin boundary in 304 stainless steel. (c) Direct transfer of dislocations across a coherent twin boundary in commercially pure α-Ti. (d) Dislocations interacting with a random high angle grain boundary in 304 stainless steel showing absorption and both forward and back emission. Arrows indicate direction of dislocation propagation

the resolved shear stress acting on the emitted dislocation system was found to be important only in that it must be of sufficient magnitude to propagate the dislocations away from the boundary. In coarse-grained systems with little prior damage, the stress needed is relatively low, diminishing the importance of Criterion 3. These criteria were originally established in coarse-grained, FCC materials under quasistatic loading conditions, though experiments have confirmed their validity in additional crystal systems [53, 54].

5.2

Temperature Effects

Increasing the environment temperature during material testing increases the likelihood of thermally activated events such as dislocation cross-slip and climb occurring and increases the rate of kinetic processes. Similarly, elevated temperatures can provide energy to overcome barriers to dislocation events associated with grain boundary interactions, such as absorption and nucleation. Kacher et al. directly compared the interaction of dislocations with a coherent twin boundary in austenitic stainless steel during in situ TEM deformation by first, observing the interaction under ambient conditions, relieving the applied stress, increasing the temperature to 673 K,

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and reapplying the stress to reinitiate the interaction [55]. They found that, at elevated temperatures, the barrier to dislocation transfer across the boundary decreased and additional dislocation systems were activated, increasing the complexity of the interaction. The interaction followed the same criteria outlined in the previous section at both ambient and elevated temperatures, suggesting that the temperature primarily influenced the rate of the interaction but not the governing mechanisms. In investigating dislocation–grain boundary interaction with random high-angle grain boundaries at elevated temperatures, Kacher et al. found that the barrier to absorption of high angle grain boundaries to impinging dislocations was significantly reduced [55]. The dislocation nucleation process at the boundary was also somewhat altered in relationship to ambient-temperature interactions, with a significant increase in partial dislocation activity observed. Figure 13a, b shows an example of dislocations interacting with a random high-angle grain boundary during deformation at 673 K. Figure 13a shows that a large number of dislocations have piled up at the grain boundary. This initial pileup occurred prior to the temperature increase; the initial stages were not captured. Diffraction contrast imaging suggests that the dislocation pileup led to a large concentration of elastic strain at the interaction point. To avoid the concentrated elastic strain and the associated stress fields, a few of the dislocations were seen to cross-slip from the dominant slip system and, after propagating a few hundred nanometers, cross-slipped back onto a parallel slip plane. The interaction of cross-slipped dislocations with the grain boundary occurred without the stress field associated with dislocation pileups. That is, the external load was the primary driver for the interaction. The absorption process into the boundary plane occurred quickly with no dislocation pileup needed to initiate the absorption process. Figure 13b shows that each absorbed dislocation resulted in the emission of a lead partial dislocation into the neighboring grain. The emitted partial dislocations did not propagate far from the boundary plane but instead were anchored by the trailing partial dislocation, resulting in an elongated faulted region extending from the boundary to the lead partial dislocation. A second high-temperature interaction, shown in Fig. 13c, d, similarly demonstrated the low-barrier of dislocation absorption into the boundary plane. The progress of a single dislocation was tracked as it approached the grain boundary (Fig. 13c). Upon reaching the grain boundary, the dislocation was absorbed and immediately after, to within the 0.1 s, a lead partial dislocation was emitted into the neighboring grain. Similar to the previous interaction, the trailing partial dislocation remained in the boundary and a faulted region extended to the lead partial dislocation. In both interactions shown in Fig. 13, the dislocation system activated at the boundary minimized |bres|. However, this minimization occurred differently at different stages of the interaction due to the elevated activity of partial dislocations. In the initial stages of the interaction, dislocation emission from the boundary was dominated by lead partial dislocations, with the trailing partial remaining in the boundary. Characterization of the Burgers vectors of the incoming and outgoing dislocations showed that, in both interactions, emission of the trailing partial dislocation significantly increased the value of |bres|. The associated large energy barrier was hypothesized to account for the elongated faulted regions extending from the boundary.

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Fig. 13 Dislocations interacting with random, high-angle grain boundaries at 400  C during in situ TEM deformation. (a, b) Images of the same interaction taken using two different diffraction conditions. Arrows indicate direction of dislocation propagation. Arrowheads in (b) indicate where lead partial dislocations have nucleated at and propagated from the grain boundary. (c, d) Two frames of a video taken 0.1 s apart. Arrow indicates direction of dislocation propagation. The arrowheads track a single interaction as a perfect dislocation is absorbed into the boundary and a lead partial dislocation nucleates at the boundary immediately after

5.3

Coherent Twin Boundaries

Coherent twin boundaries present a special case of dislocation–grain boundary interactions for a number of reasons: (1) in FCC materials, the twin plane is also a dislocation glide plane, (2) the crystal symmetry across the twin boundary can allow perfect transfer of the Burgers vector, (3) in cases when the crystal geometry does not allow perfect transfer of the Burgers vector, the residual grain boundary dislocation often takes the form of a partial dislocation glissile on the twin boundary plane, and (4) coherent twin boundaries are exceptionally low-energy, potentially increasing the barrier to dislocation transmission (as shown computationally in ([56])). As a result, geometric factors, specifically the relationship between the dislocation Burgers vector, line direction, and boundary plane, have a much larger influence in how the interaction progresses than in more general dislocation interactions with high angle grain boundaries. Figure 14 illustrates the effect that the relationship between the twin plane and the incoming dislocation character can have on dislocation–twin boundary interactions in FCC materials [55]. In this case, two different systems of perfect dislocations in an austenitic stainless steel sample impinged on a coherent twin boundary during in situ

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Fig. 14 Dislocations interacting with a coherent twin boundary during in situ TEM deformation of 304 stainless steel at 673 K. (a, b) Image of interaction site under two different diffraction conditions showing presence of distinct slip systems. (c, d) Interaction of a dislocation from system (b) showing impingement on and incorporation into the boundary. Arrows track an individual dislocation immediately before (c) and after (d) absorption into the boundary plane. (e, f) Interaction of a dislocation from system (a) showing impingement on, nucleation, and propagation from the boundary. Arrows indicate a dislocation immediately prior to being absorbed into the boundary (e) and its direction of motion after emission from the boundary (f) (Image sequences are separated by 0.1 s)

TEM deformation at 673 K. The Burgers vector of the first system dislocations, visible in Fig. 14a, allows them to either glide in the twin boundary plane or completely transfer across the boundary plane without leaving a residual twin boundary dislocation. Figure 14c, d shows that, upon impinging on the boundary plane, they were absorbed quickly and glided along the twin plane as perfect dislocations; an arrow

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tracks a single dislocation immediately before (Fig. 14c) and after (Fig. 14d) it is absorbed into the boundary plane. The Burgers vector of the second system, visible in Fig. 14b, does not allow glide in the twin plane. In Fig. 14e, f, a dislocation from the second system is seen impinging on the twin plane. The dislocation was absorbed quickly into the twin plane and immediately, to within 0.1 s, a dislocation was emitted into the neighboring grain; the arrow in Fig. 14e indicates the location of the dislocation immediately prior to being absorbed and the arrow in 14f indicates the direction of dislocation motion after emission from the boundary. As the interaction progressed, the density of extrinsic dislocations in the twin boundary continued to increase, raising the stress levels in the boundary. The effect of this buildup of dislocations is apparent when the associated stress reaches a critical level, shown in Fig. 15. Initially, the extrinsic grain boundary dislocations continued to glide away from the interaction point; this is apparent from the curvature of the individual dislocations (Fig. 15a). However, an uncharacterized structural transition took place at the interaction point that released the dislocations from the boundary plane, relieving the stress acting on the dislocations. As a result, the dislocations reverse course and rapidly exit the boundary at the interaction point; this course reversal is apparent from the reversed curvature of the individual dislocations seen in Fig. 15b. This behavior could have important implications in failure mechanisms at twin boundaries. Figure 16a demonstrates the ability of dislocations in certain systems to transfer across a twin boundary via simple cross-slip [54]. This image shows a-type screw dislocations in α-Ti gliding on a prismatic plane interacting with a {1102}-type twin boundary. The prismatic planes aligned on either side of the boundary and the crystallographic relation across the boundary allowed perfect transfer of the Burgers vector of the dislocations; this is evident from a three-dimensional dislocation model constructed from an electron tomogram of the interaction. In situ observations of the dislocation glide showed that the twin boundary posed a very low barrier to slip transmission. Some dislocation trapping in the boundary plane is evident from the model. This is most likely due dislocation constriction prior to cross-slip.

Fig. 15 Dislocation glide in the plane of a coherent twin boundary during in situ TEM deformation at 673 K. Dislocation glide reversed at a critical point between (a) and (b). 2.6 s elapsed between frames.

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Fig. 16 (a) TEM micrograph of dislocations interacting with a twin boundary in α-Ti. Arrows indicate direction of dislocation propagation. (b) Dislocation model of the interaction constructed from an electron tomogram. Coordinate systems aligned with the crystallographic crystal axes are included in each crystal region. Dotted lines indicate the slip plane trace, showing alignment at the twin boundary plane

5.4

Propagation Limited Systems: Irradiated Materials

Bombardment of metallic alloys by energetic particles is known to cause microstructural changes, including the formation of nanoscale dislocation loops distributed throughout the matrix. The interaction of dislocations with these defects are discussed in depth in an earlier section, though it is appropriate here to note that these nanoscale defects act as barriers to dislocation propagation, effectively hardening the matrix and subsequently having a significant effect on dislocation–grain boundary interactions. Figure 17 shows three separate examples of dislocation systems interacting with grain boundaries in austenitic stainless steels during in situ TEM deformation at 673 K [37, 38, 41, 57]. In Fig. 17a, a dislocation system is seen interacting with a low-angle grain boundary [37]. The intersection of the dislocation system resulted in a large strain concentration at the boundary and the emission of dislocations into the neighboring grain. Initially, perfect dislocations were emitted in two different slip systems, with one of the systems being significantly more active than the other. After the emission of ~20 dislocations, the more active system transitioned from the emission of perfect dislocations to partial dislocations. Due to the barrier presented

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Fig. 17 Dislocation–grain boundary interaction during in situ TEM deformation of ion-irradiated stainless steel samples. (a) Dislocation interactions with a low-angle grain boundaries. (b) Dislocation source widening during dislocation emission from a grain boundary (used with permission from [38]). (c) Dislocations interacting with a Σ9 grain boundary. Arrows indicate direction of dislocation motion (used with permission from [57])

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by the dispersed irradiation-induced defects, the dislocations did not propagate easily from the boundary and formed a high density near the boundary source. Figure 17b shows the emission of perfect dislocations from an uncharacterized grain boundary [38]. The dislocation emission resulted from the impingement of a single dislocation system. As the dislocations nucleated from the boundary, they remained tightly packed, resulting in large, internal pileups. Black lines mark the slip traces of the dislocations originally emitted from the boundary source. Over time, this source volume increased. This can be seen from the increased distance between slip traces at the boundary, indicated by arrowheads in Figure 17bi. In the third example, shown in Fig. 17c, dislocations impinged on a Σ9 boundary [57]. Originally, a few partial dislocations emitted from the boundary, well aligned with the incoming dislocation system. These were soon followed by the emission of multiple perfect dislocations belonging to a different slip system. Each of these perfect dislocations originated from a different location on the boundary plane and was not able to propagate far from the boundary. Two additional dislocation systems, not visible in the video frames, also activated at the interaction site, with one system propagating through the boundary plane and the other emitted back into the original grain. The activity of these additional systems was limited in comparison to the two systems visible in Fig. 17c. To explore the micromechanics occurring during dislocation–grain boundary interactions in irradiated systems, Cui et al. characterized multiple interactions in terms of grain boundary character and dislocation slip systems. Using this information, they calculated the local stress state at the boundary using a “super dislocation” model where the incoming dislocation pileup was represented as a single dislocation at the boundary. The emitted dislocation systems were analyzed in terms of |bres| and the resolved shear stress, similar to the criteria proposed by Lee et al. [53]. Cui et al. found that, while |bres| was still the most important factor, the relative importance of the resolved shear stress condition increased. Instead of easy propagation from the boundary, the dislocations required sufficient stress to overcome the barrier posed by the defect field. They confirmed this by observing that in some cases additional dislocations had to be added to the dislocation pileup, that is, the local stress increased, before dislocations were emitted from and propagated away from the grain boundary. Despite the differences seen in the grain boundary character, dislocation type, and loading conditions of the interactions shown in Fig. 17, the three interactions share commonalities general to dislocation–grain boundary interactions in irradiated systems. In each interaction, the irradiation damage was not seen to limit dislocation nucleation from the boundary. Propagation from the boundary, however, was restricted by the dispersed defect field. This barrier to propagation resulted in the formation of “reverse” pileups at the grain boundary. That is, a high concentration of dislocations develops at the emission side of the boundary in contrast to the rapid propagation from the grain boundary source commonly seen in unirradiated, coarse-grained materials. This effect has important implications in the local strain concentrations developing at boundaries and has been linked to irradiation assisted stress corrosion cracking [58]. In addition to the development of reverse dislocation pileups, each interaction shows significant source broadening at the grain boundary. In the interactions shown

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in the series of images presented in Fig. 17a and c, this broadening occurs through the emission of multiple dislocation systems from different locations on the boundary plane. In the interaction shown in Fig. 17b, tracing of the slip traces on the foil surface shows that multiple parallel slip systems were emitted from nearby locations in the boundary. This occurs despite only a single dislocation system impinging on the grain boundary. This source broadening has important implications in understanding confined plasticity and the development of dislocation channels, key aspects of irradiation induced embrittlement, and is thought to be related to the stress concentrations associated with reverse dislocation pileups.

5.5

Confined Microstructure: Ultrafine Grained and Nanotwinned Materials

Efforts to fully exploit the Hall–Petch effect have led to the development of increasingly fine-grained systems. Although the dislocation transfer mechanisms across a grain boundary in ultrafine-grained materials are not expected to differ significantly from their coarse-grained counterparts, the limited volume for the development of dislocation pileups and the attendant long-range stress fields can alter the local stress state. As the deformation of nanograined materials is largely dominated by grain boundary deformation mechanisms, the discussion here is limited to nanotwinned and ultrafine-grained systems. Under monotonic loading in an ultrafine-grained sputter-deposited Al film, Mompiou et al. characterized the stresses associated with dislocation nucleation from a grain boundary and compared them with the stress needed to propagate a dislocation through the boundary plane [59]. Using dislocation curvature, they found that dislocation nucleation from a triple junction required a resolved shear stress level of 500 MPa. In comparison, the stress needed to propagate the dislocation through the boundary plane was only 157 MPa, suggesting that nucleation, rather than propagation, was the rate limiting step in the deformation process. Mompiou et al. also investigated the dislocation evolution in ultrafine-grained Al (avg. grain size was ~500 nm) produced using equal-channel angular pressing (ECAP) under multiple loading-unloading cycles [60]. They observed that for both predominately edge and screw dislocations, the initial impingement of a dislocation at a grain boundary resulted in the absorption of the dislocation at the boundary. That is, for an undamaged grain boundary, the barrier to dislocation absorption was low. Dislocation contrast in the boundary plane post-absorption was visible for tens of seconds to minutes, after which dislocation dissolution took place. During this period, the subsequent interactions of dislocations with the grain boundary were significantly influenced by the nature of the dislocation. Predominately screw-type dislocations underwent significant cross-slip due to the repulsive stress fields surrounding the absorbed dislocations. Similar to the initially absorbed dislocation, these cross-slipped dislocations were also able to incorporate into the boundary plane. Upon unloading, the dislocations remained firmly entrenched in the grain boundaries. In contrast, predominately edge-type dislocations, which are unable to

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Fig. 18 Dislocation–phase boundary interactions in a multilayer AgCu system. (a) Twin crossing cube-on-cube boundaries. Arrows indicated twin location. Twin marked by arrows. (b) Twin or stacking fault impinging on an incoherent twin interface with shared planes resulting in the emission of perfect dislocations from the interface after clearing the planar defect. Arrow indicates dislocation propagation direction. (c) Dislocation impingement on and back emission from an incoherent twin interface. (d) Twins in Ag impinging on coherent twin interface, resulting in increased elastic strain in the Cu lathe. Twin nucleation location indicated by an arrowhead [63]

cross-slip, all reached the grain boundary at the same location, resulting in the buildup of increasingly large stress concentrations. Upon unloading, these dislocations were reemitted into the original grain from the grain boundary. This behavior continued for multiple cycles, even with continued dislocation generation. Various groups have studied the deformation of nanostructured multilayered and nanotwinned systems [61, 62]. Eftink et al. used in situ TEM deformation to investigate the interactions of dislocations with interphase boundaries in a Ag–Cu eutectic system [63]. They detailed the dislocation response at the interfaces for three different types of boundaries: cube-on-cube, incoherent twin, and coherent twin. At cube-oncube interfaces, they found that perfect dislocations piled up at the boundary, but only limited transmission occurred. Twins and twinning dislocations, however, were seen to easily propagate across the interface, leading to twin formation in the Cu layers as well as the Ag layers (Fig. 18a). Incoherent twin interfaces posed a weak barrier to perfect dislocation propagation on the twin plane, which was continuous across the interface, resulting in some dislocation pileup formation at the interface. Twins originating from

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the Ag layers and impinging on the Cu–Ag interfaces, however, did not directly transmit as transmission into the Cu layers required the leading and trailing partial dislocations to first recombine (Fig. 18b). In contrast, incoherent twin interfaces imposed strong barriers to slip transmission for dislocations on planes other than the twin plane. The activity of perfect dislocations leading to pileups at the boundaries was observed in the Ag layers, leading to absorption into the boundary, cross-slip from the pileup at the boundary intersection, and back emission from the boundary into the Ag layer, but no transmission (Fig. 18c). Interestingly, coherent twin interfaces appeared to transmit plasticity via twinning in the Ag layers and elastic strain in the Cu layers, suggesting that the layer thinness facilitated communication between the Ag layers without plastic deformation of the Cu (Fig. 18d). Eftink et al. recently combined in situ deformation experiments with molecular dynamic computer simulations of dislocation interactions with incoherent twin interfaces in a 500 nm thick Ag–Cu layered system to explain the plastic strain recovery observed experimentally in this system [64]. They demonstrated that the plastic recovery was driven primarily by the accommodation of dislocations in the Ag and Cu layers into the interfaces. The dependence of the magnitude of the plastic recovery on the total strain was shown to be due to the formation of dislocation tangles at the higher strain as these locked the dislocations in place and prevented them from returning to and being accommodated in the interfaces. This example demonstrated the use of computer simulations to shed additional insight into dislocation interactions with interfaces.

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Conclusion

TEM is uniquely able to image and characterize dislocation interactions in real time and at the necessary resolution to observe unit processes. In situ experiments, including deformation, irradiation, and heating, have been instrumental in elucidating complex dislocation interactions, contributing to our fundamental understanding of these interactions as well as aiding in material design and deployment. Advances in sample preparation techniques, TEM testing holders, and characterization capabilities promise further to maintain in situ TEM testing as a premier method for understanding dislocation interactions. Acknowledgments For the preparation of this manuscript, the authors acknowledge financial support from Georgia Tech. (JK), Los Alamos National Security, LLC, operator of the Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396 with the US Department of Energy (BPE), and US Department of Energy under award No. DE-FG02–08 ER46525 (IMR). Experimental work from the Robertson group was supported by US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science, under award No. DE-FG02–08 ER46525 (radiation damage work by Cui) and US Department of Energy Office of Basic Energy Sciences, Division of Materials Science, under award No. DEFG-02-07ER46443 (slip transfer studies by Kacher and, partially, the study by Eftink on deformation across interfaces).

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47. Robertson IM, Robach JS, Lee HJ, Wirth BD. Dynamic observations and atomistic simulations of dislocation-defect interactions in rapidly quenched copper and gold. Acta Mater. 2006;54 (6):1679–90. 48. Briceno M, Kacher J, Robertson IM. Dynamics of dislocation interactions with stacking-fault tetrahedra at high temperature. J Nucl Mater. 2013;433(1–3):390–6. 49. Dougherty LM, Robertson IM, Vetrano JS. Direct observation of the behavior of grain boundaries during continuous dynamic recrystallization in an Al-4Mg-0.3Sc alloy. Acta Mater. 2003;51:4367–78. 50. Clark BG, Robertson IM, Dougherty LM, Ahn DC, Sofronis P. High-temperature dislocationprecipitate interactions in Al alloys: an in situ transmission electron microscopy deformation study. J Mater Res. 2005;20:1792–801. 51. Kacher J, Eftink B, Cui B, Robertson IM. Dislocation interactions with grain boundary interactions. Curr Opinion Solid State Mater Sci. 2014;18:227–43. 52. Shen Z, Wagoner RH, Clark WAT. Dislocation and grain boundary interactions in metals. Acta Metall. 1988;36:3231–42. 53. Lee TC, Robertson IM, Birnbaum HK. TEM in situ deformation study of the interaction of lattice dislocations with grain boundaries in metals. Philos Mag A. 1990;62(1):131–53. 54. Kacher J, Robertson IM. In situ and tomographic analysis of dislocation/grain boundary interactions in α-titanium. Philos Mag. 2014;94(8):814–29. 55. Kacher J, Robertson IM. Quasi-four-dimensional analysis of dislocation interactions with grain boundaries in 304 stainless steel. Acta Mater. 2012;60(19):6657–72. 56. Sangid MD, Ezaz T, Sehitoglu H, Robertson IM. Energy of slip transmission and nucleation at grain boundaries. Acta Mater. 2011;59(1):283–96. 57. Cui B, Kacher J, McMurtrey MD, Was GS, Robertson IM. Influence of irradiation damage on slip transfer across grain boundaries. Acta Mater. 2014;65:150–60. 58. Cui B, McMurtrey MD, Was GS, Robertson IM. Micromechanistic origin of irradiation-assisted stress corrosion cracking. Philos Mag. 2014;94(36):4197–218. 59. Mompiou F, Legros M, Boe A, Coulombier M, Raskin JP, Pardoen T. Inter- and intragranular plasticity mechanisms in ultrafine-grained Al thin films: an in situ TEM study. Acta Mater. 2013;61(1):205–16. 60. Mompiou F, Caillard D, Legros M, Mughrabi H. In situ TEM observations of reverse dislocation motion upon unloading in tensile-deformed UFG aluminium. Acta Mater. 2012;60 (8):3402–14. 61. Wang J, Misra A. An overview of interface-dominated deformation mechanisms in metallic multilayers. Curr Opinion Solid State Mater Sci. 2011;15(1):20–8. 62. Wang J, Zhou C, Beyerlein IJ, Shao S. Modeling interface-dominated mechanical behavior of nanolayered crystalline composites. JOM. 2014;66:102–13. 63. Eftink, BP. Dislocation interactions with characteristic interfaces in Ag-Cu eutectic. In: Materials science and engineering. University of Illinois at Urbana-Champaign; 2016. 64. Eftink BP, Li A, Szlufarska I, Robertson IM. Interface mediated mechanisms of plastic strain recovery in a AgCu alloy. Acta Mater. 2016;117:111–21.

7

Multiscale Modeling of Radiation Hardening Ghiath Monnet and Ludovic Vincent

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Atomistic Simulations of Dislocation Interactions with Radiation Defects . . . . . . . . . . . . . . . 3 Scale Transition from the Atomistic to the Continuum Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Absolute Strength of Local Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Activation Energy of Unpinning from Local Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Interaction with Extended Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Validation of the Scale Transition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Interaction with Radiation Defects at the Grain Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 DD Simulations of Interactions with Local Precipitate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 DD Simulations of Interaction with Extended Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Transition to the Macroscopic Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Constitutive Equations for Radiation Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Crystalline Law Including Radiation Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Radiation Hardening at the Macroscopic Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Future Directions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Radiation Creep, Dynamic and Static Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Interaction with Dislocation Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Strain Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Dislocations Behavior in Concentrated Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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G. Monnet (*) EDF – R&D, MMC, Paris, France e-mail: [email protected] L. Vincent CEA, DEN, SRMA, Paris, France e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_8

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Abstract

All materials used in nuclear reactors experience radiation hardening, thus altering significantly their mechanical behavior. Despite the large efforts made by the scientific community to correctly predict the radiation effects on nuclear materials, many difficulties persist in modeling radiation defects. Although radiation hardening is responsible for macroscopic consequences such as embrittlement, its fundamental mechanisms prevail at the atomic scale. The fast development of simulation techniques, especially atomistic simulations, helped in exploring many features of dislocation interactions with radiation-induced defects. For a long time, the obtained results were used to validate theoretical models and to qualitatively explain and rationalize experimental observations. More recently, with the ubiquity of simulation results and techniques at different scales, quantitative physically based predictions of the mechanical behavior became a realistic objective. Efforts were made to couple simulation results at different scales through the development of scale transition methods and the construction of constitutive equations of the local mechanical behavior. The objective of this chapter is to trace the evolution and progress of this strategy, thus enabling the construction of a chain of physical knowledge across the scales. Details of simulations techniques and methods are not presented. We emphasize on basic achievements of simulations and on the treatment of results with the aim to bridge the different scales of interest for the mechanical behavior. Keywords

Defect strength · Activation energy · Radiation defects · Scale transition · Multiscale modeling

1

Introduction

Radiation is known to induce defects of several types in materials. For a review on radiation induced microstructure, readers are advised to consult textbooks like [1] or specific reviews such as [2] for reactor pressure vessel (RPV) steels and [3] for austenitic stainless steels. The defects can be of atomic size (vacancies, interstitials) and may grow up at higher radiation doses to several nanometers. In metallic alloys, irradiation induces basically the same kind of defects: dislocations loops (DLs), voids, solute clusters (SCs), and precipitations. Some defects are specific to some crystallographic structures such as staking fault tetrahedron (SFT) in faced-centeredcubic (FCC) structure. Despite their small size, mechanical tests show that radiation defects strongly alters the mechanical behavior by increasing the yield stress to the detriment of ductility, resulting in significant embrittlement. In order to rationalize these observations, dislocation interactions with radiation defects must be thoroughly studied and analyzed. The first attempts envisaged the interaction as a collision of the two static elastic fields representing the gliding dislocation and the defect. No relaxation is allowed and the main output of such

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interaction models is the so-called interaction energy. For example, the method has been applied to compute the interaction energy with voids [4] and with dislocation loops [5]. The major drawback of this approach is the treatment of the singularity of the dislocation elastic fields. Since the dislocation core structure is disregarded, an arbitrary cutoff radius is usually introduced to avoid divergence of the interaction energies. But a second serious drawback can be noticed: the interaction energy itself is of no use in evaluating quantitatively the defect strength and the resulting hardening. Another strong assumption is thus needed to evaluate the defect strength. The latter is often considered simply proportional to the interaction energy, with no further justification. In all these previous studies, given their small size, investigation of dislocation–defect interactions can hardly be envisaged within only the framework of elastic theory, that is, without accounting for the specific atomistic features, inherent to the dislocation core structure and to the specific atomic configuration of the defect. Fortunately, with the increasing computing power and the development of empirical interatomic potentials, it became possible to simulate the behavior of a large number of atoms submitted to external loading. By the creation of a lattice dislocation and of a radiation defect, the interaction can be directly observed in all necessary details using molecular statics and molecular dynamics simulations. For the sake of simplicity, all simulations at the atomic level will be designated in this chapter by atomistic simulations (AS). From the late 1990s, many ASs of dislocation interactions with radiation defects have been reported (see the pioneering work by Hu et al. [6]), revealing impressive variety of interaction mechanisms and strengths [7], and confirming the weaknesses of the elastic models of interactions. Although interactions can be characterized within arbitrary accuracy and details in ASs, due to the limitation in the simulated time (few nanoseconds) and in the number of simulated atoms (few millions) [8], it is not possible to simulate interaction with a realistic distribution of defects. AS results cannot directly provide any realistic estimation of radiation hardening. The well-known reason is that hardening is a mechanical propriety defined at the macroscopic scale, measured by mechanical tests. At best, it can be defined at the grain scale, where a large number of dislocations interact simultaneously with several millions of radiation defects. Larger scale simulations are thus required. These requirements can be achieved using techniques of simulations at the mesoscopic scale, such as dislocation dynamics (DD) [9, 10], phase field method [11], etc. Among these methods DD technique is the most mature and well adapted for investigations of the collective dislocation behavior. Nowadays the different available DD codes (see for example [12, 13]) are able to predict the behavior of a realistic dislocation distribution within volumes close to those of grains in industrial materials. However, given the current limitation in computing efficiency, only few percent in deformation can be achieved, which prevent DD simulations from predicting the stress–strain curves or the dislocation microstructures, prevailing in deformed materials. DD simulations offer thus only a prediction of the flow stress associated to a given microstructure, such as the radiation microstructure.

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Consequently, it is possible, in principle, to compute hardening induced by a specific type of radiation defects according to the following procedure: (i) perform atomistic simulations to determine the interaction mechanism and strength; (ii) deduce the corresponding interaction rules to implement in DD simulations; (iii) validate the interaction rules by checking that the obtained unpinning stress are close to that computed in AS simulations; (iii) carry out DD simulations of dislocations interacting with a realistic distribution of defects to determine the corresponding hardening; (iv) introduce the new hardening sources (radiation hardening) as a new strengthening component leading to the increase of the critical stress; (v) perform macroscopic simulations (homogenization, finite element, etc.) using the so obtained constitutive equation as a local behavior law and deduce the new yield stress of the material. The present chapter is dedicated to the description of this methodology and its application on the case of hardening induced by solute clusters. In the following, the strategy in coupling simulation results in order to predict hardening on the macroscopic level is called multiscale modeling (MM). In this strategy, the link between scales/simulations is ensured by the transfer of data produced at the lower scale towards a simulation at the larger scale. MM allows for coarse graining without loss in physical knowledge or content. MM must not be confused with the so-called multiscale simulation techniques, where link between scales is performed through direct numerical coupling of the different codes or methods, such as coupling atomistic simulations with the finite elements method (FEM) [14] or coupling DD simulations to FEM [15].

2

Atomistic Simulations of Dislocation Interactions with Radiation Defects

In this type of simulations, the trajectory of the explicit positions of every atom is integrated numerically according to a cohesive model of the crystal (usually including the atom of interest with its neighborhood). The derivation of the potential energy of the atom with respect to its position provides the net force applied on the atom. Then classical dynamics are applied to update the atom velocity and position. The cohesive model of the crystal, the interatomic empirical potential, must reproduce some fundamental properties of pure elements such as lattice parameter and elastic constants. If more than one element is considered (binary, tertiary alloys, etc.), additional requirements are much more difficult to fulfil. Only few interatomic potentials are available for nonpure materials. Many ASs have been dedicated to the study of dislocation interactions with radiation defects. For more details, one may consult the review by Bacon et al. [7]. Most of the reported investigations were performed to reveal interaction mechanisms, which allows for a qualitative classification of interaction strength and output. The general scheme of these simulation is simple as described in [16]: first the defect and the dislocation are created, then an applied shear strain (or shear stress) is imposed on the fixed layers of the simulation box in order to initiate the dislocation motion towards the defect. The interaction is usually characterized by the maximum

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stress corresponding to the unpinning moment from the defect. In molecular statics simulations (simulations at zero K), the maximum stress describes well the strength of the defect, while in molecular dynamics simulations (finite temperature), the whole loading history must be taken into account. From the reported results, the following general tendencies can be recognized [7]: (i) at equal sizes, voids are the strongest defect type [17], (ii) defect strength increases with increasing defect size [18], (iii) upon interaction, the gliding edge dislocations often suffer local climb [19], (iv) the effect of long-range stress induced by the defects is negligible compared with the contact interaction [20], (v) the defect strength usually decreases with temperature [21]. Clearly, the last three features cannot be treated within the framework of elastic theory, which confirms the limitation of elastic approaches in the determination of interaction outputs and strengths. On the other hand, the dislocation character plays an important role in BodyCentered-Cubic (BCC) [7], where dislocation cores are narrows and the screw dislocation core structure is not planner. In contrast, all dislocations are dissociated (not to the same extent) in FCC structure [22]. Given the limitations in simulation conditions in atomistic simulations, for the dislocation to sweep the whole simulation box, the mechanical loading must confer upon the dislocation a velocity of at least some meters per second. Such order of magnitude of dislocation velocity is common in FCC materials in usual mechanical tests. It is also observed for nonscrew dislocations in BCC structure. However, it is at least six orders of magnitude higher than the velocity of the screw dislocation at low and intermediate temperature in BCC materials [23]. Much more precautions must be taken in interpreting simulation results concerning the screw dislocation interaction with radiation defects in the BCC structure. While the output of interaction with most radiation defects is simple (usually shearing) interaction with dislocation loops is quite complex [7]. It depends on the crystallographic structure, loop configuration (size, Burgers vector, shape, and sometimes position as well), temperature, character of the gliding dislocation, etc. It is not easy to summarize the interaction variety. However, a general tendency can be noticed in the case of small dislocation loops (2–3 nm). Often, the interaction output is complete or partial absorption by edge dislocations [24], while screw dislocations form helical terns upon absorption [25, 26]. Since helical turns are mobile only in the direction of the Burgers vector, the screw dislocation is strongly pinned by the helical turn. Unpinning operates when helical turns are closed due to the bowing of dislocation arms on both sides. The result is the release of the dislocation loop with the same Burgers vector as the gliding dislocation. The net output of screw dislocation interactions with small loops is thus the change in the Burgers vector of the loops by acquiring that of the moving dislocation. The question is how to use results of the atomistic simulations in larger scale simulations such as dislocation dynamics? In the following section the possible links between these two scales are discussed, depending on the interaction temperature and the defect nature.

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Scale Transition from the Atomistic to the Continuum Level

Using atomistic simulation results requires specific treatment in order to extract parameters or variables of interest at the continuum scale, where only the defect strength and its evolution with temperature are of interest. Such treatment is called scale transition. An example of such successful transition is shown in Fig. 1 comparing molecular dynamics simulation and DD simulations of an edge dislocation interacting with a 2 nm void in iron at 0 K [27]. By using an appropriate strength parameter, DD simulations can provide not only the same unpinning stress (205 MPa in MS vs 200 MPa in DD) but also comparable critical dislocation shape. The first reported method (the conventional approach) attempts to associate the defect strength with the so-called critical “cusp angle,” which is the critical angle between the pinned dislocation arms on both sides of the defect. The defect is thus reduced to pinning point with no volume. Upon pinning of the dislocation, the line tension, Γ, on each side of the obstacle generates a resultant force F = 2Γ cos(φ/2), where φ is the cusp angle [27]. With an increasing line curvature, φ decreases and F increases. When the force exceeds the resistance of the obstacle, Fmax, the dislocation unpins from the obstacle. This scheme has been used in theories of alloy strengthening, for a review see [28]. It was also used for the treatment of the Orowan mechanism [29] by employing a line tension that accounts properly for the dipole interaction between dislocation arms around the particle. However, the line tension approximation suffers from serious limitations. (i) Except for materials with a vanishing Poisson’s ratio, the line tension strongly varies with the local dislocation character and the dislocation does not uniformly bow in the form of circular arc; (ii) it is not easy to extend the concept of the critical angle to obstacles of finite size; (iii) the method implies Γ can be defined in the vicinity of the obstacle, but this is uncertain because the curvature of the line is altered by the dipolar interaction; (iv) The force balance on a dislocation segment is not necessarily parallel to the segment direction [30]. With these drawbacks, the concept of the cusp angle is not easy to Fig. 1 Scale transition from atomistic simulations (MS) to continuum scale (DD) for an edge dislocation interacting with 2 nm void in iron at 0 K

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deduce from AS results to evaluate the defect strength. Even when this approach is used with precaution, the predicted results are not coherent with the AS results themselves. For example, the critical stress obtained by this approach with the φ computed in ASs is not in good agreement with the critical stress computed in the same simulations [31]. To overcome these difficulties, different approaches has been reported in the literature. In order to provide a comprehensive picture of the different treatments, defects can be classified into two defect categories. The first one is called local obstacles, designing precipitate-like defects, for which unpinning is controlled by the shearing process rather than interaction with the stress fields induced by defects. Local obstacles can only be sheared or bypassed, such as solute clusters, precipitates, voids, and bubbles. Furthermore, image forces affecting the gliding dislocation can be neglected in the case of small precipitates as pointed out by Shin et al. [32]. In contrast, the second category, referred to as extended obstacles, denotes defects with a dislocation character, such as dislocation loops and SFTs. As expected and shown in ASs, the MM of the dislocation interaction with defects depends on the category of the defect.

3.1

Absolute Strength of Local Obstacles

Absolute strength is defined at 0 K where no thermal activation can occur. Increasing temperature may lower the energies barrier for unpinning due to entropy effects. In the next section we discuss the thermal activation effects. Different approaches have been proposed for the MM of precipitate-like defects. Hafez Haghighat et al. [33] compared AS results with DD simulations (coupled with finite element method) of an edge dislocation interacting with 2 nm void. No data transfer from ASs to DD simulations has been carried out. Although the elastically predicted image force is in agreement with AS results, no qualitative agreement was reported, in the sense that the maximum stresses predicted by the two simulation techniques were not compared. In the absence of image forces, the relevant parameter to compute from AS and to use in DD simulations is the shear resistance. This approach has been first proposed by Mohles et al. [34] and later developed by Monnet et al. for the treatment of voids [27] and extended for generic precipitate states [35]. The method is based on the mechanical equilibrium of the dislocation line submitted to applied stress τapp, friction stress τf (resulting from the solid solution) and shear resistance of the defect. At equilibrium, the defect resistance is equal to the effective stress τeff applied on the pinned segment. If the defect size is D and the free defect spacing is l (see Fig. 1), the quasi-static equilibrium prevails as long as the defect provides a shear resistance τloc (the index loc refers to local) balancing the effective stress on the pinned segmentτeff. The total driving force on the dislocation [b(τapp
τf)(l þ D)] is concentrated on the pinned segment and equal Dbτeff. It must balance the pinning force of the precipitate: Dbτloc. Consequently, the current shear resistance can be given by:

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τloc ¼

 lþ D
τapp
τf D

(1)

At zero temperature, the unpinning occurs when τapp reaches its maximum value τapp,max and τloc reaches the maximum shear resistance τobs. The latter can thus be given by the simple formula: τobs ¼

 lþ D
τapp, max
τf : D

(2)

τobs corresponds to the maximum shear resistance that can be provided by the defect. When the effective stress on the pinned segments exceeds (in absolute value) τobs, the pinned segment is released by shearing the defect. τobs represents thus the net resistance of the defect, accounting for all the physical mechanisms involved in the interaction (lattice and modulus mismatch, chemical effects, etc. [28]).The corner idea of this approach is that τobs, calculated from AS, can be directly and straightforwardly used in DD simulations as a back stress prevailing in the volume representing the obstacle. Consequently, by plotting (τmax
τf) as a function D/(l þ D), for different defect sizes and dislocation lengths, the obtained slop is an estimate of the shear resistance, τobs. Available AS data of edge dislocations interacting with voids [31], Cu precipitates [31] and Cr precipitates in iron [36] are gathered in Fig. 2. The precipitates are coherent with the BCC matrix of iron, which allows gliding dislocations to shear them under high enough applied stress. As it can be clearly seen, depending on the defect nature, points are aligned along straight lines passing by 0, which indicates that τobs is a characteristic feature of the defect independent of the length of the gliding dislocation or the defect size. Points representing voids are quite separated from points representing interactions with Cu and Cr precipitates. This means that the strength of voids is much higher than strengths of other defect types. Given the slops of straight lines in Fig. 2, one can evaluate τobs for the different defects types: τvoid ¼ 4:5 GPa, τCu ¼ 2:4 GPa, τCr ¼ 2:15 GPa

(3)

The resistances of Cu and Cr precipitates are quite close and must represent usual resistance of coherent precipitates in iron.

3.2

Activation Energy of Unpinning from Local Obstacles

At finite temperature, an intrinsic stochastic feature prevail in dislocation unpinning, involving not only the unpinning stress, but the whole loading history, that is, the stress strain curve of the atomistic simulations. Different approaches have been proposed in the literature for the treatment of MD simulation outcome. Although some empirical statistical analyses have been reported (see for example [37]), most

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Fig. 2 The net driving force (unpinning stress minus the friction stress) as a function of the ratio defect size over the dislocation length. The slop provides the shear resistance of the defect. Data correspond to atomistic simulations of voids, Cu and Cr coherent precipitates in pure iron at 0 K

of interpretations are based on the transition theory [38]. According to this theory, the thermal activation rate, ω, can be given by: ω ¼ ωo exp


ΔH ðτloc Þ , kT

(4)

where ωo is a frequency factor close to the Debye frequency, ΔH the enthalpy of activation, k the Boltzmann constant and T the absolute temperature. Sometimes, the free energy (or free enthalpy) is considered to depend on temperature without further justification, but in general the energy barrier is considered independent of temperature. ΔH(τ) is the appropriate thermodynamic function when MD simulations are curried out at constant pressure, while the free energy function ΔF(τ) becomes the appropriate one in simulations performed at constant volume. In this chapter, we keep ΔH(τ) as a generic notation of the appropriate thermodynamic function. The complete thermodynamic characterization of the activation process requires the knowledge of ΔH as a function of stress. The important question is: what stress must be considered for the variation of ΔH? In some studies ΔH is considered to vary with the applied shear stress τapp. From thermodynamic standpoint, this function is not appropriate since applied stress do work on pinned and unpinned segments. Since unpinning is a local event resulting from stress concentration on the pinned segment, the relevant stress for the activation energy is the effective stress on the pinned dislocation segment, which equals the local stress τloc and can be computed from Eq. 1. The objective of the treatment of molecular dynamics results in the MM strategy is thus the estimation of ΔH from ASs, with no constrains on the interaction nature. The experimental validation of the Eq. 4 [39] requires ΔH(τloc), and hence ω, to be constant over a given period. In these conditions, the evolution of the activation probability follows the Poisson’s distribution [40], which is at the heart of Arrhenius rate processes. When MD simulation are carried out at constant applied stress τapp, and assuming the atomic system to obey the principle of ergodicity, the average activation time tact

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can be directly computed by performing several atomistic simulations under the same temperature and stress, but using different thermalization conditions. The activation rate is simply the inverse of the activation time, which by inverting Eq. 4, leads to the expression: ΔH ðτloc Þ ¼ kTlnðtact ωo Þ,

(5)

By performing MD simulations at different applied stress, ΔH can be tracked as a function of the local stress, which allows identifying all parameters in Eq. 4. In contrast, when MD simulations are carried out at constant strain rate, the stress varies with time and hence ΔH. The activation rate does no longer follow the Poisson distribution function, which enters into contradiction with Eq. 4. In principal, since τ varies during the ASs, ΔH(τloc) cannot be directly estimated. The reason is that even if ΔH can be computed from Eq. 4, there is no specific local stress τloc to associate with ΔH. In order to make progress and to use Eq. 4 again, the activation process has to be reduced to an equivalent Poisson’s distribution, as discussed by Monnet [41]. In other words, a constant stress τp (Poisson’s stress) has to be defined in order to be associated with ΔH. τp is defined by the requirement that it must provide the same activation probability (at the end of the interaction time) as that operating in the atomistic simulation with its recorded stress profile. The first method was proposed by Rodney [42]. Noticing that, in the case of linear increase of stress, the activation probability distribution is peaked around the maximum stress, Rodney considers the average of the maximum stress (or jumb stress) to be a good estimate of the Poisson’s stress. In the general case, that is, for any stress profile τ(t) recorder in the MD simulation, Monnet [41] proposed a method for the estimation of τp directly from τ(t). It requires, however, to have a rough estimation of the activation volume V, which can be approximated by V = b2D, where D is the distance over which the activation occurs. The latter can safely be taken equal to the defect size. Knowing V, τp can be given by:   kT VτðtÞ ln exp τP ¼ : V kT

(6)

All parameters in Eq. 6 are known from the MD simulation. This allows computing τp with no assumptions on the stress profile or the defect nature. Combining Eqs. 5 and 6, we obtain a the activation energy ΔH as a function of the local or effective stress τloc. As an example, this strategy has been recently applied on α’- Cr precipitation [35], known to occur in irradiated Fe–Cr alloys [43]. Data were collected from different MD simulations [36] of an edge dislocation interacting with coherent Cr precipitates of different sizes at different temperatures in iron. In this context, the relevant stress to be associated with the activation energy is the effective stress introduced in the last section and computed from the unpinning stress using Eq. 1. At 0 K, the Poisson’s stress equals simply τCr. At finite temperature, the stress profile involved in Eq. 6 corresponds to the local stress (given by Eq. 1). Since the precipitate strength scales with the defect size, the

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Fig. 3 Example of scale transition from the atomic scale to the continuum. (a) Evolution of the activation energy normalized by the precipitate size as a function of the Poisson’s stress and (b) evolution of the activation energy as a function of temperature. Results are obtained from atomistic simulations of edge dislocation interacting with Cr precipitates of different dislocation lengths and precipitate sizes

activation energy varies with defect strength. Monnet [35] has shown that the activation energy computed from MD simulations is proportional to precipitate size. The results of the MM strategy concerning this example are shown in Fig. 3. The evolution of the activation enthalpy normalized by the precipitate size is depicted as a function of the local stress in Fig. 3a. Although MD simulations used different dislocation lengths and different precipitate sizes, all points collapse into one curved line in this representation, which indicates that the methodology used here is robust. On the other hand, the activation energy is known to be proportional to temperature [39]. It is important to check if this relation is obeyed by the results shown in Fig. 3a. The activation energy computed in Fig. 3a is plotted as a function of the simulation temperature in Fig. 3b. ΔH is found, as required, proportional to temperature, which can be considered as an additional validation of the method. Consequently, the activation energy can be written as ΔH = CkT, where C is a constant representing the efficiency of thermal activation. In MD simulation conditions, C is equal to 6.7 (see Fig. 3b), which is coherent with MD simulation conditions [40] but much lower than C  25, measured in experimental tensile tests in iron [39]. This is because of the high strain rate applied in MD simulations, leading to a high activation rate observed in these simulations.

3.3

Interaction with Extended Obstacles

Interaction with defects having dislocation character such as SFTs or DLs is complex and difficult to summarize. As revealed by atomistic simulations, the interaction does not only induce pinning of the mobile dislocation, it generally induces a change in the

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dislocation shape and in the defect configuration. The catalogue of the interaction strength and mechanisms is so large that no transition method accounting for the variety of interaction results has been reported. In fact, when a dislocation encounters a SFT or a DL, the contact does not only leads to junction formation and annihilation coupled with climb and cross slip (features that can be simulated in DD simulations), but it often involves behaviors of anelastic type. The size of radiation defects is in the same order of the dislocation core extent, a region where linear elasticity does not apply. The most “achieved” treatment has been recently published by Shi et al. [44], in which authors try to reproduce MD results of edge dislocation interacting with loops in iron [45] by a strong parameterization of the DD code. MD simulations reveal that in most cases, when the dislocation touches the loop, a junction is formed. In many loop configurations, the junction sweeps the loop surface leading finally to the absorption of the loop by the creation of two super jogs on the gliding dislocation. The outcome of DD simulations was similar in most cases, which confirm the validity of the transition method employed by Shi et al. [44]. In the first DD investigations of radiation hardening, elastic interactions were fully accounted for [46], but contact reactions were always neglected. In most of the other reported simulations, the interactions revealed in MD are phenomenologically accounted for in DD simulations. For example, Ghoniem et al. [44] modeled dislocation interactions with SFT and voids by representing the dislocation by an ensemble of circular arcs bowing under the effect of applied stress. Local unpinning from a defect occurs when the critical cusp angle of dislocation arms is reached. Once dislocation in released from the defect, the latter is removed from the simulation box and the vacancy content of the defects is implicitly distributed uniformly along the arc, by operating a local climb of the arc. However, the critical angle was not computed from ASs and dislocation self-stress was not taken into account. Similar strategy has been reported by Crospy et al. [47] in which DD simulations are used to predict the flow stress of irradiated Cu micropillars. In this study SFT are considered of Orowan type and the probability of destruction of a SFT is considered to increase every time the SFT is bypassed by a dislocation. In another investigation, Terentyev et al. [48] treated DL as local obstacle and applied the strategy described in the last section to determine the activation energy as a function of the effective stress. The results are used in DD simulation to determine hardening induced by random distribution of DLs. On the other hand, Hiratani et al. [49] investigated though DD simulations thermal activation properties of dislocation interacting elastically with SFTs (no contact reaction). Here again, the activation energy was not determined from ASs. Finally, Arsenlis et al. [50] investigated dislocation interaction with perfect DLs, but the DL size was too large compared with radiation induced DLs and interactions were considered purely elastic with no input data coming from MD simulations.

3.4

Validation of the Scale Transition Method

By validation, we mean comparison (as that shown in Fig. 1) of the maximum stress and of dislocation shape. The importance of the former is evident, and the latter is

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required because if dislocation curvature predicted with the two simulation techniques is very different, the interaction statistics can be strongly altered. Using the concept of shear resistance, simulating dislocation interaction with precipitates using DD simulations is straightforward [40], as shown in the interaction scheme in Fig. 4a. At 0 K, there is no thermal activation; the precipitate is characterized by its shear resistance. When a segment touches the spatial zone representing the precipitate, it is stopped. At every DD simulation step, the effective stress on the segment τeff is compared with the obstacle shear resistance τobs. Since the latter is in the order of GPa, τeff is always less than τobs at the beginning of the interaction. When the dislocation acquires a significant curvature, τeff increases rapidly. When it exceeds τobs, the segment is allowed to shear the precipitate. The validation of the transition method is performed by imposing the same simulation conditions in MS and DD simulations and by comparing the unpinning stress and the critical dislocation shapes in both simulation techniques. In the case of edge dislocation interacting with Cr precipitates, the dislocation shapes were found comparable, similar to the case of voids shown in Fig. 1. Likewise, unpinning stresses predicted by DD simulations is in agreement with that recorded in atomistic simulations. Figure 4b shows a comparison between the unpinning stresses obtained in both simulation techniques for an edge dislocation of 21 nm length interacting with Cr precipitates at 0 K in iron. The agreement confirmed in Fig. 4b suggests that the mechanical equilibrium required in Eq. 2 is fulfilled in MD as well as in DD simulations. At finite temperature, the validation of the transition method is less straightforward, because thermal activation is not a spontaneous feature of DD simulations.

Fig. 4 (a) Scheme of dislocation pinning by two precipitates as modeled in DD simulations. A pinned dislocation segment is allowed to penetrate the precipitate only when the effective stress on it exceeds the shear resistance of the precipitate; (b) Comparison between the maximum stresses predicted by DD simulations and atomistic simulations of edge dislocation of 21 nm length interacting with Cr precipitates of different sizes in iron at 0 K [35]

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Fig. 5 Comparison between unpinning stresses predicted by AS and DD simulations of edge dislocation interacting with (a) 2 nm-Cr precipitate (dislocation length 21 nm) and (b) 2 nm void (dislocation length 42 nm) in pure iron

Time enters into account explicitly in the integration scheme. A full discussion of the basis of this integration is given in [40]. The principle of treatment is that when a segment enters in contact with the precipitate, the shearing probability becomes a stochastic process. The consideration of explicit and discrete time step, which is inherent to DD simulations, induces a binomial probability distribution, which tends to the theoretical Poisson’s process when the time step goes to zero. In practice, once the effective stress is computed on the pinned segment or node, the activation energy can be calculated, since the evolution of ΔH with the stress is known from ASs (see Eqs. 5 and 6). Then, the probability of thermal activation within the DD time step is calculated from Eq. 4. Here again, the validation of this method can be done by comparing the unpinning stress recorded in ASs and DD simulations, using the same simulation conditions. In Fig. 5, the comparison of the unpinning stresses is shown in the case of a 2 nm-Cr precipitate (Fig. 5a) and 2 nm void in pure iron (Fig. 5b). It can be seen from Fig. 5 that the scale transition method allows DD simulations to reproduce the same temperature sensitivity observed in ASs. In the same conditions, the unpinning stresses predicted by DD and ASs are similar over a large range of temperature.

4

Interaction with Radiation Defects at the Grain Scale

Once AS results are treated according to the transition method, and the transition method validated, simulations at larger scales can be carried out to predict the increase in critical resolved shear stress (CRSS) induced by irradiation. The scale of interest is the intragranular scale, called for simplicity mesoscale in the following. At this scale, it is possible to account for the dislocation–dislocation interactions and for dislocation

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interactions with a large number of defects. It is also possible to consider a realistic defect distribution, such that observed in experiment. The most popular simulation technique at this scale is DD simulations, sometimes also called DDD (for discrete dislocation dynamics). This simulation technique can be used to predict strengthening induced by a microstructural change, such as radiation defects or work hardening. Different DD codes are available, but there is a general agreement in handling dislocation motion and interaction. Dislocation lines are discretized into small segments and all stress components (applied shear stress, self-stress, image stress, friction stress, etc.) are calculated in the middle of segments (or nodes). The resultant stress, called effective stress τeff, is the driving stress for the motion of the dislocation segment. Depending on the mobility low, the position of every segment (or node) is updated, which enables the prediction of the new dislocation configuration. If a segment encounters another dislocation segment, the possible interaction is selected among the tabulated list of interactions, depending on the Burgers vectors and line directions of the interaction segments. In contrast, there is no agreement on the treatment of dislocation interactions with defects. In the following, the different DD investigations of dislocation interactions with defects are discussed.

4.1

DD Simulations of Interactions with Local Precipitate

Only few papers reported treatments with local defects at the mesoscale. Mohles and coworkers [34, 51] proposed a large number of treatments of precipitates of different strengths and nature with and without associated stress fields. Their method is based on the concept of shear resistance, discussed in the last section, but their results were not compared to atomistic results for the validation of the overall process. Once the parameters defined by the transition method are implemented, large scale DD simulations can be undertaken to predict hardening induced by a realistic distribution of defects. When defects are randomly distributed through the crystal, the average planner spacing is given by λ = (DC)
0.5, where C denotes the number of defects per unit volume. λ is an intrinsic length scale, characterizing the defect configuration (density, size, and distribution) and determining interaction statistics with the dislocation microstructure considered in the DD simulations. In order to ensure satisfactory interaction statistics, the dislocation length must be at least ten times λ and the dislocation must cross a distance of at least ten times λ. This requirement guarantees that simulation samples at least 100 interaction configurations. These requirements are minimal. By increasing further the dislocation length and the advance distance, spurious effects induced by the restriction of the random distribution are minimized. In Fig. 6(a), an example of DD simulations of dislocation interaction with Cr precipitates [35] is shown. The dislocation length is larger than 20 times the average planner spacing (λ = 0.22 μm in this case). When the interaction statistics are correctly accounted for, the stress–strain curves predicted by DD simulations must exhibit a clear plateau corresponding to the stress necessary to propagate the dislocation through the defect population. This can be seen in if Fig. 6b for different precipitate sizes and densities. To each precipitate

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Fig. 6 (a) DD simulation of an edge dislocation interacting with random distribution of Cr-precipitates in iron (D = 2 nm, C = 1022 m
3). The initial dislocation length is 5 μm, more than 20 times the average void planner spacing; (b) stress–strain curves of the simulations for different size (in nm) and density (in 1024 m
3) of precipitates

configuration, one flow stress is associated. Tow features have to be clarified here. The first one is that the stress–strain curves in the plateau region are serrated as can be seen in Fig. 6b. Every maximum is associated to a given hard configuration resulting from the distribution randomness, while the stress minima have no physical meaning since they are inherent to the limitation of the simulated volume. In real crystals, which do not exhibit size effects, all hard configurations are present with equal probability. The flow stress to be associated with hardening at the mesoscale corresponds thus to the average of stress maxima. The latter is called the critical stress in the following. The second feature is related to the fact that, in most materials, the defect family simulated in DD is not the only obstacles for dislocations motion. Unless the superposition of hardenings is proven to be linear, hardening computed in DD simulations cannot be taken as the radiation hardening at the mesoscale. This point will be discussed later in the section dedicated to the establishment of crystalline laws. At 0 K, the computed critical stress depends on the shear resistance of the defects, but also on the mobility law of the dislocation. In the athermal regime (FCC and high temperature BCC), in which dislocations are known to move quickly at low stress, the flow stress must not depend on the strain rate imposed in DD simulations, except for dynamical loading conditions, such as mechanical shocks. In contrast, when dislocations experience lattice friction or solid solution effects, the flow stress becomes a signature of radiation hardening coupled to dislocation mobility [52]. At finite temperature, when the activation energy is not too large, thermal activation allows for decreasing the critical stress. In Fig. 7(a) the evolution of the

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Fig. 7 (a) Evolution of the critical stress with temperature is DD simulations of dislocation interaction with Cr precipitates of different sizes and densities; (b) evolution of the critical stress with temperature for different average dislocation velocities propagating through Cr precipitates of 2 nm size and 4  1024 m
3 density

critical stress with temperature is shown for different precipitate configurations [35]. The athermal plateau that can be noticed on this figure is the consequence of the strong increase in ΔH observed in ASs for low stresses (see Fig. 3a). The intensity and the extent of thermal activation are quite different in ASs and DD simulations. Two possible interpretations can be advanced to explain this apparent discrepancy: (i) the average dislocation velocity in DD simulations is lower than that recorded in atomistic simulations, and (ii) the randomness of the distribution induces different interaction statistics. Temperature sensitivity of dislocation interactions with the precipitates induces strain rate sensitivity, which, at the mesoscale, manifests itself as sensitivity to the dislocation average velocity. In Fig. 7(b), the critical stress related to a given defect configuration is given as a function of temperature for three different dislocation velocities: 0.18, 1.8, and 18 m/s. As expected, the strain rate sensitivity is maximal at intermediate temperature, in agreement with experimental observations.

4.2

DD Simulations of Interaction with Extended Defects

As discussed in Sect. 3.3, ASs reveal a large variety of interaction mechanisms with defects having dislocation character as DLs or SFTs. There is no general method for scale transition available in the literature. Consequently, there is no general implementation of extended defects in mesoscale simulations, such as DD simulations. In most DD simulations of dislocation interaction with DLs and SFTs, the stress field of the defects is taken into account but not the contact reaction, which results in total or partial absorption of the defect. Only two exceptions can be noticed. As mentioned before, Shi

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Fig. 8 Hardening induced the absorption of dislocation loops of 2 nm size and different densities [53]. The dashed lines represent two different theoretical predictions (see text for more details)

et al. [44] were the first to adjust for every dislocation type at every simulation temperature a mobility low in order to reproduce AS results. In a completely different perspective, Monnet [53] investigated the effect of loop absorption, by simulating dislocation interaction with collinear loops, that is, sharing the same Burgers vector as the gliding dislocation. The absorption leads to the formation of mobile superjogs on the edge dislocation and helical turns on the screw dislocations. Taken individually, the absorption of the loop does not hinder the edge dislocation motion but induces a strong pinning of the screw dislocation. Monnet [53] pointed out that absorption of randomly distributed loops has a strong hardening effect on dislocations of all characters. In the case of screw dislocation, the relaxation of line tension due to the formation of helical turns reduces strongly the mobile section of the dislocation. When dislocation loops are initially randomly distributed, the screw dislocation acquires a complex shape with a small portion of planner sections. Unpinning from many randomly distributed helical turns requires a significantly high stress as can be seen in see Fig. 8. On the other hand, the motion of the edge dislocation leads to a global drift of DLs in the direction of motion of the dislocation, which ends up by forming clouds of heterogeneous density along the dislocation line. Segments surrounded by cloud of low density bow out forming sections of mixed character, which were found to form helical turns with the intersected DLs. Hardening induced by DL populations of 2 nm size and different densities are shown in Fig. 8. Monnet [53] noticed that hardening induced by loop absorption is larger than twice the hardening induced by impenetrable obstacles of comparable size and density.

5

Transition to the Macroscopic Scale

With the large number of input and output parameters, the main challenge in DD simulations is to express the various results into constitutive equations accounting for all relevant parameters. This is a scale transition to the continuum level, which is

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necessary to construct a crystalline law for the local behavior in simulations at larger scale, such as Finite Elements Method (FEM). In this section, first DD results on radiation hardening are expressed in terms of constitutive equations, then, these equations are gathered to construct a complete crystalline of an athermal mechanical behavior.

5.1

Constitutive Equations for Radiation Hardening

In the absence of thermal activation, DD simulations [35] show that strengthening induced by local spherical obstacles of size D, density C, and infinite strength (Orowan precipitates) can be expressed by:

ΔτOrowan ¼

!3=2 2D ln b μb 2l ln , 2l 2π l b ln b

(7)

where l is the free obstacle spacing l = [(DC)
0.5
D], and D is the harmonic mean of D and l. It is interesting to point out that the 3/2 power term on Eq. 7 results from the distribution randomness, as initially suggested by Friedel [54] and used by Bacon et al. [29]. However, ASs show that most of local radiation defects are sheared by dislocations. Consequently, they must have a finite shear resistance τobs. In this case, hardening is naturally lower than that predicted by Eq. 7. Monnet [35] showed that hardening induced by precipitates of finite strength can be given by: ( Δτprcpt ¼

τobs τ1

3=2

ΔτOrowan ΔτOrowan

τobs  τ1 , τobs  τ1

(8)

where τ1 is the shear resistance beyond which dislocations cannot shear the precipitates. It is close to 4.5 GPa in iron [35]. Eq. 8 was found in agreement with all results of DD simulations of dislocation interactions with precipitates over a large range of precipitate size, density and resistance. Figure 9 shows comparison between precipitation hardening computed in DD simulations and predictions of Eq. 8. At finite temperature, DD simulations show that Eq. 8 is still valid, but the obstacle shear resistance τobs must be replaced by an appropriate value τloc. The first attempt that can be made to estimate the latter is as follows. If the rate ω is kept constant in Eq. 4, when temperature is set to Tf, ΔH increases to ΔHf. Since ΔH is a known decreasing function of the shear resistance, the value of τloc,f can be deduced from the known profile ΔH(τloc) (see for example Fig. 3a). However, if we insert τloc,f in Eq. 8, the predicted hardening does not concur with results of DD at Tf. The error in this reasoning is that Eqs. 4 and 8 cannot be used together simultaneously, because Eq. 4 connects temperature with dislocation velocity in ASs (for periodic row of precipitates), while Eq. 8 accounts for the competition between unpinning events from randomly distributed precipitates. Instead, another equation can be proposed to deduce the value of τloc as a function of dislocation velocity v in DD simulations [35]:

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Fig. 9 DD simulations of precipitation hardening induced by 2 nm Cr precipitates in iron at 0 K [35] with two densities (1024 and 2  1024 m
3) as a function of shear resistance. The dashed lines correspond to predictions of Eq. 8

 v¼

τloc τ1

3=2 ωo lexp


ΔH ðτloc Þ , kT

(9)

This equation constitutes a scale transition between AS results and DD simulations. If a DD simulation is curried out at given dislocation velocity v, free defect spacing l, and temperature T, then the average shear resistance of precipitates τloc, can be obtained numerically by resolving Eq. 9. Precipitation hardening can then be predicted by inserting the obtained value of τloc in Eq. 8. This method was found to provide values of Δτprcpt in good agreement with DD simulation results (see Fig. 7a). Concerning hardening induced by small dislocation loops ΔτDL, DD simulations [53] show that the computed hardening is significantly larger than hardening induced by impenetrable precipitates of similar size and density. Predictions of Eq. 7 were added to Fig. 8 for comparison with DD simulationpresults. However, hardening ffiffiffiffiffiffiffi computed in DD simulations remains lower than μb DC , which is known as the Orowan stress. This can be checked on Fig. 10, where hardening was computed for several loop populations of different sizes and densities. Simulation results were fitted by a straight line passing by the origin. The slop of the dashed straight line in Fig. 10 is close to 0.5. This allows us to express hardening induced by DLs as: ΔτDL ¼ μb

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi aDL DC,

(10)

with aDL is the interaction coefficient of DL and equal approximately 0.25. Since the computed hardening induced by loop absorption is by far larger than the maximal hardening induced by local obstacles (the Orowan hardening), the details of the interaction mechanisms prior to absorption have little effect on the overall hardening. To the knowledge of author, Eqs. 8 and 10 are the only available constitutive equations for radiation hardening, confirmed by DD simulations. There is no available equation for hardening induced by SFTs or large dislocation loops.

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Fig. 10 Hardening computed by DD simulations as a function of the Orowan stress

6

Crystalline Law Including Radiation Hardening

In most of the reported investigations, the increase in the yield stress Δσ y is deduced from radiation hardening at the mesoscale Δτirr by applying the Taylor model of homogeneous deformation in all grains (see for example the work by Bergner [55]); Δσ y ¼ MΔτirr ,

(11)

where M is the Taylor factor. Δτirr can represent hardening induced by local obstacles (Eq. 8) or DLs (Eq. 10). This method implicitly assumes a linear superposition of hardenings induced by different obstacles. Recently, Queyreau et al. [56] have shown that the superposition of forest hardening with precipitate hardening is rather quadratic, that is, the increase in Δσ y is not proportional to Δτirr and must depend on the yield stress before irradiation. In a multiscale modeling scheme, one has to: (i) establish first a set of constitutive equations (called crystalline law) describing the local crystalline behavior before irradiation; (ii) use the crystalline law in macroscopic simulations (FEM, homogenization, etc.) to predict the yield stress; (iii) validate the crystalline law of the unirradiated materials by comparison with experiment tensile tests; (iv) implement radiation defects in the crystalline law and compute the new yield stress: (v) compare with experimental tensile tests of irradiated materials. This procedure is somehow heavy to undertake, but it is nevertheless necessary for the construction of physically based modeling of mechanical behavior. Several crystalline laws were reported in the literature, mostly dedicated to FCC structure. For the BCC structure, crystalline laws must account for thermal activation of dislocation motion at low temperature [57, 58]. At high enough temperature, crystal plasticity in both structures becomes temperature-independent and follows basically the same features.

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In the following we recall the basic equations of plastic flow in the athermal regime, designed for A508 Cl3 bainitic steel, used in the manufacturing of Reactor Pressure Vessel (RPV) in Pressurized Water Reactors (PWR). Radiation damage is one of the most important features of safety assessment in these reactors. For the sake of simplicity, this steel is referred to as RPV steels. First, the shear strain rate on a given system s is expressed in a viscous flow equation: 2 3n   τsapp  s 4 5 sign τs , γ_ ¼ γ_o (12) app τsc where γ_o are n are constant and τsc is the critical stress representing the stress threshold for plastic flow. It must account for all obstacles present in the slip plane of system s. In RPV steels, the critical stress is related to the following microstructure components: initial dislocation density ρdis, distribution of intragranular carbides with density Ccar and size Dcar, small grain size leading to a strong Hall–Petch effect. The Taylor equation has been confirmed to well predict forest hardening. It can be expressed as [59]: Δτsf ¼ μb

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X asj ρj ,

(13)

where asj are interaction coefficients between system s and j. On the other hand, the carbide hardening can be deduced from Eq. 7, since carbides are large incoherent precipitates forcing dislocations to bypass them. Given the typical carbide size (100 nm) and density (3  1019 m
3), Eq. 7 can be simplified to a good approximation by: Δτcar ¼ μb

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ccar Dcar :

(14)

However, the determination of the Hall–Petch contribution ΔτHP is not straightforward. It implies to use gradient plasticity or other approaches in order to account for size effects and strain heterogeneities (see for example [60]). Although the Hall–Petch effect is observed at the polycrystals level, it can be implemented in the local behavior as a back stress ΔτHP , given by: K ΔτHP ¼ pffiffiffi , d

(15)

where d is the average size of plastic confinement zones, which must lie between the size of lath blocs ( 10 μm) and lath size ( 1 μm). Upon irradiation, atom probe tomography revealed the presence of Solute Clusters (SC) as the only radiation-induced features at low radiation fluence in RPV steels [61]. But the structure of SCs is not well known. In the absence of

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characterization by atomistic simulations of dislocation interaction with these defects (no available data on their shear resistance or activation energy for unpinning), we consider these defects to have the same shear resistance as Cr or Cu precipitates in iron, that is, τSC  2.2 GPa. Using typical values for the size (D  3 nm) and density (C  1023 m
3) of SCs, we can simplify Eq. 7 by introducing an interaction coefficient aSC such that: ΔτSC ¼ μb

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aSC DSC CSC ,

(16)

with aSC is close to 0.015. Once the different components defined, we can express the critical stress as: τsc ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δτ2f þ Δτ2car þ Δτ2SC þ ΔτHP

(17)

Note that the superposition of ΔτHP to other hardening sources is not quadratic in Eq. 17. The reason is that Hall–Petch effect is expected to result from long-range back stress, which superposes linearly to hardening induced by local interactions (forest dislocations, carbides, SCs). Of course, for the unirradiated material, radiation hardening ΔτSC vanishes from Eq. 17. Consequently, the increase in the CRSS due to irradiation is given by: Δτsc ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δτ2f þ Δτ2car þ Δτ2SC
Δτ2f þ Δτ2car :

(19)

This indicates that even at the mesoscale, radiation hardening in RPV steels cannot be equal to ΔτSC. Radiation hardening is not independent of the initial critical stress, which confirms that Eq. 11 does not apply in this context. The last equation necessary for the crystalline law is that giving the evolution of the dislocation density during plastic strain. Inspired from the classical approach (see for example Meckin et al. [62]), the dislocation density on a given system s is expected to increase only when this system accommodates some plastic shear strain. 0

pffiffiffiffiffiffiffiffiffiffiffiffis ρ_ s 1 B B aself ρ þ ¼ γ_s b @ K self

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X asj ρj þ Dcar Ccar þ aSC DSC CSC j6¼s

K obs

1 C
y s ρs C A:

(20)

The first fraction on the right-hand side of Eq. 20 accounts for dislocation storage rate due to self-interactions (between dislocations belonging to the same slip system). The second fraction represents the storage rate induced by dislocation interactions of system s with all other obstacles including dislocations on other slip systems. Finally, the last term account for microstructure restauration induced by dislocation annihilation. Kself and Kobs are constant, aself the self-interaction coefficient, asi the interaction coefficients with other slip systems and ys is the annihilation distance of dislocation dipoles. All other terms are defined in the text.

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Table 1 Numerical values of the different parameters of the crystalline law used for the calculation of the macroscopic stress–strain curves in Fig. 10 Variable n γo

Signification Rate power Rate constant

Equation 12 12

Value 100 10
5 s
1

b μ Ccar Dcar DSC K d aSC aself acol asj Kself Ksobs y ρs

Burgers vector Shear modulus Carbide density Carbide size Solute cluster size Hall–Petch constant Grain size SC interaction coefficient Self_interaction coefficient Collinear interaction coefficient Other interaction coefficients Self-storage constant Obstacle-storage constant Annihilation coefficient Dislocation densityper system

13,14, .. 13,14, .. 14 14 16 15 15 16 20 20 20 20 20 20 13,20

0.248 nm 87 GPa 3  1019 m
3 100 nm 3 nm 0.17 MPa.m0.5 3 μm 0.015 0.06 0.5 0.06 60 20 2 nm 8  1012 m
2

The values of the different parameters are given in Table 1. The set of equations introduced in this section are necessary to define the local behavior in RPV steels. In the next section, this crystalline law is used to predict the macroscopic behavior before and after irradiation.

7

Radiation Hardening at the Macroscopic Level

To test the local behavior law detailed in the previous section, larger scale simulations are needed. Crystalline laws can be implemented and integrated in different simulation techniques: finite-element method (see for example [63]), fast Fourier transform [64], and homogenization techniques. The description of the mechanical behavior of nonlinear heterogeneous materials through homogenization techniques has received considerable attention for decades. The first works of Taylor [65] were already devoted to the polycrystals for which a local phase regroups all grains with identical crystallographic orientation. For such materials the self-consistent estimate, which considers that the homogeneous medium surrounding each phase is the researched equivalent macroscopic medium, is particularly well adapted. The main difference between homogenization techniques relies then in the choice for the necessary linearization formulation (secant, tangent, affine, second order) that has to be made in order to represent the nonlinear behavior of the material. In this section, results obtained by the homogenization model proposed by Berveiller and Zaoui (BZ) [66] are presented. This explicit homogenization rule has been widely used in the literature for describing

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the effective behavior of elastoplastic polycrystals. A set of 500 different grain orientations has been chosen in order to reproduce an isotropic mechanical behavior. A Small strain formulation and an explicit Runge–Kutta integration scheme with adaptative time step is used to integrate the constitutive equations of the model. The simulated macroscopic stress–strain curves for RPV steels are shown in Fig. 11. The unirradiated steel contains zero density of solute clusters. The other curves simulate irradiated steels containing different densities of SCs of 3 nm size. The shear resistance of all SCs is taken equal to 2.2 GPa, which is close to the shear resistance of Cr and Cu precipitates in iron, computed from atomistic simulations (see Fig. 2). The predicted behavior of the unirradiated steel is in agreement with experiment [57] for the yield stress ( 500 MPa) and the flow stress at 0.15 strain ( 620 MPa). This evidences the validity of the crystalline law used to describe the local behavior of unirradiated RPV steels. However, validation of the radiation hardening is less obvious because most of the published data attempt to relate the evolution of mechanical properties directly with the radiation dose and not with the microstructure evolution. Only the order of magnitude of hardening is of interest here. For 40 years exposure time in PWR, the dose is close to 6.6 1023 n.m
2. For this dose the density of SCs is approximatively 1023 m
3 [61] and the radiation hardening is close to 100 MPa [67]. These orders of magnitude are well reproduced in Fig. 11. These results are obtained using realistic values for the different parameters given in Table 1. Only the Hall–Petch contribution ΔτHP was adjusted to reproduce the yield stress of the unirradiated steel at 300 K. Of course, predictions can be refined if other parameters related to the steel microstructure and chemical composition are adjusted. For example, some characteristic values such as the initial dislocation density or the intragranular carbide density can be directly measured in transmission and scanning electron microscopes.

8

Conclusions

Multiscale modeling of mechanical behavior, in general, and of irradiated materials, in particular, provides a coherent scheme for the integration of simulations results at all relevant scales. Its primary objective is not to confirm or validate any theoretical interaction models, but to rationally integrate results of numerical simulations across the scales. In this strategy, the objective is not to explain simulation results. For example, when atomistic simulations allows for the identification of the shear resistance of a given defect, the implementation of this resistance in DD simulations or other simulations at the mesoscopic scale does not require an explanation or justification of the computed shear resistance. What matters is DD simulation can predict an unpinning stress in agreement with that recorded in atomistic simulations. This does not mean that theory is useless in this context. Theoretical models are still essential in validating simulation results and, especially, in developing methods for scale transitions. Usually the “gross” results of simulations at a given scale are unusable at other scales. Other quantities of interest at the target scale must be

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Fig. 11 Simulated stress strain curves of RPV steels obtained at 300 K by the BZ homogenization model for un irradiated and irradiated states characterized by different densities (in m
3) of solute cluster of 2.2 GPa resistance and 3 nm size

identified. For example, DD simulations allow for the prediction of hardening induced by a population of precipitates of different size and densities. But only a constitutive equation reproducing these results is of interest at the macroscopic scale. The ultimate validation of the multiscale modeling is the final comparison between the predicted value and the measured value of a relevant macroscopic mechanical property. In the case of radiation hardening, the conclusive validation is the comparison between the experimentally measured increase in yield stress and the increase in the simulated flow stress of the irradiated material (see Fig. 11). The advantage of this strategy is to minimize resorting to adjustable parameters, such as quantities related to the diffuse concept of line tension (cusp angle) or integrated properties of little help in numerical simulations (interaction energy). The difficulty in achieving such approach is the need for the construction of a full chain of simulation techniques in order to investigate the material behavior at all scales of interest. Every gap in this chain implies the use of adjustable parameters.

9

Future Directions and Open Problems

The multiscale modeling strategy attempts to gather the available physical information produced at all phenomenological scales. Despite the huge progress made in numerical simulations and modeling, several features of mechanical behavior are still difficult to model, especially those lying at the intersection of two different scales. Some phenomena are transversal by nature and require nontrivial scale coupling.

9.1

Radiation Creep, Dynamic and Static Aging

When dislocation behavior is strongly affected by the diffusion of self-interstitial atoms, solute atoms or vacancies, two different time scale have to be coupled.

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Diffusion must be predicted in the presence of heterogamous mechanical fields, such those induced by dislocations. The fundamental features of diffusion prevail at the atomic scale. But atom jumps remain rare events compared with the frequency of atomic oscillations, characterized by the Debye frequency, in the order of 1013 vibrations per second in metals of interest in nuclear applications. If we recall that the relevant time scale for radiation creep (years, even tens of years), it appears that simulation of dislocation motion coupled to diffusion remains out of reach for atomistic simulations. Phenomenological models for radiation creep and dynamical aging are available and useful for industrial purposes. But multiscale modeling of these phenomena remains a real challenge for the scientific community.

9.2

Interaction with Dislocation Loops

Dislocation loops are one of the most common features of radiation microstructure. As discussed in this chapter, the variety of interaction mechanisms is difficult to assess at the mesoscopic level. Too many parameters are needed to account for all types of interactions. When interactions involve anelastic features (dislocation climb, rotation of the Burgers vector), local rules must be explicitly implemented in DD simulations, which accentuates further the feeling of “knitting” in numerical simulations. In addition, multiscale modeling of interaction with loops requires the knowledge of the number of encountered loops. If loops are highly mobile, all loops in a thick layer below and above the slip plane will interact strongly with the gliding dislocation. Attractive loops will migrate to encounter the dislocation and repulsive loops will move away out of the layer. If the layer is of thickness κ, the effective planner density of loops encountered by the dislocation is not (DC) but (κC), which may differs by one order of magnitude. Furthermore, κ is not well known in any industrial material. Also, it is well accepted now that dislocation loops are decorated by solute clusters and atmospheres in all industrial materials. This affect hardening through two ways: (i) decoration modifies the interaction mechanisms and strength as evidenced by atomistic simulations, and (ii) decoration reduces the loop mobility which may drastically decrease the loop migration towards the gliding dislocation. If the loop motion is completely inhibited, only loops crossing the slip plane of the dislocation suffers the contact with dislocation. Decoration affects thus the interaction details and statistics. The first difficulty in modeling decoration is that decoration involves several types of solutes atoms. Atomistic simulations of several atom types are not possible currently because there is no interatomic empirical potential for modeling atomic interactions with all the involved elements. Decorations and segregations are difficult to characterize also in experiment. The most advanced technique in this domain is atom probe tomography; it does not provide the accurate positions of atoms. It may thus lead to an artificial dispersion of the segregation structure. Effects of segregation and decoration on hardening induced by dislocation loops represent a serious challenge in multiscale modeling.

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Strain Localization

It is a common challenge in all approaches dedicated to modeling the mechanical behavior. Among the manifestation of the effects of strain localization, we find the Hall–Petch effect, embrittlement due to clear channels in some deformed irradiated materials, etc. Strain heterogeneities have received a large attention from the scientific community and many dedicated theory have been developed, such as gradient plasticity, field dislocation dynamics, phase fields, etc. In these approaches, dislocations are discrete or continuously distributed in an elastic media, with no or little input information coming from lower scales. For example, it is not well known why gliding dislocations are sometimes absorbed, attracted or repelled by grain boundaries, though this feature is at the heart of embrittlement in irradiated materials. The relaxation mechanisms of the dislocation microstructure in the vicinity of obstacles (precipitates, grain boundaries, etc.) are still badly known. They include cross slip of screw dislocations, edge dislocation climb, grain boundary glide, heterogeneous dislocation nucleation at grain boundaries, etc. The atomistic features in these mechanisms cannot be neglected and their investigations by atomistic simulations require a huge amount of work.

9.4

Dislocations Behavior in Concentrated Alloys

Materials of high concentration of alloying elements, as in the case for austenitic steels used for manufacturing the internal structure in PWRs, raise some difficulties in multiscale modeling. The reason is that fluctuation in local concentrations, inherent to these types of materials, affects the local staking fault energy and, hence, dislocation dissociation and mobility, which has serious effects on dislocation behavior. For example, the activity of cross-slip is no longer a uniform property along the dislocation partial dislocations of screw character. Since all dislocations are dissociated in these steels, heterogeneous cross-slip events may occur along all dislocations pinned by obstacles. Also, solid solution induces a threshold for partial dislocation motion, which leads sometimes to the formation of extended faults. In these conditions, dislocation interactions with loops, which are already complex in pure FCC materials, becomes far more difficult to model and simulate in concentrated alloys.

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8

Atomistic Simulations of Metal–Al2O3 Interfaces Stephen Hocker, Alexander Bakulin, Hansjörg Lipp, Siegfried Schmauder, and Svetlana Kulkova

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomic and Electronic Structures of α-Al2O3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Work of Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Interfaces with bcc Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Interfaces with fcc Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Calculations of Stress–Separation Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Electronic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Bcc Metal–Oxide Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fcc Metal–Oxide Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Metal–Oxide Interface Under Tensile Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Influence of Homogenous Strain in the Interface Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Molecular Dynamics Simulations Using ab initio Based Polarizable Force Field . . . . . . . . 9 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Interfaces between metals and ceramics are of great importance for a wide range of industrial applications such as multifunctional devices, fuel cells, thermal barrier coatings, corrosion protection, and microelectronics. The technological S. Hocker (*) · H. Lipp · S. Schmauder Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected]; [email protected] A. Bakulin · S. Kulkova Institute of Strength Physics and Materials Science of SB RAS, Tomsk, Russia Tomsk State University, Tomsk, Russia e-mail: [email protected]; [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_9

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interest arises from the possibility to obtain composite materials with much better characteristics than that of both constituents. In order to control the metal–ceramic interfaces it is necessary to understand them at different scale levels. The approaches used for this goal can be ranged from models at the macroscale to first principles calculations. Atomistic theoretical studies of varied metal–ceramic systems are based on ab initio calculations or molecular dynamics simulations which allow to understand the interactions at the interface at the atomic level. Using the plane wave pseudopotential method within density functional theory, the electronic and mechanical properties of Me–Al2O3 interfaces with fcc (Ni, Cu, Pd, Ag, Pt, Au) and bcc (Nb, Mo, Ta, W) metals are investigated to find correlations between the work of separation and crystal structure, strain conditions, and electronic properties of both constituents. Stress–displacement relationships of separation perpendicular to the interface are calculated in case of Al, Cu, Ag, and Nb on the Al-terminated Al2O3(0001) surface. It is shown that obtained results such as work of separation and tensile strength can be understood from the electronic structure. The influence of misfit dislocations on the fracture behavior of an Al–Al2O3 interface is demonstrated using classical molecular dynamics simulations. The comparison with experimental data provides good agreement for both the work of separation and qualitative prediction of mechanical reinforcement.

Keywords

Metal–Al2O3 interface · ab initio calculations · Molecular dynamics simulations · Work of separation · Tensile test

1

Introduction

Metal–oxide interfaces are studied intensively in the last 2 decades because of their technological and scientific importance [1]. They have an important role in many applications due to their properties which are a combination of the properties of metals (ductility, high electrical and thermal conductivity) and ceramics (high hardness, corrosion, and wear resistance). Materials containing metal–oxide interfaces are applied within automotive industry, turbine blades, squid magnetometers, and dental implants. In microelectronics, corundum (α-Al2O3) is one of the most commonly used ceramics, for example as a substrate for integrated circuits or as a tunnel barrier for superconducting devices. As a result, Al2O3 was a subject of numerous conference papers and reviews [1–4] and remains in the focus of experimentalists and theoreticians. There is literature discussing theoretical questions about bonding between interfacial atoms as well as about numerical simulations on both atomistic [5–7] and mesoscopic scale [8, 9]. In general, composite interface properties can be determined by some factors characterizing the interactions on both scale levels. Mechanical characteristics

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of composites depend significantly on the intrinsic properties of included metal–ceramic interfaces. The atomic and electronic structures of the interfaces, including the compatibility degree of the crystal lattices of the metal and the ceramic play a very important role. The formation of metal–oxide interfaces with good mechanical properties is closely related to the formation of strong chemical bonding at the interface. Knowledge of the electronic structure of the interfaces as well as of interfacial processes and their evolution under the action of different factors such as deformation is necessary for qualitative and quantitative understanding of adhesion. The most suitable method for investigations of chemical bonding at the interface are ab initio studies which can reveal the work of separation (Wsep) for various configurations of metal films. The work of separation is defined as the energy needed to reversibly separate an interface into two free surfaces. It is necessary to emphasize that plastic and diffusional degrees of freedom are neglected. Nevertheless it is a fundamental quantity determining the strength of interfaces [5]. Comparative studies of the correlation between the work of separation and the properties of materials containing metal–ceramics interfaces revealed some general trends [7, 10, 11]. A majority of works consider the Nb(111)–α–Al2O3(0001) interface with a high adhesion energy of ~9.8–10.6 J/m2 [5, 12–15]. Despite the increasing number of publications in the last decade, the mechanisms of chemical bonding at metal–oxide interfaces remain under discussion and in the focus of theoretical investigations [16–18]. The available results of the study of the adhesion at the interfaces differ both in the values of energies and in the conclusions regarding the stability of different atomic configurations of films at the interfaces and even in mechanisms of chemical bonding. For example, the difference in the calculated energies of adhesion of the copper film from the O termination of the surface reaches ~1.4 J/m2 [19, 20] and the values on the Al termination of the surface [19, 21] substantially differ from experimental data. Nearly equal values (0.672 and 0.679 J/m2 [22]) were obtained for silver at the interface with Al and Al-enriched terminations, whereas the difference of ~1.5 J/m2 was obtained in [11]. It should be emphasized that the existing theoretical works give different values of the relaxation of interface layers, splitting of the first metal layer, and other structural and electronic characteristics. As mentioned in [23] the reasons for these discrepancies are the use of several calculation models differing in structural parameters such as the numbers of metal or oxide film layers and interfaces (one or two) or calculation details such as the approximation used for the exchange-correlation functional. It has also been shown that strain at the interface has a significant influence on the work of separation [11]. Therefore, comparative ab initio studies taking into account these influences are needed for the understanding of chemical bonding and adhesion at metal–ceramics interfaces. It must be noted that ab initio investigations are usually restricted to coherent interfaces because of the much larger number of atoms in case of configurations with misfit dislocations. There are few exceptions, for example the incoherent Cu–MgO interface was considered in [24]. Since the classical molecular dynamics method allows simulations with significantly larger structures it can be used to investigate the effect of misfit dislocations on the properties of metal–ceramics interfaces. Detailed analyses of misfit dislocations simulated by molecular dynamics were

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performed for Cu–Al2O3 [20] and Al–Al2O3 [25]. In order to investigate dynamic fracture processes at even larger length scale, calculations using the finite element method can be used. Since mesoscopic and macroscopic models need input data which so far had to be obtained by complex experiments, the effort to use ab initio or classical molecular dynamics for calculations of such data has increased significantly during the last decade. In [26, 27] crack propagation was investigated by finite element simulations using data from atomistic simulations. The fracture toughness of the Ni–Al2O3 interface [9] was investigated using a stress–displacement relationship obtained from ab initio calculations as described in [18]. First principles calculations of tensile strengths were performed for the O- and Al-terminated Cu–Al2O3 interface [16, 17] providing data usable in large-scale simulation models. The work presented in this chapter is a comparative study of Me–Al2O3(0001) interfaces with some bcc and fcc metals. Almost all calculations are performed using the pseudopotential plane wave method within density functional theory. Computational details of these ab initio calculations are described in Sect. 2. Since surfaces are very important in fracture processes, the atomic and electronic structures of differently oriented Al2O3 surfaces are discussed in Sect. 3. A detailed overview about the work of separation of bcc and fcc metal interfaces with Al2O3 is given in Sect. 4. The influence of vacancies on the work of separation is discussed as well. Stress–displacement relationships representing brittle interface fracture are described in Sect. 5. In Sect. 6 detailed analyses of charge densities and density of states are presented for bcc and fcc metal interfaces at several stages of fracture. The influence of strain in the interface plane which is of high importance in coherent structures is discussed in Sect. 7. In Sect. 8 coherent and incoherent structures are considered using the classical molecular dynamics method. A tensile test is simulated and the influence of misfit dislocations on tensile strength and fracture processes is discussed. Finally, summary and outlook are presented in Sect. 9.

2

Computational Details

Atomic and electronic structures of Me(111)–Al2O3(0001) interfaces with some fcc and bcc metals were calculated using the pseudopotential plane wave (PP-PW) method within density functional theory (DFT). Vienna ab initio simulation package (VASP [28, 29]) was used for total energy calculations. The interaction between ions and electrons was described by ultrasoft Vanderbilt pseudopotentials implemented in this package, which provide good convergence for the considered systems. The generalized gradient approximation (GGA-PW91) [30] for exchange-correlation functional was applied. The choice of GGA is due to a better description of the relevant bulk phases of metals and oxides compared to LDA. A 4
4
1 Г centered k-points mesh was used for integration over the Brillouin zone. An energy cutoff of 400 eV was used for plane wave basis. In case of the Al-terminated α-Al2O3(0001) surface, six oxygen atomic layers and 12 Al atomic layers were used. Ten Al atomic layers were sufficient in the case of the O-terminated oxide surface. Atoms of the three central atomic layers were fixed in the oxide film with interlayer distances as in bulk Al2O3, whereas atoms in the other layers were allowed to relax to reach the total-energy and residual force convergence

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criteria, 105 eV and 0.01 eV Å1, respectively. The details for modelling of (-1100) and (11-20) surface orientations will be given in Sect. 3. Several models of repeated multilayered films separated by a vacuum gap of 20 Å were used to simulate the Me–Al2O3 interfaces: the model with two interfaces was mainly used for the calculations of the work of separation apart from Nb(11-2)–Al2O3(1100) and Nb (1-10)–Al2O3(11-20) interfaces which were described by a computational cell with a larger number of atoms in comparison with that for Me(111)–Al2O3(0001). In these cases models with one interface were adopted. Me(111) slabs were aligned to both sides of Al2O3(0001) in the model with two interfaces, keeping inversion symmetry relative to the center. In most cases the Me(111) film thickness was fixed to six layers but for Nb, calculations with four to eight layers were performed as well. The lattice parameter of Al2O3 used in the calculations was a = 4.7628 Å and c = 13.003 Å. Since only coherent interfaces are considered, metal lattices were expanded or compressed to different degrees relatively to their equilibrium lattice constants: Ni (9.48%), Cu (6.34%), Ag (6.70%), Au (7.00%), Al (3.76%), Nb (2.3%), etc. For Pd (0.05%) and Pt (0.91%), there is a good agreement between the lattice parameters of metal and oxide. The considered structures differ in the location of the first metal layer on the Al2O3(0001) surface and the stacking which determines the locations of the second and third metal layers. In the case of bcc metals, the three metal configurations considered on the O-terminated oxide surface are shown in Fig. 1a–c. Four metal configurations were considered in case of the Al-terminated Al2O3(0001) surface. One is the Al-next configuration where Me atoms occupy the position of the second Al atom at the interface (Fig. 1d). The other ones are almost the same as for the O-terminated interface. The three different positions of the first fcc metal layer shown in Fig. 2 are Al-top and O-top configurations where Me atoms of the interfacial layer are located above the aluminum or oxygen sites and the three fold coordinated by oxygen configuration (H-hollow). In order to calculate stress–displacement relationships the model with one interface as described in [18] and a metal film consisting of eight (111) layers is used. In this model, atoms far away from the interface are fixed in bulk positions. Relaxation is permitted only for atoms of the two metal layers as well as the four Al and the two O layers closest to the interface. More atoms were allowed to relax in the calculations

Fig. 1 Atomic configurations of bcc metal monolayers on the O-terminated α-Al2O3(0001) surface: (a) Me in the H-hollow site, (b) Me in the Al3 site, (c) Me in the Al4 site; on Al-terminated α-Al2O3(0001) surface: (d) Me in the Al-next site. Large green and red balls correspond to Al and O atoms, small grey balls are metal atoms

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Fig. 2 Atomic structure of fcc Me(111) monolayers on the O-terminated Al2O3(0001) surface (top view): (a) Me in the Al-top configuration; (b) Me in the O-top configuration; (c) Me in the H-hollow configuration. Large red and green balls are interfacial O and Al atoms, small grey balls are Me atoms

of the influence of strain in the interface plane (Sect. 7) where relaxations of interlayer distances perpendicular to the interface plane are desired. In this case, only one layer of oxygen and one layer of bcc or fcc metal most far away from the interface are kept fixed. The work of separation is calculated from the total energies of relaxed structures according to
W sep ¼ Eme þ Eox  Eme=ox =nA

(1)

where Eme/ox is the total energy of the supercell containing the interface structure, while Eme and Eox are the total energies of the same supercell containing slabs of either metal or oxide, respectively. A is the interface area and the factor n takes into account the number of interfaces in the models used. Brittle fracture caused by load perpendicular to the interface is modelled by rigid separation of oxide and metal slabs. The interfacial distance is increased incrementally in small steps of 0.2 Å and relaxation calculations are performed for each separation distance. The stress–displacement relationships are calculated from the energies (E) and the separations (s) of the relaxed structures: σZ ¼

3

1 ΔE A Δs

(2)

Atomic and Electronic Structures of a-Al2O3 Surfaces

The composition, morphology, and surface structure of Al2O3 influence its mechanical, chemical, and electronic properties considerably. To control interfacial processes and their evolution under different factors including deformation and loading and also to gain a deeper insight into interface adhesion, it is necessary to study the electronic structure of α-Al2O3 surfaces with different orientation. Since the atomic and electronic structures of α-Al2O3(0001) have been studied extensively in the last 2 decades, the attention to surfaces of other orientations, for example (-1100) and

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(11-20), was significantly less [31, 32]. This is mainly due to a larger number of atoms per computational cell for these structures compared to the (0001) one. As a result, the computational cost considerably grows in the former case. It is well known that a difference in the surface relaxation values and surface energies exists even for the most studied (0001) surface. For example, in case of the Al-terminated (0001) surface the relaxation of the first interlayer distance (d12) varies from 69.6% to 93.8% ([13, 15] and references therein). Mostly, the theoretical values of relaxation (Δd12 = (d12–d0)/d0, where d0 is the bulk interlayer distance) are in the interval 82 to –88%, while experiments give smaller values: from 51% to 63%. For the O-terminated α-Al2O3(0001) surface, the first interlayer relaxation is much smaller: 7.1% [13] and 20.1% [14]. Note that the value of the first interlayer distance is weakly dependent on exchange-correlation potential approximations used. Atomic structures of three α-Al2O3 surfaces are shown in Fig. 3. It should be noted that only the O-terminated surfaces are shown in this figure. The (-1100) surface was simulated by a seven-layer film with four aluminum atoms and six oxygen atoms per atomic layer (Fig. 3b). The (11-20) plane was simulated by a film consisting of four aluminum layers (12 atoms per layer) and eight oxygen layers (nine atoms per layer) (Fig. 3c). The atomic and electronic structures of the (0001) surface (Fig. 3a) was considered in detail in many studies, for example in [13–15, 21, 33]. Here, only a brief description of obtained results will be done. For the Al-terminated surface, the relaxation of 83.4% for the first interlayer distance was calculated which is in good agreement with earlier works [13, 14, 21]; that is, aluminum atoms lie almost in

Fig. 3 Atomic structure of the relaxed Al2O3 surfaces: (a) (0001), (b) (-1100), and (c) (11-20) (side view) and (d) surface densities of states. Aluminum and oxygen atoms are shown by large and small circles, respectively

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the same layer as subsurface oxygen atoms. For the O-terminated surface, the relaxation of the first interlayer distance is significantly less (8.5%) that is closer to 7.1% obtained in [13]. The structure of the (1100) surface layer changes considerably under relaxation: atomic layers consisting of oxygen and aluminum atoms split (Fig. 3b). It is should be mentioned that in the earlier work [31] the initial atomic configuration of the surface layer was taken to be 2O–4Al2O–2O, whereas all aluminum and oxygen atoms belonging to the same layer were considered in the present work. Atoms in this layer relax quite differently, but they show an equal displacement of pairs of atoms. In the first layer, the sequence of atoms is 2O–2O–2O–2Al–2Al that is in line with [31], in spite of the difference in the initial configuration. It is necessary to emphasize that the initial splitting of both pairs of oxygen atoms was 0.11–0.12 Å in [31]. In the results shown in Fig. 3b the relaxation of oxygen atoms in the surface layer varies from +5.0 to 0.1%; that of aluminum atoms from 0.7 to +1.9%. The distance between oxygen atoms in the first layer reaches 0.41 Å (0.35 Å in [31]) whereas the splitting between Al layers of 0.10 Å (0.07 Å in [31]) is smaller. The distance between the first and second split atomic layers is 0.79 Å, which is somewhat smaller than the literature value 0.92 Å [31]. The second split layer has the following sequence of atoms: 2Al–2O–2Al–2O–2O. Upper aluminum atoms are shifted relative to the nearest oxygen atoms by 0.29 Å (0.20 Å in [31]), and the distance between the first pair of oxygen atoms and the second pair of aluminum atoms is 0.08 Å (0.01 Å in [31]). The last two pairs of oxygen atoms are split insignificantly: 0.01 Å versus 0.02 Å in [31]. In total, the splitting of the second layer is 0.41 Å (0.23 Å [31]) but the splitting of the third layer is almost twice as small (0.22 Å), the third layer being 1.25 Å distant from the second layer (1.21 Å in [31]). Thus, the atomic structure calculated for the (-1100) surface is in good agreement with the results of [31], in which the method of linear combination of atomic orbitals implemented in the CRYSTAL2003 code was used. For the (11-20) surface, the sequence of layers is the following: 3O–6O–6Al–6Al–6O–3O (Fig. 3c). The splitting in the two first oxygen layers is 0.30–0.32 Å but it decreases twice (0.14 Å) in the third oxygen layer from the surface. In the subsurface aluminum layer, the splitting is only 0.08 Å. For this surface, the relaxation is much less compared with the (0001) surface. The interlayer distances in the first surface layers estimated from the averaged positions of atoms in the layer change by 3.6%, 4.0%, and 9.3%. In [31], the relaxations were larger: 14%, 30%, and 7%. Figure 3d shows the total surface densities of electronic states (DOS) for the three surfaces studied. The peculiarity of the surface DOSs is the narrower forbidden band compared with that in the bulk (4.51 eV for Al-terminated (0001) surface and 5.92 eV in bulk, respectively), which can be attributed to the redistribution of the electronic structure and penetration of the surface states into this gap. In addition, the density of electronic states for the O-terminated (0001) surface has small narrow peaks which are split from the top of both valence bands. These peaks are centered at 16 eV and ~0 eV, respectively. These states originate from the surface oxygen atom. The widening of both valence bands compared with the Al-terminated (0001)

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surface is also due to surface states of oxygen. A slight shift of the center of gravity of oxygen subbands is observed for the (-1100) and (11-20) surfaces in comparison with that for the O-terminated (0001) surface. Additionally, a narrowing of the forbidden gap occurs. The surface energy of Al-terminated Al2O3(0001) is found to be 1.68 J/m2 as was mentioned earlier, which is also consistent with the available theoretical data (1.50 [34], 1.54 [35], 1.85 [32], 2.01 [21], and 2.15 J/m2 [14]) and an experimental value of 1.69 J/m2 [36]. For the nonstoichiometric O-terminated surface, the surface energy depends on the chemical potential of oxygen. The calculations show that the surface energies of the Al-terminated and O-terminated Al2O3(0001) surfaces differ considerably (Δσ = 2.12 J/m2 [15]) with the chemical potentials of oxygen in the solid and gas phases being the same. The value of the surface energy obtained in [14] for the O-terminated (0001) surface is also almost twice as high as for the Al-terminated one. The surface energies are 2.45 J/m2 for the (-1100) surface (2.44 J/m2 [31], 2.23 J/m2 [37]) and 2.17 J/m2 for the (11-20) surface (2.39 J/m2 [31], 2.50 J/m2 [37]). Thus, the calculation of surface energies confirmed that the Al-terminated Al2O3(0001) surface is the most stable, which is in agreement with the conclusion in [31]. In general, the results for the atomic and electronic structures presented in this section are in good agreement with the available theoretical and experimental data.

4

Work of Separation

4.1

Interfaces with bcc Metals

It is well known that strong adhesion exists in case of bcc Nb(111) film on the O-terminated Al2O3(0001) surface. Therefore, this system was intensively investigated in the last 2 decades, in particular in the papers [5, 13, 14, 38] and references therein. In the following, the adhesion of Me–Al2O3(0001) interfaces will be discussed for the bcc metals Nb, Mo, Ta, and W. In case of a bcc metal film the configuration in which the metal atoms occupy aluminum positions is of most interest on the O-terminated Al2O3(0001) surface. As can be seen in Fig. 1a, the metal atom of the first layer is located in the centers of the oxygen triangle (H-hollow configuration) in this case and interacts simultaneously with three surface oxygen atoms which causes strong adhesion at the interface. This configuration is found to be the most stable among the possible metal film configurations on the oxide surface, which was confirmed by the calculations for the series of bcc metals listed in Table 1. It is seen that the separation energy strongly depends on the film configuration. In the other considered configurations (Fig. 1b,c) the metal atom is located above the third aluminum layer (Al3) or above the fourth aluminum layer (Al4). Note that the first layer of Al atoms is absent in the case of the O-terminated interface. Nevertheless, a common denotation of atomic layers is used for both terminations of the Al2O3(0001) surface. The obtained values of Wsep in the case of the Me–O cleavage plane for Al3 and Al4 configurations is noticeably lower than that for the H-hollow

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Table 1 Work of separation (J/m2) at the Me(111)–Al2O3(0001) interface with bcc metals Interface Configuration Nb Mo Ta W

Me–Al2O3(0001)O H-hollow 9.80, 9.26 [39], 9.8 [13], 10.6 [14] 9.38, 9.58 [39] 10.34, 10.47 [39] 9.41, 9.76 [39]

Al3-top 8.49

Al4-top 8.69

8.93 8.89 8.89

8.33 9.19 8.48

Me–Al2O3(0001)Al Al-next 2.13 2.8 [13], 2.6 [14] 2.04 2.15 2.44

configuration, but they are much higher than that for the Al-terminated interface with Me atom in the Al-next position (Fig. 1d) and the Me–Al cleavage plane (Table 1). Only the Wsep value for the most preferred configuration of bcc metal films on the Al-terminated interface (Al-next) is given in Table 1. It should be noted that atoms of the second layer of the Me film in the case of the Al3 configuration tend to occupy the aluminum atom position (Al1) on the oxide surface and the atoms of the first and second metal layers lie almost in the same layer because the distance between them decreases considerably to ~0.1 Å in case of Nb and Ta. For the interfaces with Mo and W films, the atoms of the second layer are located by 0.31 Å and 0.27 Å lower than the atoms of the initially first metal layer. For this Al3 configuration, the interface distance between the metal and oxygen layers is about 0.2 Å larger than the value calculated for the most stable H-hollow configuration and correspondingly less adhesion at the interface is obtained. In the case of the Al4 configuration, atoms of the metal film form bonds with both oxygen and aluminum, which could enhance the binding of films with the surface. On the other hand, larger interfacial distances between oxygen and metal layers and between the first and the second layers of the metal film correlate with a decrease of adhesion for this configuration compared to the H-hollow configuration. As follows from Table 1 the Al4 configuration is energetically more favorable than the Al3 one for Nb and Ta films, whereas the opposite relation is observed for Mo and W films. This is a result of the strong rearrangement of atoms in the first layers of the metal film during interface optimization. In general, in the model with two interfaces and vacuum, Wsep values obtained for the Nb–(Al2O3)O and Nb–(Al2O3)Al systems are in good agreement with earlier results for the model without vacuum [38] and also with the results reported in [5, 13, 14, 34]. The work of separation for Nb–(Al2O3)O is by 0.54 J/m2 larger than the value obtained in [39] but Wsep for Ta–(Al2O3)O of 10.47 J/m2 is in good agreement with the value of [39], where the model of the interface without vacuum and the SIESTA code was used. It is necessary to mention that the optimization of the bcc Me–Al2O3(0001) interface structures is accompanied by a strong increase of the distance between Al and O atoms but it converges rapidly to the bulk value in the oxide (Table 2). The distances between aluminum atoms in the subinterface region remain 10–20% smaller than that in the bulk for all considered configurations of the Me–Al2O3(0001) interfaces. The first interlayer distance (Al1-O2) in case of the Al-terminated oxide surface increases also substantially but it remains smaller than

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Table 2 Interlayer distances dij (Å) and their relative changes Ddij /d0ij (%) in case of H-hollow configuration of Me–(Al2O3)O and the Al4 configuration of Me–(Al2O3)Al interfaces. In brackets the values from [13, 14] are given, d 0ij is the interlayer distance of corresponding bulk material

Atomic layers d ij Me 2 -Me 1 Me 1 -O 2 O 2 -Al 3 Al 3 -Al 4 Al 4 -O 5 O 5 -Al 6

0.63 1.14 0.91 0.39 0.91 0.85

Nb/(Al 2 O 3 ) O

Mo /(Al 2 O 3 ) O

Ta/(Al 2 O 3 ) O

Δ d ij d ij0

d ij

Δ d ij d ij0

d ij

Δ d ij d ij0

d ij

Δ d ij d ij0

0.27 1.31 0.90 0.42 0.89 0.85

-70.3 – 7.6 -14.5 6.6 1.3

0.58 1.18 0.90 0.41 0.90 0.85

-39.0 – 7.3 -17.0 7.8 1.9

0.22 1.36 0.89 0.45 0.88 0.85

-76.1 – 5.9 -9.6 5.0 1.4

a

b

- 34.4 ( - 26.3 , - 35.4 ) – 8.8 (10.1a , 8.1b) - 20.1 ( - 17.2a , - 19.6b) 8.7 (7.1a , 7.7b) 1.7 ( -0.9a , 2.4b) Nb/(Al 2 O 3 ) Al

Atomic layers d ij

Δ d ij d a

Mo /(Al 2 O 3 ) Al

0 ij

d ij b

Me 2 -Me 1 0.90 - 6.3 (3.6 , 5.3 ) 0.98 – Me 1 -Al 1 Al 1 -O 2 0.80 - 4.65 ( -2.2 a , -7.5 b ) 7.28 (8.5 a , 4.7 b ) O 2 -Al 3 0.90 Al 3 -Al 4 0.40 - 19.35 ( - 12.9 a , - 17.5 b ) Al 4 -O 5 9.18 (8.7 a , 7.3 b ) 0.91 2.19 (1.7 a , 1.3 b ) O 5 -Al 6 0.86 a b Note: – Ref. [13], – Ref. [14].

0.97 0.97 0.79 0.88 0.41 0.91 0.86

Δ d ij d

6.6 – -6.0 5.6 -16.8 8.6 2.4

0 ij

Ta/(Al 2 O 3 ) Al d ij 0.95 0.94 0.82 0.89 0.42 0.90 0.86

Δ d ij d

2.3 – -2.2 6.1 -14.9 7.6 2.2

0 ij

W/(Al 2 O 3 ) O

W/(Al 2 O 3 ) Al d ij

Δ d ij d ij0

0.93 0.98 0.80 0.87 0.43 0.90 0.86

1.1 – -4.8 4.4 -13.1 7.3 2.4

that in the bulk material, that is, the negative relaxation decreases (in absolute value) from 83.4% for the pure surface to 2.2% to –6.7% at the interface with bcc metals. For the O-terminated surface, the distance (O2–Al3) increases by 5.9% to 8.8%. Note that the tendency for a decrease in the interlayer distance (Me2–Me1) at the O-terminated interface with niobium is in good agreement with the available theoretical data. In particular, a calculated value of 34.4% (Nb2–Nb1) for the case of the O-terminated interface is in good agreement with a value of 35.4% obtained in [14] using the full-potential linearized augmented plane-wave method. For the most stable H-hollow configuration on the O-terminated interface, this distance is lowered significantly for all considered bcc metals. The distance between the first niobium layer and oxygen is also consistent with the results obtained for Nb in [13, 14]. In general, the relaxation at the interfaces considerably depends on the model of interface used in the calculations. There is a problem to determine the interfacial volume correctly in the model without vacuum because it can be over- or underestimated. This problem is eliminated in the model with vacuum between metal films, which was applied in this work. Although an increase in the number of layers in the metal film affects the interlayer distances in the film itself [15, 38], the structural parameters of oxide and interfacial layers change only slightly. Thus, the interaction of alumina with the metal films promotes the recovery of bulk distances in the subinterface region of the oxide. More details can be found in [40]. It should be noted that in the case where the cleavage plane is between the first and second metallic layers and, therefore, the uppermost layer of the O-terminated oxide becomes metallic (since Me substitutes Al), the work of separation is about half as much. The obtained value for Nb1-Nb2 cleavage plane of ~4.0 J/m2 agrees

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well with the value 3.8 J/m2 obtained in [13] but it is larger than in case of the separation of the metal film from the Al-terminated oxide. The latter can be connected to the fact that the metallic bonding in the Nb film is caused mainly by the Nb d-orbitals and, hence, is stronger than that between the Nb d-orbitals and the Al s,p-orbitals. The chemical bonding at the bcc metal–(Al2O3)O interface is associated with strong hybridization of the Me d-states with the O 2p–orbitals. It should be noted that the binding of a Me monolayer to the oxide surface is stronger than that of a thick metal film. The values of Wsep in the cases of the oxygen and aluminum uppermost layers of the oxide are 12.5 J/m2 (10.8 J/m2 [13]) and 3.57 J/m2, respectively in case of Nb film. A smaller value of Wsep was calculated in case of a Nb monolayer at the Al3-site on the O-terminated surface (10.4 J/m2). This fact can be explained by the stronger ionic bonding in the case of the Nb monolayer. The considerable drop of the adhesion at the Al-terminated interface in case of bcc metals can be explained by a decrease of ionic bonds of interface metal atoms with subsurface oxygen atoms because of increased interlayer distances between the Me and oxygen layers (Table 2). Therefore, bonding at the interface becomes largely of metallic type. In addition, several words should be said about V and Cr which are isoelectronic to bcc metals considered in this work. In order to match V and Cr films with the crystalline structure of the oxide, in-plane strain must be larger than in the case of the other bcc metals considered. This leads to a noticeable rearrangement of their crystal structures and as a result the metal films cannot be considered as bcc. For these materials calculations with misfit dislocations and reduced strain would be more convenient. However, the integrity of the crystal structure, as well as the sequence of atomic layers, is not broken for deformed Nb, Mo, Ta, and W films. Finally, the work of separation was calculated for two other orientations of Nb–Al2O3 interfaces in order to confirm that strong adhesion can be achieved in the case of Me(111)–Al2O3(0001)O only. It is well known that the Nb–α-Al2O3 interface can be described by the following orientation relationships: [0001]||[111], [-1100]||[11-2], and [11-20]||[1-10]. According to experimental data [41], Nb grows almost unstrained on any surface of the oxide and produces ordered crystalline structures. It is evident that the surface orientation of the oxide can influence considerably the bonding mechanisms of metal films with oxide. Some estimations were done in [39] for interfaces of Me with ZrO2, HfO2, and MgO. In case of the Nb (11-2)–Al2O3(-1120) and Nb(1-10)–Al2O3(11-20) interfaces, the computational cells contain a larger number of atoms than for the Nb(111)–Al2O3(0001) interface (100 and 184, respectively). The metallic films were allowed to relax on one oxide surface only whereas atoms on the opposite surface were fixed in equilibrium positions for the bulk material. For the Nb(11-2)–Al2O3(-1100) interface, ten niobium layers (30 atoms in total) were taken into account; for the Nb (1-10)–Al2O3(11-20) interface, four niobium layers with 64 atoms in total. On the Al2O3(-1100) surface, a doubled lattice of Nb(11-2) was considered, and on the Al2O3(11-20) surface, 2
4 Nb(1-10) surface cells were considered to match the crystal lattices.

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Table 3 Work of separation (J/m2) at the differently oriented Nb–Al2O3 interfaces and some structural parameters (Å): d, interatomic distance; dz, interlayer distance Configuration Nb(111)–Al2O3(0001)O Nb–Al2O3(0001)Al Nb(11-2)–Al2O3(-1100) Nb(1-10)–Al2O3(11-20)

Wsep 9.80 9.91 [15] 2.13 1.86 1.37

d(Nb1-O2) 2.11 2.23[15] 2.32 2.01–2.28 2.03–2.25

d(Nb1-Al) 3.72 3.68 [15] 2.92 2.68 2.96

dz(Nb1-O2/Al1) 1.14 1.16 [15] 1.00 1.78 1.83

Fig. 4 Distribution of charge density difference (Δρ) at the metal–oxide interface in the plane passing through oxygen atoms in the case of the (a) Nb(111)–Al2O3(0001)O, (b) Nb (11-2)–Al2O3(-1100) and (c) Nb(1-10)–Al2O3(11-20) interfaces

It is seen from Table 3 that the work of separation obtained for the Nb (11-2)–Al2O3(-1100) and Nb(1-10)–Al2O3(11-20) interfaces is much lower than that for the Nb–Al2O3(0001)O interface but even smaller in comparison with the value obtained for Nb–Al2O3(0001)Al. The explanation of this fact can be understood from Fig. 4. It is seen that a major part of niobium atoms at the Nb (11-2)–Al2O3(-1100) and Nb(1-10)–Al2O3(11-20) interfaces are separated by a distance exceeding the sum of the atomic (or covalent) radii of niobium and oxygen. As a result, these atoms cannot contribute to bonding at the interface. The bond length in some Nb–O pairs varies from 2.01 to 2.28 Å at the Nb(11-2)–Al2O3(-1100) interface and from 2.03 to 2.25 Å at the Nb(1-10)–Al2O3(11-20) interface. Thus, the bond lengths at all considered interfaces including Nb(111)–Al2O3(0001) are nearly the same. Therefore, the considerable reduction of the adhesion at these interfaces can be explained by a much lower number of Nb–O pairs involved in bonding compared with the Nb(111)–Al2O3(0001) interface, where all interfacial atoms contribute to bonding. This conclusion is also confirmed by the charge density difference distribution (Δρ = ρMe + ρoxide – ρMe/oxide) for these interfaces presented in Fig. 4. The charge density distribution demonstrates a considerably weaker bonding between interface atoms in comparison with the bonding of metals on the basal plane of oxide. In addition, a charge transfer of ~0.4e from the interfacial Nb atom to oxygen using Bader approach [42] was obtained in case of the Nb–Al2O3(0001)O interface whereas ~0.12e was obtained for the Nb (1-10)–Al2O3(11-20) interface. Moreover, the lattice mismatch between the niobium and oxygen surface structures at the (0001) interface is minor but it is considerably

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larger at the other interfaces [43]. Thus, the obtained results demonstrate that larger adhesion energies can be obtained for bcc metal films of group V–VI at the O-terminated Al2O3(0001) interface.

4.2

Interfaces with fcc Metals

In this section the trends in the works of separation for interfaces with fcc metal films are demonstrated. The overview of the results for Ni, Pd, Pt, Cu, Ag, and Au given in Table 4 shows clearly that substantially smaller values of Wsep are obtained in comparison with that obtained for bcc metal–Al2O3(0001) interfaces (Table 1) which was intensively studied in earlier papers [5, 13–15, 34, 38, 39, 44]. This trend is observed for all oxide surface terminations. It is seen in Table 4 that the highest values of Wsep were obtained in the case of fcc metal films on the O-terminated Al2O3(0001) surface, which confirm the importance of the ionic contribution in the chemical bonding for strong adhesion at metal–oxide interfaces as was shown in earlier papers [5, 7, 13, 38, 39, 45]. The considered configurations Table 4 Work of separation (J/m2) for the most stable Me(111)–Al2O3(0001) interfaces. The configurations of the fcc metal films are given in the brackets Interface Ni–Al2O3 Theory [46] Experiment [47, 48] Pd–Al2O3 Theory [44] Pt–Al2O3 Theory [33] Theory [44] Cu–Al2O3 Theory [19] Theory [21] Theory [44] Theory [46] Experiment [47, 49] Ag–Al2O3 Theory [11] Theory [22] Theory [33] Theory [44] Theory [50] Experiment [49] Au–Al2O3 Theory [11] Theory [44] Experiment [47, 49]

Al-termination 1.24 (O-top) 1.30 (O-top) 1.10–1.11 0.90 (O-top) 0.70 (O-top, Al-top) 0.74 (O-top) 0.57 (Al-top) 0.60 (Al-top) 0.47 (O-top) 1.02a 0.9 (O-top) 0.9 (O-top) 0.58 (O-top) 0.44 0.41 (Al-top) 0.33 (Al-top) 0.672 (H) 0.36 (Al-top) 0.6 (Al-top) 0.27 (H) 0.32 0.17 (Al-top) 0.29 (Al-top) 0.4 (Al-top) 0.277

Note: a – no configuration is given

Al2-termination 3.91 (H) 3.78 (H)

O-termination 5.82 (Al-top) 6.84 (Al-top)

4.66 (H)

4.80 (O-top)

4.84 (H)

5.23 (O-top)

2.50 (H)

5.79 (Al-top) 5.62a

2.66 (H)

5.94 (H)

1.80 (H) 1.83 (H) 0.679 (H)

4.03 (Al-top) 3.93 (Al-top) 4.649 (Al-top)

4.7 (H) 1.92 (H) 2.31 (H)

2.64 (Al-top) 2.78 (Al-top)

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of fcc metal films on the O-terminated Al2O3(0001) surface are shown in Fig. 2. Among the considered fcc metals the maximum values of Wsep were obtained for interfaces with nickel or copper. As seen in Table 4 the work of separation of the nickel film from the O-terminated Al2O3(0001) surface is lower by ~1 J/m2 than that obtained in [46], although the results for other Ni–(Al2O3)Al and Ni–(Al2O3)Al2 interfaces are in a good agreement with results of [46]. It should be noted that Wsep is somewhat lower in case of metals which are isoelectronic to Ni (Pd or Pt). The O-top configuration of metal films (Fig. 2b) is found to be more stable for both Ni and Pt on the O-terminated interface whereas the Al-top configuration (Fig. 2a) is stable for other considered metals. A significantly greater reduction of Wsep is found for the second isoelectronic metal series (Cu, Ag, Au), where the Al-top configuration of the metal film at the O-terminated interface is more stable among the considered configurations. This is in agreement with another theoretical study [11], whereas the H-hollow configuration was found to be most stable for Cu–(Al2O3)O in [46] and for Ag–(Al2O3)O in [50]. It should be noted that the authors of [46] emphasize the insignificant energy difference between the H- and O-top configurations. In our calculation, the difference between these configurations is ~0.20 eV/atom but it is insignificant ~0.04 eV/atom for a monolayer Cu coverage. The lowest values of Wsep were obtained for metal–oxide interfaces in case of the Al-terminated oxide interface. The results for the first series of metals (Ni, Pd, and Pt) are approximately twice as much as the values for the second series of metals, which is caused by almost filled d-bands for the second metal series. The O-top configuration is more preferable for metals of the first series and also for Cu. The energy gain in the O-top configuration for the copper film is 0.13 eV/atom compared to the Al-top configuration whereas a greater difference of 0.45 eV/atom was found in the case of the nickel film. Although in [22, 50] the H-configuration was found to be most stable for the Ag(111) film, the present calculations show that the Al-top configuration is most stable among the three considered configurations for both Ag and Au films. This conclusion is in agreement with earlier results [11, 44]. It should be mentioned that Pt in the monolayer coverage also prefers the Al-top configuration which agrees with result of [33]. In general, the results shown in Table 1 are in good agreement with experiments [47, 49]. The work of separation in the case of the Al-enriched interface (the oxide surface is terminated by a double layer of aluminum) has intermediate values, but for Pd, Pt, and Au they are close to the results obtained for the O-terminated interface. All metals at this interface prefer the H-hollow configuration, and this agrees with results for nickel, copper, silver, and gold films [11, 22, 46]. However, the difference between the energies of H-hollow and O-top configurations is small (it is 0.02 and 0.05 eV for silver and copper films, respectively). In general, the further filling of the d-band of metals from the second series favors weakening of the chemical bonding at all studied interfaces compared to the corresponding interfaces with metals of the first series. It is necessary to point out that the interfacial metal layer in the case of the Al-top configuration of films at both Al- and O-terminated oxide is split because some

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atoms of the first metal layer relax to positions of Al-atoms on the oxide surface. In the case of the Al-terminated interface, this splitting for silver and gold is 0.88 Å and 0.66 Å, respectively. The second metal layer from the interface is approximately planar. Metal atoms of the Al-terminated interface tend to keep the same interatomic distances as in bulk metals. But in case of interfacial Al and O atoms, which are located almost in one layer at the corresponding oxide surface, a substantial increase of interlayer distances is obtained. The splitting in the first metal layer in case of the Al-top configuration on the O-terminated interface is 0.46 Å and 0.57 Å for Ni and Cu, respectively which is significantly less than the values obtained for Ag and Au (0.82 Å and 1.06 Å, respectively). In case of the Al-enriched interface a planar interfacial metal layer was obtained for all considered metals. An overview of the values of interlayer distances for metals of the first series is given in Table 5. Table 5 Interlayer distances and the buckling of the interfacial metal layer (dz in Å) for stable Me (111)–Al2O3(0001) interfaces (Ме = Ni, Pd, Pt). In brackets the percental values of relaxation relative to bulk values in oxide are given Interlayer distances

Ni/(Al2O3)Al O-top

Ni/(Al2O3)Al2 H

Ni/(Al2O3)O Al-top

dz(Me) Me2-Me1 Me1-Al0 (Al1, O2) Al0-Al1 Al1-O2 O2-Al3 Al3-Al4 Al4-O5 O5-Al6

– 1.75 1.41 – 0.66 (-21.5) 0.91 (8.2) 0.28 (-41.8) 0.97 (15.4) 0.84 (0.2)

– 1.91 1.76 0.36 (-26.0) 0.95 (13.0) 0.84 (0.2) 0.52 (7.8) 0.85 (1.4) 0.83 (-1.7)

0.46 1.95 1.32 – – 1.00 (19.6) 0.27 (-44.1) 0.97 (15.1) 0.84 (-0.3)

Interlayer distances

Pd/(Al2O3)Al O-top

Pd/(Al2O3)Al2 H

Pd/(Al2O3)O O-top

dz(Me) Me2-Me1 Me1-Al0 (Al1, O2) Al0-Al1 Al1-O2 O2-Al3 Al3-Al4 Al4-O5 O5-Al6 Interlayer distances

– 2.30 2.05 – 0.49 (-42.0) 0.90 (7.4) 0.28 (-42.6) 1.00 (18.6) 0.86 (1.9) Pt/(Al2O3)Al O-top

– 2.41 1.85 0.28 (-42.0) 0.97 (17.5) 0.85 (1.0) 0.54 (11.0) 0.84 (0.0) 0.86 (2.1) Pt/(Al2O3)Al2 H

– 2.31 1.96 – – 0.91 (8.8) 0.27 (-44.1) 0.99 (17.9) 0.84 (-0.5) Pt/(Al2O3)O O-top

dz(Me) Me2-Me1 Me1-Al0 (Al1, O2) Al0-Al1 Al1-O2 O2-Al3 Al3-Al4 Al4-O5 O5-Al6

– 2.38 2.15 – 0.51 (-39.6) 0.89 (6.5) 0.27 (-44.4) 1.00 (18.6) 0.85 (1.6)

– 2.46 1.87 0.26 (-46.7) 0.99 (17.8) 0.83 (-1.0) 0.56 (14.7) 0.83 (-1.2) 0.86 (1.8)

– 2.33 1.97 – – 0.92 (9.5) 0.26 (-46.4) 0.99 (17.7) 0.83 (-1.1)

Note: in the case of the split Me1 layer, average values are given.

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In general, the work of separation may be affected by impurities, vacancies, and dislocations which are present at real interfaces and these influences have to be taken into account in comparisons with experimental and theoretical results. In the present ab initio investigations dislocations were not considered because a unit cell including misfit dislocations is beyond the computational limits of this method. In order to obtain greater insight into the nature of chemical bonding at the metal–alumina interfaces, point defects in the interfacial layers were included. A vacancy was simulated by removing one of the oxygen or metal atoms from the interfacial layers of the oxide or metal film. Obtained results demonstrate that the influence of O-vacancies on the adhesion is most pronounced. The work of separation decreases from 4.03 J/m2 to 2.01 J/m2 and from 5.79 J/m2 to 2.49 J/m2 for Ag–Al2O3(0001)O and Cu–Al2O3(0001)O interfaces, respectively. A smaller decrease to the values of 3.27 J/m2 and 4.93 J/m2 was calculated in case of Ag and Cu vacancies in the interfacial layer of the O-terminated interface. A strong influence of oxygen vacancies was also demonstrated in [51] at the Ni(Fe)–ZrO2(001)O interface. The formation of oxygen vacancies in the surface layer of the oxide substantially decreases the work of separation from 6.25 J/m2 (Ni in the next Zr-site) and 7.35 J/m2 (Ni in the O-bridge site) to 1.35 J/m2 and 2.24 J/m2 in the case of the O-terminated interface. A change of Wsep from 8.40 J/m2 (9.49 J/m2) to 2.47 J/m2 (3.08 J/m2) was calculated for the Fe/ZrO2(001)O interface for corresponding configurations of metal films. However, a minor change of Wsep (~0.1–0.2 J/m2) was found for both Me–ZrO2(001)Zr interfaces when an oxygen vacancy exists in the second layer of the oxide substrate. The partial breaking of the Me–O bonds at the Me–ZrO2(001)O interface leads to a drastic decrease of the binding energy while at the Me–ZrO2(001)Zr interface subsurface oxygen vacancies do not affect the work of separation significantly. The electronic properties of metal–oxide interfaces which allow to understand the reasons of such changes of the work of separation will be discussed in Sect. 6.

5

Calculations of Stress–Separation Relationships

In order to calculate stress–separation relationships which can be used for simulations at larger length scales such as the calculations of interface toughness described in [9] the model with one interface was applied. As in earlier work [18] all atoms far from the interface are kept fixed in their bulk positions whereas the atoms of two Me layers as well as of four Al and two O layers closest to the interface are permitted to relax. This model represents brittle fracture at an interface between bulk materials. It must be noted that the work of separation obtained with this model is slightly different compared to the values listed in Tables 1 and 4. This is partly due to the fact that a model with two interfaces was used for the calculations presented in Tables 1 and 4 and the numbers of fixed atoms were also different. Additionally the work of separation depends on the number of layers used for the metal film as well as

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Fig. 5 Energy displacement curves for the most preferred Me(111) film configurations: (a) Nb–Al2O3; (b) Al–Al2O3, Cu–Al2O3, and Ag–Al2O3. Structures 1 and 2 show the relaxed Al-top configurations for states 1 and 2 in the curves for Al and Ag

the stacking sequence of the layers. The stacking sequence determines the locations of the second and third metal layers from the interface. In case of Nb–Al2O3(0001)Al it was found that the work of separation is about 0.3 J/m2 higher for the four layer film at the Al-next position (stacking Al-top, H-Hollow) compared to the same configuration with a film of eight metal layers. The comparison of the results obtained with the same four and eight layer films but different stacking (H-Hollow, Al-top) reveals that the work of separation is about 0.7 J/m2 higher in the case of the eight layer film [23]. The influence of the stacking and the number of interfaces in the model is less pronounced in the case of fcc–alumina interfaces. The changes of work of separation due to different stacking is around 0.02 J/m2 and the decrease obtained for the model with one interface is about 0.05 J/m2 [23]. The curves showing energy changes due to normal separation are visualized in Fig. 5 for the most stable configurations of Nb, Cu, Al and Ag metal films on the Al-terminated interface. The work of separation values corresponding to the minima are 2.62 J/m2, 0.71 J/m2, 0.68 J/m2 and 0.36 J/m2, respectively. Whereas the energy–displacement relationships of Cu and Nb are continuous, the results obtained for Al and Ag show a jump at a separation of 0.6 Å. The reason is the splitting of the first metal layer as discussed in Sect. 4 caused by the relaxation of one metal atom to an Al-position on the oxide surface. This relaxation occurs for separations below 0.6 Å whereas a planar interfacial metal layer exists at larger separations. The corresponding structures are shown in Fig. 5. According to Eq. 2 the stress–separation relationships are calculated from the slopes of the energy curves. The results shown in Fig. 6 demonstrate that the highest ideal tensile strength of 26 GPa is obtained for the Nb–alumina interface (Al-next configuration) at a separation of ~1 Å. The high maximum stress in case of Nb is connected with a small interlayer distance between the metal atom of the first layer and interfacial Al or O atoms of the oxide. The Me1–Al1 distance is 0.98 Å in case of the Nb film, but it is twice more (2.03 Å) for the Cu film. Since the first metal layer is split in case of the other metals it is in the range of 0.93–2.51 Å for Al and 1.73–2.84 Å for Ag. The same trend is found for the Me1–O1 interlayer distance which is 1.76 Å and 2.44 Å for Nb and Cu, respectively. Ideal tensile strengths are much lower for the considered fcc metal–alumina interfaces, they are 7.1 GPa, 5.5 GPa, and

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Fig. 6 Stress displacement relations for the most preferred Me(111) film configurations: (a) Nb–Al2O3; (b) Al–Al2O3, Cu–Al2O3, and Ag–Al2O3

4.2 GPa at separations of 0.7 Å, 0.6 Å, and 0.6 Å for Cu, Al, and Ag, respectively. It can be seen that the order of the tensile strengths is the same as for the corresponding work of separation values whereas the separations are only slightly different from each other. The stress–displacement relationships presented in this section can be used to adjust parameters of fracture models for larger length scales [8, 9, 26, 27]. They are valid for cleavage at the interfacial plane of defect free bulk materials. It is possible to take into account the influence of vacancies or impurities by performing the corresponding ab initio calculations with these defects as described in Sect. 4. In general, the computational cell becomes too large for ab initio investigations with misfit dislocations at these interfaces. For this task classical molecular dynamics simulations which are applied in Sect. 8 are a suitable method.

6

Electronic Properties

6.1

Bcc Metal–Oxide Interface

In order to understand chemical bonding at the considered metal–oxide interfaces a detailed analysis of the electronic characteristics such as charge densities difference distribution and densities of states, N(E), is used. The charge densities difference distributions Δρ(r) for some bcc Me(111)–Al2O3(0001) systems in planes which are perpendicular to the interface and pass through the interfacial atoms are visualized in Fig. 7. In the case of the O-terminated interface it is seen that the metal atoms of both the first and second layers from the interface strongly interact with surface oxygen atoms. As was mentioned earlier the distance between these metal atoms in the film is shortened considerably at the interface which enhances the contribution of the metal atom of the second layer to the bonding at this interface. The latter leads to strengthening of bonding at the interface. Fig. 7a,b reflects the charge transfer from the metal film to the oxygen atoms: a region of charge depletion is visible near the

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Fig. 7 Distribution of the charge density difference Δρ for Ta (a, b) and W (c, d) films on the O- as well as (e, f) and (g, h) Al-terminated Me–Al2O3(0001) interface: the preferred H-hollow and Al next positions are shown, respectively. For each interface, two cross sections are given in order to illustrate bonding between atoms involved in the interaction

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Table 6 Bader charge analysis of bcc metal–Al2O3(0001) interfaces Interface Atom Me3 Me2 Me1 O2 Al3 Al4 O5 Al-bulk O-bulk

Me(111)–Al2O3(0001)O Nb Mo Ta 0.24 0.27 0.31 0.49 0.51 0.40 1.21 0.74 1.06 1.42 1.34 1.40 2.47 2.47 2.47 2.47 2.47 2.47 1.66 1.64 1.65 2.47 2.47 2.47 1.65 1.65 1.65

W 0.46 0.56 0.75 1.37 2.47 2.47 1.65 2.47 1.65

Me(111)–Al2O3(0001)Al Atom Nb Mo Me2 0.18 0.15 Me1 0.36 0.25 Al1 1.77 2.08 O2 1.60 1.57 Al3 2.47 2.47 Al4 2.47 2.47 O5 1.65 1.65 Al-bulk 2.47 2.47 O-bulk 1.65 1.65

Ta 0.08 0.37 1.84 1.58 2.47 2.47 1.65 2.47 1.65

W 0.03 0.37 2.16 1.57. 2.47 2.47 1.65 2.47 1.65

bcc metal atom and a charge accumulation region is located around the oxygen atom. It is seen from Fig. 7a that the charge accumulation is greater around oxygen atoms in the plane crossing Me1 and O2 atoms than in the plane of Me2 and O2. This is in agreement with the different interatomic distances of 2.07 Å and 2.29 Å, respectively. The estimation of charge transfer from metal of the first layer using Bader approach shows that about ~0.74–1.2e are transferred to interfacial oxygen atoms (Table 6) whereas only ~0.4–0.6e is transferred from the Me2 atom to oxygen. In general, it can be seen from Fig. 7 that ionicity is decreased at the interface with increasing filling of the d-shell of the bcc metal: the charge depletion region near Ta atoms (Fig. 7b) is increased in comparison with that for W (Fig. 7d). This conclusion is in agreement with calculated charge transfers given in Table 6. The second Al atom from the interface donates charge as in bulk due to its negligible interaction with the Me(111) film. Thus, the ionic contribution to the chemical bond ensures strong adhesion of bcc metal films on the O-terminated interface. Noticeable is that a larger charge transfer is connected with an almost unoccupied d-shell as in bcc metals of groups V and VI. In comparison, fcc metals of the end of the d-series with completely filled electronic shell make it more difficult for oxygen to get electrons for filling its p-shell. It is noted that the ionic component is most pronounced for monolayers of bcc metals as in the case of fcc metals considered in next subsection. An increase in the number of layers of the metal film is accompanied by the enhancement of metallic bonding which leads to weakening of the ionic bonding at the interface. At the Al-terminated interface (Fig. 7e,f), the ionic bonding is noticeably weakened owing to both to an increase in the first interlayer distance (Al1–O2) in the oxide and to a strong increase in the Me1–O2 distance up to 1.70–1.80 Å (Table 2). As it is seen from Table 6 substantially less charge (~0.3–0.4 el.) is transferred to the oxygen atoms at this interface. Almost the same value 0.37 el. was calculated in [34] for the Nb–(Al2O3)Al interface using Mulliken analysis. However, the value of 0.73 el [34]. which is provided by the terminating layer of Al is significantly less than that

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calculated in the present work. It is known that the Mulliken charge is less than the Bader one. As reported in [19], the charges in bulk oxide (1.57 el. (Al) and 1.00 el. (O)) are also less than those obtained in present calculations. It is known that the Mulliken charges depend on the choice of basis set. Several words should be said about the influence of thickness of the metal film on the charge transfer. The results in Table 6 were obtained using six layers bcc metal film. In case of an eight layer Nb film the charge transfer from the Nb interfacial atom was by ~0.1 el. larger than the value in Table 6: 0.45 el. and 0.48 el. were calculated in the models with two and one interface, respectively. This difference was smaller (0.03–0.04 el.) for the charge transfers from the second Nb2 atom. In contrast, the charge transfer from Al1 atom was by 0.1–0.14 el. less for a thicker Nb film. The Al and O bulk values in oxide were in agreement with available theoretical results. It should be stressed that the model with two interfaces without vacuum was used in [19, 34] with 10 to 12 layers of Nb depending on oxide termination whereas in the present estimations six layer metal films on both oxide surfaces are considered within the model with vacuum. Nevertheless the calculated charges using both approaches can be an indicator of trends in ionicity or covalency of chemical bonding at the considered interfaces. Note that the charge transfer from metal atoms decreases considerably in the depth of the metal film. In general, it is seen from Table 6, that ionicity is decreased at the interface in comparison to bulk oxide. The interfacial Al atom loses less charge than in bulk due to the metallic interaction with the Me(111) films. Note that in addition to the metallic bond between the Me1 atom and the interfacial Al1 which is not shown in Fig. 7e,g because it is out of the visualized plane, one can see the Me3-Al1 bond (Fig. 7f,h). Thus, the metallic type of bonding between aluminum and bcc metal atoms with small ionic Me1-O2 contribution plays a significant role at the Al-terminated oxide interface. Changes in the bonding character for different compositions of the interface can also be illustrated by local electronic densities of states of interfacial atoms. Figure 8 shows local DOS of Me, Al, and O atoms that are involved in the formation of the chemical bonds and located where the redistribution of charge at the metal–oxide interfaces is maximal. It can be seen that the interaction between d-orbitals of the metal and 2p-orbitals of oxygen at the O-terminated interface is strong (Fig. 8, top panel). In this case, there are small peaks in the DOS of all considered metals centered around 20 eV, which are due to the interaction with s-states of oxygen; in addition, the states of the interfacial metal atom expand to the low energy region down to 10 eV in comparison with bulk, which is due to the interaction with oxygen 2p-orbitals. Hybridization between aluminum and metal states is much less pronounced, because aluminum atoms are at distances larger than 3 Å. In general, the local DOS for Al and O exhibit the closure of the characteristic gap present in the bulk oxide owing to the interaction with the metal, which makes a noticeable contribution in the energy range from 4 to 4 eV. In the case of the Al-terminated interface, there is weak ionic bonding of metal atoms with oxygen atoms of the second layer from the interface (Fig. 7e) in addition to the metallic bonding between Al and Me (Fig. 7f). Figure 8 (bottom panel) demonstrates that the number of states of interfacial aluminum near the Fermi energy increases, whereas the interaction

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Fig. 8 Local densities of states of the interfacial atoms at the bcc Me–Al2O3 interfaces

with oxygen weakens and a decrease in the density of electronic states of oxygen in the region around the Fermi energy occurs. The latter is connected with a decrease in the overlapping of Me and O orbitals. The analysis of the structural parameters that was given in Sect. 3 and also the charge density difference and local densities of states indicate that the increase in the work of separation for the Me–(Al2O3)O interfaces is due to the ionic-covalent contribution to the chemical bond which is important for strong adhesion at the metal–oxide interface. It must be mentioned that spin polarization calculations performed for 3d metals (V, Cr) demonstrate that the adhesion energies for these metals are also high (about 10 J/m2). At the same time, very large lateral strains, 10.33% and 14.48%, make these metals less attractive for composite materials owing to poor lattice mismatch. In addition, interesting changes in the magnetic characteristics are observed in films of these metals, because as was shown in previous work [51] for Ni-Fe/ZrO2, adhesion can depend on the magnetic state of interfacial atoms. It is worth emphasizing that oxygen vacancies can strongly affect the chemical bonding at interfaces and noticeably reduce adhesion as in the case of fcc metals. Calculations performed for Nb–(Al2O3)O demonstrate that an oxygen vacancy in the interfacial layer reduces Wsep to 4.53 J/m2 in case of a thick film and less (to 6.62 J/m2) in case of a Nb monolayer. On the other hand, if the separation occurs between the first and second Nb layers, then the work of separation increases by 0.4 J/m2, because the bonding of Nb atoms at the interface is somewhat weaker whereas the metallic

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bonding in the film is stronger in this case. In the case of the Al-terminated interface, the presence of oxygen vacancies in the subinterface layer has almost no effect on the work of separation of the interface. Since the Nb atom occupying the position of the second Al atom relaxes to the oxygen vacancy, the Nb–Al distance decreases, which leads to strengthening of bonds at the interface.

6.2

Fcc Metal–Oxide Interface

The charge densities distributions shown in Fig. 9 demonstrate the interaction between interfacial atoms of the Me(111) film and Al or O atoms of the oxide which are involved in the chemical bonding at the different interfaces. It is seen from Fig. 9a that there is accumulation of the electron charge density at the Pt1–Al1 bond. Such a distribution of isolines is typical for metallic bonding at the interface. In

Fig. 9 Charge density difference Δρ = ρMe + ρoxide – ρMe/oxide on the Me(111)–Al2O3(0001) interfaces in the plane perpendicular to the interface and passing through the interfacial atoms: (a, b) Pt(111) film in the O-top configuration on Al2O3(0001)Al, plane passes through Al1–Pt1 (a) and O2–Pt1 (b) bonds; (c) Ag(111) film in the Al-top configuration on Al2O3(0001)Al, plane passes through O2, Al1 and Ag1 atoms; (d,e) Pt–(Al2O3)Al2 interface (H-hollow configuration), plane passes through Al0-Pt1 bond (d) and Al1–Pt1 bond (e); (f) the O-top configuration at the Pt–(Al2O3)O interface, plane passes through one of three O2–Pt1 bonds

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addition to the metallic type of bonding at Pt–(Al2O3)Al, a weak ionic bonding can be seen in Fig. 9b. A region of charge depletion is seen near Pt atoms, whereas charge accumulates near O atoms. The higher stability of the O-top configuration of metal films in case of Ni, Pd, Pt, and Cu can be caused by this ionic contribution to chemical bonding. The interlayer distance of 2.15 Å between the metal and aluminum layers, which is greater than the interlayer distance in oxide of 0.51 Å, correlates with weaker Pt–Al bonds in comparison with the Al–O bonds in the oxide. The increase of the interfacial distance between Al atoms of the oxide substrate and Ag (2.23–3.12 Å) or Au (2.42–3.08 Å) atoms in the split interfacial metal layer leads to weakening of the ionic component of chemical bonding as seen clearly in Fig. 9c. The latter leads to reduction of Wsep in this case. The variation in the local density of states of nonequivalent metal atoms in the interfacial layer (Fig. 10a) leads to their polarization,

Fig. 10 Local densities of electronic states of the interface atoms at the Pt(111)–Al2O3(0001) (top panel) and at the Ag(111)–Al2O3(0001) (bottom panel) interfaces. Symbols Ag1, Ag3 correspond to atoms of the first split interfacial layer of the Ag film. An additional Al atom in case of the double Al layer termination of the oxide surface is denoted by Al0

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which, along with hybridization of the s,d-orbitals of Ag(Au) with the s,porbitals of Al, can provide the preference of the Al-top configuration. The same can be attributed to the Al(111) film as well. In general, it is possible to distinguish the following types of chemical bonding at the Al-terminated interface: the metallic Me–Al bond, polarization of atoms in the metal film, and the weak ionic Me–O bond. For an Al-enriched interface, the metal bonding is strengthened due to interaction of Pt interfacial atoms with Al atoms of the first (Fig. 9d) and second (Fig. 9e) layers from the interface but ionic contribution to bonding is almost absent. In other words, the metal bonding is strengthened for the Al-enriched interface owing to a greater overlap of the Me electronic states with states of the interfacial double layer of Al atoms that is shown in Fig. 10b. It should be noted that only one Pt atom lies in the plane shown in Fig. 9d,e but other Pt atoms are also involved in the chemical bonding by the same mechanism. It is seen that for the interface with a double Al layer the oxygen DOS shifts away from the Fermi level (EF) which reduces the overlap of Me–O states (Fig. 10b). Such trends were obtained for all considered metals. Moreover, the local densities of states of metals from the first series are located closer to the Fermi level than the metals of the second series (Fig. 10), which causes their higher chemical activity at interfaces. Finally, on the O-terminated interface, the ionic bonding caused by the charge transfer from the metal to the oxide substrate prevails (Fig. 9f) over other possible contributions. The charge densities difference distribution (Fig. 9f) demonstrates the accumulation of the charge near interfacial oxygen atoms and its depletion near Pt atoms. The polarization of metal atoms is also visible in Fig. 9. A strong hybridization of oxygen orbitals with Pt or Ag states and also a substantial shift of oxygen DOS towards EF for both Pt–(Al2O3)O and Ag–(Al2O3)O interfaces can be seen in Fig. 10c. Moreover, in this case the densities of states at the Fermi level are increased significantly in comparison with those for the Al- or Al2-terminated interfaces. Interaction of Me–O states leads to the appearance of the low-lying Me states: a small peak in the Me DOS lies exactly in the region of 20 to 16 eV where the O s-band is located (Fig. 10c). These trends are demonstrated in Fig. 11 as well, where the local DOS of the interfacial Me and O atoms are given for both series of metals. A strong overlapping of O and Me states provides higher values of adhesion at the O-terminated interface. It is necessary to point out that calculations of Me–oxide interactions in monolayer and submonolayer coverage demonstrate a more pronounced ionic character of chemical bonding at interfaces [15, 33] than in the case of thick films. Since the atoms in the metal film tend to form metallic bonds and, therefore, bonding is strengthened compared to the monolayer case, they interact less strongly with interfacial or subinterfacial oxygen atoms. This leads to weakening of the chemical bonding at the metal–alumina interface. The influence of interface defects on the charge distributions at the O-terminated Ag–Al2O3(0001) interface is presented in Fig. 12. It is seen that the partial breaking of Me–O bonds effects dramatically on the chemical bonding at this interface: the charge depletion region near the interfacial Ag layer in Fig. 12b is substantial less pronounced in comparison with the ideal interface (Fig. 12a). Figure 13a shows the significant change in the DOS of the alumina interfacial layer in the case with

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Fig. 11 Local DOS of the interfacial Me and O atoms at Ме/(Al2O3)O

Fig. 12 Charge density difference Δρ for the Al-top configuration of Ag(111)–Al2O3(0001)O interface in the plane perpendicular to the interface and passing through the interfacial atoms: (a) ideal interface; (b) interface with oxygen vacancy; (c) interface with Ag vacancy

oxygen vacancy: a shift of DOS away from the Fermi level in comparison with the ideal interfacial layer given by grey background in this figure and also a decrease of the overlapping of metal–oxygen states. In any case the DOS structure of this interfacial layer is quite different from the DOS of the bulk-like region in the oxide which reflects the interaction at the interface. Contrary to the case of the oxygen vacancy, the vacancy in the Ag film, especially in the layer far from the interface does not have a significant influence on the charge distributions (Fig. 12c) or the densities of states of oxide interfacial atoms (Fig. 13b) and the adhesion energy changes only slightly in this case. Electronic states located in the region 7 to

Fig. 13 LDOS for Ag–(Al2O3)O (a, b) and Al–(Al2O3)O (c, d) interfaces with O-vacancy in the interfacial layer (a, c) and with Me1-vacancy in the interfacial layer of the metal film (b, d). Metal atoms are located in the Al-top positions on the O-terminated interface. Symbols I and I + 1 mean the interfacial layer and second metal layer from the interface, C denotes bulk-like layer. LDOSs for ideal interfaces are shown by grey background

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–5 eV partly vanish but new states with energies of 2 eV appear resulting from the redistribution of electronic charge within the interfacial split Ag layer. The redistribution of metal atom states from the second layer can be also seen in the top panel of Fig. 13b. Almost the same trend can be seen in case of the Al–Al2O3(0001)O interface also (Fig. 13c,d). In the latter case the change in the DOS structure of the alumina interfacial layer is less pronounced because Al atoms from the metal film partly restore the bonds in the oxide. More details about structural and electronic properties of fcc metal–alumina interfaces can be found in [38, 45].

6.3

Metal–Oxide Interface Under Tensile Test

In order to understand the changes in chemical bonding during fracture processes, analyses of charge densities distributions and densities of states are performed for some of the Al-terminated interfaces for which the stress–separation relationships were calculated (Sect. 5). The metal films are located in their most preferred configurations, namely Al-next for Nb and Al-top for Ag. The charge density difference distribution calculated for the interface with Nb is visualized in Fig. 14. Remarkable is the occurrence of a charge depletion region near interfacial atoms (Fig. 14b) which increases steadily with increasing separation. At the separation corresponding to maximum stress, bond breaking takes place and finally the materials are separated completely (Fig. 14d). It can be seen from Fig. 14 that the metallic Al–Nb bonds weaken much faster than the ionic Nb–O2 bonds. The Bader analysis of charge transfer

Fig. 14 Charge density difference distribution at the Al-terminated Nb(111)–Al2O3(0001) interface for different displacements of the Nb(111) film relative to oxide: (a) 0 Å, (b) 1.14 Å, (c) 2.08 Å, (d) 3.12 Å. The change of metallic Nb-Al bonding is shown in the top panel whereas the change of Nb-O bonding is given in the bottom panel

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Fig. 15 Total charge density distribution for different displacements of metal film relative to oxide: (a, e) 0 Å, (b, f) 0.62 Å (c, g) 0.83 Å (d, h) 1.66 Å at the Al-terminated Ag(111)–Al2O3(0001) interface in different planes across Ag and Al interface atoms. Top: Across the atom located in the first split Ag layer but far from oxide. Bottom: Across the atom of the first split Ag layer which relaxed close to the oxide

from the Nb atom confirms decreasing values with increasing separation. It is 0.07 el. at the separation shown in Fig. 14b and almost zero in the situation shown in Fig. 14c. In case of the visualization of the total charge density as shown for the Ag–Al2O3 interface in Fig. 15, the charge depletion region increasing with separation can be seen very clearly. Additionally, this result demonstrates metallic-covalent bonding indicated by the accumulation of isolines between interfacial Ag atoms which are located far from the oxide in the first metal layer and Al atoms of the oxide (Fig. 15, top). An ionic contribution to bonding of the metal atom which relaxes to the Al-next position on the oxide surface in case of the Al-top configuration and the oxygen atom can be seen in Fig. 15, bottom. However, the Bader charge analysis reveals that this interaction is much weaker than in bulk oxide. A charge of 0.25e is transferred from this Ag atom. As discussed in a comparative study [23], the charge transfer is significantly larger (1.04e) in case of the Al–Al2O3(0001)Al interface. In-depth understanding can be obtained from the analysis of densities of states of interfacial atoms upon separation. In Fig. 16 (top panel), the DOS difference is presented for interfacial atoms in case of the Nb(111)–Al2O3(0001)Al system. A strong shift towards the Fermi level (EF) can be seen in the DOS of interfacial oxygen atoms at the separation corresponding to maximum stress and the fully separated system. Negative peaks of ΔN(E) centered at 8.5 eV occur for Al atoms, whereas more electronic states are present in the range 7.7 to –2.0 eV. Oxygen states with energies lower than 3.6 eV mostly disappeared but new states are located within the range of 3.6 to –1.4 eV. As a result it can be concluded that the chemical reactivity of these atoms is increased in the separated system. Caused by the interaction of interfacial Nb atoms with atoms of the oxide metal-induced gap states occur in the energy region 1.5 eV to 3.5 eV at equilibrium state. Increased

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Fig. 16 Difference of local DOS for Nb (Cu), Al and O interfacial atoms at Nb (111)–Al2O3(0001)Al (top) and Cu(111)–Al2O3(0001)Al (bottom) interfaces. The upper and lower panels correspond to the differences between DOS for critical (maximum stress) and equilibrium systems and for fully separated and equilibrium systems, respectively

separation leads to a gap of about 5.0 eV as obtained in the calculations of the alumina surface (Sect. 3). It can be seen in Fig. 16b that the states of Al interfacial atoms for the separated system are decreased considerably in this energy range. The unoccupied states of Nb atoms are changed more pronounced than their occupied states. It is seen in Fig. 16a that there is a sharp negative peak centered at 3.5 eV and the number of electronic states increases near the Fermi level. However, the density of states at EF increases only slightly. A narrowing of the band for Nb atoms which are close to the cleavage plane is visible in Fig. 16a also. The DOS of the Nb atom which is the second from the interface looks as in bulk due to a large interlayer distance in the Nb(111) film and changes insignificantly in the process of brittle cleavage, and therefore it is not shown in Fig. 16. Although the curves of the DOS differences of the Cu(111)–Al2O3(0001)Al look quite different (Fig. 16, bottom panel), the trends found in case of Nb are also observed in this case. A significant difference is that the d-band is fully occupied in this case and therefore, only s,p-electrons contribute to N(EF) which leads to the lower value in comparison with Nb. Changes of DOS for Cu interfacial atoms (Fig. 16a, bottom panel) are less pronounced in comparison with Nb ones: the depletion of states can be seen in the range 8 to –3.5 eV but new states are located from 3.5 eV up to 1.0 eV. The metal-induced gap states are less pronounced in

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case of other fcc metals also [23]. In general, the shape of the DOS of Al and O interfacial atoms for the fcc metal–Al2O3(0001)Al systems with a large separation (>3.1 Å) looks as in the case of a free oxide surface. From this analysis, it is possible to conclude that atoms in the surface region after fracture are more chemical reactive and their electrons have a higher energy than those at the interface before fracture.

7

Influence of Homogenous Strain in the Interface Plane

Due to lattice mismatch there is always homogenous strain in coherent interface structures. Ab initio calculations can be performed using an averaged lattice constant. Another approach is to keep the lattice constant of the stiffer material, namely the Al2O3 in metal–oxide interfaces. Several studies [11, 46] showed that the work of separation obtained from these two models differ significantly. As varying strains are present in experimental situations, it is important to investigate the influence of strains in the interface plane. Work of separation and tensile strengths presented in the previous sections were calculated by adjusting the metal lattice constant to the Al2O3 lattice. Based on these structures additional calculations with 3% strained in-plane lattice constants were carried out for Nb, Al, Cu, and Ag metal films. As can be seen in Fig. 17 the work of separation of Ag (111)–Al2O3(0001)Al increases by 0.20 J/m2 at a strain of 3% and decreases by

Fig. 17 Normal stress (top panel) calculated along [0001] direction at Al–(Al2O3)Al (left) and Ag–(Al2O3)Al (right) interfaces and corresponding energy–displacement curves (bottom panel) with 3% strain in the interface plane

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Fig. 18 Effect of interface plane strain on the electronic properties of Al–Al2O3: (a) total charge density; (b) local DOS of interfacial atoms

0.16 J/m2 at a strain of +3%. Corresponding values for Al and Nb are in the same range; they are 0.43 (0.26) J/m2 and 0.14 (0.29) J/m2 for Al–Al2O3 (Nb–Al2O3) at strains of 3% and +3%. In case of Cu–Al2O3 the change of the work of separation is only 0.02 J/m2 and 0.04 J/m2 at strains of 3% and +3%. In contrast to the other systems considered, a planar interfacial layer is present is Cu–Al2O3. The situation at the interfacial layer is connected to the observed strain dependency. In the cases where a metal atom relaxes to an Al-next position the decrease of interlayer distances caused by the expanded lattice impedes the relaxation to the oxide surface. Compression which leads to larger interlayer distances enhances the displacement of the metal atom towards the oxide surface and consequently the work of separation is increased. It must be stated that this effect can change the cleavage plane from the interface to the plane between first and second layers of the metal film. This can be seen in the charge density distribution shown in Fig. 18a, bottom panel where the formation of a void between atoms of the first and the second metal layer is observed. The depletion region at the interface increases in case of expansion Fig. 18a, top panel and decreases in the compressed structure. Local densities of states (Fig. 18b) are also helpful for understanding the changes of the work of separation due to strain. A shift towards the Fermi level is observed in case of compression which enhances the strength of the interface. Since the width of the pvalence band of interfacial Al and O atoms is slightly increased more overlapping states contribute to interfacial bonding. Thus, the strained condition of both metals and oxide at the interface can influence on Wsep which has to be taken into account in comparisons with experiments.

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Molecular Dynamics Simulations Using ab initio Based Polarizable Force Field

Since the ab initio calculations discussed before are limited to several hundreds of atoms it is hardly possible to investigate the influence of large scale defects such as misfit dislocations. Another point is that the calculations of the stress–displacement relationships described in Sect. 5 are static but in real systems fracture is a dynamic process. Bond breaking occurs not simultaneously in the whole interfacial area. Instead, it is often observed in experiments that cracks nucleate at defects and propagation depends strongly on types and concentrations of defects in the crack plane. A simulation method which allows calculations with large scale defects is classical molecular dynamics simulation. Within this method the trajectories of atoms are calculated by numerically solving Newton’s equations of motion. The forces between the atoms are determined from an interaction potential. The development of a suitable interaction potential for a specific system is a challenging task. In general, potentials are adjusted to match important physical properties such as work of separation or interfacial distances in case of potentials for interface structures. But it must be noted that usually not all physical properties can be described correctly by the potential. As an example of a simulation of dynamic interfacial fracture the Al-terminated Al–Al2O3 interfaces with and without misfit dislocations are considered in this section. Interaction potentials of [52, 53] are used for the Al metal film and the oxide. In the Tangney–Scandolo type potential model used for alumina the oxygen atoms are polarizable and the dipole moments are determined self-consistently from the electric field of surrounding charges and dipoles. The potential was adjusted to match ab initio calculated forces, energies, and stresses of several configurations with surfaces and strains. More details are explained in [53] and references therein. It should be noted that this potential is able to describe the alumina surfaces very well. For example, the (0001) surface energy calculated with the potential is 1.59 J/m2 which is close to the ab initio result of 1.68 J/m2 reported in Sect. 3 and also in the range of theoretical data (1.50–2.15 J/m2 [14, 21, 32, 34, 35]) and consistent with the experimental value of 1.69 J/m2 [36]. Additionally, the strong relaxation of the Al-surface layer to almost the same distance from the surface as the first oxygen layer was also observed with the potential. In case of the (1100) interface the potential is able to demonstrate the splitting of surface layers as described in Sect. 3. In the interface simulations presented in this section, the interaction of Al atoms of the metal film with both atom types in the oxide is modelled by an embedded atom method (EAM) potential as described in [52]. Whereas exactly the potential functions of [52] were used for the fcc metal film, the Me–Al and Me–O interactions are optimized with the same approach as used for the alumina potential [53]. The potential was adjusted to match forces, energies and stresses of reference structures consisting of Al-terminated Al-top, H-Hollow, and O-top configurations of the Al–Al2O3(0001)Al interface with variable interfacial distances and strains. Within this optimization, the transfer functions of the Al and O-atoms of the oxide were kept zero because it is assumed that the ions of the oxide do not contribute to the metallic electron gas of the

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embedded atom model. The resulting potential obtained in this optimization is shown in Fig. 19. It can be seen that bonding at the interface is dominated by the attraction of metallic Al and the Al atoms of the oxide. Work of separation values and interatomic distances calculated for the Al–Al2O3(0001)Al interface using this potentials and metal films of eight layers are of the same magnitude as obtained in ab initio calculations. The most stable configuration is H-hollow with a work of separation of 0.59 J/m2 and the same value of 0.29 J/m2 was found for both Al-top and O-top configurations. As discussed in literature both H-hollow and Al-top configurations were obtained as the most stable ones in ab initio studies [22, 23, 38]. In order to simulate tensile tests of systems with and without misfit dislocations the following structures were generated: A coherent O-top Al–Al2O3(0001)Al structure with 24 Al2O3 and Al lattice constants in [-2110] direction and eight Al2O3 lattice constants and 24 Al lattice constants in [0-110] direction. The structure with misfit dislocations contains the same oxide structure but only 23 Al lattice constants in both [-2110] and [0-110] directions. In [0001] direction both structures contain 11 Al metal layers, 12 Al layers, and 12 oxygen layers. In total, the structures contain around 12,000 atoms. In order to equilibrate the structures at the desired temperature of 300 K and zero pressure, simulations in the isothermal–isobaric ensemble are performed for 30,000 time steps of 1 fs. Thereafter tensile loading is applied. In each time step the sample is stretched uniaxially normal to the interface by a factor of

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Fig. 20 (a) Stress–strain curves of tensile tests. (b), (c) Configurations at 7% strain during tensile tests. Colors denote atom types in the oxide (Al: red, O: blue) and crystallographic structures in the metal (fcc: yellow, hcp (stacking faults): green, other (dislocations or interfaces): grey). (b) Structure with misfit dislocations, (c) coherent O-top structure

1106, whereas the pressure in both perpendicular directions is kept zero by the pressure control. Periodic boundary conditions are applied in all directions. As can be seen in the stress–strain curves obtained from the tensile tests (Fig. 20a) the stress increases almost linearly up to 3.9 GPa in case of the structure with misfit dislocations and reaches 4.2 GPa in case of the coherent O-top structure. It must be noted that these values are close to the ideal tensile strength of 4.3 GPa [23] calculated ab initio for the O-top structure using the rigid displacement approach described in Sect. 5. However, in the molecular dynamics simulation, the drop in the stress after the maximum does not concur to interfacial fracture of the coherent O-top structure. Instead, stacking faults surrounded by Shockley partial dislocations nucleate at the interfaces and propagate through the aluminum slab. As can be seen in Fig. 20b,c stacking faults (atoms colored green) and dislocations (noninterfacial atoms colored grey) are obtained in both structures. The main difference regarding these defects is that nucleation occurs in the whole interface area in case of the coherent O-top structure whereas it is limited to the locations of the misfit dislocations in case of the incoherent structure. Another remarkable difference is that the nucleation of crystallographic defects in the aluminum slab of the structure with misfit dislocations concurs with the propagation of an interfacial crack. As shown in Fig. 20b interfacial fracture occurred almost along the whole interface except of the region where stacking faults and dislocations can be seen in the aluminum. In contrast, the corresponding configuration of the coherent O-top structure at 7% strain (Fig. 20c) shows no interfacial crack. The higher stress value at 7% strain confirms that there are more intact bonds at the interface in case of the coherent structure whereas the value close to zero obtained for the structure with misfit dislocations is typical for fracture. Finally, interfacial crack propagation also occurs in case of the coherent structure but as indicated in the stress–strain curve this happens at quite high strains of ~8%.

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This investigation of the effect of misfit dislocations on the behavior of Al–Al2O3(0001)Al interfaces during tensile tests clearly shows that these defects have an influence on both the tensile strength and the fracture behavior. In general, the generation of stacking faults surrounded by Shockley partials in the ductile material is a concurrent effect to interfacial fracture. Misfit dislocations can support the formation of interfacial cracks, whereas homogenous strains as present in coherent structures can favor the generation of stacking faults and dislocations leading to a shift of interfacial fracture to higher strains.

9

Summary and Outlook

Ab initio methods such as plane wave pseudopotential method within density functional theory as applied in the studies presented in this chapter are an excellent tool for investigations of metal–oxide interfaces. They can be used to understand specific interface configurations with various types of bonding as well as for the determination of input data for material modelling on a mesoscopic scale. Studies of Me(111)–Al2O3(0001) interfaces with fcc (Ni, Cu, Pd, Ag, Pt, Au) and bcc (Nb, Mo, Ta, W) metals revealed the most stable positions of the metal films on the Al2-terminated, the Al-terminated, and the O-terminated oxide surface. In case of the Al2-terminated surface metallic bonding is found and the most stable configuration for all considered fcc metal films is the H-hollow position. At the Al-terminated surface the most stable positions are O-top for Cu, Ni, Pd, and Pt, Al-top for Ag, Au, and Al. These differences originate from varying bonding types at the interface. In case of Ni and isoelectronic elements an ionic contribution of metal and second layer oxygen atoms is found, whereas the metal–oxygen distance is increased in case of Ag, Au, and Al. Instead of ionic contribution hybridization of metal–aluminum orbitals takes place with Ag, Au, and Al. The Al-next configuration is most stable at the Al-terminated bcc-metal–Al2O3 interface. Similar to the results with Al and Ag it is found that Nb atoms of the interfacial layer relax to the positions of the next Al atoms on the oxide surface. In general, the values of the work of separation for the Al-terminated interfaces are much lower than for the Al2- and O-terminated interfaces. The reasons obtained in the presented investigations are weak chemical bonding due to hybridization, weak atomic polarization and negligible charge transfer. These effects are most pronounced for fcc metals, whereas in case of bcc metals the work of separation is increased. However, the values for the O-terminated interface are much larger for both fcc and bcc metals. Increased values for the Al-terminated interface such as in the case of Nb–Al2O3 can be explained by more pronounced charge transfer and ionic contribution to interfacial bonding. Charge transfer is enhanced for elements with partially occupied d-shell, whereas negligible charge transfer is obtained for elements with fully occupied d-shells such as Cu and Ag. At the O-terminated interfaces strong charge transfers take place. The ionic bonds cause large values of work of separation. It is found that metal atoms occupy Al positions on the oxide surface which leads to a

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splitting of the interfacial metal layer. The Al-top position is the most stable configuration for Al, Cu, and Ag at this interface. Due to the significant contribution of metal–oxygen bonds to chemical bonding at the interface, oxygen vacancies lead to a strong decrease of the work of separation. This effect can also be seen in the local densities of states; the overlap of metal and oxygen states is decreased due to a shift of oxygen states away from the Fermi level. Vacancies in metal films have less effect on bonding and lead to slightly decreased work of separation values. Work of separation data obtained by ab initio calculations can be used as input data for material simulations on a mesoscopic scale. For such applications the tensile strength which can also be determined using ab initio calculations is another required parameter. It is obtained from structures with incrementally increased distances of Al2O3 and metal at the interface. The tensile strengths and corresponding displacements are calculated from the slopes of the energy–separation curves. It is found for Nb, Al, Cu, and Ag metal films that an increase in the work of separation is connected to an increase of the tensile strengths whereas the corresponding displacements are hardly dependent on the work of separation. A remarkable result obtained in these calculations is that strain in the interface plane can have a strong influence on both work of separation and tensile strength. This effect is most pronounced when metal film atoms occupy the next Al position on the oxide surface such as in case of Al and Ag. It must be noted that the influence of strain must be taken into account in comparisons with experiments. Real interfaces can contain differing strains and various additional defects such as impurities or misfit dislocations. Due to the large number of atoms needed for calculations of structures containing misfit dislocations ab initio methods are often not suitable for such investigations. An applicable atomistic method for these cases is the classical molecular dynamics simulation which allows studies of dynamic fracture processes. For the Al-terminated Al(111)–Al2O3(0001) interface molecular dynamics simulations with interaction potentials based on ab initio data revealed that misfit dislocations can have influence on both the tensile strength and the processes of crack propagation and defect nucleation. As shown in this chapter atomistic calculations are a powerful tool to investigate specific influences on the work of separation and tensile strength. In order to perform more realistic calculations on a larger length scale, it is necessary to develop simulation models which use atomistic results for adjusting some model parameters, whereas other parameters are used for taking into account the various situations which can exist at real interfaces. A challenge in future modelling of interfaces is to identify the most important influences on interface strength and to transfer the atomistic data to models at larger length scales.

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51. Eremeev SV, Schmauder S, Hocker S, Kulkova SE. Ab-initio investigation of Ni(Fe)/ZrO2(001) and Ni-Fe/ZrO2(001) interfaces. Surf Sci. 2009;603(14):2218–25. 52. Streitz FH, Mintmire JW. Electrostatic potentials for metal-oxide surfaces and interfaces. Phys Rev B. 1994;50:11996–2003. 53. Hocker S, Beck P, Schmauder S, Roth J, Trebin H-R. Simulation of crack propagation in alumina with ab initio based polarizable force field. J. Chem. Phys. 2012;136:084707(1–9).

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Multiscale Simulation of Precipitation in Copper-Alloyed Pipeline Steels and in Cu-Ni-Si Alloys Dennis Rapp, Seyedsaeid Sajadi, David Molnar, Peter Binkele, Ulrich Weber, Stephen Hocker, Alejandro Mora, Joerg Seeger, and Siegfried Schmauder

Contents 1 Introduction and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Multiscale Simulation of Precipitation in Copper-Alloyed Pipeline Steels . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Sequential Multiscale Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Obstacle Fields Based on Experiments, KMC, and PFM Simulations . . . . . . . . . . . . . . 2.4 Obstacle Strength from Molecular Dynamics Shear Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Particle Strengthening from Dislocation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Macroscopic Damage Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Precipitation in a Copper Matrix Modelled by Ab Initio Calculations and Atomistic Kinetic Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D. Rapp (*) · P. Binkele · S. Hocker · S. Schmauder Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected]; stephen. [email protected]; [email protected] S. Sajadi · D. Molnar · A. Mora Institut für Materialprüfung, Werkstoffkunde und Festigkeitslehre (IMWF), University of Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected] U. Weber Materials Testing Institute University of Stuttgart (MPA Stuttgart, Otto-Graf-Institut (FMPA)), Stuttgart, Germany e-mail: [email protected] J. Seeger Wieland-Werke AG, Ulm, Germany # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_10

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Abstract

In crystalline solids, the formation of precipitates is caused by the clustering of individual solute atoms dissolved in the matrix. This can happen, if the formation of clusters is energetically favorable relative to the dissolved state of atoms randomly distributed within the matrix. The clustering is driven by the stochastic process of diffusion and depending on diffusion speed, solute concentration, and mixing energies, atom clusters will form and grow, shrink or reach an equilibrium state after a long time of diffusion. As the presence of precipitates can greatly alter the mechanical properties of materials, the simulation of this nanoscopic process starting from the alloy composition up to the final distribution of precipitate numbers and sizes is key for computational design of alloys with desired mechanical properties. For achieving this, the atomistic kinetic Monte Carlo (AKMC) simulation method is used to mimic the stochastical diffusion process. For later stages of precipitation involving bigger clusters, the continuum phasefield method (PFM) can be used to accelerate the simulation. In this chapter, we will present two successful examples for the simulation of precipitation: One coupled approach of AKMC and PFM that was used for predicting the precipitation in copper-alloyed iron and another example of AKMC in the multicomponent Cu-Ni-Si system.

1

Introduction and Outlook

The atomistic kinetic Monte Carlo simulation is used to mimic the self-ordering of the dissolved atoms within a crystal lattice. This simulation considers the atoms to be placed on fixed lattice points with random jumps between atoms and empty lattice sites. The probability of these random jumps is derived from the energy associated with each jump. This means that jumps that decrease the overall energy are more likely to occur than jumps that increase the energy. Depending on whether the formation of aggregates of the same atomic species is favorable, this will over time result in a transport of solute atoms towards clusters. This simple simulation method is already sufficient to simulate the formation of precipitates in the Fe-Cu material system, where experimentally small coherent copper precipitates are observed to be formed within the iron matrix. Building upon this, we will present an approach that derives the macroscopically observable precipitation strengthening only from the alloy composition and temperature history of the material. This approach connects multiple simulation methods working on different length and time scales and was presented completely in Molnar et al. [1] for the first time. We included this work as an example for the full multiscale approach for coherent precipitates in this chapter with permission of the original publisher John Wiley & Sons Inc. As a second step in the multiscale framework, the growth of precipitates is also considered using a phase-field approach that does not intrinsically consider the crystalline lattice. With this extension, the limitation of the simulation time by the simulation system size was overcome and later ageing states were successfully recreated in accordance with experimental observations.

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The second example presented herein will be the Cu-Ni-Si + X material system, in which the focus lies more on how to consider this multicomponent material system. The aim of this second investigation was to find out, which alloying element X could be used to improve the electrical and mechanical properties of the Cu-Ni-Si material system, where the known Ni-Si precipitate types are used for strengthening. The atomistic kinetic Monte Carlo simulations also revealed that for the investigated elements Cr, Fe, Al, and Mg the atoms are located at different positions: Mg and Al atoms are preferably located at the interface of Ni-Si precipitates and Cu matrix, Cr atoms diffuse into Ni-Si precipitates, and Fe atoms form Fe clusters surrounded by Ni-Si shells. Cr addition leads to a significantly reduced fraction of atoms dissolved in the matrix which indicates an increase of electrical conductivity whereas Mg addition results in a high misfit strain at the Cu/Ni3Si interface which can contribute to increased strength. This second example was presented in Sajadi et al. [2] and is reproduced with permission from John Wiley & Sons Inc. likewise. These two examples show the application of lattice-based Monte Carlo methods to the problem of precipitate formation. One main obstacle for other material systems is that with increasing precipitate size the energetic cost of maintaining the coherent embedding becomes too high and a structural transformation of the individual precipitates takes place, leading to the formation of incoherent or semi-coherent precipitates with misfit interfaces. This can not be simulated using a fixed-lattice approach, but rather requires a flexible approach that can consider different lattice types at the same time. Pure off-lattice Monte Carlo methods on the other hand are currently too slow to simulate the process of precipitate formation in crystalline systems where millions of diffusion steps have to be taken until the first signs of precipitate formation become visible. Thus, for future investigations of incoherent systems, we will try to develop a multi-lattice kinetic Monte Carlo method that provides more configurational freedom to accommodate the different lattice structures of matrix and precipitates but still avoids the computational cost of sampling the energy landscape for basins and transition between them, which is the main expense in off-lattice Monte Carlo methods. We hope to reach sufficient ageing states using this approach in incoherent systems and to provide a working tool for alloy development regardless of precipitate lattice type.

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Multiscale Simulation of Precipitation in Copper-Alloyed Pipeline Steels

2.1

Introduction

Recently multiscale simulations have become more and more important in the field of computational engineering. Especially in materials modelling, the aim is a transition from an empirical material description towards simulation-based design of new materials with tailored high-tech properties thus enabling a computational materials design Cluster of Excellence in Simulation Technology (SimTech), University of Stuttgart [3]. In this context, the term multiscale may refer to length scales

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as well as to time scales while each involved scale increases the complexity of the system to model. Typically, multiscale modeling approaches are either of a simultaneous [4, 5] or of a sequential type [6–13] where the latter is applied in this work which is based on parts of Molnar [14]. Both of the mentioned scale types, i.e., length scales and time scales, have to be considered in the case of the mechanical properties of steels that undergo a change during thermal treatment or during service due to an evolving underlying micro and nano structure. This work focuses on copper-alloyed α-iron as material system. Copper-alloyed steels are applied in a wide range of applications, e.g., as pipe material in power stations [15, 16]. The alloying with copper increases the flow stress partly due to the effect of solid solution hardening. The greater part of the strengthening is due to small copper particles blocking dislocations that enable plasticity [17, 18], hence due to the effect of particle strengthening [8]. The small copper particles form during thermal treatment or during service at elevated temperatures (> 300
C) yielding a strengthening in the material [8]. Along with the strengthening, a toughness reduction can be observed [15, 16, 19]. In contrast to the desired strengthening, the toughness reduction is critical due to safety reasons since the material may suddenly fail by fracture with less or without prior plastification. The nanoscale copper particles and the macroscopic specimen size give the problem a multiscale character. Much research has been performed with respect to the Fe-Cu system and particle strengthening in materials. The process of Cu precipitation within the α-Fe matrix has been investigated experimentally by means of small angle neutron scattering (SANS) and high resolution electron microscopy (HREM) [19–21] and was modelled using a kinetic Monte-Carlo approach [22, 23]. The simulated interactions of dislocations with solute Cu atoms [24] as well as with Cu precipitates [25–29] confirm the experimentally found strengthening effect in Fe-Cu-alloys [8]. From experiments [30–32], it is known that Cu precipitates with radii below approximately 2 nm remain coherently embedded within the surrounding α-Fe matrix before undergoing a structural transition during growth from the bcc to the fcc crystal structure with two intermediate crystal structures (9R, 3R), i.e., bcc ! 9R ! 3R ! fcc. Further experimental investigations on the structural transition of larger Cu precipitates can be found in Othen et al. [33], Habibi-Bajguirani and Jenkins [34], Habibi [35], and Monzen et al. [36, 37]. On the modelling side, the structural transition has been investigated mainly by means of Molecular Dynamics simulations [38–43]. In Goodman et al. [44], Fe-1.4 at.% Cu-samples aged at 500
C showed the highest stength for particles with diameters of 2.4 nm, thus long before the end of the precipitation process. Russel and Brown (RB) introduced a dislocation theoretical approach [45] considering the energy of dislocation segments inside and outside of a precipitate, respectively. With the adjustment of several parameters to their experimental data, their analysis suggested a peak strengthening at a radius of 1.3 nm being in good agreement with the value found in Goodman et al. [44]. In Kizler et al. [8] and Weber [16, 46], experimentally obtained information on precipitate mean radii and precipitate volume fractions of the Cu-alloyed steel 15 NiCuMoNb 5 (WB36) were taken in order to calculate the strengthening with the RB-theory. In turn, the obtained strengthening levels were added to the plastic part of a reference stress

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strain curve of an experimental tensile specimen to represent a virtually aged specimen. With this modified stress strain behavior, damage mechanical calculations with C(T)-25 specimen were performed with the Finite-Element Method (FEM) applying the Rousselier Damage model Rousselier [47] yielding in total a multiscale approach consisting of experimental information, dislocation theory and Damage Mechanics (DM). In Singh and Warner [13], Molecular Dynamics simulations and Dislcoation Dynamics simulations were performed in order to provide the strengthening levels of GP-zones at different states of thermal ageing while the thermal ageing was calculated via an analytical approach. In order to predict macroscopic damage behavior due to an underlying nanostructure for specific test cases, it is desirable that as many parts of the multiscale approach as possible consist of numerical simulations which may account for scenarios where analytical approaches fail due to their implicit assumptions. Therefore in this work, a sequential multiscale approach is presented incorporating five simulation methods (Fig. 1), each applied on its appropriate length and time scale. Simulations with the kinetic Monte-Carlo (KMC) method and the Phasefield Method (PFM) will provide precipitation states at different stages of thermal ageing. With Dislocation Dynamics (DD) simulations, the strengthening in the material due to the obtained precipitation states will be calculated after a calibration of the DDsimulations by means of Molecular Dynamics (MD) simulations with respect to shear stress over the ratio between obstacle diameter and spacing. Finally, with the DD-strenghtening levels, FEM-based Damage Mechanics (DM) simulations will yield the macroscopic damage properties of, e.g., C(T)-25 specimen. Thus, parameters, i.e., output values of each simulation method, will be transferred from one simulation method to the next one in order to bridge length and time scales yielding the macroscopic damage behavior based on an underlying nanostructure of Cu precipitates at defined thermal ageing states. In the PFM simulations, the Cu particles are treated as coherently embedded in the matrix material since the investigated particle radii are only a few nanometers in size [48]. In Sect. 2.2, the sequential multiscale modelling approach will be explained in detail. Section 2.3 will describe the generation of particle fields based on experimental data as well as on KMC and PFM simulations. In Sect. 2.4, Molecular Dynamics (MD) simulations are performed in order to calibrate Dislocation Dynamics (DD) simulations. The strengthening levels for several thermal ageing states based on experiments and on KMC and PFM simulation data will be obtained in Sect. 2.5. In Sect. 2.6, the effect on the macroscopic damage behavior due to the strengthening levels obtained from DD simulations will be simulated by means of FEM simulations including the Rousselier damage model. Finally in Sect. 2.7 a conclusion will be given.

2.2

Sequential Multiscale Approach

The bridging from the atomistic to the macroscale is achieved by sequentially coupling five simulation methods just as mentioned in the previous section. In the following, the modelling schemes of each of the applied simulation methods are

time scale

Molecular Dynamics (MD)

Phase-field Method (PFM)

Discrete Dislocation Dynamics (DDD)

method

method

Ab-Initio (AI)

Kinetic Monte Carlo (KMC)

Finite-Element Method (FEM)

precipitate formation

AI

se

e qu

n

nm

l tia

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KM

C

c ltisD u m M

m ale

further coarsening of precipitates

M PF

DD

D

µm

EAM potentials interface energies ...

od

in ell

g

strengthening effects due to dislocation movement

obstacle strength

coarsened precipitates mean radius volume fraction

strengthening

M FE

mm

length scale

precipitate configuration mean radius volume fraction

crack resistance i curve for compari rision with experiiments

Fig. 1 Scheme and use case of the simulation methods applied for the sequential multiscale modelling with their typical length- and timescales and their respective transfer parameters for linking of these methods

ns

μs

ms

s

days

years

multiscale simulation-based design of materials

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described and the output parameters that serve as input parameters for the next simulation method will be denoted. The simulation conditions of the MD, DD, and FEM simulations will be chosen to correspond as effectively as possible to the experimental conditions in Uhlmann [15] and Weber [16] where the damage mechanical properties as well as the micro and nanostructures of the pipe material 15 NiCu-MoNb 5 (WB36) were analyzed for two different states of thermal ageing, i.e., for a reference state (E60A) and after thermal ageing for 57,000 h at 340
C (E60B). Performed tensile tests at 90
C showed a strength increase of approximately 100 MPa of state E60B with respect to state E60A. SANS measurements confirmed the assumption of additionally formed precipitates. This detailed information will allow for comparisons between the simulations and the corresponding experiments. The KMC and PFM simulations represent a test scenario for predictions. Further details will be given throughout the text.

2.2.1 Kinetic Monte Carlo (KMC) Method: Early Precipitation The Kinetic Monte-Carlo (KMC) method for simulating the nucleation, growth, and early particle coarsening of Cu precipitates in α-Fe is based on Soisson et al. [23]. In this method, a thermally activated vacancy jump mechanism yields a cluster formation of Cu atoms within a rigid bcc iron matrix. Bulk behavior is approximated by applying periodic boundary conditions. In Binkele [22], this method has been extended to a quaternary system with the additional alloying elements Ni and Mn, already giving a perspective for future developments of the multiscale approach presented here. The total number of lattice sites is N = NFe + NCu + NV where V denotes the vacancy. The jump rates of the vacancy are calculated via


 ΔE Fe,V Γ Fe,V ¼ νFe exp  and kBT
 ΔE Cu,V , Γ Cu,V ¼ νCu exp  kBT

(1)

(2)

where νFe and νCu denote the attempt frequencies. ΔEFe, V and ΔECu, V are the activation energies depending on the local surrounding of the vacancy by taking first and second nearest neighbors into account. For each of the eight possible jumps on the bcc lattice, its jump probability is calculated by the corresponding jump rates. One of these jumps is then selected using a random number and the jump, i.e., the position-exchange of the vacancy with one of its neighbors, is performed. During a KMC simulation, typically 1011–1012 vacancy jumps are performed. Further details can be found in Soisson et al. [23], Schmauder and Binkele [18], Binkele [22], Soisson and Fu [49, 50], and Hocker et al. [51]. Output parameters of the KMC method at different states of thermal ageing are: • Atom configuration, i.e., position and type of each atom • Mean precipitate radius Rðt Þ • Precipitate concentration and precipitate volume fraction fprec(t)

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In this work, mean precipitate radii and precipitate volume fractions are taken from the KMC results obtained in Molnar et al. [52] for the case of 2 at.% Cu in α-Fe. The simulation samples are cubic lattices with N = 4,194,304 lattice sites yielding an edge length of approximately 36.7 nm. 61011 KMC steps are performed at a temperature of 700
C. The final configurations of the KMC simulations serve as starting configurations for the PFM simulations in order to simulate further particle coarsening. The details of the KMC ! PFM transfer can be found in Molnar et al. [52]. Furthermore, information can be transferred to DD simulations in order to generate realistic particle fields, i.e., obstacle fields for dislocations. In total, the following parameter transfers are made: KMC ðlastconfigurationÞ ! PFM ðfirstconfigurationÞ Molnar et al ð2012bÞ   KMC Rðt Þ, f prec ðt Þ ! DD ðobstaclefieldsÞ

(3) (4)

2.2.2 Phase-Field Method (PFM): Particle Coarsening In contrast to the KMC method, atoms are not resolved in the Phase-field Method (PFM). Instead, a distinction is made between Cu and Fe phases by using an order parameter, the so called Phase-field parameter ϕ which may vary continuously between zero (pure Cu) and one (pure iron) yielding a diffuse interface between Fe phase and Cu phase. The propagation of the diffuse interface between two phases leads to the phase evolution of the system Nestler and Choudhury [53] due to driving forces that can be derived from a functional that contains the interface properties as well as the thermodynamic energies of the pure phases Nestler and Choudhury [53]. In Choudhury and Nestler [54], a grand potential functional was introduced which in turn was applied in Molnar et al. [52] in order to simulate the coarsening of Cuparticles in α-Fe. A detailed introduction into the Phase-field Method is given in Janssens et al. [55] and a historical overview can be found in Nestler and Choudhury [53]. Further phenomena that can be simulated with the Phase-field Method are described in, e.g., Boettinger et al. [56], Chen [57], Thornton et al. [58], SingerLoginova and Singer [59], Moelans et al. [60], and Nestler and Choudhury [53]. Output parameters of the PFM at different states of thermal ageing are: • Particle configuration, i.e., position and composition of each particle • Mean particle radius Rðt Þ • Particle concentration and particle volume fraction fprec(t) In this work, mean particle radii and precipitate volume fractions are taken from the PFM results obtained in Molnar et al. [52] for the case of 2 at.% Cu in α-Fe. The PFM simulation boxes are cubic containing N = 2,097,152 grid points yielding an edge length of approximately 36.7 nm, i.e., each grid point in PFM corresponds to two atoms in the KMC simulations. The PFM simulations are carried out for a temperature of 700
C. Further details can be found in Molnar et al. [52]. Particle

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configurations can be, e.g., transferred back to the atomistic regime in order to perform nano tensile tests by means of molecular dynamics simulations Molnar et al. [61]. Here, similarly to the KMC results, PFM results will also serve for the generation of realistic particle fields for DD simulations. To summarize, the following parameter transfer is made:  PFM

 Rðt Þ, f prec ðt Þ ! DD ðobstaclefieldsÞ

(5)

2.2.3 Molecular Dynamics Simulations (MD): Atomistic Strengthening In MD simulations, the forces between atoms are derived from an interatomic potential and Newton’s equations of motion are used. By integrating the equations of motion, the trajectories of each atom can be followed in time. By this means, simulation samples can be deformed, e.g., in order to perform nano tensile tests Hu et al. [62], Saraev et al. [63], Molnar et al. [64], Hocker et al. [65], Molnar et al. [52, 61] or nano shear tests Nedelcu et al. [66], Osetsky and Bacon [29], Lasko et al. [67], Kohler et al. [27, 28], Bacon and Osetsky [25], Schmauder and Kohler [24], Grammatikopoulos et al. [26]. In this work, the Molecular Dynamics (MD) simulations are carried out with the IMD-code Stadler et al. [68] applying the EAM potential introduced in Bonny et al. [69] which contains the interactions between Fe, Cu, and Ni, thus also giving a perspective for future applications of KMC and MD, as well as of PFM on precipitation in this ternary system. Nano shear tests with a single infinite dislocation and Cu precipitates of different sizes are performed according to Bacon and Osetsky [25] and Drotleff [70] using a   (ycuboidal simulation sample with surface normals in ½112 (x-direction), ½110 direction), and [111] (z-direction). An edge dislocation is implemented into the simulation sample since this dislocation type dominates at the onset of plastic deformation in bcc metals [71, 72] before screw dislocations start to govern the plastic deformation behavior due to their lower mobility. The edge dislocation glides   plane and the glide direction is ½111. Periodic boundary conditions are in the ½110 applied in x- and z-directions. A free surface in y-direction is necessary, otherwise a second edge dislocation would be created at the y-border of the simulation box due to different numbers of atom layers below and above the gilde plane. First, the simulation samples are relaxed and then heated up to 90
C in order to meet the experimental testing conditions. During shearing, in each MD step, two free boundary layers (y-direction) are sheared by moving one layer by 0.0025 Å with respect to the other one. The simulation conditions are summarized in Table 1. Output parameters of the MD simulations for different precipitate sizes are: • The critical resolved shear stress (CRSS) τCRSS that has to be exceeded in order to have the edge dislocation pass the particle. • Information on the dimension of the simulation sample in order to calculate the inter-particle distance Lpart due to the periodic boundary conditions as well as the particle diameter Dpart.

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Table 1 Simulation parameters and boundary conditions of the MD simulations. Further information regarding the EAM potential, the glok-relaxator, and the NVT-ensemble can be found in Bonny et al. [69] and IMD [73], respectively Quantity Edge lengths of the simulation box (x,y,z) Number of atoms Cu precipitate radii Temperature Fe-Cu potential Relaxation Ensemble Boundary conditions MD time step MD steps

Value 16.1 nm, 18.2 nm, 38.1 nm 953,304 0.28 nm to 1.43 nm 90
C (during NVT) EAM bonny et al. [69] Gloki IMD [73] NVT IMD [73] Periodic in x, z 1 fs 25,000 (glok), relaxation 50,000 (NVT), heating 1,800,000 (NVT), shear test

The transfer parameter to DD simulations is the shear resistance τshear of a copper particle which is estimated according to Queyreau et al. [11] and based on Monnet [10] as the slope of the critical resolved shear stress τCRSS versus the ratio of particle diameter Dpart and the inter-particle distance Lpart. To summarize, the following parameter transfer is made:    MD τCRSS Dpart =Lpart ! DD ðτshear Þ

(6)

2.2.4 Dislocation Dynamics (DD): Mesoscale Strengthening The Dislocation Dynamics (DD) simulations treat dislocations as a chain of line segments while single atoms are not resolved. Introductions to Dislocation Dynamics and the theory of dislocations can be found in Hull and Bacon [74] and Hirth and Lothe [75]. The DD simulations are carried out using the microMegas (mM) code Devincre [76] and simulation samples with surface normals in h100i directions for which displacement controlled tensile tests are perfomed. The simulation box is cubic with egde lengths of 1.174 μm. A single infinite edge dislocation is placed on a  -direction. A strain rate of 50 s1 is chosen since test (011) plane moving in ½111 simulations showed that a further reduction of the strain rate increases the computation time but hardly affects the simulation results Molnar [14]. The simulation is performed for a temperature of 90
C and the shear modulus at 0 K is estimated as G = 84.94 GPa (experiment: E = 204.8 GPa Uhlmann [15]) assuming a Poisson’s ratio of 0.3. The shear resistance of the Cu particles τshear is derived from MD simulations. Particle fields, i.e., obstacle fields that interact with the dislocation are generated based on the mean particle radius Rðt Þ and the precipitate volume fraction fprec(t) obtained from KMC and PFM simulations as well as from experimental data [15, 16, 22]. Knowing thus the mean particle radius and the precipitate volume

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fraction, the number of precipitates within the DD simulation volume can be calculated. A random distribution of the small particles (< 4 nm) is chosen due to the fact that no information on the particle distributions is known from available experiments. Furthermore, test simulations with particle and radius distributions based on KMC simulations did not show significant deviations from the DD strengthening levels when choosing random particle distributions possesing a mean radius of the corresponding KMC sample since the particle volume fraction was the same in both cases and is obviously the dominating factor in the present task. The tensile stress of a polycrystal is estimated via the well-known equation σ DD ¼ M τDD ,

(7)

where M is the Taylor factor and τDD is the resolved shear stress acting on the glide plane. Russel and Brown stated that values for M between 2.2 and 3.1 are applicable and found the best fit to their experiments with M = 2.5 [45]. Therefore, this value is also taken in this work for comparison reasons. Nevertheless, a comparison is also made with a Taylor factor of M = 3.067 which is typically used for bcc metals and the 110 {110} h111i glide system Rosenberg and Piehler [77]. The DD strengthening levels obtained for thermally aged structures are related to reference structures corresponding to initial conditions of the experiments (e.g., E60A), i.e., strengthening levels Δσ DD will be obtained, characterizing the change in strength of a thermally aged structure with respect to an initial structure. To summarize, the following parameter transfer is made: DD ðΔσ DD ðKMC=PFM=experimentalobstaclefieldsÞÞ ! FEM ðstress-strainrelationÞ

(8)

2.2.5 Finite Element Method (FEM): Macroscopic Damage Behavior The macroscopic damage mechanics simulations are carried out using the FEMsoftware ABAQUS 6.12 following the approach in Weber [46]: A User-Subroutine (UMAT) describing the Rousselier damage model [47, 78] is applied in order to account for the ductile damage observed experimentally in C(T) experiments with the material WB36 [15, 16]. The macroscopic failure was found to originate from elongated MnS-particles ( 10 μm) having a weak connection to the surrounding iron matrix, thus being a source for void nucleation, growth, and coalescence ([46], p. 101). Within the Rousselier model, the flow function is given by Φ¼


 σv σH þ σ k Q f exp  σ0, 1f ð1  f Þσ k

(9)

where σ v is the equivalent von Mises stress, σ H denotes the hydrostatic stress, and σ 0 the flow stress. σ k and Q are material constants and f is the void volume fraction with an intitial value of f0, the volume fraction of potential void initiating particles. Damage in the material is considered as severe, when f exceeds a critical value fc.

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Table 2 Parameters for the FEM simulations with the Rousselier model [46] Initial void volume fraction f 0 Material constant σ k Material constant Q Mean void distance lc Critical void volume fraction fc

0.13% 420 MPa 2 Batisse et al. [79] 0.1 mm 0.05 Seidenfuß [80]

Severely damaged elements are removed from the stiffness matrix [46]. In Weber [46], the Rousselier parameters have been adjusted to have the simulations fit to the experimentally derived ones from unnotched tensile test specimen as well as to the C (T)-25 specimen of the material WB36 at the reference state E60A. The C(T)-25 specimen was modelled in 2D plane strain in order to account for the predominance of this state throughout the thickness of the samples. The dimensions of the specimen as well as the initial crack length were adjusted to the experimental C(T)-25 specimen of the reference state E60A. Furthermore, it was shown that the same parameters yielded a very good agreement between experiment and simulation for the case of the sample E60B which had been thermally aged for 57,000 h at approximately 340
C. The Rousselier parameters as well as the modelled C(T)-25 specimen of the reference state E60A obtained in Weber [46] will also be applied in this work. The Rousselier parameters are summarized in Table 2. The initial void volume fraction f 0 in Table 2 accounts for the elongation of the MnS particles via pffiffiffiffiffiffiffiffiffiffiffi f 0 ¼ K MnS  f 0 with K MnS ¼ 1:9: (10) KMnS is calculated by taking the ratio of largest to smallest width of elongated particles analyzed in Uhlmann [15]. Further information on the Rousselier model such as the evolution equation of the void volume fraction can be found in Rousselier [47], Seidenfuß [80], Weber [46], and Rousselier [81]. In order to obtain a virtually aged C(T)-25 specimen in this work, strengthening levels Δσ DD obtained from DD simulations are added to the plastic part of the stressstrain relationship which is an input parameter of the FEM simulations, i.e.,     σ p1,B ep1 ¼ σ p1,A ep1 þ Δσ DD ,

(11)

where σ p1, B(ep1) and σ p1, A(ep1) denote the plastic parts of the stress strain relationsships of the virtually aged state (B) and the reference state (A), respectively. In Kizler et al. [8] and Weber [16, 46], this procedure was successfully applied with strengthening levels from the analytical theory of Russel and Brown [45], i.e., Δσ RB instead of Δσ DD. C(T)-25 specimen corresponding to serveral thermal ageing states have been generated and the damage mechanics simulations yield force-crack opening displacement (F-COD) curves as well as crack resistance (J  Δa) curves, both characterizing the damage behavior of the material. To summarize, the following experimental comparisons will be made:

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253

FEM ðF  CODÞ $ CðTÞ experiment ðF  CODÞ

(12)

FEM ðJ  ΔaÞ $ CðTÞ experiment ðJ  ΔaÞ

(13)

Obstacle Fields Based on Experiments, KMC, and PFM Simulations

2.3.1 Obstacle Fields Based on Experiments The obstacle fields for the DD simulations are generated according to the procedure described in Sect. 2.2.4 based on particle mean radii R as well as precipitate volume fractions fprec known from experiments on the material WB36 [8, 16, 22, 46]. Besides the sample pair E60A and E60B, two more sample pairs will be considered. The corresponding information on the initial and final precipitate states is given in Table 3. As can be seen in Table 3, each final state has newly formed particles (Distribution 3) with mean radii between 1.29 nm and 1.9 nm. Since their volume fractions are close to the ones of the original particles which are still present (Distribution 2), a significant effect on the material strength is expected. In fact, in Table 3 Mean particle radii R and precipitate volume fractions fprec of initial (A) and final (B) ageing states obtained from experiments of three different sample pairs (E60A Weber [16], E59 ([22], p. 40), E59W ([46], p. 95)) of the material WB36. The strength increase between the initial and final states calculated with the RB-theory are 88 MPa (E60), 97 MPa (E59) and 91 MPa (E59W). The tensile tests showed an increase of approximately 100 MPa for E60 and approximately 110 MPa for the latter two samples [8, 16, 22, 46] Initial

R

Distribution 1 0.95 nm

Distribution 2 3.87 nm

Distribution 3 –

Sample 1

(E60A) Final

fprec R

0.0029% 0.95 nm

0.402% 3.87 nm

– 1.9 nm

1

(E60B) Test case 1

fprec R

0.0029% 0.95 nm

0.402% 3.87 nm

0.36% 1.9 nm

1

(Cu in precipitates) Test case 2

fprec R

0.0029% 0.95 nm

0.402% 3.87 nm

0.16% 1.9 nm

1

(Cu and Mn in precipitates) Initial

fprec

0.0029%

0.402%

1.08%

R

1.05 nm

2.79 nm

–

2

(E59A) Final

fprec R

0.0043% 1.05 nm

0.43% 2.79 nm

– 1.29 nm

2

(E59B) Initial

fprec R

0.0043% 1.14 nm

0.43% 4.59 nm

0.36% –

3

(E59WA) Final

fprec R

0.0043% 1.14 nm

0.431% 4.59 nm

– 1.54 nm

3

(E59WB)

fprec

0.0043%

0.431%

0.36%

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tensile test experiments, all sample pairs showed a strength increase of approximately 100 MPa to 110 MPa going from the initial to the final state [8, 16, 22, 46]. In addition to the distributions known from experiments, DD simulation samples for two test scenarios (test case 1 and test case 2) are generated. For the material WB36, it is known that the alloy contains approximately 0.56 at.% Cu which is less than the total precipitate volume fractions found for the final precipitation states. Therefore, it has to be assumed that small amounts of atoms of further elements such as Mn and Ni are also present in the precipitates. As an approximation, the same obstacle strength is assumed for possible mixed Cu-(Ni-Mn) precipitates as for pure Cu precipitates. In this way estimates can be made for the test scenarios where only Cu atoms entirely precipitate (test scenario 1) or where the Cu and Mn entirely precipitate (test scenario 2).

2.3.2 Obstacle Fields Based on KMC and PFM Smulations The Kinetic Monte-Carlo (KMC) simulation results as well as the Phase-field method (PFM) simulation results are taken from Molnar et al. [52] where the latter ones were obtained by the group of B. Nestler (KIT). In this multi-timescale approach, samples with 2 at.% Cu in α-Fe were aged at 700
C. During the KMC and PFM simulations a precipitation state with a mean particle radius of 2.5 nm was achieved starting from a solid solution. Due to this high temperature, an amount of approximately 0.54 at.% Cu remains solved in the α-Fe matrix ([22], p. 37). Thus for the DD simulation samples, a precipitate volume fraction of fprec(t) = 1.46% is assumed and particle mean radii Rðt Þ between 1.04 nm and 2.5 nm will be used for the generation of ten obstcale fields that can be related to the mean particle radii of the KMC and PFM ageing times in Molnar et al. [52].

2.4

Obstacle Strength from Molecular Dynamics Shear Tests

The obstacle strength, i.e., the shear resistance τshear for the DD simulations is derived from MD simulation results as described in Sect. 2.2.3. Cu particles with diameters between 0.56 nm and 2.86 nm (i.e., radii between 0.28 nm and 1.43 nm) are taken into account in this study. Figure 2 shows a snapshot of an MD simulation with a Cu particle with a radius of 1.43 nm (Dpart = 2.86 nm). Fe atoms being on bcc lattice positions are not visible while those Fe atoms which deviate from the bcc lattice are colored by applying a coordination analysis available in the IMD-code Stadler et al. [68]. By this means, the downward moving bowed-out dislocation which is blocked by the precipitate becomes visible. After cutting, one half-sphere of the precipitate is sheared by one Burgers vector with respect to the other half-sphere of the precipitate, leaving a step at the particle’s surface. Analyzing the shear stress during the simulation for different precipitate sizes (see Fig. 3) shows that the blocking capability of the precipitate becomes stronger with increasing particle diameter. When the dislocation approaches the particle, it is first attracted yielding a drop of the shear stress. After reaching and entering the particle, the dislocation bows out until the critical resolved shear stress (CRSS) is exceeded

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.1

[111]

[111]

18

nm

.2

255

nm

38.1 nm

38.1 nm

Step

[112]

[110] 16.1 nm

[112]

18.2 nm

[110]

Fig. 2 A coordination analysis shows Fe atoms not being in perfect bcc structure and thus indicating the dislocation. The Cu particle has a diameter of 2.86 nm. Left: Perspective view.  -direction. Right: View along the dislocation line Middle: View onto the glide plane along the ½110 direction ½112 . By cutting the precipitate, a step on the precipitate surface is created. The visualization is performed with a viusalization tool written at the IMWF that combines awk-scripts and the ray tracer POVRay (http://povray.org/)

particle diameter 2.86 nm

shear stress

1.16 nm 0.56 nm

critical resolved shear stress

2.30 nm 1.72 nm

time

Fig. 3 Left: MD shear tests of Fe simulation samples containing an edge dislocation and a Cu particle of different sizes. Without particles, the shear stress, i.e., the Peierls stress remains small. Right: Almost linear dependence of the critical resolved shear stress τCRSS with respect to the ratio of particle diameter Dpart and inter-particle distance Lpart

resulting in the detachment of the dislocation from the precipitate. The slope of the critical resolved shear stress τCRSS versus the ratio of particle diameter Dpart and the inter-particle distance Lpart is obtained by fitting a straight line. (see Fig. 3, right). The shear resistance (= slope of the straight line), i.e., the obstacle strength of the Cu particles is found to be τshear = 1.018 GPa. The straight line intersects the y-axis at a stress level of 0.007 GPa which is taken as the Peierls’ stress.

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Particle Strengthening from Dislocation Dynamics

The DD simulations are performed as described in Sect. 2.2.4 with particle fields, i. e., obstacle fields generated on the basis of experimental data (see Sect. 2.3.1) as well as of KMC and PFM precipitation simulations (see Sect. 2.3.2). Figure 4 shows the DD results for particle distributions based on information from experiments with the sample E60 (containing particles with R = 3.87 nm and additonally particles with R = 1.9 nm in the case of thermal ageing. For further details on the distributions see the entries for sample 1 in Table 3). A Taylor factor of 2.5 is used in order to compare the strengthening levels with the RB-theory. Furthermore, the strengthening levels are shown relative to the initial state E60A which, therefore, posseses an assumed strengthening value of 0 MPa. This procedure highlights the strength increase due to the additionally precipitated particles. The strengthening level calculated by DD for the thermally aged sample E60B (Δσ DD = 93 MPa) is in very good agreement with the one predicted by the RB-theory (88 MPa) as well as with the experimental value (100 MPa) from Fig. 4. The non-smooth behavior of the DD flow curve is due to the finite number of particles taken into account in the DD simulation (see Fig. 4, right). Nevertheless, the deviations between the strengthening obtained with the DD calculations and the strengthening values derived from experiments or the RB-theory are relatively small. The advantage of DD over the straightforward theory of Russel and Brown is the possibility to account for realistic precipitate distributions (e.g., radii and spatial distributions) directly obtained from, e.g., KMC simulations and using CRSS values calculated with MD simulations. Additionally DD provides the ability to consider plastic deformation by dislocation motion in detail, allowing the simulation sample A (E60A) simulation sample B (E60B) Cu and Mn in precipitates (1.48 at.%)

experiment E60B Russel & Brown

thermally aged (E60B, 0.76 at.%) Cu in precipitates (0.56 at.%) initial (E60A, 0.40 at.%) 1.174 μm

1.174 μm

Fig. 4 Left: Stress-strain curves of DD simulations assuming different fractions of precipitated copper in order to approximate the material WB36 (E60A and E60B) as well as test cases (0.56 at.% Cu and 1.48 at.% Cu). A Taylor factor of 2.5 is assumed for comparison with the result obtained by the theory of Russel & Brown (lower black solid line). The simulation result (93 MPa) for the sample E60B is in good agreement with the experimental value of 100 MPa. Right: Simulation samples corresponding to the thermal ageing states E60A and E60B of the material WB36. An infinite edge dislocation (periodic boundary conditions in all dircetions) is moving on a single glide plane and is blocked by the Cu particles having radii according to Table 3, i.e., R = 3.87 nm for both states and additionally particles with R = 1.9 nm for E60B. View along the glide plane normal. Visualization with MATLAB (mathworks.de)

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description of phenomena such as work hardening caused by dislocation-dislocation interaction. Also the effect of different initial dislocation densities can be taken into consideration to predict the strength of materials. Furthermore, the multiscale approach is easily extendable to further compositions of alloying elements, whereas the RB theory is applicable for binary systems only. Besides the validation of the multiscale procedure with the thermally aged state E60B, the two test scenarios 1 and 2 can be considered as predictions, i.e., as virtually designed materials. With the DD simulations, the effect of particle fields with respect to size variations on the strength increase can be estimated. Test scenario 1 with the small volume fraction of Cu (see Table 3) yields a strengthening of only 53 MPa suggesting that there has to be an additional effect causing the experimentally found strengthening value of 100 MPa. Test scenario 2 can be considered as a worse case scenario yielding a strengthening of 195 MPa. The fact that this is a worse case scenario is due to the severely reduced fracture toughness as will be shown in the next section. Figure 5 shows the simulated DD strengthenings of the three samples of Table 3, the corresponding strengthenings calculated applying the RB-theory, the experimentally observed values, as well as the strengthenings of the test scenarios 1 and 2 for sample 1. In the left part of the figure, a Taylor factor of 2.5 is applied while in the right part a Taylor factor of 3.067 is used, affecting the DD-simulated and the RB-values. The reason for showing both versions is discussed in Sect. 2.2.4. With both Taylor factors a sufficiently good agreement between experiments and DD simulations can be achieved. All in all, the procedure of calibrating the DD simulations with MD simulations while incorporating the experimental conditions yields an efficient

M = 2.5

M = 3.067

sim (Cu+Mn)

sim (Cu+Mn)

exp sim RB

exp RB sim

exp RB sim

sim RB exp

RB exp sim

RB exp sim

sim (only Cu) sim (only Cu)

Fig. 5 Comparison of the results for the strengthening obtained by DD simulations (sim) with the experimental values (exp) and the ones predicted by the RB-theory (RB) for the three different simulation samples mentioned in Table 3. The values are compared using a Taylor factor of M = 2.5 (left) and using one of M = 3.067 (right), with the former corresponding to the value used in the RBtheory and the latter one being the standard value for dislocations in bcc materials with slip on {110} planes Kocks [82]. For the first sample, two additional scenarios corresponding to test case 1 (Cu) and test case 2 (Cu + Mn) were also considered

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1.46 at.% Cu thermal ageing: 700°C tensile test: 90°C M = 2.5

1.04 nm 1.32 nm

transfer parameters: Cu atom positions KMC

1.64 nm 2.11 nm PFM 2.50 nm

transition from atomistic to mesoscale simulations

thermal ageing with KMC ( ) and PFM ( )

thermal ageing with KMC

Fig. 6 Left: schematic description of the multi-timescale approach introduced in Molnar et al. [52]. First, KMC simulations yield nucleation, growth, and early particle coarsening of Cu precipitates. The very last KMC configuration file is tranferred to PFM simulations where further coarsening is simulated in order to achieve larger particle mean radii. Right: DD strengthenings Δσ DD based on the particle mean radii and the Cu precipitate concentration of the KMC and PFM simulations in relation to the KMC and PFM ageing times

Table 4 Strengthening levels Δσ DD obtained with DD simulations for different thermal ageing states from KMC and PFM simulations. The strengthening value is shown relative to the state E60A known from the experiment. The volume fraction of particles is fprec = 1.46 at.% Cu KMC ! PFM I R 7[nm] Δσ [MPa]

(KMC) II 1.04 224

(KMC) III 1.32 210

(PFM) IV 1.64 202

(PFM) V 2.11 195

(PFM) 2.50 185

predictivity with respect to the DD strengthening levels. Now strengthening levels for different ageing stages, which were obtained from KMC and PFM, can be calculated relative to the initial state E60A, i.e., Δσ DD. These strengthenings are shown in the right part of Fig. 6. The graph contains information on the particle mean radius and precipitation times, both values obtained from KMC and PFM simulations in Molnar et al. [48], as well as on the DD-derived strengthening level Δσ DD. During thermal ageing, the strengthening effect of the precipitates is decreasing, i.e., the simulation samples correspond to the overaged regime. The left part of Fig. 6 shows the multitimescale approach and the KMC ! PFM parameter transfer described in Sect. 2.2.1. From the KMC and PFM simulated ageing stages, the strengthening levels shown in Table 4 will be transferred to the macroscopic Damage Mechanics (DM) simulations along with strengthening levels corresponding to sample 1 in Table 3.

2.6

Macroscopic Damage Behavior

The macroscopic FEM-based Damage Mechanics (DM) simulations are performed with C(T)-25 specimen according to Sect. 2.2.5. The strengthening levels Δσ DD are taken from the DD simulations of the previous section. First, a validation is made in Fig. 7 by comparing the simulated force-displacement curves as well as the crack resistance curves of the initial and final states with the experimental ones. A good agreement

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F

60,0 mm

CT specimen (E60A)

E60B (sim) E60A + 110 MPa E60A + 93 MPa

259

E60A (sim, U. Weber, 2006) E60A (exp) E60A (sim) COD

a = 26,337 mm W = 50,5 mm

+93 MPa E60B (sim) +110 MPa

62,5 mm

E60A (sim) E60A (sim, U. Weber, 2006) E60A (exp)

crack growth Δa

Fig. 7 Validation of the multiscale approach with C(T)-25 specimen. Left: Force (F) versus crack opening displacement (COD). Using the Rousselier parameters from Table 2, the curve from the experiment of the initial state E60A (exp) can be matched exactly with the FEM-Simulation (E60A (sim)). Adding the strengthening level of 93 MPa obtained from DD-simulations to the E60A stressstrain curve yields a F-COD curve in good agreement with the FEM-simulated thermally aged state (E60B (sim)) that is based on the experimentally determined stress-strain relationship of the E60B tensile specimen. Adding 110 MPa yields an even better agreement. Right: J-integral versus crack growth Δ a, i.e., the crack resistance curve. The same conclusions as for the F-COD curve hold. The inlet shows a sketch of the C(T)-25 specimen used in the FEM-analysis

between experiment and simulation with (Δσ DD = 93 MPa) is found although a strengthening of 110 MPa, which is a test simulation, would give an even better result: Here it is worth to remind that the 93 MPa strength increase is a result of a Taylor factor of 2.5. Choosing a Taylor factor of 3.067 would yield a strength increase of 114 MPa and would slightly overestimate the ideal value of 110 MPa. After validation, prediction of possible material behaviors can be made. Therefore, the DD strengthening levels based on the KMC and PFM simulations as well as on the experimental test scenarios are used to investigate their effect on the damage behavior. Additionally, possible cases where the Cu concentration is even below the one in material WB36 (0.56 at.%), i.e., 0.1 to 0.3 at.% Cu are taken into account. For the corresponding DD simulations a mean particle radius of 1.9 nm has been assumed. The results of the FEM simulations are shown in Fig. 8. As expected the F-COD curves corresponding to the small Cu concentrations (0.1 to 0.3 at.% Cu) lie below the reference F-COD curve (E60A (sim)). With increasing strengthening the maximum force also increases up to 68 kN while the plateau region becomes shorter. For even higher strenghening values, the maximum force decreases again together with an elongation of the plateau region. This means in terms of finite elements that severe damage of an element is reached faster after a severe damage of the previous element indicating a reduction of ductility. Analyzing the corresponding J-integrals, i.e., the crack resistance curves, it appears that the material becomes less and less ductile with increasing strengthening levels thus confirming the interpretation of the F-COD curves. The reason for the ductility reduction is the increased yield strength since more energy is stored in the elastic deformation of the alloy. Hence, regions near the crack tip tend to fail at lower plastification. Furthermore, the limit of the Rousselier damage model describing ductile damage behavior is assumed to be nearly reached for very large strengthening levels (Δσ DD  200 MPa) where the ductility in this material is severely reduced.

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E60A + 202 MPa E60A + 210 MPa E60A + 224 MPa

E60A + 195 MPa E60A + 185 MPa E60B (sim) E60A + 93 MPa E60A + 53 MPa

0.1 at. % Cu 0.2 at. % Cu 0.3 at.% Cu E60A (sim) +53 MPa +93 MPa E60B (sim)

E60A (sim) 0.3 at. % Cu 0.2 at. % Cu 0.1 at.% Cu

+185 MPa +195 MPa +202 MPa +210 MPa +224 MPa crack growth Δa

Fig. 8 Prediction of macroscopic damage behavior. Left: Force (F) versus crack opening displacement (COD). Right: J-integral versus crack growth, i.e., the crack resistance curve. The strengthening levels for the input stress-strain relationship are taken from DD simulations based on KMC and PFM precipitation states (+185 MPa. . . + 224 MPa, Table 4) as well as on experimental information (+53 MPa, +93 MPa, 0.1 at.% Cu. . .0.3 at.% Cu)

2.7

Conclusions

In this work five simulation methods (KMC, PFM, MD, DD, FEM) were coupled sequentially via transfer parameters in order to simulate the macroscopic damage behavior due to an underlying nanostructure of Cu precipitates. Each simulation method was applied on its appropriate length scale. The computational results were validated by comparisons with experiments and simulations were performed in order to predict the damage behavior of virtually designed materials. Although the coupling scheme works well, there are still improvements possible. At the moment, a reference stress-strain relationship from experiments is necessary along with information on the corresponding precipitation state in order to design precipitation states relative to a cleaned reference state. In future work, the whole stress-strain relationship could be obtained computationally by means of DD simulations, including a large number of dislocations with their interactions as well as grain boundaries. Further elements could be incorporated in the study and obstacle strengths of different precipitate compositions could be taken into account as well as the internal structural change of the precipitates during their growth.

3

Precipitation in a Copper Matrix Modelled by Ab Initio Calculations and Atomistic Kinetic Monte Carlo Simulations

3.1

Introduction

Copper-based alloys have been extensively utilized in electronic industries since they possess a combination of excellent electrical conductivity, improved mechanical properties, and reasonable prices. Precipitation strengthening is usually the main strengthening mechanism [83]. Ternary Cu-Ni-Si is a typical precipitation

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strengthened alloy and is frequently applied in lead frames, contact wire clamps, and wear-resistant parts [84–86]. Precipitates in Cu-Ni-Si alloys have been investigated repeatedly since their discovery in 1927 by Corson [87]. Spinodal decomposition of the supersaturated solid solution into periodic (Ni,Si)-rich and (Ni,Si)-poor regions is the key mechanism during the early stages of aging treatment [86]. Ordered metastable phases (Cu,Ni)3Si and β-Ni3Si with DO22 and L12 structures form within (Ni,Si)-rich regions as aging advances. These phases are ordered fcc with cube-oncube orientation relationship (OR) corresponding to the Cu matrix Lei et al. [88]. Through further aging, coherent δ-Ni2Si precipitates with orthorhombic lattice structure nucleate within the (Cu,Ni)3Si matrix [86, 89]. Jia et al. [84] determined (100)δk{001}m, [100]δkh110im and [010]δkh110im as the ORs for δ-Ni2Si precipitates. Due to the orthorhombic lattice structure and orientation of δ-Ni2Si, dislocations cannot shear these precipitates and dislocations only bypass them; as a result, the Orowan mechanism is the main reason of strengthening in these alloys [90]. Achieving improved mechanical properties while maintaining reasonable conductivity is required in modern applications in electronic industry. Small additions of elements, e.g., Cr, Mg, Al, Fe, and others to CuNiSi alloys have shown improvements in the desired properties of the alloy: Thermal aging of Cu-Ni-Si-Cr leads to an increase in both hardness and electrical conductivity [91]. Both properties were also increased with Fe additions of only 0.3 at% Suzuki et al. [92]. Mg improves strength and provides better stress relaxation resistance by decreasing inter-precipitate spacing and through the Mg atom drag effect on dislocation motion [93, 94]. A similar result was obtained in case of Al addition; it was observed that Al promotes the precipitation and enhances the stress relaxation resistance [90]. These elements segregate at different locations and show different behavior during aging which make them interesting for further investigations. Moreover, there are enough experimental investigations to be compared with the simulation results for verification. A detailed investigation of the locations of the dissolved impurities after aging is needed to understand the causes of increased strength or conductivity. These studies reveal whether there are different interface compositions or additional precipitates. For instance, ordered fcc (Ni,Cr,Si)-rich precipitates were found in Cu-Ni-Si-Cr [91] and preferential segregation of Mg atoms between the precipitates and the matrix was identified in the case of Cu-Ni-Si-Mg [95]. The atomistic kinetic Monte Carlo method (AKMC) based on a vacancy diffusion model has shown promising results in simulations of precipitation processes [22, 23]. Considering several parameters (e.g., aging temperatures, types of alloying elements, and concentrations) in the simulations is economically favorable compared to high-priced experimental studies. To our best knowledge this method was not applied to Cu-Ni-Si alloys up to now. It was applied in many investigations of precipitation in Fe-Cu [18, 23, 49, 96, 97], e.g., Fe-Cu-Ni-Mn [51] and Al-Zr-Sc [98]. It was shown that this method allows accurate simulations of the formation, growth, and Ostwald ripening of Cu precipitates in Fe during thermal aging. The objective of this work is a detailed investigation of the precipitation process in Cu-Ni-Si with additions of Cr, Mg, Al, or Fe. The number of precipitates and mean radii are determined in dependence of aging temperature and concentrations of

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alloying elements. In order to estimate the electrical conductivity of these alloys, the number of impurities dissolved in the matrix is determined after thermal aging. The simulations also reveal preferred locations of impurities, e.g., within NiSi-precipitates or at Cu-NiSi interfaces. Alloying elements at the interfaces can enhance brittle fracture of these interfaces, and at the same time they can lead to higher strength by increased coherency hardening. In order to estimate these effects, density functional theory-based ab initio calculations of misfit strain and work of separation are performed considering coherent Cu/Ni3Si interfaces with impurities, e.g., Cr, Fe, Mg, and Al. Ab initio calculations are also used to calculate the mixing energies which are the crucial input parameters needed for the AKMC simulations.

3.2

Computational Details

3.2.1 Kinetic Monte Carlo Method In the present simulations, an atomistic kinetic Monte Carlo algorithm (AKMC) is applied on a rigid fcc lattice with L = 64 lattice constants. Thus, approximately one million (N = 4 L3) lattice sites are available which are occupied by Cu, Ni, Si, X, and one vacancy, where X can be Cr, Mg, Al, or Fe. Periodic boundary conditions are used in all directions. Atom movement is simulated via thermally activated vacancy diffusion. In this algorithm, jump rates for an atom of type xj into a vacancy are given by:
 ΔExj ,ν Γ xj ,ν ¼ νxj exp  kT xj  fCu, Ni, Si, Xg

(14)

where Γ xj ,ν is the jump rate, νxj is the attempt frequency for a jump of an atom of type xj, k is the Boltzmann constant, T is the temperature, and ΔExj ,ν is the activation energy for an atomic jump. Activation energies strongly depend on the local atomic arrangements and are calculated separately for each first neighbor atom vacancy exchange. This energy can be calculated as follows: m m X X ΔExj ,ν ¼ Esp,xj  nx j x k e x j x k  nνxk eνxk k¼1 k¼1   xj , xk  fCu, Ni, Si, Xg m : numberofatomtypes

(15)

where Esp,xj is saddle point energy, exj xk is atomic pair interaction energy, eνxk is vacancy interaction energy, and nxj xk is the occupation number. The chemical binding between atoms is described by first neighbor pair interactions. Hetero-atomic pair interactions exj xk are determined by mixing energies ωxj xk : ωx j x k ¼

 z1  exj xj þ exj xk  2exj xk 2

(16)

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263

where z1 = 12 is the number of first nearest neighbors in an fcc lattice structure. Atom jumps are performed using the rejection-free residence time algorithm [23]. Depending on the first neighbors, jump rates are calculated for each of the 12 nearest neighboring atoms of the vacancy. Weighted by its probability, one of these twelve possibilities is chosen by random. The time associated with the Monte Carlo steps is given by: t MC ¼

z1 X

!1 Γ x j ,ν

(17)

j¼1

Another key parameter which affects the precipitation kinetic is the vacancy concentration. In real systems, the local vacancy concentrations are determined by the vacancy formation energy, which depends on the solute concentration and distribution. Therefore, the total vacancy concentration evolves with the state of the system [49]. Thus, to relate the physical time to the time obtained in the Monte Carlo simulations, the vacancy concentration in the simulation box and in a real system have to be taken in to account. Under the assumption that the vacancyvacancy interaction is neglected, the physical time t is given by: t ¼ t MC

C MC ν ðCuÞ C eq ν ðCuÞ

(18)

eq where C MC ν ðCuÞ and C ν ðCuÞ are the vacancy concentrations in the simulation box and in the copper matrix for a real alloy at the equilibrium, respectively. Since only one vacancy is introduced in the simulation box, the vacancy concentration in the 3 simulation box is C MC ν ðCuÞ ¼ 1=L , and the vacancy concentration at the equilibrium is given by:

C eq ν ðCuÞ

¼ exp 

E Fνxj kT

! (19)

A so-called symmetrical model exj xj ¼ exk xk was applied in the present simulations, in which only the hetero-atomic pair interactions exj xk determine the environments are preferred. As a result, the following relations hold: The homo-atomic pair interactions are estimated from the cohesive energy of the pure copper (3.49 eV): E coh,Cu ¼ 

z1 ex x 2 jj

(20)

The binding between atoms and the vacancy is described by the interactions between vacancy and the first neighbors eνxj , and is related to the cohesive energy and the vacancy formation energy of pure copper (1.28 eV) as:

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E Fν Cu ¼ Ecoh,Cu  z1 eνxj

(21)

The saddle point energies E sp,xj are assumed to be constants and are calculated by knowing the vacancy migration energy of pure copper (0.70 eV) as: EM ν Cu ¼ E sp,xj  ðz1  1Þeν,Cu  eνxj  ðz1  1ÞeCu,xj

(22)

Also attempt frequencies are kept constant νxj ¼ νCu ¼ 0:6254  1015 s1 . With these choices, differences between cohesive energies and vacancy binding energies of Cu, Ni, Si, and X are neglected and vacancy trapping effects are suppressed. Vacancy trapping effects and detailed analysis of diffusion kinetics have been investigated by Soisson and Fu [49] for FeCu. It should be noted that the model has some limitations. Due to the fixed fcc lattice it cannot be used to simulate precipitates of other crystallographic structures. Furthermore, ordered structures on the fcc lattice such as L12 cannot be obtained because it requires to take into account second and third neighbors, energetic anisotropies, and elastic energies. Nevertheless, the model is able to reveal which elements form precipitates and which elements prefer to segregate to matrix/precipitate interfaces.

3.2.2 Ab Initio Calculations The density functional theory-based plane-wave code (VASP) [99–102] was used to calculate mixing energies. Four different cube types have been prepared. First was a cube with the fcc crystal structure of pure copper, and second was a cube with the crystal structure of pure solute element xj. According to the crystal structures, cubes with different numbers of lattice sites (i.e., 108, 64, 54, and 48 lattice sites for fcc, diamond, bcc, and hcp, respectively) were prepared. In all cases, the lattice constant was determined by relaxation of the cube with volume adjustment of the simulation cell. The third cube was made by replacing one copper atom with a solute atom in the first cube, and the fourth cube was made by replacing two neighboring copper atoms with two different solute atoms. Following calculations were performed with a constant volume. All cubes were relaxed regarding positions of contained atoms. In all calculations, a 7  7  7 k Monkhorst Pack [103] mesh was adapted; projector augmented wave pseudopotentials [104, 105]; and the PerdewWang 91 generalized gradient approximation [106, 107] were used; and the plane-wave cutoff energy is kept constant at 500 eV. From the obtained energies of the relaxed structures and number of first neighbors z1, the mixing energies are calculated:

 107 1 E108Cu þ E nxj  E107Cu,1xj 108 n    

¼ z1  E106Cu,1xj ,1xk þ E 108Cu  E107Cu,1xj þ E107Cu,1xk ωCu,xj ¼

ωx j x k

(23) (24)

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  Theseenergies  actually correspond to the formation energies of single ωCu,xj or multiple ωxj xk substitutional point defects as found in Eq. 6 of Jelinek et al. [108].

3.3

Results

3.3.1 Ab Initio Results After obtaining the energies of the relaxed structures, mixing energies can be calculated by applying Eq. 23 and Eq. 24. Table 5 shows the mixing energies between Cu, Ni, Si, Cr, Fe, Al, and Mg. Calculated mixing energies between copper and other elements are in agreement with the phase diagrams of these binary systems, since it is proposed for binary systems that, with mixing energies ωxj xk < 0 the system has a tendency to form precipitates, with ωxj xk ¼ 0 atoms are ideally dissolved, and with ωxj xk > 0 there is a tendency to form superstructures Soisson et al. [23]. High positive ωNi, Si in the copper matrix indicates the existence of Ni-Si precipitates in Cu-Ni-Si system which agrees with the aging process results previously presented in this paper and the thermodynamic description of the Cu-Ni-Si system by Miettinen [109]. 3.3.2 Kinetic Monte Carlo Results Kinetic Monte Carlo simulations up to 1011 steps were performed at several temperatures from 450 to 900 K. For all temperatures, the number of dissolved atoms were not changed in case of simulations with more steps which indicates that 1011 steps are sufficient to reach the equilibrium volume fraction of precipitated atoms (composition of precipitates and solubility of the foreign atoms do not change in further calculations). Precipitation in Cu-Ni-Si was simulated using different concentrations of Ni and Si. In order to investigate the effect of a fourth element in the precipitation process, simulations were performed with 2.5 at% Ni and 1.25 at% Si and different concentrations of Cr, Fe, Al, or Mg. The concentration of Ni and Si were chosen by averaging over some experimentally investigated materials [95, 110–112]. As explained in Sect. 3.2.1, cohesive energy, vacancy formation energy, vacancy migration energy, and diffusion constant of copper were used for all elements in the alloy. Ab initio results, which are listed in Table 5, are used for AKMC simulations. Cu-Ni-Si Figure 9 shows the AKMC simulation results for Cu-2.5at%Ni-1.25at%Si after 1011 steps aged at 450, 600, 750, and 900 K. The obtained Ni-Si precipitates shown in the Table 5 Calculated mixing energies (in eV) between Cu, Ni, Si, Cr, Fe, and Mg in a copper matrix determined by ab initio calculation ω Cu Ni Si

Ni 0.07 – –

Si 0.23 1.80 –

Cr 1.96 3.64 1.48

Fe 0.99 0.40 1.77

Mg 0.02 0.77 1.39

Al 0.51 1.43 0.73

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Fig. 9 Obtained simulation results after 1011 steps utilizing the mixing energies of Table 5 resulting from ab initio calculations. Aged at (a) 450 K, (b) 600 K, (c) 750 K, and (d) 900 K. Ni: blue, Si: yellow

900K 750K Num. of prec. Mean r

600K

450K

12 10 8

10

6 4

5

Mean radius (Å)

Number of precipitates (×102)

15

2 0 10–6

0 10–1

104 Time (s)

109

Fig. 10 Number of precipitates and their mean radii in the Cu-2.5at%Ni-1.25at%Si ternary system during aging at several temperatures

figure indicate that the results of simulations are in good agreement with the aforementioned experimental works. The influences of aging temperatures on the mean radii and number of precipitates are shown in Fig. 10. Increasing the aging temperature from 450 K to 900 K decreases the required time for nucleation, growth, and coarsening processes from 1010 s to 1 s. The mean radius of precipitates increases significantly from 5 Å at 450 K to 11 Å at 900 K, and the number of precipitates decreases from 1400 to 400. The volumes of the 450 K and 900 K annealed precipitates are approximately four times and 40 times of the unit cell of orthorhombic δ-Ni2Si (a = 4.99 Å, b = 3.72Åand c = 7.06 Å), respectively. A long period of time is needed to reach the maximum mean radius at 450 K whereas at 900 K the mean radius of 11 Å is obtained in less than a second. Furthermore,

Multiscale Simulation of Precipitation in Copper-Alloyed Pipeline Steels and. . .

Fig. 11 Number of dissolved nickel and silicon atoms in the Cu-2.5 at%Ni-1.25at%Si ternary system during aging at several temperatures

Number of dissolved atoms (×104)

9

3

900K

750K

600K

267

450K

Ni 2

1

Si

0 10–6

10–1

104

109

Time (s)

Table 6 Ni/Si ratio in precipitates after 1011 steps for several aging temperatures in the Cu-2.5at% Ni-1.25at%Si ternary system Temp (K) Ni/Si in the precipitates

450 1.83

600 1.64

750 1.48

900 1.37

Table 7 Ni/Si ratio in precipitates after 1011 steps obtained from the simulations with different concentrations of Ni and Si in the simulation box Ni/Si in the simulation box Ni/Si in the precipitates

0.5 0.8

1.0 1.0

2.0 1.5

3.0 1.8

6.0 3.0

decreasing the aging temperature leads to a decrease in the number of dissolved atoms. As shown in Fig. 11, the number of dissolved Si atoms is negligible for temperatures below 750 K whereas the number of dissolved Ni atoms decreases from 12,950 to 2870 (from 50% to 10% of the total number of Ni atoms in the simulation box) in the temperature range considered. Since the number of dissolved atoms in the copper matrix is correlated to the electrical resistivity of the alloy [23], a better electrical conductivity is expected at lower aging temperatures. In summary, for technological applications, an intermediate temperature is preferable since the number of dissolved atoms is clearly too high and the aging process is extremely fast and hardly controllable at 900 K whereas it is very slow and impractical at 450 K. Moreover, the chemical composition of precipitates after aging is analyzed. Table 6 illustrates that the Ni/Si ratio in the precipitates is in the range of 1.4 to 1.9 for different aging temperatures. Furthermore, different concentrations of Ni and Si in the simulation box lead to changes of the Ni/Si ratio in the precipitates after aging. Table 7 shows the changes of Ni/Si ratio in the precipitates versus Ni/Si ratio in the simulation box. Starting the simulations with a ratio of Ni/Si = 6.0 in the simulation box, results in a Ni/Si ratio of 3.0 in the precipitates after aging. Hence, the chemical formula of Ni3Si can be predicted for the precipitates. However, the crystallographic structures of β-Ni3Si or δ-

268 1200

L

1100 1000

993 °C

900

(Cu,Ni)

(Cu,Ni)+Ni1.5Si

800

797 °C

700 (Cu,Ni)+Ni2Si

Phase diagram

600

Simulation results

Fig. 13 Number of precipitates and their mean radii during aging at 750 K for Cu-2.5at%Ni-1.25at%Si1.25at%X, X {Al, Mg, Fe, Cr}

Number of precipitates (¥ 102)

500 100 Cu 0 Ni 0 Si

25 20

80 Cu 13.3 Ni 6.6 Si

90 (at%)

Cr Fe Mg Al

9 8 7 6

15

5 4

10 5 0 10–4

3 Num. of prec.

Mean radius (Å)

Temperature (°C)

Fig. 12 The simulation results compared with the experimental solubility limits extracted from a vertical section of the Cu-Ni-Si ternary phase diagram [113]

D. Rapp et al.

2

Mean r

1 0 10–2

100

102

Time (s)

Ni2Si were not observed in the simulations. This is due to our model which is optimized for long simulation times by using a fixed fcc lattice and taking into account first nearest neighbors only. But nevertheless, the formation of Ni-Si precipitates with Ni/Si ratios close to those of precipitates which are known from experiments can be simulated. The simulation results have been compared with the experimental solubility limits Villars and Okamoto [113] by simulating different concentrations of (2Ni, Si) at several aging temperatures spaced 50 K apart from each other (see Fig. 12). There is found a close agreement of the simulated solubility limit with the experimental phase diagram. Cu-Ni-Si-X To analyze the effect of the fourth element on Cu-2.5at%Ni-1.25at%Si, several concentrations of Cr, Fe, Al, and Mg were added. Chromium and iron cause an increase in the number of precipitates and a decrease in the mean radii (see Fig. 13). Mg and Al do not have significant effects on the number of precipitates and their

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269

Fig. 14 Precipitates in Cu-2.5at%Ni-1.25at%Si-1.25at%Cr after 1011 steps aged at 750 K. Ni, Si, and Cr atoms are shown in blue, yellow, and red spheres, respectively. Chromium atoms are found inside the Ni-Si precipitates

Fig. 15 Precipitates in Cu-2.5at%Ni-1.25at%Si-1.25at%Al after 1011 steps aged at 750K. Ni, Si, and Al atoms are shown in blue, yellow, and red spheres, respectively. Most of the Al atoms are dissolved in the copper matrix and small percentages of them are accumulated at the interface of NiSi precipitates and Cu matrix

mean radii. It was found that atoms of these four elements are located at different positions according to atom types and concentrations: (a) Chromium has a negative mixing energy with copper and high positive mixing energies with nickel and silicon. These values are used in the AKMC simulation of Cu-Ni-Si-Cr for several concentrations of Cr and temperatures, and results are extracted after 1011 steps. As shown in Fig. 14, no Cr atoms stay dissolved in the matrix. For any concentration of Cr, these atoms move into Ni-Si precipitates and, therefore, precipitates consist of Ni, Si, and Cr which strongly agree with the experimental results of Cheng [91]. (b) In the Cu-Ni-Si-Al system, there are positive mixing energies for Cu-Al and NiAl, and the negative one for Al-Si. AKMC simulation results after 1011 steps for several concentrations of Al and aging temperatures show that most of the Al atoms are dissolved in the copper matrix; however, some of them are accumulated at the interface of Ni-Si precipitates and the Cu matrix (see Fig. 15). (c) The calculated mixing energy for Cu-Mg is positive and nearly zero; magnesium has a positive mixing energy with Ni, and a quite negative mixing energy with Si. AKMC simulations lead to similar results as in the case of Cu-Ni-Si-Al: most Mg atoms stay dissolved in the matrix but there is a tendency for Mg atoms to accumulate at the interface of Ni-Si precipitates and the copper matrix. (Fig 17)

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D. Rapp et al. 20 Number of X atoms at the interface (×102)

Cu-Ni-Si-Mg 15

Cu-Ni-Si-Al

10

5

0 10–4

10–2

100

102

Time (s)

Fig. 16 Concentration of Al and Mg atoms at the closest distance from precipitates at the interfaces of Ni-Si precipitates and the copper matrix. Concentration of X at the interface is calculated by dividing the whole number of Mg or Al atoms at the interfaces by the whole number of these atoms in the simulation box

Figure 16 illustrates the concentration of Al or Mg atoms at the closest distance from precipitates at the interface. The number of Al atoms at the interfaces is slightly higher than the number of Mg atoms at the interfaces. The concentrations of Mg and Al at shells with the thickness of two lattice constants around the precipitates is also calculated. For these calculations, the average numbers of Mg or Al atoms in shells around ten precipitates are compared to the same volume in the matrix. An increase of concentration by 16% and 19% was observed for Mg and Al in the shells, respectively. The segregation of Mg atoms at the interface of Ni-Si precipitates and the Cu matrix has been observed by S. Lee [95]. The concentration of Mg at the interface is different from our simulation results since initial concentrations, aging temperature, and aging time are not the same. Moreover, precipitates in Lee’s experiments consist of more Ni atoms, according to the mixing energies Mg is attracted by Ni and not by Si and, therefore, it can be understood that more Ni-rich precipitates are likely to have more Mg at the interface (Fig. 17). (d) Iron in copper has a high negative mixing energy. Several concentrations of Fe were added to Cu-2.5at%Ni-1.25at%Si and aging of the alloy was simulated by AKMC simulations. All Fe atoms take part in the formation of precipitates and almost no iron atoms stay dissolved in the matrix. In contrast to Cr, Fe atoms form a precipitate core consisting mostly of Fe. This core is surrounded by a shell consisting of Ni and Si (see Fig. 18). Compared to the Ni-Si precipitates without Fe addition, this shell is more Ni-rich. To analyze the effect of small concentrations of the fourth element on the electrical conductivity, 0.25at% of the elements mentioned above were added to

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Fig. 17 Precipitates in Cu-2.5at%Ni-1.25at%Si-1.25at%Mg after 1011 steps aged at 750 K. Ni, Si, and Mg atoms are shown in blue, yellow, and red spheres respectively. Most of the Mg atoms are dissolved in the copper matrix and small percentages of them are accumulated at the interface of NiSi precipitates and Cu matrix

Fig. 18 Precipitates in Cu-2.5at%Ni-1.25at%Si-1.25at%Fe after 1011 steps aged at 750 K. Ni, Si, and Fe atoms are shown in blue, yellow, and red spheres, respectively. Fe atoms form a precipitate core surrounded by a shell consisting of Ni and Si

Fig. 19 Number of dissolved nickel, silicon, and X atoms after 1011 steps in Cu-2.5at% Ni-1.25at%Si(0.25at%X) aged at 750 K

Cu-Ni-Si-Al Cu-Ni-Si-Mg Cu-Ni-Si-Fe Ni

Cu-Ni-Si-Cr

Si

X

Cu-Ni-Si 0

4000

8000

12000

Number of dissolved atoms

Cu-2.5at%Ni-1.25at%Si and the alloy is aged at 750 K. Cr has the best effect on the electrical conductivity of the alloy since the number of dissolved atoms is lowered to half compared to Cu-Ni-Si (see Fig. 19). Almost no dissolved Cr atoms are left since all Cr atoms go to precipitates. The addition of Fe decreases the number of dissolved atoms to 80% of that in Cu-Ni-Si, and similar to Cr almost no dissolved Fe atoms are left in the Cu matrix. Additions of the same concentration of Mg or Al do not have much effect on the number of Ni or Si dissolved atoms compared to Cu-Ni-Si.

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3.3.3 Ab Initio Investigation of a Cu/Ni3Si Interface Lattice misfit is defined as the mismatch in the arrangement of lattice sites of different phases at a flat phase boundary [114]. Having a slight misfit increases the stress field around the coherent precipitate, this stress field hinders dislocation movement and increases the strength of the alloy by the misfit hardening mechanism. Hence, ab initio investigations of the misfit strain can be helpful to investigate the strength of materials. β-Ni3Si is one of the strengthening precipitates in the early stages of aging in CuNi-Si material systems. This is an ordered fcc phase with cube-on-cube orientation relationship between the precipitates and the copper matrix. To calculate the misfit strain, the Cu/Ni3Si interface was modeled by a super-cell containing 10 layers of Cu (40 copper atoms) and 10 layers of Ni3Si (30 nickel atoms and 10 silicon atoms). A schematic sketch of the super-cell is shown in Fig. 20. Periodic boundary conditions were used, the Cu/Ni3Si slabs are separated by a vacuum of thickness larger than 10 Å. The lattice is distorted to the average of the lattice constants of Cu (3.61 Å) and Ni3Si (3.51 Å) in the directions parallel to the interface so that coherency is achieved. Because of L12 structure of Ni3Si, there can be two possibilities for a Ni3Si interface, one with Ni and Si at the interface (see Fig. 20a) and one with only Ni at the interface (see Fig. 20b). In order to estimate the stability of these interfaces, the work of separation was calculated for both cases. In the present study, the concept of Griffith rupture work is employed to evaluate the binding strength (indicating the bonding energy) of the interface. By definition, the work to cleave an interface is referred to as Griffith rupture work, which can be calculated by the following equation according Hu et al. [115]:

Fig. 20 Scheme of the Cu/Ni3Si interface which is used to calculate the work of separation and the misfit strains. (a) Ni and Si at the interface with the work of separation Wsep = 2.96 eV and (b) only Ni at the interface with the work of separation Wsep = 3.61 eV. The atoms surrounded by circle are replaced by several foreign atoms in calculations. Copper, Nickel, and silicon atoms are shown in white, blue, and yellow spheres, respectively

9

Multiscale Simulation of Precipitation in Copper-Alloyed Pipeline Steels and. . .

W sep ¼

E Cu þ E Ni3 Si  ECu,Ni3 Si A

273

(25)

where ECu and E Ni3 Si are the total energies of the Cu and Ni3Si slabs with free surfaces, respectively, ECu,Ni3 Si is the total energy of the Cu,Ni3Si slab embedded in vacuum, and A is the area of the interface. All systems were calculated under exactly the same conditions, and all atoms were fully relaxed. A work of separation (Wsep) of 3.61 eV for the interface containing purely Ni and 2.96 eV for the interface containing Ni and Si was obtained. Based on this result, the structure with purely Ni at the interface is chosen for all calculations with impurities. In order to check the result of the AKMC simulations that Mg and Al atoms are preferentially segregated at the interface, additional calculations with one Cu or Ni bulk atom substituted by Al or Mg were performed. The comparison of free energies of the relaxed structures revealed that all interfacial sites are energetically favorable compared to the bulk sites. This was found for both Al and Mg and confirms the results obtained in the AKMC simulations. As a result, impurities were introduced by substituting one atom at one of the interfacial sites, e.g., Cu1, Cu2, Ni1, or Ni2 by Al or Mg (see Fig. 20b). The lattice misfit can be calculated according to Novikov [114]: δ¼

d  d0 d0

(26)

where d and d0 are the inter-atomic distances of atoms in Cu1 and Cu2 (or Ni1 and Ni2) sites of relaxed structures with and without foreign atom. Misfit strain and work of separation results are shown in Table 8. It can be seen that Mg impurities decrease the work of separation. The decrease is more pronounced in case of Mg at Ni sites with a value around 2.8 J/m2, whereas at Cu sites, the Wsep is decreased to 3.47 (Cu1) and 3.36 (Cu2) J/m2. In regard of these values compared to the work of separation of the interface without impurities (3.61 J/m2) and the energy needed to create two free surfaces of copper (3.6 J/m2 Tyson and Miller [116]), it can be stated that Mg impurities slightly increase the risk of brittle interface fracture. However, it can be assumed that brittle fracture is not a significant failure mechanism. The decrease of Wsep with Al impurities at Ni sites is smaller compared to Mg at these sites. It is 3.16 and 3.30 J/m2 which is quite close to the value of the interface without impurities. In case of Al at Cu sites, the Wsep even increases to values around 3.9 J/m2 and, therefore, reduces the risk of brittle interface fracture. Table 8 Calculated work of separation and misfit strain at Cu,Ni3Si interface determined by ab initio calculation

W sep mJ2 δ[%] Mg Al Mg Al Cu1 3.47 3.95 1.78 0.79 Cu2 3.36 3.91 1.98 0.99 Ni1 2.76 3.16 0.99 1.78 Ni2 2.83 3.30 1.58 0.20

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Mg impurities at the interface lead to a positive misfit strain in the range of 1–2%. Higher values were obtained for Mg at Cu sites with a maximum of 1.98% for the Cu2 site. The lowest value of 0.99 was found for the Ni1 site. Much lower misfit strains were calculated in case of Al impurities. Al at Cu sites leads to a positive misfit strain of 0.79 and 0.99%, whereas the misfit strain values for Al at Ni sites were found to be negative with a minimum of 1.78% for Al at the Ni1 site. In order to estimate the influence of Al or Mg impurities on strain at the interface with regard of the misfit strain values, it has to be taken into account that the lattice is distorted to achieve a coherent Cu/Ni3Si interface. As mentioned before, the coherency is achieved by a stretch in Ni3Si and a compression in Cu. Therefore, an increase of strain at the interface for all interfacial sites substituted by Al can be noted. In case of Mg, the strain at the interface increases in case of Cu substitution only. But due to the highest misfit strain obtained for these cases, the influence on the strain at the interface is expected to be strong.

3.4

Summary and Conclusions

In order to understand the effects of alloying elements on the electrical conductivity and the strength of copper-based alloys, simulations of precipitation in Cu-Ni-Si alloys with Cr, Fe, Al, or Mg addition were performed. The kinetic Monte Carlo method on a rigid fcc lattice and a vacancy diffusion model were adapted. Important input data for this method, namely, the mixing energies, were calculated by density functional theory-based ab initio calculations. Furthermore, ab initio calculations were used to estimate the influence of Al or Mg at Cu/Ni-Si interfaces on precipitation strengthening and fracture toughness. The precipitates obtained in kinetic Monte Carlo simulations of Cu-Ni-Si-Mg and Cu-Ni-Si-Al consist of Ni and Si. In agreement with experimental investigations of Cu-Ni-Si-Mg [95], an accumulation of Mg at the interfaces between copper matrix and Ni-Si precipitates was found. In the Cu-Ni-Si-Al alloy, the accumulation of Al at the interface was even more pronounced. Precipitates consisting of Ni, Si, and Cr were obtained in the simulations of Cu-Ni-Si-Cr which agrees well with the existence of cubic Ni-Si-Cr precipitates found in experimental studies [91]. In case of Cu-Ni-Si-Fe, the simulations revealed the formation of Fe precipitates which are surrounded by a shell consisting of Ni and Si. A strong dependency of the type of alloying element on the number of dissolved atoms in the copper matrix was found. Whereas most Mg or Al atoms stay dissolved in the matrix, almost no dissolved Cr or Fe atoms were observed. Additionally, Cr and Fe significantly reduce the number of dissolved Ni atoms. This effect is most pronounced in case of Cr addition where the number of dissolved Ni atoms is about half of the value obtained for Cu-Ni-Si. Since a main part of the electrical resistivity in copper-based alloys originates from impurities dissolved in the matrix, it can be concluded that alloying with Cr leads to a strong increase of the electrical conductivity. Fe addition also improves the electrical conductivity, whereas for Al or Mg additions, an electrical conductivity slightly below the value for Cu-Ni-Si is expected.

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Ab initio calculations of misfit strain and work of separation were performed for structures containing the coherent Ni-rich Cu/Ni3Si interface with Mg or Al impurities. The calculations confirmed that for both impurity elements all interfacial sites are energetically more favorable than bulk sites in Cu or Ni3Si. Impurities lead to a decrease of the work of separation by 0.2–0.9 J/m2 besides for Al at Cu interfacial sites where an increase of 0.3 J/m2 was obtained. In most cases, the work of separation values are close to the value for the interface without impurity. Therefore, fracture at these interfaces is not expected to be a significant failure mechanism. The misfit strain obtained for Mg impurities is in the range of 12%. Al impurities at Cu sites lead to a smaller misfit strain and for Al at Ni sites, a negative misfit strain was obtained. In the coherent Cu/Ni3Si structure, Ni3Si is stretched and Cu is compressed due to the difference in lattice constants. Hence, it can be noted that Mg increases the strain at the interface in case of Cu substitution because its positive misfit strain requires more compression of Cu. Al also causes a positive misfit strain at Cu sites but a negative one at Ni sites which indicates increase of the strain at the interface in all interfacial sites (e.g., Cu1, Cu2, Ni1, or Ni2). Since all interfacial sites are energetically favorable for Al and Mg impurities, it can be concluded that both Al and Mg lead to a higher strength by increased coherency hardening. The investigations performed in this work clearly show that the atomistic kinetic Monte Carlo method supported by ab initio calculations is suitable to study the precipitation in copper-based alloys. Despite of the simplified model used, a reasonable agreement with experimental results was obtained. Furthermore, this work provides understanding of the effects of specific alloying elements on the electrical conductivity and the strength and, therefore, it can contribute to the aim of targeted alloy development. Acknowledgments Support from the DFG through Collaborative Research Centre SFB 716, Project B.2 is gratefully acknowledged. The detailed description of each example in the chapter is reproduced from recently published works of the authors with kind support of the involved publishing company John Wiley & Sons, Inc. The original sources are Molnar et al. [1] and Sajadi et al. [2].

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Atomistic Simulations of Hydrogen Effects on Lattice Defects in Alpha Iron

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Shinya Taketomi and Ryosuke Matsumoto

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Hydrogen Trap Energy of the Trap Sites in Alpha Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Calculation of Hydrogen Trap Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Elastic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Stacking Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Free Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Hydrogen Trap Energy of Trap Sites in Alpha Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Hydrogen Effect on Individual Lattice Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hydrogen Occupancy at Trap Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hydrogen Effect on Vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hydrogen Effect on the Cohesive Energy of Grain Boundaries . . . . . . . . . . . . . . . . . . . . . 3.4 Hydrogen Effect on Dislocation Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Interaction Between Hydrogen and Individual Lattice Defects . . . . . . . . . . . . . . . . . . . . . . 4 Lattice Defects and Hydrogen Embrittlement Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Interaction Between Many Lattice Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Cracks and Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Supersaturation of Vacancy Concentration Associated with Plastic Deformation . . . 4.4 Reduced Lattice Defect Energy (Defactant-Like Effects) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary, Future Directions, and Remaining Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S. Taketomi Department of Mechanical Engineering, Saga University, Saga, Japan e-mail: [email protected] R. Matsumoto (*) Department of Mechanical Engineering and Science, Kyoto University, Kyoto, Japan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_11

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Abstract

Solute hydrogen atoms degrade the strength of materials. This phenomenon, termed as hydrogen embrittlement (HE), has been a matter of concern for various industrial applications for more than a century. In recent years, HE has to be addressed because of the need for more efficient storage and transport of hydrogen. In this chapter, we present an overview of the current state of knowledge of the interaction between hydrogen and lattice defects. In Sect. 2, the hydrogen trap energy of various trap sites in alpha iron is reviewed and summarized. In Sect. 3, first, the hydrogen concentration around the defects is outlined based on the evaluation of the occupancy at each trap site. Subsequently, the effect of hydrogen on the stability and the kinetics of the lattice defects that trap hydrogen atoms are reviewed. In Sect. 4, mesoscopic calculations of the complex interactions among hydrogen-affected lattice defects are reviewed. Finally, the current state of knowledge of hydrogen effects on lattice defects and future directions are discussed. Alpha iron is considered because it is a basic steel component, and steel is a potential material for hydrogen storage and transport systems from engineering and economic viewpoints. Keywords

Hydrogen embrittlement · Molecular dynamics · Density functional theory · Lattice defect · Mechanical property · Deformation

1

Introduction

It is well known that the strength of metals decreases in the presence of solute hydrogen [1–4]. This is known as hydrogen embrittlement (HE) and causes serious problems in the design and operation of various mechanical components. In recent years, hydrogen has attracted attention as a clean energy source. To use hydrogen as an energy source, we have to store and transport it. In addition, hydrogen is stored at a high-energy state (i.e., high-pressure gas or liquid state). Thereby, the chances of having failure of mechanical components owing to HE are high. Although the mechanism of HE has been studied for more than a century, the overall embrittlement process has not yet been clarified even for simple material systems. Several embrittlement mechanisms for metals have been proposed. Hydrogen-enhanced decohesion (HEDE) [5, 6] is the earliest one and is based on the decrease of cohesive energy in the presence of hydrogen, i.e., cracks basically propagate in a brittle manner. Hydrogen-enhanced localized plasticity (HELP) [7–10] is another mechanism wherein hydrogen locally enhances the plastic deformation, i.e., the dislocation movement is enhanced at the crack vicinity. These two mechanisms have long been debated. Recently, M. Nagumo proposed the hydrogen-enhanced strain-induced vacancy (HESIV) mechanism [11, 12], in which hydrogen enhances the formation and coalescence of vacancies, resulting in

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ductile fracture. R. Kirchheim also proposed the defactant concept [13] to describe the hydrogen stabilization of lattice defects. In any case, an understanding of the interaction between the solute hydrogen atoms and lattice defects is required to clarify HE. The difficulty in understanding HE is attributed to the nature of the solute hydrogen atoms. Hydrogen atoms have high diffusivity within metals, and their distribution rapidly changes according to the distributions of the lattice defects and stress field. Hydrogen atoms also undergo complex interactions with the various lattice defects; thus, the direct evaluation by experiments of hydrogen distribution and its effect on lattice defects is extremely difficult. Thereby, atomistic and quantum-level simulations have been used to analyze the HE mechanisms. Several studies on the atomistic simulations of the effect of solute hydrogen atoms on the strength of materials have been published; however, none of them has quantitatively evaluated the hydrogen concentration under actual environmental conditions. In this chapter, the hydrogen concentration near defects is evaluated. Subsequently, the effect of hydrogen on the stability and kinetics of individual lattice defects are reviewed. Next, the results of mesoscopic calculations of the complex interactions among hydrogen-affected lattice defects are reviewed. The subject material is alpha iron because alpha iron is a basic steel component and steel is a potential material for hydrogen storage and transport. The chapter comprises the following sections. Section 2: The hydrogen trap energies of various trap sites, i.e., elastic strain field, vacancies, edge and screw dislocations, stacking faults, grain boundaries (GBs), and free surfaces, in alpha iron are reviewed. Then, the trap sites are ranked according to the magnitude of the trap energies. The obtained hydrogen trap energies in alpha iron are compared with those in aluminum. Section 3: First, the hydrogen occupancy at each trap site under hydrogen gas conditions is obtained; then, the hydrogen effect on the stability and kinetics of each lattice defect is reviewed. It is evident that solute hydrogen affects the lattice defects; this is a function of the hydrogen concentration and deformation condition. Section 4: Experimentally observed hydrogen embrittlement is the product of the interaction among lattice defects during deformation and fracturing. In this section, these phenomena and the competition among the various microscopic mechanisms are discussed based on large-scale molecular dynamics (MD) and discrete dislocation dynamics (DD) simulations, where the hydrogen effects on each individual lattice defect are modeled based on Sect. 3. Section 5: The conclusions are summarized, and future directions as well as challenges are briefly discussed. The relation among sections and subheads is shown in Fig. 1.

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Fig. 1 Relations among sections and subheads

2

Hydrogen Trap Energy of the Trap Sites in Alpha Iron

2.1

Calculation of Hydrogen Trap Energy

Lattice defects are known to trap solute hydrogen atoms. Despite the progress in this matter, directly observing the hydrogen distribution in metals is still difficult because of the ultralow concentration of hydrogen atoms, their high diffusivity, and their small size and mass. This section reviews the interaction energy between solute hydrogen atoms and various trap sites obtained by atomistic and quantum-level simulations. The binding energy between a solute hydrogen atom and a trap site is called hydrogen trap energy and is E

Trap


 1 ¼ EHOS  EdþH ðxÞ  Ed  EH2 , 2

(1)

where Ed and Ed + H(x) are the energy of the system before and after the introduction of a hydrogen atom at position x, respectively, EH2 is the energy of a hydrogen molecule, and EHOS is the heat of solution into a material (0.27 eV; a hydrogen atom into a tetrahedral site (t-site) in alpha iron). For hydrogen atoms, considering the zero-point energy (ZPE) as a quantum effect owing to its small mass is crucial. The trap energies calculated via density functional theory (DFT) are all ZPE-corrected if no supplementary information is given. On the contrary, the trap energies calculated using interatomic potentials are not ZPE-corrected because the reference physical

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properties used to develop the interatomic potential include ZPE. The details of the calculation procedure of the hydrogen trap energy using DFT and interatomic potentials have been previously determined [14–16].

2.2

Elastic Strain

It is generally believed that hydrogen atoms accumulate in high hydrostatic stress regions, such as ahead of mode-I crack tips and the tensile side around an edge dislocation. Several previous studies have included the hydrostatic stress gradient in the hydrogen diffusion equation as a hydrogen accumulation term [17–20]. This procedure well predicts the hydrogen diffusion in fcc and hcp metals. However, it has long been known that interstitial atoms that occupy octahedral sites (o-site) in the bcc lattice, such as carbon atoms in alpha iron, strongly interact with shear deformations [21, 22]. The hydrogen trap energy owing to both volumetric and shear strain was evaluated using the Vienna ab initio simulation package (VASP) [23–25] in Ref. [15]; shear strain (0.01) along a specific direction changes the most stable occupation site of a solute hydrogen atom from t-site to o-site. Furthermore, it was shown that despite that the trap energy arising from volumetric strain ev = 0.05, which is almost equal to the elastic strain around a dislocation core (r > 0.5 nm), is 0.154 eV, the shear strain yields maximum of ~ 0.2 eV depending on the direction. Although the energy difference between sites is only 0.03 eV for undeformed lattices, the difference increases under shear strain, e.g., ~0.42 eV when the shear strain is0.05 and ~0.1 eV when 0.01 [15]. Thus, elastic strain makes the potential surface of hydrogen diffusion uneven, and the diffusion coefficient and diffusion pass are significantly affected by shear strain, as directly confirmed near the core of edge dislocations by MD simulations [16].

2.3

Vacancies

Vacancies in alpha iron produce trap sites with extremely strong trap energy and attract multiple hydrogen atoms at nearby sites under hydrogen gas conditions, as shown in Sect. 3.1. Thus, calculating the hydrogen trap energy by sequentially introducing hydrogen atoms based on a stable order is essential. Tateyama et al. first performed extensive analyses of the hydrogen and (mono) vacancy interaction using DFT calculations and reported the hydrogen trap energy for up to six hydrogen atoms (0.56–0.04 eV) [26]. Furthermore, they showed that two hydrogen atoms are trapped at opposing o-sites in a vacancy and the hydrogen-vacancy complex (VH2) forms planar or linear structures. In Ref. [27], using VASP [23–25], the following values are reported for the first to sixth hydrogen atom trapped at o-sites in vacancies: 0.64, 0.66, 0.34, 0.30, 0.21, and -0.16 eV. These values are in good agreement with the calculations [26] and experiments (0.63 and 0.43 eV) [28].

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Dislocations

Dislocations are known to interact with the solute hydrogen atoms. Especially, elastic theory suggests that edge dislocations strongly interact with hydrogen atoms responding to the hydrostatic stress field, which is known as the Cottrell atmosphere. Dislocations are believed to keep moving in the Cottrell atmosphere, and this could be one of the hydrogen transport mechanisms within a metal. Thus, the interaction between solute hydrogen atoms and dislocations could play an important role in HE, i.e., plastic deformation and hydrogen transport. On the contrary, the development of computational resources and analytical techniques has contributed to the investigation of the interactions between dislocations and hydrogen. Atomic-scale investigations directly evaluate the effect of the dislocation core on the hydrogen trap and have suggested that the core structure could be the strongest hydrogen trap site [29]. In alpha iron, the major slip planes are {110} and {112}, and the slip direction is for both. Taketomi et al. calculated the hydrogen trap energy of typical trap sites within the core structure for {110}< 111> and {112} edge dislocations [30] using molecular statics calculations based on the embedded atom method developed by Wen et al. [31]. The maximum values are the same for both core structures, i.e., 0.49 eV; however, the distribution of trap sites differs. The detailed distribution of hydrogen trap energies around the {112} edge dislocation is also given by Taketomi et al. [16]. The trap energy distribution around the core structure, i.e., the elastic strain field, is striped owing to the effect of shear strain on the bcc structure (see Sect. 2.2). These characteristics yield planar hydrogen diffusion in MD calculations. Conversely, the hydrogen trap energy for the screw dislocation core is estimated at 0.26 eV by first-principles calculations [32] and ~0.30 eV by the path-integral MD method [33].

2.5

Stacking Faults

It has been reported that hydrogen reduces the stacking fault energy, thereby increasing the width of the dislocation cores. Seki et al. evaluated the hydrogen trap energy at stacking faults and reported the hydrogen trap energies in {110}< 111> and {112} systems: 0.12 and 0.02 eV, respectively [14]. The interaction energy between hydrogen and stacking faults is extremely weak; thus, it is considered that hydrogen does not accumulate at stacking faults and the width of the dislocation core structure in alpha iron does not change except in extreme cases, i.e., very high hydrogen concentration.

2.6

Grain Boundaries

Riku et al. [34] estimated the hydrogen trap energy of GBs for symmetrical tilt GBs (STGBs) and twist GBs in alpha iron using atomistic simulations

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with a similar procedure as that in Ref. [16]. They showed that the GB energy and hydrogen trap energies show good correlation, because high-energy GBs tend to have large gaps (free volume) and hydrogen atoms are more stable in the large gaps. This tendency was also confirmed using DFT calculations for two symmetric tilt GBs: high-energy Σ3(111) STGB (ϕ = 70.5 ) and stable Σ3(112) STGB (ϕ = 109.5 ). Hydrogen trap energies of 0.49 eV are reported for the Σ3(111) STGB and 0.34 eV for the Σ3(112) STGB [35]. The non-ZPE-corrected value for the Σ3(111) STGB is 0.45 eV [36].

2.7

Free Surfaces

The hydrogen effect on the surface energy has long attracted interest because it is important in fracturing. The trap sites on the free surfaces, where hydrogen atoms are adsorbed, are classified according to the adsorption position, for example, onefold (1F), twofold (2F), threefold (3F), and fourfold (4F) sites. The hydrogen trap energies are summarized in Table 1. It is confirmed that the maximum hydrogen trap energy of the free surfaces is almost 0.75 eV and independent of the surface index.

2.8

Hydrogen Trap Energy of Trap Sites in Alpha Iron

The hydrogen trap energies of various trap sites in alpha iron are shown in Fig. 2. For reference, the hydrogen trap energies of various trap sites in aluminum are also shown in Fig. 3 [38]. It is quite interesting that when various trap sites are ordered

Table 1 Hydrogen trap energy of free surfaces [eV] Adsorption site Trap energy

{100} 4F 0.67a

2F-1 0.73b

{112} 2F-2 0.71b

4F 0.73b

a

Ref [37] Ref [14] c Ref [36] b

Fig. 2 Hydrogen trap energy of various trap sites in alpha iron [eV]

{110} 3F 0.84b

{111} 0.79c

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Fig. 3 Hydrogen trap energy of various trap sites in Al [eV]

according to the strength of the hydrogen trap energy, as shown in Figs. 2 and 3, the sequence is very similar even for different materials, i.e., free surface > vacancy > grain boundary > edge dislocation. Thus, it is suggested that there is a correlation between the local free volumes at trap sites and trap energy besides the chemical interaction with a third element. However, although the energy of the hydrogen molecule is most stable for aluminum, there is near screw dislocations in alpha iron. Thus, at thermal equilibrium, the hydrogen concentration in alpha iron is higher than in aluminum, and lattice defects in alpha iron trap more hydrogen atoms.

3

Hydrogen Effect on Individual Lattice Defects

3.1

Hydrogen Occupancy at Trap Sites

The basic steps of hydrogen embrittlement are believed to involve changes in the stability and mobility of lattice defects owing to the presence of hydrogen. Thus, capturing the fundamentals of the hydrogen effect on isolated lattice defects is important. To analyze the hydrogen effect on each lattice defect using atomistic and quantum-level simulations, allocating hydrogen atoms at stable sites with appropriate probability that corresponds to real conditions is essential. The procedure to evaluate the hydrogen occupancy at each trap site using the hydrogen trap energies, which were summarized in Sect. 2, is given. The relation at thermal equilibrium between hydrogen occupancy at a trap site with hydrogen trap energy ETrapi and hydrogen occupancy at the t-site in the undeformed lattice, ci and ct-site, respectively, is expressed as follows [39]:
Trap  ci ctsite E i ¼ exp , 1  ci 1  ctsite kB T

(2)

where kB is the Boltzmann constant and T is the absolute temperature. For hydrogen gas conditions of temperature T and pressure p, the ct-site is given by the following empirical relation [40]:

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Fig. 4 Hydrogen occupancy vs. hydrogen trap energy for various hydrogen gas pressures at room temperature (300 K)

ctsite ¼ 0:9686  10

 3440 p exp  : T

6 pffiffiffi




(3)

By defining p and T, the hydrogen occupancy at trap site ci is calculated as a function of the trap energy ETrapi using Eqs. 1 and 2, as shown in Fig. 4. The hydrogen occupancy at lattice sites in metals can also be quantitatively evaluated under hydrogen atmosphere using first-principles calculations and lattice vibration analysis [41, 42] over a wide range of hydrogen conditions where experimental data are unavailable [43]. In this study, a similar relation between the hydrogen occupancy at the t-site and temperature and pressure, as shown in Eq. 3, is presented for alpha iron. As shown in Fig. 4, ci rapidly increases for ETrapi > 0.4 eV under various hydrogen gas pressures at room temperature. For example, at T = 300 K and p = 70 MPa, components exposed to high hydrogen gas pressures, such as pipes and tanks in fuel cell vehicles, ci > 0.5 for ETrapi > 0.41 eV is obtained. Therefore, the free surfaces (ETrap ~0.75 eV [14, 36, 37]) and vacancies (ETrap = ~0.65 eV [26, 27]) trap hydrogen atoms with high probability at hydrogen gas conditions. The edge dislocation cores (ETrap ~0.44 eV [16]) and high-energy GBs (ETrap = ~0.49 eV [35]) also interact with hydrogen atoms. Conversely, the hydrogen concentrations around screw dislocation cores (ETrap < 0.30 eV [32, 33]), low-energy GBs (ETrap ~ 0.34 eV [35]), and stacking faults are relatively small. Using computations, the lattice defects that trap hydrogen atoms are defined, and the distribution of hydrogen around each lattice defect is determined using random numbers and occupancy rates based on the distribution of the hydrogen trap energy around lattice defects. Thus, constructing atomistic models with realistic hydrogen distributions and concentrations for specific hydrogen gas pressures and temperature conditions is possible. Many theoretical calculations of hydrogen-lattice defect interactions have used extremely high hydrogen concentrations and artificial hydrogen distributions. In the following sections, the hydrogen effects on vacancies (Sect. 3.2), GBs (Sect. 3.3), and edge dislocations (Sect. 3.4) at practical hydrogen

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concentrations are reviewed. These defects are related to the typical hydrogen embrittlement mechanisms, i.e., HESIV [11, 12], HEDE [5, 6], and HELP [7–10].

3.2

Hydrogen Effect on Vacancies

Using the hydrogen trap energy of a vacancy in Sect. 2.3 and the evaluation procedure of the hydrogen occupancy at each trap site in Sect. 3.1, we confirm that two hydrogen atoms are trapped by a vacancy under a wide range of hydrogen gas conditions. Matsumoto et al. estimated the apparent vacancy formation energy and the mobility of a vacancy when a vacancy traps two hydrogen atoms in hydrogen gas environment [27]. The apparent lattice defect energy is evaluated by the following equation: w=o_H

_H  W Ww f f



 X  Trap c i Ei þ μ b ,

(4)

i

where Ww_Hf and Ww/o_Hf are the formation energy of the lattice defect with and without hydrogen, respectively. μb is the change in configurational entropy and is evaluated by the following equation:
 ctsite μb  kB Tln : 1  ctsite

(5)

At T = 300 [K] and p = 70 MPa, ct-site = 8.5  108 and μb is 0.42 eV. Furthermore, using Ww/o_Hf = 2.14 eV, ETrap1 = 0.64 eV, and ETrap2 = 0.66 eV, Ww_Hf is estimated to be 1.68 eV. At thermal equilibrium, the concentration of vacancy is evaluated by the following equation:
 Wf C0 ¼ exp  : kB T

(6)

The reduction in the vacancy formation energy from 2.14 to 1.68 eV owing to the presence of hydrogen increases the vacancy concentration from 1.12  1036 to 6.00  1029. Although hydrogen increases seven orders of magnitude the vacancy concentration, the latter remains very low. Thus, the change in the vacancy concentration at thermal equilibrium does not affect the mechanical behavior of alpha iron. The mobility of a vacancy is estimated by calculating the activation energy for migration using the climbing image nudged elastic band method [44, 45] and VASP [23–25]. The activation energy increases from 0.62 to 1.07 eV when a vacancy traps two hydrogen atoms. This change in the activation energy reduces the diffusion frequency from 1.71  102 to 6.42  106 s1. Unlike the thermal equilibrium concentration of vacancy, the reduction in diffusion frequency may significantly affect the mechanical properties of alpha iron under hydrogen gas because the vacancy

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diffusion is assumed to be zero and the vacancy concentration in a specific plane where many dislocation activities occur is supersaturated, as discussed in Sect. 4.3.

3.3

Hydrogen Effect on the Cohesive Energy of Grain Boundaries

It has been experimentally reported that the strength of materials in hydrogen environment improves by controlling the distribution of the misorientation angles of GBs [46]. The hydrogen effect on the GBs in alpha iron has been evaluated by using interatomic potentials [34] and first-principles calculations [35, 36, 47]. Yamaguchi and Gesari evaluated the hydrogen binding energy and cohesive energy of GBs using DFT calculations and demonstrated that hydrogen embrittles GBs [36, 47]. Matsumoto et al. [35] estimated the hydrogen concentration at GBs in alpha iron under hydrogen gas using DFT calculations and obtained the cohesive energy of GBs by assuming that the hydrogen atoms trapped in GBs are adsorbed on the halves on the generated two surfaces. From the DFT calculations, the reduction in cohesive energy was estimated to be approximately 25% for Σ3(111) STGB (ϕ = 70.5 ) and almost zero for Σ3(112) STGB (ϕ = 109.5 ). In contrast, Yamaguchi et al. revealed that the hydrogen effect is much pronounced when considering mobile hydrogen that diffuses to the surface during fracturing [48].

3.4

Hydrogen Effect on Dislocation Mobility

Dislocation movement is the fundamental process of plastic deformation. Therefore, it has attracted much attention with respect to the HE process and mechanism, e.g., HELP. Taketomi et al. reported that the Peierls potential for the {112} edge dislocation is reduced in the presence of low hydrogen concentrations, 0.49/nm hydrogen atoms per unit length of dislocation line, i.e., approximately 300 K, 0.01 MPa hydrogen gas conditions from Sect. 3.1 [49]. Moreover, the interaction among neighboring dislocations, i.e., shear stress field, is almost the same even though the hydrogen concentration increases up to 7.35/nm [49]. To take into account the competition of dislocation movement and hydrogen diffusion, Taketomi et al. calculated the stress-dependent thermal activation energies for hydrogen diffusion and dislocation movement and estimated the dependence of the dislocation velocity on the applied shear stress [50]. The results demonstrated that the dislocation mobility increases (softening) with decreasing applied stress, whereas the dislocation mobility decreases (hardening) with moderate stress. For high applied stress, although the dislocation is pinned by hydrogen atoms at the beginning, the dislocation moves separately from hydrogen and does not interact during steadystate movement. Interestingly, stress-dependent softening or hardening only occurs at very low hydrogen concentrations. Taketomi et al. reported the hydrogen concentration effect on the Peierls potential [51], and the results suggested that only the hardening is dominant at high hydrogen concentration. Softening and hardening are also reported by Itakura et al. for screw dislocations using DFT calculations [31].

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3.5

Interaction Between Hydrogen and Individual Lattice Defects

In this section, the procedure to evaluate the hydrogen occupancy at each trap site is first introduced. Then, it is shown that vacancies, grain boundaries, and edge dislocations can trap hydrogen atoms. Subsequently, the hydrogen effects on these lattice defects are reviewed.

4

Lattice Defects and Hydrogen Embrittlement Mechanisms

4.1

Interaction Between Many Lattice Defects

Experimentally observed hydrogen embrittlement results from the interaction among lattice defects during deformation and fracturing. In this section, we review the phenomena caused by the interaction among the hydrogen-affected lattice defects and the competition among the various microscopic mechanisms using large-scale MD and DD simulations, where the hydrogen effects on each individual lattice defect are modeled based on Sect. 3.

4.2

Cracks and Dislocations

The contrasting behavior for individual dislocation movements is suggested by the atomistic calculations, as reviewed in Sect. 3.4. However, plastic deformation is attributed to the multiple slips of numerous individual dislocations. In particular, the dislocations generated from a crack tip dominate the crack propagation and fracture process. Taketomi et al. performed simple discrete DD calculations around a mode-II crack tip [52] based on the results obtained by the atomistic calculations to evaluate the many-body effects of dislocations around a stress singularity of a crack tip. In that study, the effect of solute hydrogen on dislocation emission is also taken into account that hydrogen enhances the dislocation emission from a crack tip [53]. Thus, the multiple effects of hydrogen concentration and applied stress conditions affect the dislocation movement ahead of the crack tip. For low hydrogen concentrations (CH = 0.49/nm along the dislocation line), the dislocation slip region is extended under small applied stress intensity factor rate conditions comparable to the absence of solute hydrogen atoms. The slip region decreases with increasing applied stress rate; however, it does not show significant differences compared to the no hydrogen conditions. On the contrary, for the high hydrogen concentration case (CH = 1.24/ nm along the dislocation line), the dislocation slip region is restricted near the crack tip owing to the hardening of the dislocation movement even when the solute hydrogen enhances the dislocation generation at the crack tip. The trend of dislocation localization is always realized for all ranges of applied stress intensity factor rate conditions. Considering the effects of hydrogen concentration and applied stress conditions on the dislocation movement, the elementary process of HE fracturing in an ideal material is summarized. The dislocation generation from a crack tip is the

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Fig. 5 Schematic of the transition of the HE fracturing mechanism for different hydrogen concentrations and applied stress intensity factor rate conditions for ideal materials

start of the plastic deformation and fracturing of a cracked ideal material. The dislocation slip behavior around a crack tip is enhanced at low hydrogen concentrations and low applied stress intensity factor rate conditions. The number of sliding dislocations increases in such conditions; thus, the plastic deformation around a crack tip increases and the fracture mechanism of HELP or HESIV would be dominant. The crack will blunt owing to the increased plastic deformation around a crack. At high hydrogen concentrations, in contrast, only hardening occurs independent of the applied stress intensity factor rate conditions. Therefore, the dislocation slip is restricted and the number of generated dislocations decreases. The crack shape becomes sharp and the hardening leads to brittle fracturing HEDE so far. The relations of the HE mechanisms for simplified ideal materials are shown in Fig. 5. The results strongly suggest that the HE mechanism changes depending on the combination of environmental and mechanical conditions even in the same material.

4.3

Supersaturation of Vacancy Concentration Associated with Plastic Deformation

Takai et al. showed that the vacancy concentration increases owing to plastic deformation under hydrogen conditions [54]. Jog dragging by a screw dislocation under shear stress was believed to be one of the mechanisms for the vacancy multiplication owing to plastic deformation. Matsumoto et al. estimated the vacancy distribution behind a jog, which is moving at constant velocity, using the hydrogenaffected physical properties of a vacancy obtained in Sect. 3.2 and the diffusion

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equation [27, 55]. According to this report, although the vacancy distribution spreads behind the jog for the no hydrogen case, the vacancy does not show distinct diffusion and stays in the form of a line behind the jog even for slow dislocation velocity (108 m/s). The jog dragging of the screw dislocation is one of the examples for vacancy generation. This analysis suggests that supersaturated vacancies would locally accumulate during plastic deformation in HE.

4.4

Reduced Lattice Defect Energy (Defactant-Like Effects)

The complex interactions among the lattice defects and the effect of hydrogen atoms on them affect HE. MD simulations can model the complex interactions among the various lattice defects. The effects of hydrogen mostly become large at small deformation speeds because the hydrogen distribution follows the stress distribution at low deformation rates and responds to the defect nucleation and motion [56–58]. Hydrogen diffusion in materials is definitely extremely fast compared to the diffusion of other elements, and it is well known that the hydrogen distribution follows the movement of lattice defects and the stress distribution during deformation. Nevertheless, the diffusion is too slow to model by direct MD simulations. Matsumoto et al. developed the interatomic potential with pseudohydrogen effects for alpha iron under hydrogen gas at 70 MPa and 300 K based on what is discussed in Sects. 2 and 3 [14] and analyzed the activity of many lattice defects and the interaction among them using large-scale MD simulations [59]. The interatomic potential incorporates the thermodynamic effects of hydrogen on the lattice defect energies, and thus, calculations of the hydrogen movement are not required. The kinetics of hydrogen and the time-dependent phenomena related to hydrogen diffusion are important in some cases; however, in this study, as a limiting case, the hydrogen-related phenomena appearing under equilibrium hydrogen distribution were examined. MD simulations of (i) crack propagation, (ii) nanoindentation, and (iii) tensile loading of a polycrystalline nanorod are performed, and the effects of hydrogen on the deformation and fracturing under these boundary conditions are discussed. The results are also compared for the cases of with and without hydrogen using the Mendelev potential for alpha iron [60]. The following conclusions were reached: (i) Crack growth simulations: The crack propagation behavior in alpha iron simulated with pseudohydrogen effects indicated that the material was extremely brittle, whereas crack propagation simulated without hydrogen effects occurred at a higher stress intensity factor after the emission of dislocations. This result corresponds to HEDE caused by the reduction in the surface energies. (ii) Nanoindentation: For the alpha iron simulated with pseudohydrogen effects, the pop-in load significantly decreased. Furthermore, dislocation loops were

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297

generated toward the four bottom corners. These phenomena are occasionally referred to as HELP. In addition, as the indentation progressed, several vacancies were generated throughout the sample, which agrees with the HESIV mechanism. (iii) Tensile loading of a polycrystalline nanorod: For alpha iron simulated with pseudohydrogen effects, the sample ruptured along GBs in a brittle manner, whereas in the simulations without hydrogen effects, intergranular fractures did not occur, but a large amount of dislocation activity was observed. The intergranular fracturing occurred because hydrogen significantly reduces the cohesive energy of GBs when the maximum effects of hydrogen mobility are incorporated. Although the MD simulation incorporated the hydrogen mobility effects through the interatomic potential with pseudohydrogen effects, it was shown that depending on the boundary conditions and initial material conditions, well-known hydrogen-related phenomena, i.e., HELP, HESIV, and HEDE, potentially appear despite the use of the same interatomic potential, i.e., same material system. Thus, the reduction in the defect formation energies leads to the well-known hydrogenrelated phenomena, and the defactant-like effect of hydrogen plays a significant role in HE.

5

Summary, Future Directions, and Remaining Problems

In this chapter, recent studies that treat the HE of alpha iron from an atomistic viewpoint are reviewed. In Sect. 2, the hydrogen trap energies of various trap sites, i.e., elastic strain field, vacancy, edge and screw dislocations, stacking faults, grain boundaries, and free surfaces, in alpha iron are reviewed first. In Sect. 3, the hydrogen occupancy at each trap site under hydrogen gas conditions is first examined; then, the hydrogen effect on the vacancy, grain boundary, and edge dislocations that are lattice defects that trap considerable hydrogen atoms under such conditions is reviewed. In Sect. 4, the interaction among hydrogen-affected lattice defects and the competition among the various microscopic mechanisms are discussed. As shown in this chapter, the evaluation of the hydrogen concentration in defects under given boundary hydrogen concentrations is critical to the atomistic and quantum simulations. Moreover, the HE is strongly affected by environmental (hydrogen concentration) and mechanical (deformation rate, applied stress, stress concentration) conditions. Thereby, we note that HE varies depending on the conditions even in the same materials. To fully understand hydrogen embrittlement, considering the competition of the various effects of hydrogen on lattice defects, including the time-dependent hydrogen diffusion, is critical. Lattice defects, which are dominant, are controlled by the boundary conditions, initial structure, and degree of deformation. Furthermore, the effect of added elements should also be clarified in a sequential manner.

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47. Gesari S, Irigoyen B, Juan A. Segregation of H, C and B to Σ=5 (013) α-Fe grain boundary: a theoretical study. Appl Surface Sci. 2006;253:1939–45. 48. Yamaguchi M, Kameda J, Ebihara K, Itakura M, Kaburaki H. Mobile Effect of hydrogen on intergranular decohesion of iron: first principles calculations. Philo Mag. 2012;92:1349–69. 49. Taketomi S, Matsumoto R, Miyazaki N. Atomistic simulation of the effects of hydrogen on the mobility of edge dislocation in alpha iron. J Mater Sci. 2008;43:1166–9. 50. Taketomi S, Matsumoto R, Miyazaki N. Atomic study of the competitive relationship between edge dislocation motion and hydrogen diffusion in alpha iron. J Mater Res. 2011;26:1269–78. 51. Taketomi S, Matsumoto R, Miyazaki N. Molecular statics study of the effect of hydrogen on edge dislocation motion in alpha-Fe. In: Somerday BP, Sofronis P, editors. Hydrogen-materials interactions: ASME, 2 Park Avenue, New York, NY 10016, USA; 2014. p. 765–70. 52. Taketomi S, Imanishi H, Matsumoto R, Miyazaki N. Dislocation dynamics analysis of hydrogen embrittlement in alpha iron based on atomistic investigations. In: Proceedings of 13th International Conference Fracture, ICF13, Beijing; 2013. p. 5721–9. 53. Taketomi S, Matsumoto R, Miyazaki N. Atomistic study of the effect of hydrogen on dislocation emission from a mode II crack tip in alpha iron. Int J Mech Sci. 2010;52:334–8. 54. Takai K, Shoda H, Suzuki H, Nagumo M. Lattice defects dominating hydrogen-related failure of metals. Acta Mater. 2008;56:5158–67. 55. Karthikeyan S. Evaluation of the jogged-screw model of creep in equiaxed gammma-TiAl: identification of the key substructural parameters. Acta Mater. 2004;52:2577–89. 56. Kotake H, Matsumoto R, Taketomi S, Miyazaki N. Transient hydrogen diffusion analyses coupled with crack-tip plasticity under cyclic loading. Int J Press Vess Pip. 2008;85:540–9. 57. Doshida T, Nakamura M, Saito H, Sawada T, Takai K. Hydrogen-enhanced lattice defect formation and hydrogen embrittlement of cyclically prestressed tempered martensitic steel. Acta Mater. 2014;61:7755–66. 58. Matsuoka S, Tanaka H, Homma N, Murakami Y. Influence of hydrogen and frequency on fatigue crack growth behavior of Cr-Mo steel. Int J Fract. 2011;168:101–12. 59. Matsumoto R, Seki S, Taketomi S, Miyazaki N. Hydrogen-related phenomena due to decreases in lattice defect energies—molecular dynamics simulations using the embedded atom method potential with pseudo-hydrogen effects. Comput Mater Sci. 2014;92:362–71. 60. Mendelev MI, Han S, Srolovitz DJ, Ackland GJ, Sun DY, Asta M. Development of new interatomic potentials appropriate for crystalline and liquid iron. Philos Mag. 2003;83:3977–94.

Molecular Dynamics Simulations of Nanopolycrystals

11

Christian Brandl

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nanocrystalline Sample Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Time and Length Scale Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Interatomic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Analysis of the Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mechanical Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Deformation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Grain Boundary-Mediated Mechanisms (Intercrystalline Mechanisms) . . . . . . . . . . . . . 4.2 Dislocation-Mediated Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Concluding Remarks and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Competitive or Cooperative Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Size Scaling of Dislocation Strain and Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

302 303 303 305 306 306 308 310 311 316 323 323 323 325 326

Abstract

Nanopolycrystals are polycrystalline metals with an average grain size below 100 nm and exhibit extraordinary strength values. In contrast to the coarsegrained polycrystals, the confinement by grain boundaries of the plastic deformation in the grains approaches limits, where the conventional theories break down. In the grain size regime 10–20 nm, molecular dynamics simulations play a crucial role to elucidate the possible and surprising deformation mechanisms in nanocrystalline metals although the MD method uses assumptions, which ad hoc do not allow for a straightforward extrapolation to experimental conditions. This C. Brandl (*) Institute for Applied Materials (IAM-WBM), Karlsruhe Institute of Technology, EggensteinLeopoldshafen, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_12

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chapter presents the nanocrystalline-specific methods for MD simulations with their subtleties and summarizes the deformation mechanisms in nanopolycrystals with their signatures on the (experimental) deformation behavior. A comprehensive understanding on the interplay of the deformation mechanisms is suggested in this chapter, which is closed with a list of still-open issues in the field.

1

Introduction

In polycrystalline metals, the grain size is a prominent and obvious microstructural length scale. The ability to tailor microstructural length scales, such as grain size, down to nanometer regime ( and

b2 ¼ ½ < α  α1 >

(1)

Similarly, dislocation with Burgers vector can dissociate according to < 111 > ¼ b3 ¼

b1

þ b3 þ

b1

ð1  αÞ < 110 >

where b1

¼

½ < αα1 >

and

(2)

In Eqs. 1 and 2, α varies between 0 and 1. In the specific case where α = 1, a screw dislocation on the {110} plane with Burgers vector dissociates in two partials with Burgers vector b1 separated by an antiphase boundary {110}.

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Fig. 6 Theoretical prediction of dislocation dissociation in B2 structure using the γsurface. (a) and (b) dislocation, respectively

Figure 6 shows the general schematic diagram for the dissociation of dislocation with Burgers vectors and on the plane {110}.

3.2.1 g-Surface in B2-Structure According to the data obtained by first-principles, and reported in [47], α is expected to be around 0.7 in both B2-Mg-Ag and B2-Mg-Y. As a consequence, the screw dislocation is expected to dissociate in two partials according to Eq. 1, and the screw dislocation on the (110) plane is expected to dissociate in three partials according to Eq. 2. To check the transferability of the semi-empirical potentials for which the ground state is a B2 structure when considering equal stoichiometry between Mg and alloying element, i.e., for Y and Ag, we calculated the γ-surface on the {110} plane, as plotted in Fig. 7. According to the γ-surfaces reported in Fig. 7a, screw dislocation is expected to dissociate in two partials according to Eq. 2 with α = 1. Such an analysis is not in agreement with the first-principles predictions of Tang et al. [47], which, according to the γ-surface, predicts the dissociation in three partials. For Mg-Ag, the γ-surface in the (110) plane obtained with the Mg-Ag potential proposed by Groh [10] is in agreement with the firstprinciples data of [47].

12

Modeling Dislocation in Binary Magnesium-Based Alloys Using Atomistic. . .

distance along , Å

a

349 0.8

3.5

0.7

3

0.6

2.5

0.5

2

0.4

1.5

0.3

1

0.2

0.5

0.1

0

0

1

2

3

4

0

5

distance along , Å

b 3.5

1 0.9

3 distance along , Å

0.8 2.5

0.7 0.6

2

0.5 1.5

0.4

1

0.3 0.2

0.5 0

0.1 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

distance along , Å

Fig. 7 (110) γ-surfaces in (a) B2-Mg-Y and (b) B2-Mg-Ag predicted by the Kim et al. [29] and Groh [10] potentials, respectively. The energies are given in J/m2

3.2.2 Dislocation in B2-Mg-Ag To validate the accuracy of the potential to model dislocations in B2-Mg-Ag, largescale molecular statics calculations were performed to reveal the structure of screw dislocations with Burgers vectors and on the plane {110}. After introducing a screw dislocation in a cylinder of radius R by displacing the atoms according to the displacement field of a screw dislocation derived from linear elasticity theory, the potential energy of the system was minimized while freezing the location of the atoms in the outer layers of the cylinder. The core structure of the screw dislocation with Burgers vector analyzed using the different displacement method displayed in Fig. 8 suggests the dissociation in two partials in agreement with the prediction derived from the (110) γ-surface. It should be noted that similar structure was recovered when the initial screw

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Fig. 8 Structure of the screw dislocation. Dark and light gray atoms represent Mg and Ag, respectively. The differential displacements exhibit the dissociation of a screw dislocation with Burgers vector in two partials

dislocation with Burgers vector was set on a plane (100). Since the Mg-Ag potential proposed by Groh [10] recovers accurately the experimental elastic constants and the stable stacking fault involved in the dissociation of the screw dislocation (see the g-surface in Fig. 7b), the separation distance of a21/2 between the two partials measured in Fig. 8 may be an accurate prediction of the model. Furthermore, by performing a series of molecular statics calculations under constant shear stress, a critical resolved shear stress for the screw dislocation of 250 MPa was found. Similarly, large-scale molecular statics calculations were performed to identify the structure of the {110} screw dislocation. For this configuration, a highly nonplanar core structure was found, and a critical resolved shear stress of 1.25 GPa at T = 0 K was predicted. Since the critical resolved shear obtained at 0 K for slip is lower than the one of slip, this result partially confirms the ability of the potential to accurately predict slip behavior in B2-Mg-Ag. However, a detailed analysis of the slip behavior at finite temperature is still needed to validate the ability of the potential in predicting the to slip transition occurring at low temperature as reported experimentally.

4

Summary and Future Directions

In this chapter, we presented an overview of the interactions between solute elements and microstructural defects such as stacking faults and dislocations with Burgers vector in the basal and prismatic slip systems using semi-empirical potentials for binary systems Mg-X (X = Al, Ca, Gd, Li, Sn, Y, Cu, Fe, Si, Ag). From the data reported in the previous sections, one can conclude on the accuracy and transferability of the available potentials for the Mg-X systems as follow: • Out of the ten potentials listed in Table 1, only one, the Mg-Ag potential proposed by Groh [10], fully captures the interaction between Ag solute element and

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351

stacking faults in the basal and prismatic planes. Such a result clearly demonstrates that a large material database that includes more information than only thermodynamics data and elastic constants has to be considered when optimizing new semi-empirical potentials for the Mg-X binary systems. • When considering solid solution strengthening, all the potentials listed in Table 1 qualitatively capture the interactions between solute elements and dislocations from the basal and prismatic slip systems. However, the magnitude of the maximum solute resistive force is overestimated by factor ranging from two to ten when compared to the first-principles data [37]. In addition, the chemistry of the alloying element on the strengthening obtained by first-principles, i.e., Li strengthened the least the basal slip system, while Ca strengthened the most the basal slip system, is not accurately captured by the existing semi-empirical potentials listed in Table 1, which make them not predictable for the design of new binary systems. However, since the interactions between solute elements and dislocations in the basal and prismatic slip systems are qualitatively captured, these potentials can be used to validate/disprove the scaling law between the strengthening and the maximum resistive force. When considering Mg-X systems for biomedical applications, only two systems are currently available in the modified embedded-atom method, i.e., Mg-Ca and MgAg. Since first-principles data including thermodynamics, elastic constant, γ-surface in the B2 structure, as well as changes in stacking fault energies due to the presence of solute element on the plane adjacent to the fault, a similar strategy to model these elements in the modified embedded-atom method than the one followed in [9] is envisioned. Finally, a basic understanding of the corrosion mechanisms occurring in magnesium-based alloys as well the interplay between the chemistry of the corrosion and the degeneration of the mechanical integrity is needed in order to control the time frame of the degradation of a magnesium-based implant. Although many earlier works have used classical, short-range (modified) embedded-atom method-based potentials, the chemistry of the corrosion and its consequences on the degeneration of the mechanical properties involve many complex physical and chemical processes between multiple neutral and ionized chemical species. To capture such processes accurately, a generalized and unified framework that treats multi-body component, metallic, covalent, ionic, and mixed bond materials, reliably and accurately is needed. Within this idea, Lee et al. [48] and Kong et al. [49] proposed simultaneously an extended version of the modified embedded-atom method that can describe metallic, covalent, and ionic bonding by combining the original second neighbors formulation of the modified embedded-atom method and the charge equilibration method. Since the Mg potential of Wu et al. [9] proposed in the modified embedded method is thought to be valuable for realistic atomistic studies of general deformation and failure problems, this potential can be applied without modification in the extended framework of Lee et al. [48] or Kong et al. [49] to develop semi-empirical potentials for the binary system Mg-Cl and ternary system

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Mg-O-H. Thus, the newly developed potentials will be valuable in view of revealing the degradation mechanisms in pure magnesium, as well as the coupling between the degradation of magnesium and the degeneration of the mechanical integrity. Revealing numerically the degradation mechanisms in pure magnesium will pave the way to control the degradation rate in a specific time frame by an alloying strategy. Acknowledgments The author acknowledges the MIRACLE Project at the University of Basel funded by the Werner Siemens Foundation, Zug/Switzerland.

Annex A Generalized Stacking Fault Energy Calculation The energy-displacement curve, also referred in the literature as generalized stacking fault energy curve (GSF), cannot be measured experimentally except for a single point known as the intrinsic stacking fault energy, γs. Since the GSF energy curve is an important property that controls the slip behavior, it is common to use the energydisplacement curve predicted by density functional theory to validate the ability of a semi-empirical potential in modeling dislocation core properties. As two sets of atomic planes glide rigidly with respect to each other, the energy per area of the faulted plane increases to γus and then decreases. For slip along the < 1010 > direction in the basal plane, it exists a position at which is the lattice is stable although the crystal is not its bulk equilibrium structure. Such stable configuration is an intrinsic stacking fault characterized by its energy γs. A representative energydisplacement curve obtained for slip along in the basal plane is given in Fig. 9 with the corresponding packing of basal planes and γs and γus labeled. Although several methods are available in the literature to calculate the generalized stacking fault energy, since the stable and unstable stacking fault energies in the basal and prismatic planes are not sensitive to the method, the slab method was used in this work. Based on the slab method, two rigid blocks of atoms are displaced with respect to each other in the plane of the fault. After each increment of displacement, the forces in the direction normal to the fault are relaxed using a conjugate gradient method. The atomic configuration is relaxed when the forces are lower than 105 eV/Å.

Annex B Interaction Energy Between Dislocation and Solute Element The interaction energy between a solute element A and a dislocation is calculated as the difference between the total energy of a system with a solute element located close to the dislocation and far from the dislocation given by the following formula: EAb, x ¼ EAx , dislo  Ef

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Fig. 9 The GSF energy curve for the direction in the basal plane in a hcp crystal. The packing of basal planes is given in the perfect crystal and with displacement of 0.17 and 0.33

where EAx , dislo represents the substitutional formation energy at the location x in the vicinity of the dislocation, and Ef is the bulk substitutional energy. In the limiting case, when the substitution occurs far away from the dislocation core, EAx , dislo = Ef, and thus the binding energy tends to 0. The substitutional formation energy at the location X in the vicinity of the dislocation is given by EAx , dislo ¼ Etotaln1 Mg

A1 , dislo

Mg

A  Etotal þ EMg c  Ec




Mg A , dislo where Etotaln1 1 is the total energy of the ensemble when one magnesium atom is substituted by a solute element A in a dislocated crystal, EMg total is the total energy of a A pure and dislocated Mg crystal, and EMg and E are the cohesive energies for Mg and c c the solute element A, respectively. The bulk formation energy is calculated using the following equation: Mg

EAf ¼ Etotaln1 Mg

A

X1

Mg X  EMg total þ Ec  Ec




where Etotaln1 1 is the total energy of the ensemble when one magnesium atom is substituted by a solute element A in a perfect crystal and EMg total is the total energy of a pure Mg perfect crystal. The resistive force is given as the absolute value of the binding energy gradient along the dislocation slip direction. Assuming the slip direction is along the xdirection, the resistive force is obtained as

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ΔEb F¼ Δx In this work, a dislocation was introduced in a crystal by displacing the atomic locations according to the dislocation displacement field derived from the isotropic linear elasticity theory. A cylinder of radius 80 Å was considered. Periodic boundary conditions were imposed along the dislocation line direction, whereas nonperiodic boundary conditions were applied in the other two directions. Atoms located in an inner cylinder with 65 Å of radius were allowed to relax in all three directions to minimize the potential energy of the system. The minimization of the potential energy was performed using a conjugate gradient algorithm. The atomic configuration is relaxed when the forces are lower than 105 eV/Å. A process window around the dislocation core was defined, in which substitution of Mg atom by solute element A occurs. Energy minimization was performed using the (M)EAM implementation of the LAMMPS package [50]. The Mg-Ca, Mg-Li, Mg-Sn, Mg-Y, and Mg-Al of Lee and coworkers were provided by B.J. Lee. The Mg-Li potential proposed by Karewar et al. [11] was provided by S. Karewar. The Mg-Al potentials proposed by Liu and Adams [25] and Mendelev et al. [12] were downloaded from Interatomic Potential Repository Project of the NIST.

References 1. Witte F. The history of biodegradable magnesium implants: a review. Acta Biomater. 2010;6:1680–92. 2. Friedrich HE, Mordike BL. Magnesium technology: metallurgy, design data, application. Berlin: Springer; 2006. 3. Hench LL, Polak JM. Third-generation biomedical materials. Science. 2002;295:1014–7. 4. Zreiqat H, Howlett CR, Zannettino A, Evans P, Schulze-Tanzil G, Knabe C, et al. Mechanisms of magnesium-stimulated adhesion of osteoblastic cells to commonly used orthopaedic implants. J Biomed Mater Res. 2002;62:175–84. 5. Harrison R, Maradze D, Lyons S, Zheng Y, Liu Y. Corrosion of magnesium and magnesiumcalcium alloy in biologically-simulated environment. Prog Nat Sci Mater Int. 2014;24:539–46. 6. Zheng YF, Gu XN, Witte F. Biodegradable metals. Mater Sci Eng R. 2014;77:1–34. 7. Wu Z, Curtin WA. The origin of high hardening and low ductility in magnesium. Nature. 2015;526:62–7. 8. Kim Y-M, Kim NJ, Lee B-J. Atomistic modeling of pure Mg and Mg-Al systems. CALPHAD Comp Coupl Phase Diagrams Thermo. 2009;33:650–7. 9. Wu Z, Francis MF, Curtin WA. Magnesium interatomic potential for simulating plasticity and fracture phenomena. Model Simul Mater Sci Eng. 2015;23:015004. 10. Groh S. Modified embedded-atom potential for B2-MgAg. Simul Mater Sci Eng. 2016;24:065011. 11. Karewar SV, Gupta N, Caro A, Srinivasan SG. A concentration dependent embedded atom method potential for the Mg-Li system. Comput Mater Sci. 2014;85:172–8. 12. Mendelev MI, Asta M, Rahman MJ, Hoyt JJ. Development of interatomic potentials appropriate for simulation of solid-liquid interface properties in Al-Mg alloys. Phil Mag. 2009;89:3269–85. 13. Sun DY, Mendelev MI, Becker CA, Kudin K, Haxhilali R, Asta M, et al. Crystal-melt interfacial free energies in hcp metals: a molecular dynamics study of Mg. Phys Rev B. 2006;73:024116.

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14. Witte F, Hort N, Vogt C, Cohen S, Kainer KU, Willumeit R, et al. Degradable biomaterials based on magnesium corrosion. Curr Opinion Solid State Mater Sci. 2008;12:63–72. 15. Xiao W, Zhang X, Geng WT, Lu G. Atomistic study of plastic deformation in Mg-Al alloys. Mater Sci Eng A. 2013;586:245–52. 16. Yi P, Cammarata RC, Falk ML. Atomistic simulation of solid solution hardening in Mg/Al alloys: examination of composition scaling and thermo-mechanical relationships. Acta Mater. 2016;105:378–89. 17. Shen L, Proust G, Ranzi G. An atomistic study of dislocation-solute interaction in Mg-Al alloys. IOP Conf Series Mater Sci Eng. 2010;10:012177. 18. Shen L. Molecular dynamics study of Al solute-dislocation interactions in Mg Alloys. Interaction Multiscale Mechanics. 2013;6:127–36. 19. Kim K-H., Jeon JB., Kim NJ., Lee B-J. Role of yttrium in activation of slip in magnesium: an atomistic approach. Scr Mater 2015; 108:104–8. 20. Nahhas MK, Groh S. Atomistic modeling of grain boundary behavior under shear conditions in magnesium and magnesium-based binary alloys. J Phys Chem Solids. 2018;113:108–18. 21. Miyazawa N, Yoshida T, Yuasa M, Chino Y, Mabushi M. Effect of segregated Al on {10–12} and {10–11} twinning in Mg. J Mater Res. 2015;30:3629–41. 22. Bhatia MA, Mathaudhu SN, Solanki K. Atomic-scale investigation of creep behavior in nanocrystalline Mg and Mg-Y alloys. Acta Mater. 2015;99:382–91. 23. Reddy R, Groh S. Atomistic modeling of the effect of calcium on the yield surface of nanopolycrystalline magnesium-based alloys. Comput Mater Sci. 2016;112:219–29. 24. Karewar S, Gupta N, Groh S, Martinez E, Caro A. Srinivasan S.G. Effect of Li on the deformation mechanisms of nanocrystalline hexagonal close packed magnesium. Comp Mater Sci. 2017;126:252–64. 25. Liu W-Y, Adams JB. Grain-boundary segregation in Al-10%Mg alloys at hot working temperatures. Acta Mater. 1998;46:3467–76. 26. Zhou XW, Johnson RA, Wadley HGN. Misfit-energy-increasing dislocations in vapor-deposited CoFe/NiFe multilayers. Phys Rev B. 2004;69:144113. 27. Pei Z, Zhu LF, Friák M, Sandlöbes S, von Pezold J, Sheng HW, et al. Ab initio and atomistic study of generalized stacking fault energies in Mg and Mg-Y alloys. New J Phys. 2013;15:043020. 28. Kim Y-M, Jung I-H, Lee B-J. Atomistic modeling of pure Li and Mg-Li system. Model Simul Mater Sci Eng. 2012;20:035005. 29. Kim K-H, Jeon JB, Lee B-J. Modified embedded-atom method interatomic potentials for Mg-X (X = Y, Sn, Ca) binary systems. CALPHAD Comp Coupl Phase Diagrams Thermo. 2015;48:27–34. 30. Groh S. Mechanical, thermal, and physical properties of Mg-Ca compounds in the framework of the modified embedded-atom method. J Mech Behav Biomed Mater. 2015;42:88–99. 31. Moitra A, Kim S-G, Horstemeyer MF. Solute effect on basal and prismatic slip systems of Mg. J Phys Condens Matter. 2014;26:445004. 32. Vitek V. Intrinsic stacking faults in body-centred cubic crystals. Philos Mag. 1968;18:773–86. 33. Tsuru T, Udagawa Y, Yamaguchi M, Itakura M, Kaburaki H, Kaji Y. Solution softening in magnesium alloys: the effect of solid solutions on the dislocation core structure and nonbasal slip. J Phys Condens Matter. 2013;24:02202. 34. Alam M, Groh S. Dislocation modeling in bcc lithium: a comparison between continuum and atomistic predictions in the modified embedded atoms method. J Phys Chem Solids. 2015;82:1–9. 35. Rice J. Dislocation nucleation from a crack tip: an analysis based on the Peierls concept. J Mech Phys Solids. 1992;40:239–71. 36. Groh S, Alam M. Fracture behavior of lithium single crystal in the framework of (semi-) empirical force field derived from first-principles. Model Simul Mater Sci Eng. 2015;23:045008. 37. Yasi JA, Hector LG Jr, Trinkle DR. First-principles data for solid-solution strengthening of magnesium: from geometry and chemistry to properties. Acta Mater. 2010;58:5704–13.

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38. Muzyk M, Pakiela Z, Kurzydlowski KJ. Generalized stacking fault energy in magnesium alloys: density functional theory calculations. Scr Mater. 2012;66:219–22. 39. Shang SL, Wang WY, Zhou BC, Wang Y, Darling KA, Kecskes LJ, et al. Generalized stacking fault energy, ideal strength and twinnability of dilute Mg-based alloys: a first-principles study of shear deformation. Acta Mater. 2014;67:168–80. 40. Zu G., Groh S. Effect of segregated alloying element on the intrinsic fracture behavior of Mg. Theo Appl Frac Mech. 2016;85:236–45. 41. Fleischer RL. Substitutional solution hardening. Acta Metall. 1963;11:203–9. 42. Labusch R. A statistical theory of solid solution hardening. Phys Status Solidi. 1970;41:659–69. 43. Leyson GPM, Curtin WA. Friedel vs Labusch: the strong/weak pinning transition in solute strengthened metals. Phil Mag. 2013;93:2428–44. 44. Groh S, Marin EB, Horstemeyer MF, Bammann DJ. Dislocation motion in magnesium: a study by molecular statics and molecular dynamics. Model Simul Mater Sci Eng. 2009;17:075009. 45. Yamagata T. Correlation between characters of dislocations and operative slip systems in CsCl type intermetallic compounds. J Phys Soc Japan. 1978;45:1575–82. 46. Yamaguchi M, Umakoshi Y. The deformation behaviour of intermetallic superlattice compounds. Prog Mater Sci. 1990;34:1–148. 47. Tang P-Y, Wen L, Tong Z-F, Tang B-Y, Peng L-M, Ding W-J. Stacking faults in B2-structured magnesium alloys from first principles calculations. Comput Mater Sci. 2011;50:3198–207. 48. Lee E, Lee K-R, Baskes MI, Lee B-JA. Modified embedded-atom method interatomic potential for ionic systems: 2NNMEAMþQeq. Phys Rev B. 2016;94:144110. 49. Kong F, Longo RC, Zhang H, Liang C, Zheng Y, Cho K. Charge-transfer modified embeddedatom method for manganese oxides: nanostructuring effects on MnO2 nanorods. Comput Mater Sci. 2016;121:191–203. 50. Plimpton S. Fast parallel algorithms for short-range molecular dynamics. J Comput Phys. 1995;117:1–19.

Atomistic Simulation Techniques to Model Hydrogen Segregation and Hydrogen Embrittlement in Metallic Materials

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Douglas E. Spearot, Rémi Dingreville, and Christopher J. O’Brien

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Proposed Mechanisms for Hydrogen Embrittlement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Role of Atomistic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Survey of Prior Molecular Dynamics Simulation Studies of H Embrittlement . . . . . . . . . . . . 2.1 Atomistic Simulation Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hydrogen-Influenced Crack-Tip Behavior in Single Crystals . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hydrogen-Induced Grain Boundary Decohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Atomistic Simulation-Inspired Models for H Embrittlement . . . . . . . . . . . . . . . . . . . . . . . . 3 Hydrogen Segregation at Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Site-Energy Selection Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Statistical Mechanics Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Discussion and Comparison of Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Steady-State Crack Propagation Approach to Study H Embrittlement . . . . . . . . . . . . . . . . . . . . 4.1 Creation of a Hydrogenated Grain Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Steady-State Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Cohesive Zone Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Grain Boundary Decohesion for Σ 3 Grain Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Dislocation Nucleation During Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Influence of Hydrogen on Decohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D. E. Spearot (*) University of Florida, Gainesville, FL, USA e-mail: dspearot@ufl.edu R. Dingreville · C. J. O’Brien Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected]; [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_14

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Abstract

Hydrogen embrittlement is an important phenomenon where the mechanical properties of a metallic material are degraded in the presence of hydrogen, sometimes leading to a change in the failure mode of the metallic material. Although mechanical failures due to hydrogen embrittlement have been observed for over a century, the atomic-level mechanisms associated with the hydrogen embrittlement process are still under debate. In this chapter, atomistic simulation efforts focused on hydrogen segregation and hydrogen embrittlement are reviewed. Atomistic simulation methods provide a nanoscale modeling technique capable of studying the role of hydrogen atoms on dislocation nucleation, crack propagation, and grain boundary decohesion. Examples are provided in this chapter of the use of a site-energy selection method to study hydrogen segregation and molecular dynamics simulations to study hydrogen-induced grain boundary decohesion in nickel. Grain boundary strength and work of separation in the presence of segregated hydrogen are computed from the molecular dynamics simulations. Subsequently, this data may be used in higher length scale models and simulations of the hydrogen embrittlement process.

1

Introduction

Hydrogen embrittlement (HE) refers to the degradation of mechanical properties in metals, often leading to premature fracture, due to the introduction and subsequent diffusion of hydrogen. Hydrogen embrittlement may promote sudden and catastrophic failure of structural components under normally safe working loads. The deleterious effects of hydrogen in metals have been well recognized for more than a century [1]. For example, it has been reported that approximately 25% of equipment failures in the petrochemical industry can be attributed to hydrogen effects [2]. The main source of hydrogen leading to HE generally comes from manufacturing processes (e.g., welding or heat treatment), cleaning operations like prickling or phosphating, electrochemical processes used in the application of protective coatings, or corrosive and hydrogen-rich environments (e.g., storage equipment). Upon exposure, hydrogen atoms first accumulate and adsorb onto the metal surface giving rise to a high concentration gradient between the surface and the bulk. This high concentration on/just beneath the surface (“chemisorbed hydrogen”) provides one necessary driving force for the hydrogen to absorb and diffuse into the metal. Temperature gradients and hydrostatic stress field gradients also provide necessary driving forces. Upon absorption, hydrogen occupies and diffuses between interstitial lattice sites and can be trapped to various microstructural sites such as solute atoms, defect clusters, dislocations, grain boundaries, and precipitate/matrix interfaces. Various mechanisms take actions at these sites leading to HE. Due to both its importance in industrial applications and advances in modeling and characterization techniques [3], a large number of studies have shown that hydrogen embrittlement occurs in a wide range of structural materials including steel [4], aluminum [5], zirconium [6], and titanium [7], to name a few. Due to the

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high mobility of hydrogen in iron compared to other structural materials, HE can be particularly severe in steel. Many variables, such as hydrogen pressure, temperature, strain rate, crystal structure, and microstructural features (impurities, dislocations, grain boundaries, etc. that provide localized stress concentrations), affect material susceptibility to HE. For example, increasing the hydrogen content or the temperature increases HE susceptibility, although there can be a saturation threshold in both cases above which little to no change in the degree of embrittlement is observed. In addition, HE susceptibility decreases with increasing strain rate [8]. Other impurities in materials can enhance HE, but other impurities are not necessary for embrittlement to occur [9]. Indeed, microstructures with strong traps (such as large inclusions) can act as preferential sites for high concentrations of hydrogen and as preferred sites for crack initiation.

1.1

Proposed Mechanisms for Hydrogen Embrittlement

Despite the conspicuous amount of work on the topic, the mechanisms proposed to explain premature failure due to hydrogen are still under debate [10]. The main quandary regarding the mechanisms is in determining their role in the process of HE and when they occur in association with one another. Their relative importance depends on the different loading modes, hydrogen charging conditions, temperature, and how the evolving microstructure interacts with the hydrogen solutes. Nevertheless, the basic underlying principle consists of hydrogen atoms accumulating in local regions with high stress concentrations leading to an acceleration of microcrack propagation. Four general mechanisms have been proposed and are briefly described in the following sections: (i) formation of a hydride phase, (ii) enhanced decohesion (interatomic bond weakening) along grain boundaries, (iii) enhanced local plasticity, and (iv) an adsorption mechanism [10, 11].

1.1.1 Hydride Formation and Cracking A mechanism based on the formation and fracture of hydrides at crack tips was first proposed by Westlake in the late 1960s [12]. As illustrated in Fig. 1, this mechanism consists of the repeated sequence of (i) diffusion of hydrogen to the crack tip where the hydrostatic tensile stress is high (Fig. 1a), (ii) brittle hydride phase nucleation and

Fig. 1 Hydride formation and cracking: (a) hydrogen diffuses to the hydrostatically stressed region at the crack tip; (b) brittle hydride is formed; (c) hydride fracture and crack propagation.

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growth (Fig. 1b), and (iii) cleavage fracture of the hydride phase (Fig. 1c). Obviously, the nucleation and growth of the hydride phase can only occur for temperatures and loading rates for which hydrogen has sufficient time to diffuse to the crack tip and for which the hydride phase is stable [13]. Hydride fracture typically applies only to materials in which hydride formation is favorable. This predominantly occurs in group-Vb transition materials and their alloys such as V, Zr, Nb, Ta, and Ti [12, 14].

1.1.2

Hydrogen-Induced Decohesion (HID) and Hydrogen-Enhanced Decohesion (HEDE) First proposed by Pfeil [15], this mechanism considers the charge transfer and weakening of interatomic bonds between hydrogen and the host metal [16]. The decohesion occurs as a sequential separation of atoms ahead of the crack tip (Fig. 2) as a result of the hydrogen-induced lowering of the surface energy of atomic planes [17, 18] encouraging a more cleavage-like failure. However, due to the long-ranged atomic interactions, the separation process can be more complex requiring coupled atomic shuffling to enable a critical crack-tip-opening displacement to be achieved [19]. This mechanism can occur only if the energy stored within the system is below the value at which dislocation emission occurs. Decohesion mainly occurs at specific sites ahead of a crack tip as a result of both a high concentration of hydrogen and a high concentration of stresses. The hydrogen trapped at specific sites along grain boundaries can also be sufficient to weaken atomic bonds in a metal leading to decohesion. Although direct experimental evidence of the HID/HEDE mechanism is yet to be obtained [20], its role in HE is supported by observations of failure without significant plastic deformation [21], reduction of fracture strength [22], and thermodynamic considerations [23, 24]. Fig. 2 HEDE mechanism: hydrogen atoms weaken the interatomic bonds.

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1.1.3 Hydrogen-Enhanced Localized Plasticity (HELP) Via fractography observations, hydrogen-enhanced localized plasticity (HELP) was first proposed by Beachem [25]. The HELP mechanism considers HE to be caused by an increase in plasticity in a localized region ahead of a zone with high hydrostatic stresses (e.g., crack tip). The high stress fields promote the localization of high concentrations of hydrogen, which in turn facilitates an increase in dislocation nucleation and mobility by shielding elastic obstacles (Fig. 3). Additionally, dislocation motion is favored by hydrogen-induced reduction of the stacking fault energy allowing for an increase in separation distance of Shockley partial dislocations [26]. Transmission electron microscopy (TEM) observations of slip bands and localized plastic-like deformation at crack tips in the presence of hydrogen [27, 28] have offered clues on the connection between the HELP mechanism and HE. 1.1.4 Adsorption-Induced Dislocation Emission (AIDE) The adsorption-induced dislocation emission (AIDE) mechanism involves both the nucleation and subsequent motion of dislocations away from the crack tip and also the formation and growth of microvoids (Fig. 4). The adsorption of hydrogen at surface and subsurface trapping sites promotes the nucleation of dislocations at crack surfaces but also weakens the interatomic bonds over several atomic distances. The nucleation of dislocations from the crack surface causes a back-stress hindering more dislocations to nucleate. In turn, this favors the formation and growth of microvoids ahead of the crack tip. These voids can form at slip band intersections or at secondphase particles present in the plastic zone. The subsequent crack growth occurs not only through the emission of dislocations but also through the growth and coalescence of these microvoids ahead of the crack tip. Crack paths resulting from the AIDE process can be intergranular or transgranular depending on where dislocation and void activities occur. The AIDE mechanism has been experimentally observed through metallography and fractography where dimpled fracture surfaces have been

Fig. 3 HELP mechanism: hydrogen atoms increase dislocation nucleation and mobility leading to a localized region of high dislocation density ahead of the crack tip.

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Fig. 4 AIDE mechanism: hydrogen atoms promote dislocation emission leading to the formation and growth of microvoids ahead of the crack tip.

observed [29, 30]. Other observations supporting the adsorption mechanism include surface reconstruction in the presence of hydrogen [31, 32].

1.1.5 Combination of Mechanisms Even though no singular or unified mechanism for HE has been proven, hydrogen plays somewhat different roles in each proposed mechanism to promote subcritical fracture. It is therefore most likely that these mechanisms can occur simultaneously and even support each other [11]. For example, microvoid coalescence can be caused by either HELP, via slip-plane intersections, or HEDE, via hydrogen at interstitial sites. Alternatively, decohesion produced from HEDE can act as a crack-tip sharpener, allowing for dislocation emission in the AIDE mechanism from the sharp crack tip. Since many combinations can be composed to form the effects of HE during fracture, it is hard to experimentally identify precisely which type of mechanism or combination of mechanisms caused a fracture in a material suffering from hydrogen exposure.

1.2

Role of Atomistic Simulations

Experiments can be performed on both macroscopic specimens and in situ at the scale of individual dislocations to study how H influences plasticity and fracture. Prior experimentation has provided important observations related to the proposed models at macroscopic and microscopic length scales. However, the underlying atomic processes are still regularly debated, and cooperation between mechanisms remains uncertain. Fundamentally, each model for hydrogen embrittlement argues that hydrogen atoms influence decohesion and/or plasticity at the atomic level. To provide this atomic-scale resolution, atomistic simulations are necessary to complement experiments [3]. Thus, the primary objective of this chapter is to provide a review of select atomistic simulation efforts focused on the effects of hydrogen in metallic materials. Generally, the advantage of atomistic simulations is that individual mechanisms can be isolated in a way not possible with experiments. Simulations may be performed to

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study the role of hydrogen atoms on the stacking fault energies in materials, the grain boundary energies, the discrete motions of hydrogen atoms and their interaction with trapping sites, or the mechanisms associated with dislocation nucleation from crack tips. In addition, atomistic simulations provide data for higher length scale simulations (discrete dislocation dynamics, constitutive models for finite element simulations, etc.) and provide basis and/or parameterizations for mathematical models for material behavior. Thus, atomistic simulations are often foundational in multiscale modeling efforts focused on hydrogen embrittlement. This chapter is organized as follows. Section 2 provides a brief review of the atomistic simulation method, including the interatomic potentials commonly used to model Ni-H and Fe-H systems, and summarizes select results regarding the role of hydrogen in embrittlement mechanisms. Section 3 provides a discussion of atomistic modeling techniques used to model hydrogen segregation, including advantages and disadvantages of using site selection or statistical approaches to construct atomistic simulation models with hydrogen segregated to grain boundaries. An example is provided of the use of a site-energy selection technique to model hydrogen segregation at a nickel grain boundary. Sections 4 and 5 provide an example of the role of hydrogen on grain boundary decohesion during steady-state crack propagation. The peak strength and work of separation of a grain boundary as a function of hydrogen coverage are extracted using a statistical mechanics based approach.

2

Survey of Prior Molecular Dynamics Simulation Studies of H Embrittlement

This section presents a brief review of the atomistic simulation method, including a focused discussion of the interatomic potentials commonly used to model Ni-H and Fe-H systems. This is followed by a review of prior atomistic simulation efforts focused on (i) hydrogen-induced changes to crack-tip behavior in single crystals, (ii) crack propagation along grain boundaries in H-charged samples, and (iii) a new model for H embrittlement motivated by atomistic simulations.

2.1

Atomistic Simulation Background

Atomistic simulation refers to a suite of computational techniques used to model the interaction and configuration of a system of atoms [33, 34]. This suite includes molecular mechanics (energy minimization) calculations, molecular dynamics (MD) simulations, and atomistic Monte Carlo (MC) simulations. For problems that involve defects with long-range distortion fields, such as dislocation nucleation and crack propagation, the use of ab initio methods based on density functional theory is generally prohibited because the computational cost of the method prevents the user from modeling a sufficient volume of material around the defect without influence from the boundary conditions. Thus, interatomic potential based

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(sometimes referred to as “classical”) atomistic simulations are commonly used to study the role of hydrogen on plasticity and fracture. In the classical atomistic simulation framework, each atom is represented as a point mass in space, and an interatomic potential is defined to provide a model for the potential energy of a system of atoms. The majority of atomistic simulations focused on hydrogen embrittlement employ the embedded-atom method (EAM) or modified embedded-atom method (MEAM) interatomic potentials. Daw and Baskes [35, 36] developed the EAM to describe atomic bonding in metallic systems, with an initial focus on face-centered cubic (FCC) transition metals [37]. To approximate the potential energy of a set of atoms, the EAM includes both pair interactions between nuclei of atoms i and j and the embedding energy as a function of the local background electron density around atom i: U¼

X i

   1 X X   Gi ρh, i r ij þ ϕ r ij : 2 i jðj6¼1Þ ij

(1)

Here, Gi is the embedding energy functional, ρh,i is the background electron density function at the host site due to neighbors of atom i, ϕij is the pair interaction function, and rij is the distance between atoms i and j. In the EAM, the embedding energy is assumed to depend solely on the local background electron density provided by the surrounding atoms and its lower derivatives, which is calculated using a nondirectional summation of the atomic electron densities: ρh, i ðr ij Þ ¼

X jðj6¼iÞ

ρj ðr ij Þ,

(2)

where ρj is the electron density provided by atom j to host site i. This summation is performed over neighboring atoms within a specified cutoff distance. Since the background electron density is a local quantity, the EAM is applicable for examining systems with crystalline defects, such as dislocations and grain boundaries. For modeling Ni-H, EAM potentials have been developed by several authors including Daw and Baskes [35, 36], Ruda et al. [38], and Angelo et al. [39, 40]. The parameters used to define the EAM functions are determined via fitting the atomistic simulation results to critical material properties. For example, the EAM potential for Ni developed by Daw and Baskes [35, 36] was fit to the lattice parameter, elastic constants, surface and point defect energies, and the energy difference between phases. For Ni-H, Daw and Baskes [35, 36] included the heat of solution and migration energy of H in Ni. Angelo et al. [39, 40] used the same basic properties but add the stacking fault energy in Ni and adsorption energies for H on Ni surfaces to the fitting database. They showed that their EAM potential is capable of predicting the energy associated with trapping of H at dislocation cores and at planar defects in Ni, and thus their potential provides a more accurate description of the role of H on plasticity and fracture. For modeling Fe-H, the modified embedded-atom method potential is commonly employed [41]. The primary difference between the MEAM and the EAM is that in the MEAM, the background electron density summation (Eq. 2) is augmented by

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Atomistic Simulation Techniques to Model Hydrogen Segregation and Hydrogen. . . 365

additional angular-dependent terms. This enables more accurate modeling of nonclosed-packed metals, especially regarding the prediction of surface energies and stoichiometric compounds [42]. For example, Lee and Jang [41] used ab initio and experimental data on the dilute heat of solutions of H in BCC and FCC phases of Fe, vacancy binding energies, and a hypothetical Fe-H hydride phase to parameterize a MEAM potential for the Fe-H system. In the atomistic Monte Carlo method, atoms are perturbed from their current configuration (either in position or by adding/removing an interstitial H atom for the hydrogen segregation problem), and the interatomic potential is used to compute the energy difference between original and perturbed states. Using the magnitude of this energy difference, the new configuration is either retained or discarded, as will be explained in detail in Sect. 3. Thus, in the Monte Carlo method, configurations are sampled until equilibrium is reached, and properties are computed as averages over a number of equilibrium configurations. For energy minimization and molecular dynamics simulations, it is necessary to calculate the atomic force on every atom in the model. The force vector on a given atom, Fi, is related to the interatomic potential, U, via: Fi ¼ 

@Uðr N Þ : @ri

(3)

Here, ri is the atomic position vector, and the interatomic potential U is computed using all neighbor atoms N with separation distance rN. During energy minimization, atoms are iteratively moved using the force vectors as directional guidance to search for the minimum energy configuration of the ensemble of atoms [43]. Typically, either steepest decent or nonlinear conjugate gradient methods are used to search for the atomic positions associated with the minimum potential energy of a system. In the molecular dynamics method, the evolution of atomic positions and velocities in time is described using a prescribed set of equations of motion. Employing the concept of an extended system [44], the equations of motion can be coupled to external temperature and pressure baths [45], allowing for MD simulations to be performed at a desired temperature and pressure (or applied stress). Properties computed from MD simulations should be determined using averages over a very large number of atoms or a large number of time steps.

2.2

Hydrogen-Influenced Crack-Tip Behavior in Single Crystals

One of the first applications of the EAM was to study hydrogen embrittlement. Daw and Baskes [35, 36] created an initial crack between (111) planes and studied the reduction in the stress necessary for crack propagation when loaded in the [111] direction due to the presence of hydrogen. These initial simulations were very small ( 150 atoms), and thus periodic boundary conditions and confinement due to the loading surfaces precluded dislocation nucleation from the crack tip, demanding that brittle behavior occurs. They reported that their EAM potential correctly predicted

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that H would be energetically bound to the defect sites and that a single H atom could reduce the fracture stress by as much as 15%. Hoagland and Heinisch [46] developed a quasi-3D cylindrical model of a crack tip to study the influence of a single H atom on mode I crack propagation in bulk Ni. Two different lattice orientations were studied: (i) a brittle model with [100] crack propagation  and (010) crack plane and (ii) a ductile model with [111] crack direction on the 010 plane. For the brittle model, the crack extended without dislocation emission and the presence of H caused the crack to extend at a lower value of stress intensity, KI, than without hydrogen. Crack extension was driven primarily by a significant decrease in energy as the H atom escaped from a point ahead of the crack tip to the free surface through crack extension and to a lesser extent by the dilatational stress field due to the H interstitial in the Ni lattice. For the ductile model, crack extension again was driven by a decrease in energy as the H atom moved from the crack-tip region to a free surface in the wake of the crack. However, for this orientation, dislocation emission also occurred from the crack tip, with a partial dislocation emerging from the crack surface containing the H atom first, implying that H can enhance dislocation emission from the crack tip if positioned appropriately relative to the activated slip systems. Extending beyond a single H atom, several authors have performed molecular dynamics simulations to study the role of H concentration on fracture. Hu et al. [47] focused on HE in α-Fe, while both Xu et al. [48] and Wen et al. [49] studied HE in Ni. Qualitatively, each study showed that the fracture response of a single-edge notch specimen became more brittle with increasing H charging. This was caused by “suppressed and localized” plastic deformation [48]. The models employed by Hu et al. [47] and Xu et al. [48] were very small, and periodic boundary conditions were not used; thus, the initial crack length was a significant fraction of the total simulation cell dimensions, promoting surface image force effects. Acknowledging the length scale issue, Wen et al. [49] performed similar simulations using a much larger nonperiodic simulation cell and considered different H distributions, as shown in Fig. 5. When H was located along a {100} plane in front of the crack tip in Ni, a reduction in ductility was observed, and they concluded that since H was not distributed along the active {111} slip planes, that dislocation nucleation from the crack tip during displacement-controlled loading was not influenced by the H. Thus, the reduction in fracture strength must be due to a weakening of the cohesive force between atomic bonds. When H was seeded along the active {111} slip planes, dislocation emission was enhanced compared to that in a H-free sample. Regardless, Wen et al. [49] concluded that fracture was still caused by a weakening of Ni bonds leading to microvoid nucleation and coalescence along the slip planes. More recently, Solanki et al. [50] examined the effect of H on elastic modulus, surface energies, and dislocation nucleation from a mode I crack tip. They reported that the elastic modulus of Ni and the surface energies of low-index Ni {100}, {110}, and {111} planes decrease with increasing H concentration. Thus, based on these two results, the stress intensity necessary for cleavage decreases with increasing H concentration. During mode I loading of the crack tip, the presence of H ahead of the crack tip lowered the stress necessary for dislocation nucleation relative to a H-free model.

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Atomistic Simulation Techniques to Model Hydrogen Segregation and Hydrogen. . . 367

a

b D.I. 2.00

1.00 0.58 0.00

c

d

Fig. 5 Dislocation emission from a crack tip in specimens charged with 8000 H atoms [49]: (a) at octahedral sites ahead of the crack tip, (b) along {111} slip planes, (c) randomly distributed in a small region around the crack tip, and (d) randomly distributed throughout the entire model (Reprinted from [49] with permission from Elsevier).

Concurrently, Song et al. [51] developed an analytical model for crack-tip behavior which considered the competition between the critical stress intensity necessary for cleavage and the critical stress intensity necessary for dislocation emission. As input to this model, the fracture energies for cleavage (surface energies) and the stacking fault energies for dislocation nucleation (unstable) were assumed to be functions of hydrogen concentration. Atomistic simulations were performed to determine how hydrogen concentration influences these fault energies in the Ni-H system; their results are shown in Fig. 6. Surface energies decreased with increasing H coverage, in agreement with [50], while unstable and intrinsic stacking fault energies both increased with increasing H concentration. This implies that dislocation nucleation would require a higher applied stress intensity factor with increasing H concentration, in disagreement with prior simulation results [46, 50]. Song et al. predicted that a transition from ductile to brittle material behavior would occur at a critical hydrogen concentration and that the specific concentration depends on the cleavage plane. They assessed their predictions via direct simulation using a coupled atomistic/discrete dislocation (CADD) simulation method at low temperature. Their simulation results showed that the role of H could not be completely described via the brittle-to-ductile transition model. For example, for orientations where crack-tip dislocation nucleation was favorable and at a H concentration predicted by the model to be brittle, dislocation nucleation still occurred preventing brittle fracture.

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a

4800 4000

Γ (mJ/m2)

Fig. 6 (a) Surface energies and (b) stacking fault energies as a function of hydrogen concentration from Song et al. [51]. Surface energies are computed using two different approaches which provide very similar results at low H concentrations (Reprinted from [51] with permission from IOP Publishing).

3200 2400

{100} Jiang and Carter {111} Jiang and Carter {110} Jiang and Carter {100} Van der Ven et al. {111} Van der Ven et al. {110} Van der Ven et al.

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2.3

Hydrogen-Induced Grain Boundary Decohesion

Since grain boundaries serve as energetically preferential sites for H impurities, several authors have recently explored the role of segregated H on grain boundary embrittlement. Some of the first work in this area was done by Chandler et al. [52] using Monte Carlo and molecular dynamics simulations. Specifically, Monte Carlo simulations were used to randomly introduce H into low-angle and high-angle grain boundary models. Then, MD simulations were used to impose a tensile strain until void nucleation occurred. They reported [52] that dislocation nucleation and work hardening prior to void nucleation were not greatly influenced by the presence of hydrogen. However, the void nucleation stress was significantly reduced by the presence of H, on the order of 30% lower, with voids nucleating at the trap regions for H in the grain boundary. Thus, they concluded that a HEDE model was appropriate to describe the role of H on void initiation at grain boundaries. More recently, Kuhr et al. [53] studied the role of hydrogen on the stress-strain response of nanocrystalline Ni samples. They reported that H can perturb the structure of the grain boundaries and that diffusion of H along the grain boundary planes can promote clustering in pockets of grain boundary free volume, as shown in

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a 2 3 4 5 6

b

Burgers vector magnitude (lattice units) 1.22 1 0.75

/2

0.5

/6 0.25

0.136

Fig. 7 (a) Clustering of H atoms along a selected grain boundary and (b) crack initiation at a triple junction with Shockley partial dislocations shown in yellow from Kuhr et al. [53] (Reprinted from [53] with permission from Elsevier).

Fig. 7a. During high-rate tensile deformation, samples which contained H had a higher yield strength (via 0.2% offset method), in agreement with the increase in dislocation nucleation energy barrier reported by Song et al. [51], but once dislocation nucleation began at approximately 2.0% strain, dislocations were emitted from the majority of grain boundaries at a faster rate in H-containing nanocrystalline samples. Cracks were observed to nucleate along grain boundaries starting at the triple junctions between grains, as shown in Fig. 7b. Kuhr et al. concluded that H does not influence crack initiation, since crack initiation appears at approximately the same strain level regardless of H concentration, but the crack propagation rate was increased due to H coverage along the grain boundary. This is in disagreement with the work of Chandler et al. [52] where initiation occurred at much lower stresses in the presence of H. Ultimately, the conclusions drawn by both works must be placed in the context of grain boundary structure; the details of the specific grain boundary where crack/void initiation occurs clearly influences the role of H on the stress necessary for initiation and the rate of propagation. Also recently, Wang et al. [9] performed molecular statics (energy minimization) calculations on 50 different types of (100) symmetric tilt grain boundary and free surface models to determine H trapping sites and the evolution of grain boundary and free surface energies as a function of H coverage. Then, taking a classical view that the tendency for intergranular fracture can be determined by evaluating the difference between grain boundary and free surface energies, an “ideal” work of

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separation was computed to assess the sensitivity of intergranular fracture on grain boundary structure and hydrogen coverage. Using a Σ 5 {210} grain boundary as one example, their calculations showed that the grain boundary cohesive energy decreases rapidly as hydrogen pressure is increased and saturates to a value around 64% of the cohesive energy of the grain boundary without hydrogen, for experimentally realistic hydrogen charging conditions. However, they also show that a wide range of cohesive energy reductions can be observed depending on grain boundary type/structure. Ultimately, Wang et al. [9] argued that while mapping lattice trapping sites is extremely useful, their ideal work of separation model cannot be directly compared with experiments as it does not consider local plastic deformation. Dislocation nucleation from the grain boundary and slip transfer across the grain boundary must be considered as both mechanisms modify the H environment, in particular slip transfer which can increase H coverage at the grain boundary. Thus, they postulate that HELP effects on slip transfer at grain boundaries is a controlling factor in grain boundary decohesion.

2.4

Atomistic Simulation-Inspired Models for H Embrittlement

The work by Song et al. [51] motivated the introduction of a new model for hydrogen embrittlement wherein hydrogen segregation to the crack tip under a state of stress is sufficient to suppress dislocation emission from the crack tip [54, 55], as schematically shown in Fig. 8a. In this model, segregation of H to the crack tip due to an applied stress intensity leads to a very high concentration of H at the crack tip, essentially creating a brittle nanohydride region. Song and Curtin initially showed that this model is capable of capturing embrittlement in Ni-H for both single crystal and grain boundary models (where the crack tip is aligned with the grain boundary plane). Using an estimation of the nanohydride size, L, Song and Curtin presented [54] an atomistically computed H embrittlement map, shown in Fig. 8b, which predicted the transition from crack-tip dislocation emission to cleavage fracture as a function of applied stress intensity. In addition, Song and Curtin presented [54] a kinetic model for the formation of the nanohydride that accounted for the stress distribution around the crack tip, the bulk hydrogen concentration, and the driving force for H migration to the crack tip. Combined, this model agreed very well with embrittlement data from CADD simulations. The concept of embrittlement due to the aggregation of H at very large concentrations around a crack tip was extended to α-Fe [55]. Their kinetic model provided predictions for embrittlement as a function of bulk H concentration, temperature, and loading rate in agreement with experimental data on steels. In addition to a new model, Song and Curtin performed [56] molecular dynamics simulations of dislocation pileups in α-Fe to show that H does not increase dislocation mobility or reduce dislocation-dislocation interactions in a pileup, as often stated in the HELP mechanism to explain the observation of significant plasticity local to the fracture surface during hydrogen embrittlement experiments. Song and Curtin showed that the hydrogen cloud migrates with an edge dislocation core under

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Atomistic Simulation Techniques to Model Hydrogen Segregation and Hydrogen. . . 371

Fig. 8 (a) Formation of a hydrogen-rich nanohydride region at the crack tip causes brittle crack advance. (b) Deformation mechanism map from Song et al. [54] (Reprinted from [54] with permission from Elsevier).

a H accumulation

b 2.1

Cleavage/ advance

Cracks with the same R=0.5 nm

1.8

K1 (MPa√m)

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A0>2.72×10

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0.9 A0 0:

(7)

For the deletion of a particle: 8 < 1    Λ3 N ΔE þ μ P¼ exp  : V kT

if ΔE þ μ  0, if ΔE þ μ > 0:

(8)

The temperature of the solute reservoir and the volume of the solute are accounted for through the de Broglie wavelength of the solute (Λ). For further details, consult Ref. [66]. Being an interstitial solute, a chemical potential (μ) can be chosen for hydrogen that is used to govern overall concentration and segregation site occupancy. The chemical potential can be varied in order to examine the effects of increasing segregation to grain boundaries. Alternatively, it may be necessary for a MD/MC investigation to model the results of a specific experiment that was performed on a sample that was charged to yield a specific bulk concentration of hydrogen. In such cases, the chemical potential may be chosen to be the value that results in the desired hydrogen concentration in a single crystal at the experimentally relevant temperature [67, 52, 68]. Traditional Monte Carlo modeling [66], as described above, has been used extensively in simulations of segregation (e.g., see Refs. [69, 70]). Breaking with tradition, a hybrid technique was employed by Solanki et al. [50] that included traditional MC with displacements and insertions/deletions in addition to accounting for straining and permitting relaxation due to hydrogen addition to the system with NPT (constant number of atoms, pressure, and temperature) MD. However, there is an increasing use of a hybrid grand canonical Monte Carlo/molecular dynamics approach that replaces the random displacements with a number of MD time steps (equivalently for models at 0 K, energy minimizations [71]) between insertion/deletion of solute (such an approach is implemented in Sandia National Laboratories’ LAMMPS software [72] by Sadigh et al. [73]). Examples of studies employing this approach may be found in Refs. [67, 52, 68]. This approach serves to more efficiently sample larger regions of phase space than applying random displacements to individual atoms and permits the observation of coordinated motion among a large number of atoms [66].

3.3

Discussion and Comparison of Methods

The choice of a site-energy selection or statistical mechanics approach to appropriately model segregation is not a matter of choosing between the most physically “realistic” or computationally efficient approach. Selection of the appropriate

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method to populate a sample with hydrogen must be made with the purpose of the study in mind. Both statistical mechanics and site-energy selection approaches involve their own sets of assumptions that compromise their ability to accurately model the physics of a system. Although MC methods are more consistent with the equilibrium thermodynamics of segregation, the MC technique does not represent an actual physical process; hydrogen may be added or removed from anywhere in the simulation domain without having to undergo diffusive motion. However, over MD time scales of a few nanoseconds, even hydrogen may not diffuse very far, and MC methods offer a computationally efficient way to approximate equilibrium conditions. Perhaps the simplest solution is to simulate natural solute trajectories and leverage the accelerated diffusion along grain boundaries to distribute the hydrogen throughout the sample [53]. This technique is of questionable utility given that the generally accepted notion of accelerated diffusion comes with many caveats including a large dependence on boundary type (e.g., tilt versus twist) [64, 65]. This is not the principle limitation with this approach given that, even with accelerated diffusion, MD time scales of a few nanoseconds will not permit long-range diffusion of the solute. Manual selection of sites is a simple technique that employs intuitive notions of segregation behavior to choose likely sites for segregation. The most appropriate use of this technique is to gain a mechanistic understanding of fracture or dislocation nucleation at crack tips [46]. Solanki et al. [74] employed this technique to elucidate the type of defect (single or double H in a vacancy and single or double H in an interstitial site) and how the cohesive energy of grain boundaries in α-Fe is affected by the distance of these defects from the grain boundary. A limitation of manual selection when populating a boundary with hydrogen is that common sense metrics guiding site selection may not be reliable. It is often suggested that the free volume of the boundary or of a site is a reliable predictor of the degree of segregation to a boundary. However, this is not always the case. Although free volume is a well-defined thermodynamic state variable, in practice, it is challenging to calculate the free volume of a grain boundary as atomistic approaches typically employ measurement of local atomic volume with a Voronoi construction. At finite temperatures, the measurement requires extended time averaging to calculate the atomic volume. Although the use of free volume at specific sites in a grain boundary may serve as a reasonable guide to placing interstitial solutes, there are limitations to using the free volume to predict the degree of segregation to a specific boundary. Site segregation energy calculations offer a systematic alternative to manual site selection but are limited in that they must be conducted at 0 K to avoid spurious fluctuations in segregation energy that occur at finite temperature. Such fluctuations are difficult to account for when applying an absorption isotherm or when describing occupancy with relationships such as Eq. 4. However, calculations of site segregation energy determined by independently sampling hydrogen segregation energies at interstitial sites in a system do not necessarily account for more complex structural changes that have been observed in grain boundaries in the presence of solute.

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Modeling efforts have demonstrated that both interstitial [67, 60] and substitutional [75] solute segregations can result in changes in grain boundary structure. Iterative site segregation models [54] account for the observation that hydrogen segregation changes the local stress field that can effect where additional hydrogen will preferentially segregate. Computationally, it is very efficient but still suffers from the necessity of performing calculations at zero temperature. Examining boundary structure at finite temperature may be critically important due to the possibility of temperature-induced structural transitions [76, 77, 61]. Overall, the degree of segregation has not been paid sufficient attention in atomistic studies. Atomistic studies of grain boundary segregation often report the degree of hydrogen segregation in terms of a hydrogen coverage on the grain boundary. This approach has the advantage of being straightforward, but connections to experimental values of concentration are challenging, which are typically given as atomic fraction (e.g., parts per million). Statistical mechanical approaches can do much better. With a chemical potential chosen for hydrogen, an equilibrium concentration can be established that is consistent with experimental results.

4

Steady-State Crack Propagation Approach to Study H Embrittlement

The following sections provide an example of the use of MD simulations to study hydrogen-induced decohesion during steady-state crack propagation. A Ni Σ3(112)h 110i grain boundary is selected in this example; grain boundaries with Σ3 misorientations are commonly observed in metallic materials that have been subjected to grain boundary engineering processing routes [78]. The Σ3(112)h110i grain boundary is selected in this example as it provides sufficient free volume for H segregation compared to the coherent Σ3 grain boundary. The simulation approach is composed of three steps: (i) creation of a hydrogenated grain boundary using a MC-guided site-energy selection technique, (ii) introduction and propagation of an atomically sharp crack, and (iii) data mining to extract the tensile strength and work of separation of the grain boundary. All simulations are performed using LAMMPS [72] with an EAM potential for Ni-H [39, 40].

4.1

Creation of a Hydrogenated Grain Boundary

The lowest energy hydrogen-free Ni Σ3(112)h110i symmetric tilt grain boundary structure is determined using an automated approach where one grain is incrementally translated relative to the other, sampling a large number of initial positions, and subjected to energy minimization at each translation under stress-free boundary conditions. The translation corresponding to the minimum energy grain boundary structure is selected. Simulation models with periodic boundary conditions are created that contain four layers following the methodology proposed by Yamakov et al. [79, 80], as

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shown in Fig. 10. Hydrogen is added at the primary grain boundary between grain 1 and grain 2. The boundaries between the absorbing layers and the primary grains as well as the grain boundary between the two absorbing layers across the periodic boundary act as obstacles to prevent dislocations from passing through the periodic boundary [79, 80]. In this example, a simulation model with dimensions lx = 914.523 Å, ly = 1102.309 Å, and lz = 99.561 Å is employed containing approximately 9.39 million atoms. This simulation model size is sufficient so that the crack can propagate more than five times its initial length without self-image force interactions through the periodic boundaries. Hydrogen atoms are added at interstitial sites within the primary grain boundary (between grains 1 and 2), thus generating a hydrogenated model in which H atoms have completely segregated to the grain boundary. These sites and the equilibrium concentration of H for the Σ3(112)h110i grain boundary are determined using Monte Carlo simulations performed at 300 K. The MC simulations were performed using a smaller model for 100,000 steps per atom where 50% of the steps attempted to add or remove H. These simulations determined that the equilibrium H concentration for the Σ3(112)h110i symmetric tilt grain boundary is approximately 300 appm based on a chemical potential = 2.35 eV, which corresponds to a saturation grain boundary coverage of approximately 0.132 H/Å2 (66.4% of favorable sites are populated with H). In this example, a range of H concentrations are studied: 100% H saturation (0.198 H/Å2) refers to the configuration where every favorable site in each structural unit at the grain boundary is occupied by a H atom; 75% (0.149 H/ Å2), 66.4% (0.132 H/Å2), 50% (0.099 H/Å2), and 25% (0.050 H/Å2) models are created by randomly removing H atoms from the fully populated grain boundary model. Thus, the H grain boundary coverages examined span the equilibrium grain boundary coverage determined via Monte Carlo simulations.

4.2

Steady-State Crack Propagation

Once the grain boundary model is constructed with a desired H coverage, energy minimization is performed to allow the grain boundary structure to relax to accommodate the added H. Then, the model is equilibrated to a temperature of 300 K under a hydrostatic tension σ h for 100,000 ps in the isobaric-isothermal (NPT) ensemble. As a result, the hydrogenated grain boundary model is strained in tension to provide a sufficient driving force for crack propagation. In the case of the Σ3(112)h110i symmetric tilt grain boundary, σ h is set to 10 GPa. Although local stresses near the grain boundary will be non-hydrostatic, dislocation emission is not observed under this prestress condition. This approach was initially developed to model steady-state crack propagation in pure Al by Yamakov et al. [79, 80]. Once the system is thermodynamically equilibrated, the simulation is switched to the isovolume-isothermal (NVT) ensemble, preserving the displacements associated with the hydrostatic prestress. Then, an atomically sharp crack of length Lcrack is introduced in the center of the primary grain boundary, as shown in Fig. 10. The atomically sharp crack is introduced by screening the interatomic interactions

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Fig. 10 Steady-state crack propagation model showing the initial crack size and the definition of the cohesive zone volume elements. This model is adapted from that of Yamakov et al. [79, 80] and used in work by Barrows et al. [81] and recently in Dingreville et al. [82].

between atoms in grain 1 and grain 2 along Lcrack. For the Σ3(112)h110i symmetric tilt grain boundary in this work, the screened region is set to Lcrack = 50 Å based on preliminary simulations to find a compatible combination of initial crack length and hydrostatic prestress appropriate for crack propagation. With the imposed strain due to the prestress σ h and initial crack Lcrack, the system is evolved for 40,000 ps which allows both crack tips (left and right directions) to propagate for approximately 100 Å in each direction along the grain boundary.

4.3

Cohesive Zone Volume Elements

The peak stress and work of separation for the grain boundary are extracted from atomistic forces and displacements ahead of each crack tip by tracking the evolution of a set of cohesive zone volume elements (CZVEs) during steady-state crack propagation [79, 80]. Specifically, before the atomically sharp crack is introduced, each atom within a distance h on either side of the primary grain boundary is assigned to a three-dimensional rectangular volume element (CZVE), as shown in Fig. 10. The bounds of the CZVEs are determined by partitioning the prestressed system near the grain boundary into 4n zones with 2n zones above and 2n zones

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below the primary grain boundary, resulting in n CZVE pairs along the left and right crack propagation directions. Cohesive zones above and below the initial 50 Å crack are omitted and not used to determine the peak tensile strength and work of separation of the grain boundary. The state of each CZVE at position x along the grain boundary and time t is defined by two state variables: (i) the average normal stress, σ yy(x,t), computed using the virial stress definition, and (ii) the crack-tip opening displacement, λ (x,t), computed by tracking the change in the Y-position of the center of mass of two vertically neighboring cohesive zones (normal to the grain boundary plane) as the crack propagates. Thus, each CZVE state is represented as a point (λ, σ yy) at every snapshot in time. Using a statistical mechanics approach, in the limit of steady-state crack propagation that occurs over an infinitely long time over an infinitely long interface, all realized CZVE states will produce a density of states distribution ρ(λ, σ yy) that is a continuous function independent of time. An example of this distribution is shown in Fig. 11. As the crack propagates along the grain boundary plane, the raw density of states data includes information related to the decohesion response of the grain boundary but also data associated with non-decohesion behavior such as the elastic response of the grain boundary to the hydrostatic prestress. The extracted traction-separation relationships should correspond uniquely to the decohesion behavior of the grain boundary without bias from the imposed prestress. To limit the set of (λ, σ yy) data to that exclusively associated with the decohesion response of the grain boundary, the density of states is filtered to include only CZVEs that are at or slightly ahead of the propagating crack tip. This requires a method to track the current position of the crack tip. To do this, a crack-tip opening displacement threshold is defined, above which a cohesive zone pair is considered open, meaning that the crack is within or has propagated through this pair of cohesive zones. For the data presented in Sect. 5, a cohesive zone pair is considered open if the normal opening displacement is greater than δ = 1.5 Å. This threshold has been chosen to be greater than the Burgers vector of a Shockley partial dislocation in Ni (1.437 Å), and thus a CZVE pair would not be

Normal Stress (GPa)

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considered open after a single partial dislocation emission event. Figure 11 shows the effect of the data deconvolution process. Using data from two CZVEs ahead of the crack tip at each snapshot in time, the (λ, σ yy) data points for CZVE pairs far ahead of the crack tip (a majority of which remain at elastic equilibrium at the hydrostatic prestress of 10.0 GPa) are separated from the grain boundary decohesion data and thus are not used to determine peak strength and work of separation properties of the Σ3 (112)h110i grain boundary. Finally, note that the example shown in Sect. 5 focuses only on normal opening displacements during crack propagation.

5

Grain Boundary Decohesion for S 3 Grain Boundary

The methodologies in Sects. 3 and 4 are employed to create a hydrogenated Σ3 (112)h110i bicrystal grain boundary model and to extract the traction-separation relationship from ρ(λ, σ yy) as a function of H coverage. Insights on plasticity and decohesion mechanisms in the vicinity of the crack tip and their relation with the work of separation for the Σ3(112)h110i symmetric tilt grain boundary are discussed in this section.

5.1

Dislocation Nucleation During Crack Propagation

A qualitative analysis of left and right crack tips during steady-state crack propagation shows that varying the coverage of H at the grain boundary influences the structure of the grain boundary and modestly influences dislocation emission from the crack tip. Figure 12 shows dislocation activity from the crack tip during propagation. First, the structure of the grain boundary is modified by the presence of H. For example, in Fig. 12a, the discrete extended stacking faults associated with the hydrogen-free grain boundary structure are observable, while in Fig. 12e, the grain boundary structure is uniformly distorted along the grain boundary plane. Second, due to the crystallographic orientation of each lattice relative to the left and right crack tips, asymmetric dislocation nucleation occurs. The left crack tip emits the same twin dislocation in all systems, whereas the density and type of dislocations nucleated from the right crack tip vary with H coverage. This asymmetry is in agreement with work of [83]. The crack-tip velocity shown in Fig. 13a, computed by linearly interpolating the crack-tip position data as a function of time, shows asymmetry depending on the crack propagation direction. Specifically, when propagating to the left (x direction), the crack-tip velocity increases with increasing H concentration. However, when propagating to the right (þx direction), the crack-tip velocity decreases with increasing H concentration. Interestingly, the crack-tip velocities are converging toward each other as H coverage is increased, as shown in Fig. 13b. This is rationalized by examining the structure of the grain boundary ahead of each crack tip. As shown in Fig. 12, hydrogen modifies the structure of the Σ3(112)h110i grain

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Fig. 12 Dislocation activity during crack propagation colored by the centrosymmetry parameter at 10,000 fs with (a) 0%, (b) 25%, (c) 50%, (d) 66.4%, (e) 75%, and (f) 100% H coverage at the primary grain boundary. Hydrogen saturation at the grain boundary influences both the structure of the grain boundary and dislocation nucleation during crack propagation (Adapted from Barrows et al. [81]).

boundary, mitigating the influence of the extended stacking faults (Fig. 12a) and the orientation of the Σ3(112)h110i structural units relative to the activated slip systems in each lattice. Finally, note that both left and right crack-tip velocities are reasonably constant over the simulation time, supporting the assumption of steadystate crack propagation necessary for extracting traction-separation decohesion relationships.

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Fig. 13 (a) Crack-tip position as a function of time. Propagation of the right crack tip is positive, and propagation of the left crack tip is negative relative to the grain boundary model coordinate system. (b) The crack-tip velocities have an asymmetric dependence on H coverage, caused by the change in grain boundary structure as H atom coverage is increased (Adapted from Barrows et al. [81]).

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One quantitative approach to assess the effect of varying H coverage on crack propagation is to extract the grain boundary peak strength and work of separation. Together, these parameters capture the traction-separation relationship. Figure 14a, b shows the decohesion curves for crack propagation to the left and to the right along the Σ 3(112)h110i grain boundary, extracted from the density of (λ, σ yy) states. The peak stress for the Σ 3(112)h110i symmetric tilt grain boundary shows a general decrease with increasing H coverage, as shown in Fig. 14c, d. The work of separation (the work required to separate the grain boundary into two free surfaces) can be computed by integrating the traction-displacement curve (a maximum opening displacement of 15 Å is selected for this integration). As shown in Fig. 14c, d, the

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work of separation for the Σ 3(112)h110i symmetric tilt grain boundary generally decreases with increasing H saturation until a minimum is reached around the equilibrium H saturation of 66.4% H coverage. Then, the work of separation modestly increases for the 100% H coverage which is oversaturated with hydrogen compared to equilibrium determined by Monte Carlo calculations. It is recognized that statistical variability exists within molecular dynamics simulations. To assess the extent of this variability, three simulations with different random realizations of H coverage at the grain boundary and different random initialized thermal velocities are conducted for each H coverage. The standard deviations associated with the computed values for peak stress and work of separation show that statistical variability has negligible impact on the observed trends for peak stress and work of separation as a function of H coverage for the Σ 3(112)h110i grain boundary. Although only one grain boundary is examined in this chapter, aspects consistent with both HID and HELP can be identified in the separation response. The left crack tip nucleates the same twinning dislocation during crack propagation, regardless of H saturation; yet, the velocity of the crack increases, and the work of separation and peak strength decrease with increasing H coverage. These observations are consistent with the theory of the HID mechanism and are consistent with prior results for low H coverages. Plasticity nucleating from the right crack tip appears to decrease as H coverage increases from 0% H to 66.4% H (equilibrium saturation) and then begins to increase as H coverage increases from 66.4% H to 100% H, with different slip systems activated at different concentrations of H. Thus, for this propagation direction, both HID and HELP may be appropriate to describe crack propagation depending on H saturation. Ultimately, a high-throughput analysis of a wide range of grain boundaries must be conducted, as proposed in [82], before quantitative conclusions can be reached regarding the contribution of HID and HELP mechanisms and the role of grain boundary structure.

6

Summary and Future Directions

Decades of research on hydrogen embrittlement have provided important observations related to fracture in different materials and hydrogen environments. Based on these observations, explanations for hydrogen embrittlement have been proposed by several authors [10], including models that (1) envision hydrogen as an impurity that weakens interatomic bonds (HID/HEDE), (2) regard hydrogen as a modifier for dislocation behavior in the neighborhood of a crack (HELP), or (3) propose that hydrogen diffuses to the crack tip to form hydride phases that assist crack propagation. Such mechanisms are challenging to isolate and study experimentally; thus, atomistic simulations are necessary to elucidate the exact role of hydrogen on embrittlement. This chapter provides a review of select atomistic simulation methods used to model hydrogen segregation to grain boundaries (and other defects) and to study the role of hydrogen on plasticity and decohesion mechanisms during fracture. This chapter provides a detailed example, based on work in [60, 81], of hydrogen segregation to a Σ 3 grain

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boundary and crack propagation along the Σ 3 grain boundary. In the case of crack propagation, the role of H atoms at the grain boundary is quantitatively assessed via extraction of peak tensile strength and work of separation. Clearly, there is still debate in the literature regarding the exact role of HID/ HEDE, HELP, and other proposed mechanisms in hydrogen embrittlement. In reality, the true answer is likely that a combination of different mechanisms is active during embrittlement, with a balance that is sensitive to the metallic material and the local atomic environment. For example, local grain boundary structure will effect intergranular fracture, and lattice dislocation density will effect the ability of a transgranular crack from nucleating dislocations during propagation. The interaction of defects which contain H atoms at their trapping sites [9] is an additional critical concern yet to be elucidated by either experiments or atomistic simulations. Through high-throughput calculations, atomistic simulations can provide unique details regarding the role of H on plasticity and fracture. Of course, significant care must be exercised to understand how the results of atomistic simulations are influenced by the abilities (or lack thereof) of the interatomic potentials used and the nanoscale size and nanosecond time scales inherent to many of the atomistic simulation approaches. The Monte Carlo techniques discussed in Sect. 3 provide a route to study segregation of H in the absence of a time scale. The statistical mechanicsbased approach to extract traction-separation relationships in Sects. 4 and 5 provides a statistical rather than deterministic technique that addresses some aspects of time scale. Ultimately, the long-term goal is to sufficiently understand the mechanisms associated with HE so that new alloys or new nanostructures/microstructures can be designed to mitigate or intelligently control the deleterious effects of hydrogen. Achievement of this goal would positively impact existing petrochemical industries and importantly support the development of future industries related to hydrogen fuel cells and other clean energies. Acknowledgments This work is supported by the Laboratory Directed Research and Development Program at Sandia National Laboratories, a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy’s National Nuclear Security Administration under contract DEAC0494AL85000.

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Modeling and Simulation of Bio-inspired Nanoarmors

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Stefano Signetti and Nicola M. Pugno

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Biological Armors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Atomistic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Models for Predicting Multiscale Properties of Hierarchical Materials . . . . . . . . . . . . . . 3.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Meshless Methods and Peridynamics Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Examples of Bio-Inspired Structures for Impact Protection and Energy Absorption . . . . . 4.1 Ceramic-Composite Armor Under Ballistic Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hollow-Cylindrical-Joint Honeycombs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S. Signetti (*) Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy Currently at: Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea e-mail: [email protected] N. M. Pugno (*) Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy School of Engineering and Materials Sciences, Queen Mary University of London, London, UK Ket-Lab, Edoardo Amaldi Foundation, Italian Space Agency, Rome, Italy e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_15

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Abstract

The exploitation of bio-inspired solutions and of novel nanomaterials is gaining increasing attention in the field of impact protection. Indeed, especially for advanced applications, there is a growing pressure towards the reduction of the weight of protective structures without compromising their energy absorption capability. The complexity of the phenomena induced by high-energy contacts requires advanced and efficient computational models, which are also fundamental for achieving the optimum, overcoming the limits of experimental tests and physical prototyping in exploring the whole design space. At the same time, the modeling of bio-inspired toughening mechanisms requires additional capability of these methods to efficiently cover and merge different -and even disparate- size and time scales. In this chapter, we review computational methods for modeling the mechanical behavior of materials and structures under high-velocity (e.g., ballistic) impacts and crushing, with a particular focus on the nonlinear finite element method. Some recent developments in numerical simulation of impact are presented underlining merits, limits, and open problems in the modeling of bio-inspired and nanomaterial-based armors. In the end, two modeling examples, a bio-inspired ceramic-composite armor with ballistic protection capabilities and a modified honeycomb structure for energy absorption, are proposed.

1

Introduction

The understanding of impact behavior of materials and structures is of extreme importance in a wide range of engineering and technological applications. Among all, the protection of space vehicles from the impact of micrometeoroids or orbital space debris [1] represents one of the most challenging applications due to the high impact velocities of the smallest micrometric debris (the velocity of low Earth orbiting satellites is about 7–8 km/s, but the relative impact speed may be higher) and to their consequent high penetration capability. Other important requirements in space structures are an extreme resilience of vital systems, since multiple faults are not allowed in space missions (Fig. 1), and the limit of the overall mass due to related rocket transportation capabilities and costs. These engineering goals are always in competition, with the result that a compromise with the acceptable level of risk of failure must be reached. Thus, the design does not limit to the mere maximization of protection, which would be straightforwardly achievable with a massive armor. For some decades, the answer to these tasks has been – and still largely is – the adoption of composite materials [2, 3] based on the combination of synthetic fibers (e.g., Kevlar®, Dyneema®) and thermoset resin that has allowed to effectively reach protection levels and low weights previously unimaginable with metallic targets. However, the aging and degradation of these materials, especially in extreme environments, must be properly assessed [4, 5]. When dealing with high penetrating projectiles, a hard ceramics front layer may also be employed: impactors are first blunted and worn down by the ceramic which also spreads the load over a larger area; then, the composite

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Fig. 1 Damage on the International Space Station by micrometeoroid impact. (a) Astronaut Scott Parazynski during extravehicular activity (EVA) on 3rd November 2007 to make a critical repair on a perforated Solar Array Wing. During the spacewalk, the astronaut also cut a snagged wire risking 100 V electricity. (b) Image of a hole by orbital debris in a panel of the Solar Max experiment. The impact craters on the aluminum exterior ISS handrails for spacewalks may have particularly sharp edges, representing a real risk of damage to the gloves of pressure suits. A new generation of composites based on bio-inspired and 2D nanomaterials will be fundamental for reducing risk of fatal hazard in long human space missions, e.g., to Mars (Images: NASA Orbital Debris Program Office)

tough backing layers absorb the residual kinetic energy by fiber/matrix failure and delamination [6–9]. Nowadays, in the era of nanomaterials we are moving towards atomistic scale two-dimensional (2D) materials, like graphene, coupling high resistance [10] and flaw tolerance at the nanoscale [11, 12], which could represent a breakthrough in protection levels [13, 14]. Complementary or alternatively, superior level of protection may be pursued through smart structural solutions to be employed even with traditional materials above mentioned, with all the advantages that this option implies in terms of consolidated manufacturing techniques and costs. Nature, having worked over the ages for optimizing defensive mechanisms against predators’ attacks or impact loads, is one of the most inspiring sources in this sense [15]. Upon impact, several complex physical phenomena take place: elastic-plastic deformation and wave propagation, fracture and fragmentation, heat generation (by yielding and friction), changing of material properties due to strain-rate effects up to phase change. Their occurrence and magnitude depend on the impact velocity that may be very low or up to extreme values (
3 km/s for hypervelocity impact), with increasing challenges for armors protective capabilities as well as for their accurate modeling and design. The theoretical description of the basic aspects of impact mechanics [16–19] has reached a level of advanced maturity only singularly but when coupled, due to the severe mathematical complexity, it is in a sort of stalemate. With high speed calculators and the development of computational methods (e.g., finite element method (FEM), meshless methods), simulation [20] has become the favorite design tool, allowing optimization studies. Furthermore, technological and economic limits in large-scale production of nanomaterials and the difficulties in their manipulation or in their structural arrangement into complex bio-inspired structures require a systematic and reliable design process able to provide a tentative target

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optimum. The large variety of parameters to be considered in the study of toughening mechanisms in biological materials due to their heterogeneity, the numerous levels of hierarchy, and the complexity of the constitutive laws (also strain-rate dependent) make experimental tests scarcely viable [21]. Moreover, with mere experiments is nearly impossible to investigate the whole design space. The traditional stand-alone experimental approach for armor design according to the philosophy “add material until it stops” it is not viable any more since the addition of mass can result in even suboptimal configuration [22]. In this scenario, computational models can be powerful tools for the design of new energy absorbing materials. Although important progress has been made in the past decades to simulate damage and failure processes taking place at impact, penetration, and fragmentation, much work remains to be done. The advent of nanomaterials and bio-inspiration is further questioning the capabilities of these tools and stimulates the research in this field. In this chapter, we review computational methods for modeling the impact behavior of materials and structures under high-velocity (e.g., ballistic) impacts, severe compression and contact. In Sect. 2, we review some of the main studies regarding biological and bio-inspired nanoarmors highlighting common protection mechanisms in Nature. In Sect. 3, we put a light on the main computational methods for modeling the mechanics of biological and bio-inspired (also hierarchical) materials for impact protection, with a particular focus on the finite element method. In the end, in Sect. 4, we present in detail a modeling example of bio-inspired ceramiccomposite armor with ballistic protection capabilities, highlighting the aspect of energy absorption scalings, and an analysis on modified honeycomb structures for energy absorption under crashing, showing the key role of simulations for these studies. This review is intended to help the readers to identify starting points for research in the field of modeling, simulation, and design of bio-inspired and hierarchical energy absorbing materials and structures.

2

Biological Armors

Many current-day animals possess armor, whose scope is to provide defense from the puncturing teeth of their predators. These include mammals (e.g., armadillo and pangolin [23]), reptiles (e.g., alligators, crocodiles, and turtles [24]) and various fishes [25, 26]. Despite the wide variation in the structure and material composition, there are distinct common aspects: the armor is generally composed of discrete rigid plates connected to the body and to each other by soft collagen fibrils and muscular tissue, which serve as back substrate [15]. This solution is able to provide an effective protection together with the required flexibility [27] for locomotion. Among these, the dermal structure of the Arapaima gigas fish is one of the most widely studied bioarmors in literature due to its unique characteristics [25, 26] (Fig. 2a). Its scales are composed of inner layers of mineralized collagen fibrils arranged in lamellae forming a Bouligand pattern [28] and of a highly mineralized outer layer which both dissipate energy by fracture mechanisms. There are at least three different orientations of collagen layers providing a certain grade of isotropy in

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Fig. 2 Examples of natural armors and energy absorbing structures. (a) Multilayer and hierarchical structure of the Arapaima gigas scales. (b) Block microstructure of the nacre and example of bio-inspired composite produced via additive manufacturing showing extreme toughness a flawtolerance characteristics. The crack follows a long path through the specimen thus dissipating large amounts of energy (Image of 3D printed composite adapted from Ref. [42] with permission). (c) Red-eared slider turtle (Trachemys Scripta Elegans) outer shell showing zig-zag suture between adjacent non-overlapped scales (Micro-CT images adapted from Ref. [24] with permission). (d) Hierarchical foamy peel of the Pomelo (Citrus Maxima) (SEM image of the peel adapted from Ref. [85] with permission)

the tensile and bending response of the armor. Tensile tests of notched Arapaima gigas scales [25] have shown how the layers of collagen fibrils separate with some of them that fracture, while others remain intact. This represents an effective crack bridging, a further extrinsic toughening mechanism [29] which is widely exploited in many cross-ply fiber-reinforced polymers [30]. Other kinds of common toughening mechanism rely on the optimized hierarchical structure. For example, the mineral layer of alligator scales [31] exploits the presence of voids with optimal disposition and density that are able to deviate the crack pattern, thus increasing the toughness. Probably, the most known mechanism of this type belongs to nacre [32] (Fig. 2b). Its microstructure is mostly made of microscopic ceramic tablets densely packed and bonded together by a thin layer of biopolymer. Material properties are properly calibrated and coupled in synergy with the structural arrangement, so that the crack propagation is constricted within the polymer phase by continuing to change the direction of propagation. When the crack

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has completely passed through, a further reservoir of toughness is available due to the interlocking between the lamellae. This hierarchical structure coupled with material of different characteristics (e.g., stiffness) is another common characteristic in natural structures and has been demonstrated to be the key for such extreme toughness and flaw tolerance [33]. The importance of material mixing has been demonstrated to be fundamental through numerical simulations showing that hierarchy per se is not beneficial for increasing strength and toughness, but must be necessarily coupled with material heterogeneity within the same hierarchical lever or at different scales [34]. These concepts have been widely exploited for the realization of bio-inspired composites with enhanced toughness [35]. The effective protection of these armors does not rely only on the optimized hierarchical microstructure but also the macroscopic arrangement has an important role, especially in mitigating the pain in the inner tissues of the animal. The scalebased structure has been implemented in different animals to provide the required flexibility with specific variants. In the armadillo carapace [36], the elements are hexagonal in the pectoral part and not overlapped, with collagen fibers connecting to the adjoining osteoderms. In the alligator gar and Senegal bichir, the bony scales have some overlap and the exposed (non-overlapped) regions are covered with ganoine [31]. In the red-eared slider turtle (Trachemys Scripta Elegans), sutures forming a zig-zag pattern ensure a minimum of non-bone area and flexibility [37] (Fig. 2c). These hierarchical sutures provide also a way of energy dissipation through local deformation and friction and could represent a further source of optimization in bio-inspired armors [38]. The analysis of scales subjected to transverse compression and the evaluation of the distribution of stresses towards the substrate have been carried out in a fundamental work by Vernerey and Barthelat [39]. They concluded that the scale mechanism provides a strain-stiffening structure, and this is a strategy to prevent structural damage and failure. Comparing different scales, ligament rigidity, and grade of overlap they demonstrated how it is possible to obtain a wide spectrum of constitutive responses, optimized for the specific load. It is interesting to mention that the scale armor concept has been used since antiquity by humans. Individual elements (in metal or leather) sewn or laced to a backing in a form of overlapping rows resembling the scales of a fish/reptile have been extensively used by Roman, Bizantine, and Japanese warriors. Different kinds of loads correspond to different structures. While against puncturing predators’ teeth and high concentrated loads, the presence of voids generally represents a weakness in the armor, a porous structure may be beneficial under distributed loads, allowing high energy dissipation through the activation of buckling deformation mechanisms in the struts of the lattice. This solution is particularly frequent in non-animal armors where the problem of flexibility and armor ergonomics (also thickness) is of second importance. A meaningful example is the hierarchical structure of the foamy peel of the Pomelo fruits (Citrus maxima) [40] that are able to withstand impact forces resulting from falls of over 10 m (100–200 J) without damage. The fruit toughness is due to the graded hierarchical fiber-reinforced composite foam (Fig. 2d). The foams struts, which are cells from the biological point of view, consist of liquid-filled cores and shells (cell walls) with relatively high strength. Recently, an effective aluminum-based

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cast composite inspired by this structure has been proposed [41]. In Sect. 4, an example of analogous honeycomb-like lattice is presented, highlighting in detail the various dissipation mechanism in this kind of structures. When dealing with bio-inspired armors, complex architectural geometries, multiscale fracture and instabilities phenomena may arise through the different hierarchical levels. These must be properly modeled in order to achieve reliable predictions on their behavior and on their protective capability and to perform optimization analyses as well. In the next section, computational models to describe the mechanics of hierarchical material for armors and to model impact phenomena at different size scales are discussed.

3

Computational Methods

A clear understanding of the constitutive behavior of biological hierarchical material presented in the previous section and of their structural arrangement – or of their bio-inspired engineered counterpart as well – is fundamental for their implementation in an impact simulation model. Computational methods for the characterization of hierarchical materials and for the modeling of their impact mechanics need to span the various size and time scales of the problems involved. These can be divided schematically into three broad categories: 1. nanoscale methods such as density functional theory (DFT) and molecular dynamics (MD) to achieve characterization of the basic one-dimensional (1D) and two-dimensional (2D) constituents of hierarchical composites and evaluate the role of defects at this scale; 2. micro- or mesoscale fiber bundle model (FBM), lattice spring model (LSM), discrete and meshless methods to reconstruct the role of hierarchy and material mixing on the multiscale mechanical properties of composites, also including statistics; 3. meso- and macroscale finite element methods and discrete/meshless methods to model complex mechanical problems at the continuum level in solids. Figure 3 depicts the overall scenario of computational methods in mechanics for an ideal multiscale characterization of materials for impact analysis. The methods are here briefly discussed in the following subsections.

3.1

Atomistic Simulations

The employment of nanomaterial, such as carbon nanotubes or graphene flakes, in hierarchical bio-inspired composites requires the full understanding of the mechanical behavior starting from the lowest dimensional level (Fig. 4). Molecular dynamics is a simulation technique that consists of numerically solving the classical Newton’s equation of motion for a set of atoms, which are characterized by their position, velocity, and acceleration. After the definition of the initial conditions of the system (temperature,

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Fig. 3 Computational approaches to perform multiscale characterization of hierarchical biological and bio-inspired materials for application in armors and to model impacts at different scales. Region of applicability in spatial and timescales are indicated. Characteristic simulation are shown for the three main dimensional scales: (1) nanoscale DFT simulation of graphene production by supersonic beam epitaxy of a fullerene molecule (Image adapted under CC-BY 4.0 license from Ref. [43]); (2) mesoscale hierarchical lattice spring model (HLSM) simulation to investigate the tensile and fracture properties of a matrix embedding rigid inclusions (Image adapted under CC-BY 4.0 license from Ref. [44]), and macroscale FEM impact simulation of a steel fragment penetrating a Kevlarbased multilayer composite armor

number of particles, density, simulation time step, etc.), the initial equilibrium of the system is found and then the perturbation to be studied is introduced into the system. Each atom is considered as a classical particle that obeys Newton’s laws of motion in relation to the interaction with other atoms which are defined by the so-called interatomic potentials (or force fields) that describe attractive and repulsive forces in between pairs and larger groups of atoms [45]. Potentials may be defined at many levels of physical accuracy; those most commonly used are based on molecular mechanics which can reproduce structural and conformational changes but usually cannot reproduce chemical reactions. When finer levels of detail are needed, potentials based on quantum mechanics (DFT) are used; some methods attempt to create hybrid classical/ quantum potentials where the bulk of the system is treated classically but a small region is treated as a quantum system, usually undergoing a chemical transformation. DFT framework, based on quantum theories of electronic structure, is currently the most commonly employed quantum mechanics method, which has evolved into a powerful tool for computing electronic ground-state properties of a large number of

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Fig. 4 Schematical representation showing the merging of different computational methods at different scale levels for the characterization of biological and bio-inspired hierarchical materials for impact simulations HLSM, HFBM and MD (HLSM image adapted from Ref. [44] under the Creative Commons BY 4.0 terms, MD image adapted from Ref. [69], with permission)

nanomaterials. The entire field of DFT method relies on the theorem that the groundstate energy of a many-electron system is a unique and variational functional of the electron density, and this conceptual proposal is implemented in a mathematical form to solve the Kohn-Sham (KS) equations. Due to the level of complexity of this method, the related computational cost limits the analyses to systems of few hundreds of atoms. A further method to overcome system size limitation called density functional tight binding (DFTB) consists of a series of models that are derived from a Taylor series expansion of the KS-DFT total energy. The basic advantage is that the terms appearing in the total energy expression are parametrized to reproduce accurately high-level electronic structure calculations for several different bonding conditions and can be calculated in advance, saving then in computational cost. The DFTB method has been applied to study large molecules (e.g., biomolecules), clusters, nanostructures, and condensed-matter systems with a wide range of elements. Atomistic method does not limit to the characterization of mechanical properties of 1D and 2D materials [46]. Impact-like phenomena such as the graphene synthesis via C60 supersonic beam epitaxy [47] (Fig. 3) or the hypervelocity impact properties of 2D materials armors against nanoscopic projectile [48] have been investigated via DFT while with MD it is possible to perform simulation of the impact of even microscale projectiles on graphene sheets [13, 49].

3.2

Models for Predicting Multiscale Properties of Hierarchical Materials

Various multiscale models have been developed to capture the mechanisms involved in the optimization of global material mechanical properties starting from nanoscale.

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One example is represented by the so-called fiber bundle models (FBM) [50] which are particularly appropriate for the simulation of fibrous materials, often occurring in biomaterials as seen in Sect. 2. With this approach the material structure at a certain size scale is modeled as an ensemble of fibers arranged in parallel (same level) and in series (different levels) subjected to uniaxial tension, with statistically Weibulldistributed yield and fracture strengths or strains. Usually, an equal-load-sharing hypothesis is adopted [51], while fibers fracture stresses are redistributed uniformly among the remaining in the bundle. Heterogeneous media are modeled by assigning different mechanical properties to the fibers of each bundle. A hierarchical extension is represented by the Hierarchical Fiber Bundle Model (HFBM) [51], whereby the input mechanical behavior of a subvolume or “fiber” at a given hierarchical level is statistically inferred from the average output deriving from reiterated simulations at the lower level, down to the lowest hierarchical level (Fig. 4). Results from this and other numerical implementations of HFBM show that specific hierarchical organizations can lead to increased damage resistance (e.g., selfsimilar fiber-reinforced matrix materials) or that the interaction between hierarchy and material heterogeneity is fundamental since homogeneous hierarchical bundles do not display improved properties [34]. The effect of defects at the different scale levels can also be considered. Similar approaches, appropriate for 2D or 3D simulations, are the lattice spring models (LSM) or random fuse models [51], which provide a continuum description of the media through a network of discrete elements (springs). These have been used to simulate plasticity, damage propagation, and statistical distributions of avalanches of fracture events in heterogeneous materials [52]. The hierarchical lattice spring model (HLSM) extends the classical LSM [44] (Fig. 3). Other analytical theories such as the quantized fracture mechanics (QFM) [53] or atomistic methods such as MD can be coupled with these multiscale approaches, for instance, to determine constitutive laws at the lower scale as a function of atomic structure, defect content, or molecular organization. Both theoretical models and the previously described numerical methods have shown that reinforcement organization in biological or bio-inspired composite materials can increase damage tolerance, avoiding direct crack path propagation and drastically improving the global response. Studies have focused on the influence of the structure, reinforcement shape, aspect ratio, dispersion, organization, and of mechanical properties of the constituents at various scale levels, iteratively deriving higher scale mechanical properties from lower ones, until a global material response is obtained [54]. The combined multiscale use of different computational techniques such as HFBM and HLSM has also proved to be successful in reproducing the macroscopic behavior of artificial nanocomposites such as gelatin-graphene oxide fibers [55]. Mesoscale models allow the design of composite materials exhibiting tailored fracture properties, drawing inspiration from mineralized biological composites [42]. Another common property of biomaterials that can be easily studied and simulated is the self-healing and its effects on the elastic, fracture, and fatigue properties of materials. Self-healing can be incorporated in HFBM/HLSM models by replacing

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fractured fibers with intact ones (simulating the process of healing) with custom mechanical properties, volume fractions, replacement rates, and locations as damage evolves during simulations. The main control parameter is the healing rate defined as the ratio of the number of healed and fractured fibers in a given fixed time interval. Both distributed and local healing processes can be simulated, in case fractured fibers are replaced either over the whole structure or at specific locations where damage is accumulated, respectively [56]. Thus, HFBM and HLSM are useful in providing advanced constitutive response, including fracture, damage, and self-healing of biological and bio-inspired materials to be used as input in FEM or meshless simulations that can, therefore, be limited to the upper scale (Figs. 3 and 4). Thus, a series of parametric studies, each replicating thousands of experiments, can be performed efficiently.

3.3

Finite Element Method

One of the most widely used computational methods at the meso- and macroscale is the finite element method. The finite element formulation of the problem results in a system of algebraic equations, yielding approximate solutions at a finite number of points over the continuum domain. To solve the problem, it subdivides a large problem into smaller sub-domains that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function. Avoiding to describe in detail the theory behind the formulation of basic and advanced nonlinear finite element, which is beyond the scope of this chapter and that can be found in several fundamental books [57, 58], we here introduce some common issues in the modeling of high-velocity impact and large deformation problems, which are addressed later in the modeling examples. Nowadays, commercial software offers robust nonlinear FEM tools for the analysis of these types of large-scale problems at an acceptable computational cost, implementing advanced and highly optimized contact algorithms capable of simulating high-energy impact conditions [59]. To solve time-dependent ordinary and partial differential equations with finite element analysis, either an explicit or an implicit solution scheme can be used. The first usually represents an advantage in high-velocity impact simulations, and actually is the one employed, since the equation of motion is solved step by step by computing nodal accelerations rather than displacement, thus saving computational time and memory for the determination and allocation of the stiffness matrix. The advantage is more important as the number of degrees of freedom of the model increases. Moreover, this method is much more stable under severe materials and geometrical nonlinearities (such as the ones deriving from contacts in soft media). However, this scheme has its counteracting disadvantage: the solving technique is only conditionally stable, which means that a sufficiently small time step must be guaranteed. If the solution becomes unstable, the error will rapidly increase at every

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time step and the solution will become invalid. An explicit method usually needs to have 100–10,000 times smaller time steps than an implicit technique, which is unconditionally stable, to avoid this kind of errors. The time step Δt for this method is limited by the time that the elastic wave, that arises from the loading, takes to transmit through the smallest element in the mesh of the model: Δt ¼

d min c

(1)

where dmin is the smallest distance between any two nodes in an element and qffiffiffiffiffiffiffiffiffiffiffiffi E c ¼ ρð1v 2 Þ is the sound speed in a 2D material, here assumed for the sake of simplicity linear, elastic, and isotropic (thus defined by the elastic modulus E and the Poisson’s ratio v). This distance generally corresponds to the edge with minimum length in solid element and in thin-shell element, to the thickness in thick-shell element formulation, and to the length of a monodimensional beam element. Thus, finer discretization results in increased computational time both due to the higher degrees of freedom to be computed and to the higher number of iterations which are inversely proportional to the time step Δt. As a consequence, it is very difficult to couple disparate scale levels in the same simulation since the element size and the time step will be governed by the lowest dimensional scale. From Eq. 1 it emerges that Δt  (E/ρ)1/2, thus bulk materials coupling extreme high modulus and low density may represent an issue in terms of computational cost. On the other hand, materials with high specific modulus are the ideal candidates for impact protection since the energy dissipation capability of a material under ballistic impact can be qffiffiffi σe E [60, 61], which is assessed by the magnitude of the Cuniff’s parameter U ¼ 2ρ ρ the product of the material-specific dissipated energy times the elastic wave speed in the considered medium, and ε is the ultimate strain of the material. Common time steps for impact simulations on systems at the centimeter scales are of the order of 108 s which represents a limit for the characteristic time of the physical phenomenon to be modeled. However, this problem is partially mitigated due to the fact that the maximum characteristic time of a high-velocity impact event is of the order of few milliseconds, due to its intrinsic nature. When dealing with low velocity impacts, a common practice is to fictitiously increase the mass of the system (generally known as “mass scaling”) in order to maximize the minimum time step required for stability. However, this practice must be properly evaluated and only exploited when the kinetic energy of the system is sufficiently low to be considered negligible in the specific problem, such as in quasi-static simulations. Given the fine discretization that may be required in order to accurately simulate impact phenomena, usually under integrated elements are preferred over fully integrated formulations to reduce the computational cost of the model and compensate the previous issues. For the function to be integrated to find the solution of the finite element problem, a number of points are calculated, known as Gaussian coordinates, whose position within the element is optimized for the highest grade of precision of the quadrature rule used to approximate the integrals. For each of

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these points, the function is multiplied by an optimized weight function. Then these are added together to calculate the integral. Reduced integration uses a lower number of Gaussian coordinates when solving the integral. Clearly, the more Gaussian coordinates for each element, the more accurate the answer will be, but this has to be weighed up against the cost of computation time. For example, the stiffness matrix has to be calculated in just one integration point of the element in case of “one-point quadrature rule” rather than the four of a 2  2 Gauss integration. The use of fewer integration points should produce a less stiff element. This sometimes is beneficial since it counteracts the overstimated stiffness of some element formulations, and in some particularly nonlinear problems such as plasticity, creep or incompressible materials the slight loss of accuracy is counteracted by the improvement in approximation to the real experimental behavior. Sometimes the reduction of the stiffness matrix leads to its singularity generating the so-called “sporious” or “zero-energy” modes of deformation, also known as hourglass modes. These modes are associated to null energy and thus can easily propagate through the mesh, producing meaningless results. They typically manifest as a patchwork of zig-zag or hourglass-like element shapes (hence the name), where individual elements are severely deformed, while the overall mesh section can be nearly undeformed. It is quite common to experience severe hourglassing that may be visually apparent without magnification of the displacements. Generally, this kind of modes are mostly prone to be generated by concentrated loads or contact pressures. Hourglass can be faced in several methods: by inserting an artificial stiffness to the hourglass deformation modes (the default way utilized in static/ quasi-static problems), by inserting an artificial viscosity (preferred for dynamic and high-velocity impact problems), by using fully integrated elements (but more expensive and less robust), or by refining the mesh (computationally expensive). Sometimes proper boundary conditions can avoid the formation of these modes due to enforced displacement compatibility. The basic hourglass control methodologies have been pioneered by Belytschko and coworkers [62]. The employment of the first two solutions, which have a negligible additional computational cost, requires a careful check in order to verify that the fictitious forces introduced to contain spurious mode of deformation are associated to a contained work that could drag physical energy from the system. Consequently, according to a widely acknowledged “rule of thumb,” the hourglass energy must be lower than 10% (but the lower is better) of the strain energy to consider the simulation accurate. This condition must hold for the whole model and for each of its subparts, identified by different structural elements, material model or element formulation. Some of the remaining issues of grid-based methods in modeling impact and large deformation problems are related to dealing with material separation (fragmentation) and capturing inhomogeneities in the deformation, leading to fracture and material failure. One possible solution to treat fracture are erosion algorithms: in these approaches, the elements of the model are deleted from the simulation when the material reaches the imposed failure condition in a prescribed number of integration points. The energies (deformation, kinetic, etc.) associated to the deleted elements are properly stored and accounted in the energy balance, but these portions

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of material are not able to interact any more with the rest of the medium. Stress concentration arises around the created discontinuities and fractures can then nucleate and propagate as subsequence of element deletions. It is clear how the propagation pattern of fracture is highly dependent on the size and geometry of the mesh, which requires a proper convergence analysis. While this approach can be acceptable in impact simulations of ductile materials, leading to satisfying results for sufficiently fine meshes, it represents an intrinsic strong limit when dealing with brittle materials or in those models whose scope is to characterize the fracture parameters (e.g., toughness) by subjecting the specimen to tension, without imposing a priori a defect in the structure. To a certain extent, these problems are addressed by meshfree (or meshless) methods reviewed next.

3.4

Meshless Methods and Peridynamics Implementation

Mesh-free methods are numerical techniques in which there is no fixed connectivity between the discretization nodes, and they are advantageous when simulating impact failure, penetration, and fragmentation. The level of nonlocality of interaction is defined by the “horizon” radius δ which defines the nodes within the generated spatial sphere to be assigned as neighbor of the reference node. Mesh-free methods can be developed for continuum (solids and fluids) or for particle-based (granular materials) formulations. A new nonlocal method for modeling continuous media, the so-called peridynamics, has been specifically proposed [63] for modeling multiple interacting fractures in the dynamic regime. Unlike the partial differential equation of the standard theory, the integral equations of peridynamics are applicable even when cracks and other singularities appear in the deformation field. Thus, the continuous and the discontinuous media can be modeled with a single set of equations with the capability to model the spontaneous formation of cracks. Peridynamics naturally leads to a meshless framework which is well suitable for the simulation of high-energy impacts involving penetration and fragmentation [64]. This has already been implemented in the acknowledged molecular dynamics code LAMMPS (Large Atomistic Massive Molecular Package Software) [65] enabling simulations at mesoscopic or even macroscopic length and timescales [66]. Examples of impact of a rigid sphere on a brittle solid show the formation of shear cone matching well with the experimental evidence (Fig. 4). Peridynamics especially suits for the modeling of elastic-brittle materials. However, successful simulations have already been performed on viscoelastic materials, thin membranes, and rods [67]. Since the theory is relatively recent and both the theoretical and the numerical frameworks are still limited to few works, its effective application to a wide class of biomaterials (e.g., hyperelastic like) has yet to be performed. However, the literature in the field is in constant and high-rate update. Recently, the coupling with a finite element scheme has been succesfully applied also to impact simulations [68], showing the concrete possibility of a real multiscale approach from atomistic to continuum.

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Examples of Bio-Inspired Structures for Impact Protection and Energy Absorption

In this section, two meaningful examples of energy absorbing structures are presented. The first is a multilayer ceramic-composite armor inspired by the structure of dermal armor scales presented in Sect. 2. This represents an example of structure which asbsorbs kinetic energy by material failure. The second example is a modified honeycomb concept, whose toughness relies on the combination of severe plastic deformation and buckling instability.

4.1

Ceramic-Composite Armor Under Ballistic Impact

Ceramic armors are used for the containment of blast fragments and were developed specifically for the prevention of projectile penetration, thanks to a high hardness and compressive strength with a sensible advantage of light weight (ρ = 2500  3000 kg/m3) with respect to metallic materials. Generally, they are coupled with a backing multilayer structure of fiber/matrix composite materials with the scope of absorbing the residual kinetic energy of the generated fragments [6]. This solution is very similar to the scale structure of natural armors presented in Sect. 2 and represents an example of engineering bio-inspiration. One of the first questions that may arise is why the backing collagen layer of those scales is constituted by a multilayer rather than a bulk of soft materials, considering that the orientation arrangement provides overall isotropic properties. Moreover, the growth of a multilayer structure is energetically more demanding, thus not optimized from the mere biological point of view and some structural advantages should explain this particular solution. In order to answer this concern, we can consider a multilayer medium made up of N plies of thickness t. For the sake of simplicity, we consider here a linear elastic and isotropic material. We can describe the penetration phenomenon of a mass m by means of a simple model based on conservation of momentum. Considering just one layer and a rigid projectile, the impact kinetic energy is dissipated in a volume defined by the layer thickness t and the projectile imprint radius r:  1  2 m v0  v2res ¼ ησπr 2 t 2

(2)

where v0 is the initial projectile impact velocity, vres its residual value after penetration, and σ is the impact strength of the material. η is a corrective factor, here assumed equal to unity, which takes into account dissipation phenomena not considered here such as projectile damage, delamination, and damage of the target outside the projectile imprint area. Generally, material strength shows strain-rate dependency, that in the most simple way can be taken into account as a quadratic form of the impact velocity:

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 σ ¼ σ 0 a0 þ a1

v v

 v 2  þ a2  , v

(3)

pffiffiffiffiffiffiffiffiffiffi where v ¼ σ 0 =ρ [70] is the reference velocity, function of the static monoaxial compression strength of the material σ 0 and of its density ρ in the undeformed state; a0, a1, and a2 are dimensionless coefficients that modify the plate strength taking into account the projectile shape effects and friction [71]. The linear term is usually neglected in accordance with experimental results [3]. Eq. 2 can be applied for a sequence of N layers, assuming as initial impact velocity on the i-th layer, the projectile residual velocity vi-1 after the passage through the (i-1)-th layer. After some algebraic calculations, the velocity profile is given by the following relation [22]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #" )ffi u   2 ( " 2 # 2t u v σ πr v 0, i i i-1 vi ¼ t i exp 2a2, i  2 a0, i þ a2, i   a0 , i , a2 , i vi m v

(4)

i

which is valid when the expression under the square root is positive or equal to zero. If the previous condition is not satisfied, it means that the projectile has been stopped by the i-th layer and the corresponding final depth of penetration H can be calculated as follows, accordingly to Eq. 4: "  2 # X i1 m vi a2 , i v i - 1 2 H¼ ln 1 þ tn, þ 2a2, i σ 0, i πr 2 a0, i vi n¼1

(5)

Consistently, the higher impactor mass m or velocity v0 (thus higher impact kinetic energy) results in higher perforation, while higher target impact strength σ and impactor imprint radius r to lower depth of penetration. Note that with Eq. 4 it is possible to determine the velocity profile of the projectile into the target. The model is also in accordance with the predictions of the Cuniff’sp criterion ffiffiffiffiffiffiffiffiffiffi [60] providing lower residual velocities and depth of penetration as v ¼ σ 0 =ρ increases. A natural consequence of the strain-rate dependency is that the backing layers are expected to dissipate a lower amount of energy since Kabs  σ  v2. Thus, it emerges from the strain-rate formulation and the multilayer impact model that the addition of material, net of other dissipation mechanisms, may be suboptimal. Considering the specific absorbed energy per layer, this concept of scaling can be analytically rationalized as follows: K abs ¼ kN α : N

(6)

A scaling exponent α > 0 indicates a synergistic behavior in which single layers interact to mutually enhance their specific contribution. For α = 0, the total absorbed energy is simply the sum of single-layer contributions, while for α < 0 a suboptimal behavior is identified in which the available toughness of the materials is not fully exploited. The latter condition, although may be not intuitive, has been experimentally

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observed in ref. [72] -but there not interpreted- on multilayer composite targets where the sign of α appeared to be related to the adhesive properties between the layers. In order to investigate this aspect, numerical simulation is very useful. With respect to experiments, it is possible to have precise measurement on the key variables such as the absorbed energy or the depth of penetration in the bulk, which otherwise would require sophisticated setup or time-consuming procedures. Moreover, it is possible to have full control on the interlayer adhesive properties and their variation in order to perform a precise sensitivity analysis, univocally correlating the input with the output. With simulation the amount of energy dissipated by different phenomena and different layer can be easily discriminated, giving a comprehensive understanding of the mechanics of impacts. Last but not least, it allows to simulate a sufficient high number of combinations, without time and cost demanding physical prototyping. To argue the role of interfaces, we here present a model of a typical ceramic-composite armor. The ceramic part of the studied target is made of silicon-carbide (B4C) modeled with the Johnson-Holmquist JH-II material model [73] implementing damage and strain-rate effects, whose parameter can be found elsewhere [74]. The model allows for progressive damage, taking into account residual material strength and compressive bulking. The thickness ratio of the two-component composite plate was set according to the results of ref. [75] where it was concluded upon experimental results that the optimum ratio in order to maximize the energy absorption, i.e., the plate ballistic limit, is given by the following relation t1/t2 = 4  ρ2/ρ1 where t1 and t2 are the ceramic and composite thicknesses, respectively, ρ1 and ρ2 the corresponding material densities. For the Kevlar multilayer composite backing, thick-shell elements were used (Fig. 5). This kind of elements are more suitable with respect to solids to capture the bending of thin parts with a limited number of through-thickness element and can be used in those situation where the aspect ratio of the element does not allow the use of classical thinshell element formulations. Since in these elements the thickness is a geometrical quantity, unlike in the classical thin-shell formulations, it is possible to compute with accuracy the element deformation in the out-of-plane, dimension (thickness), making them ideal for treating the high compressive contact stresses generated by transverse impact. In the layered variant of the thick-shell formulation, it is possible to assign an arbitrary number of integration point in the element thickness and attribute to each of them a different constitutive law (woven textile or matrix) according to the volumetric fraction of the composite ply (Fig. 5). For modeling both the fiber and the matrix fractions, an orthotropic material model was used which allows to set the behavior in tension, compression, and shear along the two orthogonal in-plane directions and the out-of-plane one. The model implements a linear-elastic branch followed by a nonlinear post-peak softening behavior: the residual strength in compression, tension, and shear can be defined as a fraction of the maximum material peak stress allowing for post-peak dissipation capability. The equations which define the failure criteria used for the model are presented in detail in the paper by Matzenmiller and coworkers [76]. The adhesive contact interactions between the different plies are implemented with a so-called tiebreak contact. Considering a couple of adjacent nodes belonging to two subsequent layers, these are initially bonded together and the contact interface can sustain tractions. A stress-based constitutive law is implemented to define the

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Fig. 5 Schematic representation of the modeling of the composite part of the analyzed armor. The real woven fabric immersed in the thermoset resin is modeled as a continuum equivalent medium. The two phases of the composite, fibers and matrix, are considered in the model ply introducing to each through-thickness integration different material models or properties according to the volume fraction of each phase. As a consequence of the production techniques, the ply can be seen as formed by an inner core with the properties of the woven and the outer part filled by the matix, as confirmed by the SEM photograph. Qualitative representations of the stress-separation law of the adhesive interface (bottom-left) including friction and of the constitutive behavior of the woven textile phase (bottom right) used in the simulation are also depicted

constitutive behavior of the interface. The adhesive interface fails when the following condition is satisfied:

s⊥ σ⊥

2

þ

2 sk ¼1 σk

(7)

where s⊥ and sk are the current normal and tangential stress between two welded interface nodes, while σ ⊥ and σ k are their corresponding limit values, which may be different defining an elliptic domain. To fully define the constitutive behavior of the interface, it is also necessary to set a critical separation δ0 for the two opening modes of the interface, after which the force can decrease until a failure nodal separation δf, accounting for progressive damage (Fig. 5), or suddenly drop to zero (δf = δ0). Note that Eq. 7 is analogous to a cohesive zone model where stresses are placed by the current and critical fracture energies G. Here, the stress-based solution is used due to the fact that in the dynamic regime it is not straightforward to experimentally compute the critical fracture energies and thus perform a calibration of the FEM model. Once the nodes have

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Fig. 6 Results of the FEM simulation of a steel fragment impacting a ceramic-composite armor at 350 m/s. (a) Picture of the armor after complete penetration (only one quarter modeled due to symmetry) showing also generation of fragments, thanks to the implementation of an erosion algorithm. Note how the mesh density is variable, having smaller element dimension under the impact region to help in containing hourglass modes and obtain a more accurate response in relation to contact stability. The radius of the analyzed plate is  6 times the projectile radius: this lower-bound is acknowledged to be sufficient to avoid edge effects in high-velocity impact phenomena (Image adapted from ref. [22] with permission). (b) Typical energy output from impact simulations showing how the total energy of the system is preserved. The loss of kinetic energy of the projectile (Kabs) goes into target deformation (internal strain energy) and contact dissipation by interface failure and friction (contact energy) and kinetic energy of the armor layers. Hourglass is within the 10% of the system internal energy thus does not affect the results significantly. Note how the ceramic absorbs a limited amount of kinetic energy (first jump in the internal and kinetic energy) with respect to the backing layers (second jump) being its primary scope to damage the projectile rather than absorb energy

separated, the contact locally switches to a penalty algorithm and the layers can mutually interact with friction. The law used in the contact model to compute the current kinetic friction coefficient as a function of the local static and dynamic values, μS and μD respectively, assumes the expression μ ¼ μS þ ðμS  μD Þevrel =vcrit and is a function of the relative sliding velocity vrel of the nodes-segment in contact. Figure 6a shows a snapshot of the perforated target under the impact of a steel fragment moving at 350 m/s. A typical energy output of the simulations is depicted in Fig. 6b showing the perfect conservation of the total energy of the system, depicting the different forms of dissipations and the limitation of hourglass energy within acceptable values. The model is exploited to perform an extensive simulation campaign to investigate the scaling in energy absorption of the composite backing multilayer. Figure 7 shows the results of the variation of the scaling exponent α (Eq. 6) as a function of the two parameters defining the adhesive properties, normalized with respect to the homogenized tensile strength of the layers σ. Each of the point in the graph is obtained by best fit of the curve Kabs/N  N built on a basis of six simulations with N = 2, 4, 6, 8, 10, 12. The results clearly show how the scaling exponent can be made positive and even maximized by properly tuning the adhesive properties. A very weak interface (limit case of no adhesive interaction) may yield to α 0 in case that a synergistic coupling does not occur. Negative values are obtained also for very strong interactions σ k/σ, σ ⊥/σ ! 1. This can be explained with the fact that the strong interaction limits

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Fig. 7 Role of interface properties on the variation of the exponent α of the specific absorbed energy scaling power law (Eq. 6) in perforated multilayer composite panels. The adhesive normal and shear limit stresses, σ ⊥ and σ k respectively, are normalized to the homogenized tensile stress of the plate. Results of impact FEM simulations show the existence of optimal interface parameters that increase and maximize the scaling of specific energy absorption (i.e., maximize α in the scaling law KNabs  N α ), influencing the interaction of layers and their failure behavior. This explains controversial experimental observation of α for plates with different curing conditions (pressure and temperature) during the production process [72]. Each point of the graph was computed extracting the scaling exponent from sets of 6 simulations with different number of layers N resulting in overall 726 simulations. Such a number of experimental trials would be extremely difficult to perform (Image adapted from Ref. [22] with permission)

the deformation of the layers making the overall armor stiffer and with a monolithic behavior. As can be seen from Fig. 7, when the interface is very strong the bullet creates a perfect hole with amplitude nearly equal to the projectile radius while the rest of the plate is nearly undeformed, thus does not dissipate much energy. The optimal solution is obtained for intermediate situations where the tailored adhesive properties allow the layers to deform and spread damage over a wider area. From this results it is possible to understand the importance of the adhesive in governing the deformation of the target rather than dissipating energy by delamination, which is a little fraction of the energy dissipated by permanent deformation and fracture (Fig. 6b). This result can rationalize experimental observation reported, for example, in ref. [72] where cured multilayer during the production process showed both negative and positive scaling, while non-cured plates (no adhesion between the layers) all showed suboptimal configuration for increasing areal density.

4.2

Hollow-Cylindrical-Joint Honeycombs

We here present examples of simulations of modified honeycomb structures made of a metallic alloy, subjected compressive crushing and experiencing yielding, elastic-

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plastic instability, and fracture. These structures could in principle be realized also with other materials (e.g., traditional composite materials, graphene-based composite, silk-like artificial materials). However, the main aim of this part, rather than simulating the behavior of a specific material, is to show the importance of simulation in integrating – and also partially substituting – experimental analyses and prototyping. Hollow-cylindrical-joint honeycombs represent a modification of traditional honeycombs where the joints formed by the intersection of converging walls are placed by hollow cylinders (Fig. 8c). These kind of structures belong to the family of center-symmetrical hexagonal honeycombs [77] and have been recently proposed [78, 79] as possible and effective modification of conventional honeycomb structures (Fig. 8a, b) to further increase their specific mechanical properties, i.e., the yield strength (σ/ρ), stiffness (E/ρ), and toughness. It has analytically been proved that these characteristic quantities can be maximized, on an equal mass basis, in relation to specific geometric sizing of the radius r of the cylinders and of the length l of the walls (Fig. 8). In particular, an optimal (maximum) value of the mechanical properties was analytically predicted for aspect ratios r/l  0.3 [78] (Fig. 8c). The analytical models [78, 79] which describe the mechanics of these modified honeycombs are based on geometrical considerations and on the elastic theory of plates and shells and, thus, are limited to the description of the elastic regime. The high complexity of phenomena within the material, which experiences yielding, elastic-plastic instabilities, and fracture under large strain and strain-rate, in addition to the geometrical nonlinearity introduced by the complex 3D geometry, requires the use of advanced simulation, in order to thoroughly describe the mechanical behavior of such honeycombs up to crushing. Further advantages are associated with the virtual modeling: the closed cell structure of the honeycombs does not allow to directly visualize the formation and the evolution of plastic folds and fractures during the experiments. Moreover, simulations can go beyond the mere measurement of the force displacement curves being able to measure other important quantities such as local stresses and strain (and thus underline with precision yielded and damaged regions), to visualize the formation of shear bands, and to precisely

Fig. 8 From traditional to modified honeycombs. (a) Natural honeycomb structure of a beehive. (b) Reference model of a conventional regular hexagonal honeycomb structure. (c) Model of a hollow-cylindrical-joint honeycomb structure. Geometrical characteristics of the honeycombs are identified in the figure

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control and evaluate the role of different frictional properties (static and dynamic coefficient of friction) at the contact interfaces on the overall behavior. A series of five specimen with different r/l from 0 to 0.5 and same mass was experimentally tested. Defining ρh/ρbulk as the ratio between the honeycomb density and the density of the constituent bulk material, the thickness of each samples to keep their mass constant can be calculated as a function of r/l according to the relation ρ/ρbulk = 1.155(t/l)2 + [2.528(r/l) + 1.155](t/l) [78]. The samples utilized for this study [79] were made using 6061-T4 aluminum alloy for the bulk constituent material (ρbulk = 2700 kg/m3) being the relative apparent density of the honeycomb ρh/ρbulk = 0.1 for all the tested samples. We here report the results of monoaxial compression experiments made with a 1000HDX Instron Universal Testing Machine (ITW, USA) with loading capacity of 1000 kN. The loading rates before and after the initial yield of the samples are of 1 mm/min. Conventional quantities are defined for describing the constitutive response of the honeycomb. The compressive stress is σ h = F/A, where F is the force recorded in the load cell of the testing machine, and A is the projected convex hull area of the honeycomb samples on the plane perpendicular to the loading direction; the corresponding compressive strain is εh = Δh/h, where Δh is the variation of height of the specimen. The constitutive relationship of the aluminum alloy used in the experiment was characterized by tensioning a round dog-bone specimen with circular cross-section of diameter d = 10 mm up to failure. The determined mechanical properties extracted from the stress–strain curve were: the Young’s modulus E = 68 GPa, the yield strength σ u = 287 MPa, the ultimate stress σ u = 318 MPa, and the failure strain εf = 0.121. Although the curve can be approximated with a bilinear elasticplastic relationship, the experimentally derived curve was directly employed. Prior to yielding, the material is assumed linear elastic with εy = σ y/E. It must be mentioned that due to the size-scale effect, in the FEM model the plastic strain in σ the input curve εp1 ¼ ε  Ey was scaled from the nominal one measured from the εp1, FEM dog-bone test by εp1 ¼ dt since t d. Thus, also the ultimate strain εf, FEM is scaled accordingly. This operation is necessary, at least for metallic alloys such as the one considered here, otherwise one would obtain unnatural brittle behavior in the simulations very far from the experiments if size-scale effects are not considered. Regarding the choice of elements for this kind of problems, hexahedron 1-point integration solid element (constant stress) represents an effective solution. This solution is efficient and accurate and works very well for severe deformations. For plasticity problem, at least 3 integration points should be present through the minimum dimension of the structure, thus 3 elements of this type should be used through the thickness t. Besides the saving in computational time with respect to the 8-point fully integrated element, this choice may avoid element locking. Hourglass must also be monitored with attention and mitigated, if necessary, for this elements. The use of more element through the thickness may help in this sense. The von Mises criterion was employed for yielding. Material fracture is treated via an erosion algorithm with the elements that are deleted from simulation when either one between the principal and deviatoric strain reaches the limit εf,FEM. Contact

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interactions were implemented between the steel plates and the honeycomb. Selfcontact within the honeycomb parts must also be introduced in order to properly account for material densification during the crushing process. Static and dynamic coefficients of friction were, respectively, set to be 0.610.47 for the honeycombrigid steel plate contact and 1.351.05 for the self-contact, which are common values for aluminum–steel and aluminum–aluminum surfaces. Figure 9 reports the results in terms of stress–strain curve (quantities computed as in experiments). Since it is a displacement control simulation, the load F to compute the stress σ h is here computed from the normal component of the contact force at the contact surface between the honeycomb and the steel plate, which by equilibrium is equal to the external applied load. The lattices Young’s modulus, computed as σ h,y/εh,y is nearly constant, in accordance with the fact that the ratio Eh/ρh is a material constant [80]. The failure mechanisms obtained for the shell-plate assembled honeycombs (Fig. 10) represent a transition between the two limit behaviors of sample 1 (r/l = 0, classical hexagonal honeycomb) and sample 5 (r/l = 0.5, degenerated plates) shown in Fig. 11. The classical honeycomb, after the elastic-plastic instability of the plates, which can be approximated with the simple case of the Euler’s column

Fig. 9 (a) Stress–strain curves of the five simulated samples with different r/l ratios (dashed lines) and comparison with experimental results (solid lines). For r/l = 0.5, four states are highlighted corresponding to the images of Fig. 11b. Three simulation states for the optimal honeycomb r/l = 0.3 are depicted corresponding to yielding, minimum of bearing capacity, and complete fracture. (b) Specific yield strength (filled dots) and absorbed energy (empty dots) as a function of r/l computed from FEM simulations. Results show how the lattice is optimized for r/l  0.3 providing the higher yield strength and energy absorption (Image adapted from Ref. [79] with permission)

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Fig. 10 (a) Experimental and simulated collapse modes of the out-of-plane loaded honeycombs with r/l = 0.2 - 0.3 - 0.4. (b) Snapshots of FEM simulations (at εh = 20%) show the collapse mechanisms of the cylindrical shell-plate joint with detail of variable number of foldings for different r/l due to the mutual level of restrain between cylinders and plates and the different thickness of the honeycombs to provide the same mass (Image adapted from Ref. [79] with permission)

with both clamped edges, reaches the failure due to the formation of a sub-horizontal fracture approximately at h/2. For sample 5, the collapse of the structure is caused by the multiple folding of cylindrical shell and does not experience fracture. For single cylindrical tube, several works explain its collapse mechanisms, according to thickness, diameter, and length [81]. From Fig. 11a it can be seen how simulations are capable of capturing the transition between the two limit structures, with increasing number of folds and decreasing fold wavelength as r/l increases. This behavior can be imputed to the fact that for maintaining the same mass along all specimens the wall thickness t decreases with increasing r/l, yielding towards a more ductile behavior. Figure 11b gives a measure of the capability of simulation to catch large displacement deformation with the simulation deformed shape that can be nearly perfectly superposed to experimental pictures, and being able to predict the number of folds. The behavior under compression in the orthogonal (in-plane) direction was also studied in an analogous way and can be found in the related work [79]. The results indicate that the combination of hollow cylindrical shells and plates forms a new periodic assembly with better mechanical property and energy absorption capability with respect to conventional honeycombs. As shown, the developed numerical model was able to well describe the experiments, both in predicting the constitutive curves of the honeycomb behavior under compression and of the energy absorbing capabilities. Thus, the models could be used in predicting the performance of honeycombs of different geometries, further optimizing natural solutions. This concept is not limited to the here presented material but may be used to generate new

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Fig. 11 (a) Collapse modes transition from sample 1 (r/l = 0, unmodified honeycomb) to sample 5 (r/l = 0.5, full cylindrical joint honeycomb, restrain point between the cylinders highlighted). Contour plot of plastic strain is superposed to simulated honeycomb images. (b) Visual comparison between crushing experiment (left) and simulation (right) on sample 5 at four deformation levels (see graph of Fig. 9a) showing very good agreement in the formation of folds (highlighted by the arrows) due to elastic-plastic instabilities. For each state, the number of folds n is indicated (Image adapted from Ref. [79] with permission)

crashworthy lattices at different scales, ranging from macroscopic composite sandwich panels to material structuring at the nanoscale [82, 83].

5

Conclusion

Herein, we have discussed the importance of bio-inspired structures and nanomaterials to achieve a further breakthrough for a new generation of tougher armors. Some examples of energy absorbing multilayer and lattice structures were shown, discussing the importance of simulation in solving specific problems, in understanding natural solutions and their further optimization according to engineering needs. Although numerical codes become more and more sophisticated, many problems related to the accurate prediction of real material behavior upon impact and penetration remain to be solved. The introduction of these new materials design and concepts arises further concerns regarding the ability of current computational methods to effectively model and predict at a reasonable computational cost the impact mechanics of such protective structures, in order to compensate the limits of physical prototyping and related experimental testing. Future modeling and simulation tools will have to be able to treat concurrently and in a unified manner the many failure mechanisms present at impact and penetration, such as dynamic fracture and fragmentation, generation of dislocations, and shear bands. Some of them, such as the peridynamics, are promising in this direction. Anyhow, it must be mentioned that another crucial limit in the capability of computational analysis for the class of problems discussed here –but also for many others (e.g., fluid dynamics, astrophysics)– is the state-of-the-art in the current hardware computational architectures.

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Indeed, a really significant advance in the capabilities of computational analysis, in terms of model degrees of freedom and time scale, will necessarily require a breakthrough in the hardware conception [84]. Anyway, we have shown in this chapter how with the current resources it is possible to obtain significant results when different methods and theories are coupled in a synergistic way, overcoming the current limits in performing truly single-framework multiscale simulations. Acknowledgments NMP is supported by the European Commission H2020 under the Graphene Flagship (WP14 “Polymer Composites,” no. 696656) and the FET Proactive (“Neurofibers” no. 732344).

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Thermal Vibration of Carbon Nanostructures

15

Lifeng Wang, Haiyan Hu, and Rumeng Liu

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Thermal Vibration of CNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Stochastic Vibration of a Cantilever Euler Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Timoshenko Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Comparison Between Different Beam Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Thermal Vibration of a SWCNT with Quantum Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Thermal Vibration of a SWCNT Predicted via Beam Models . . . . . . . . . . . . . . . . . . . . . . . 3.2 Comparisons Between Classical Theory and Quantum Results . . . . . . . . . . . . . . . . . . . . . 4 Thermal Vibration of a SWCNT Predicted by Semi-quantum MD . . . . . . . . . . . . . . . . . . . . . . . . 4.1 MD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nonlinear Thermal Vibration of SWCNTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Thermal Vibration of a Simply Supported SWCNT with Thermal Stress . . . . . . . . . . . 5.2 Coupling Between Flexural Modes in Thermal Vibration of SWCNTs . . . . . . . . . . . . .

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L. Wang (*) State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China e-mail: [email protected] H. Hu State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China e-mail: [email protected] R. Liu State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Jiangnan University, Wuxi, China e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_16

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6 Thermal Vibration of a RSLGS with Quantum Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 RMS Amplitude of a RSLGS via a Nonlocal Elastic Plate Model . . . . . . . . . . . . . . . . . . 6.2 MD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Plate Model with Initial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Thermal Vibration of a CSLGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 RMS Amplitude of a CSLGS via a Circular Plate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Comparisons Between Classical Theory and Quantum Results . . . . . . . . . . . . . . . . . . . . . 8 Open Problems and Future Researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The chapter presents the study on thermal vibration of nanostructures, such as carbon nanotube (CNT) and graphene, as well as the basic finding for the relation between the temperature and the root-of-mean-square (RMS) amplitude of the thermal vibration of the carbon nanostructures. In this study, the molecular dynamics (MD) based on modified Langevin dynamics, which accounts for quantum statistics by introducing a quantum heat bath, is used to simulate the thermal vibration of carbon nanostructures. The simulations show that the RMS amplitude of the thermal vibration of the carbon nanostructures obtained from the semi-quantum MD is lower than that obtained from the classical MD, especially for very low temperature and high-order vibration modes. The RMS amplitudes of the thermal vibrations of the single-walled CNT (SWCNT) and graphene obtained from the semi-quantum MD coincide well with those from the models of Timoshenko beam and Kirchhoff plate with quantum effects. These results indicate that quantum effects are important for the thermal vibration of the SWCNT and graphene in the case of high-order vibration modes, small size, and low temperature. Furthermore, the thermal vibration of a simply supported SWCNT subject to thermal stress is investigated by using the models of planar and non-planar nonlinear beams, respectively. The whirling motion with energy transfer between flexural motions is found in the SWCNT when the geometric nonlinearity is significant. The energies of different vibration modes are not equal even over a time scale of tens of nanoseconds, which is much larger than the period of fundamental natural vibration of the SWCNT at equilibrium state. The energies of different modes become equal when the time scale increases to the range of microseconds. Keywords

Thermal vibration · CNT · Graphene · Quantum effects · Nonlinear vibration

1

Introduction

The development of nanotechnology enables one to construct small devices based on the carbon nanostructure activated by the thermal fluctuation. One example is the Brownian ratchet, where a particle moves unidirectionally over a switchable anisotropic potential [1]. Thermal fluctuations play an important role in the resonant

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423

Fig. 1 Carbon nanotube and graphene (a) Carbon nanotube and (b) graphene

properties of low-dimensional carbon nanostructures when they serve as nanoscale devices [2, 3]. Thus, the thermal vibration of carbon nanostructure gives rise to a great challenge to the development of nano-resonator-based nanosensor and has drawn considerable attention. The thermal vibration of two kinds of carbon nanostructure, i.e., carbon nanotubes (CNTs) and graphene, as shown in Fig. 1, is studied in this chapter. Interest in graphene and CNTs has kept growing since their discovery [4, 5]. Both graphene and CNTs exhibit superior mechanical and electronic properties over any known materials and hold substantial promise for new superstrong composite materials, among others [6]. The need of modeling and analyzing the thermal vibration of nanostructure has called much attention. The thermal vibration can serve as a source of self-excitation in determining the natural frequencies and the resonant properties of the carbon nanostructures. On the other hand, thermal noise is well known to be a ubiquitous and fundamental limit for the sensitivity of nano-mechanical resonator-based nanosensors. Thus, the thermal vibration of nanostructures has been studied by means of continuum mechanics, molecular dynamics (MD) simulation, and experiments. For example, Treacy et al. estimated the Young’s modulus of freestanding CNTs by measuring the amplitude of their intrinsic thermal vibrations using a transmission electron microscope [7]. Krishnan et al. presented the relationship between the size, the Young’s modulus, and the stand deviation of the thermal vibration amplitude at the tip of a CNT under a given temperature by using the model of an Euler beam [8]. In addition to experimental methods, MD and continuum models [9, 10] are very important tools to study thermal vibration in CNTs. The past years have seen more studies on the thermal vibration of CNTs by using both continuum mechanics and MD simulation. The vibration amplitude of a nanobeam due to thermal noise can be determined by using the law of energy equipartition. That is, each degree of freedom of the system in thermal equilibrium possesses the same average kinetic energy kBT/2, where kB = 1.38
1023JK1 denotes the Boltzmann constant and T represents temperature. Feng and Jones calculated the free thermal vibrations of cantilevered CNTs by using MD simulation and explained the resulting power spectral density of the tip displacement with statistical mechanics and beam theory [11]. Later, Feng and Jones found that the quality factor of the cantilever CNT is independent of its length and that the intrinsic signal-to-noise ratio for a cantilever CNT improves with an increase of length [12]. Wang and Hu studied the thermal vibration of a double-walled CNT

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by using a model of double Euler beams with van der Waals interaction between layers taken into consideration and then validated the results by using the MD simulations [13]. Barnard et al. simulated the behavior of suspended CNT resonator over a broad range of temperature [9]. They found that thermal fluctuations induce strong coupling between resonant modes. Hence, the nonlinearity may have key effects on the dynamic behavior of thermal vibration of carbon nanostructures. Most recently, Moser et al. studied CNT mechanical resonators at a cryostat temperature of 30 mK using an ultrasensitive method based on cross-correlated electrical noise measurements, in combination with parametric downconversion [14]. Hence, the quantum effects should be considered in dealing with the thermal vibration of CNT and graphene, especially for the case of low temperature and high vibration modes. The primary objective of this chapter is to study the thermal vibration of CNT and graphene. For this purpose, Sect. 2 presents the RMS amplitude of the stochastically driven vibration of a Timoshenko beam, which will be used to model single-walled CNTs (SWCNTs). Next, in Sect. 3, the RMS amplitude of the thermal vibration of a SWCNT is studied via the model of Timoshenko beam with quantum effects take into consideration. Then, Sect. 4 gives the MD simulation for the thermal vibration of a CNT with quantum effects. Section 5 presents the nonlinear thermal vibration of SWCNTs. Section 6 gives the thermal vibration of a rectangular single-layered graphene sheet (RSLGS) with quantum effects. Section 7 presents the thermal vibration of a circular single-layered graphene sheet (CSLGS) with simply supported and clamped boundary. Finally, the chapter ends with Sect. 8 of some open problems and future researches.

2

Thermal Vibration of CNT

2.1

Stochastic Vibration of a Cantilever Euler Beam

This section starts with the dynamic equation of an Euler beam with uniform cross section placed along direction x in the frame of coordinates (x, y, z), with w(x, t) being the displacement of section x of the beam in direction z at the moment t. Upon the assumption of small amplitudes, the motion of the above Euler beam is governed by the following partial equation of the fourth-order [15] ρA

@ 2 wðx, tÞ @ 4 wðx, tÞ þ EI ¼0 @t2 @x4

(1)

where E is Young’s modulus, ρ is the mass density, A is the cross section area of the beam, and I is the moment of inertia for the cross section. The boundary conditions for a cantilever Euler beam with one end fixed and other end free are

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wð0, tÞ ¼ 0,

@wð0, tÞ ¼ 0, @x

425

@ 2 wðL, tÞ ¼ 0, @x2

@ 3 wðL, tÞ ¼0 @x3

(2)

The solution of the nth mode of the cantilever Euler beam reads [15] ~ n ðxÞe jωn t ¼ Dn f n ðxÞejωn t wn ðx, tÞ ¼ w " # Dn cos αn x  coshαn x sin αn L  sinhαn L ¼ (3) ejωn t sin αn x  sinhαn x 2 þ cos αn L þ coshαn L pffiffiffiffiffiffiffi def 2 where j  1 , αn 4 ¼ ρA EI ωn , αnL = 1.8751, 4.6941, 7.8548, 10.9955, 14.1372,   , and Dn is a constant decided from initial conditions. The total energy En corresponding to the nth mode can be found by calculating the elastic energy at the instant of maximal deflection when the cantilever Euler beam is momentarily stationary for ejωn t ¼ 1 " Eelastic n

EI ¼ 2

ðL  0

@wn @x

2 # dx

¼ t¼0

EILD2n α4n 8

(4)

From the law of energy equipartition, there is an averaged kinetic energy of kBT/2 per degree of freedom for all of the relevant lateral vibration modes. Because there are both elastic and kinetic energies in each degrees of freedom for all of the relevant lateral vibration modes, then on average, hEni = kBT holds for each vibration mode. It is straightforward to show that each mode of a stochastically driven oscillator has a Gaussian probability profile, and the standard deviation of the vibration amplitude of the free tip of the Euler beam is given by [15] ~ 2n ½L ¼ D2n ¼ w

8kB T EILα4n

(5)

Then, the RMS amplitude of the nth mode at x reads pffiffiffi 2 ^ n ðxÞ ¼ D n j f n ðx Þj w 2

(6)

As all vibration modes are independent, their contributions can be added incoherently. To average coherently over all the vibration modes, one can simply add the variances σ 2n ðxÞ to get the other Gaussian distribution with the standard deviation given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X ^ ðxÞ ¼ ^ 2n ðxÞ w w n¼1

(7)

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Hence, the RMS amplitude of the thermal vibration of the Euler beam at x can be obtained.

2.2

Timoshenko Beam Model

The model of Timoshenko beam with the rotary inertia and the shear deformation taken into account has been widely used to analyze the wave propagation and vibration of CNTs [16, 17]. In this subsection, the dynamic equation of a Timoshenko beam placed along direction x in the frame of coordinates (x, y, z) is established for w(x, t) being the displacement of section x of the beam in direction z at the moment t [15]:   @2w @φ @ 2 w  þ βG ¼0 @t2 @x @x2    2  @2φ @w @ φ ρI 2 þ βAG φ   EI ¼0 @t @x @x2 ρ

(8a)

(8b)

where E is Young’s modulus, G = E/2(1 þ μ) is the shear modulus, μ is the Poisson’s ratio, ρ is the mass density, A is the cross section area of the beam, I is the moment of inertia for the cross section, φ is the slope of the deflection curve when the shearing force is neglected, and β is the form factor of shear depending on the shape of the cross section. The boundary conditions of a cantilever Timoshenko beam clamped at x = 0 and x = L are wð0, tÞ ¼ 0,

φð0, tÞ ¼ 0,

@wðL, tÞ  φðL, tÞ ¼ 0, @x

@φðL, tÞ ¼0 @x

(9)

To study the free vibration of the Timoshenko beam, let the dynamic deflection and slope be ~ jωt , w ¼ we

e ejωt φ¼φ

(10)

^ is the deflection amplitude of the beam and φ ^ represents the slop amplitude where w of the beam due to bending deformation alone. Let ξ ¼ x=L

(11)

Substituting Eq. (10) into Eq. (8), one obtains ~ @2w @e φ ~¼0 þ b2 s 2 w L @ξ @ξ2 s2

 ~  e 1 @w @2φ e¼0  1  b2 r 2 s 2 φ þ 2 L @ξ @ξ

(12a)

(12b)

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427

where b2 ¼

ρAL4 ω2 , EI

r2 ¼

I , AL2

s2 ¼

EI βAGL2

(13)

In the case of [(r2  s2)2 þ 4/b2]1/2 > (r2 + s2), the solutions for Eq. (12) read [18] ~ ¼ C1 coshbα1 ξ þ C2 sinhbα1 ξ þ C3 cos bα2 ξ þ C4 sin bα2 ξ w

(14a)

e ¼ C01 sinhbα1 ξ þ C02 coshbα1 ξ þ C03 sin bα2 ξ þ C04 cos bα2 ξ φ

(14b)

  

1=2 o1=2 1 n  α1 ¼ pffiffiffi  r 2 þ s2 þ r 2  s2 þ 4=b2 α2 2

(15)

where

In the case of [(r2  s2)2 þ 4/b2]1/2 < (r2 + s2), then Eq. (14) is replaced by [18] ~ ¼ C1 cos bα01 ξ þ jC2 sin bα01 ξ þ C3 cos bα2 ξ þ C4 sin bα2 ξ w

(16a)

e ¼ jC01 sin bα01 ξ þ C02 cos bα01 ξ þ C03 sin bα2 ξ þ C04 cos bα2 ξ φ

(16b)

  

1=2 o1=2 1 n α01 ¼ pffiffiffi r 2 þ s2  r 2  s2 þ 4=b2 2

(17)

where

In Eqs. (13 and 15), only half of the constants are independent since they are related to Eq. (12) as the following: C01 ¼

b α21 þ s2 C1 L α1

(18a)

C02 ¼

b α21 þ s2 C2 L α1

(18b)

C03 ¼  C04 ¼

b α22  s2 C3 L α2

b α22  s2 C4 L α2

(18c) (18d)

For the case of [(r2  s2)2 þ 4/b2]1/2 > (r2 + s2), the natural frequency of the cantilevered beam yields the following equation [18]: h  i 2 bð r 2 þ s 2 Þ 2 þ b2 r 2  s2 þ 2 cos hbα1 cos bα2   1=2 sin hbα1 sin bα2 ¼ 0 1  b2 r 2 s 2 (19)

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Solving Eq. (19) for b gives an infinite numberqofffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bn, n ffi= 1, 2, . . ., which determines the nth-order natural frequency ωn ¼ EIb2n =ρAL4 of the Timo~ n and φ e n for the cantilevered shenko beam in this case. The nth normal modes w beam are [18] ~ n ¼ Dn f wn ðξÞ w ¼ Dn ½coshbn αn1 ξ  λn ζ n δn sinhbn αn1 ξ  cos bn αn2 ξ þ δ sin bn αn2 ξ e n ¼ Dn f φn ðξÞ φ   θn sinhbn αn1 ξ  cos bn αn2 ξ þ θn sin bn αn2 ξ ¼ H n coshbn αn1 ξ þ λn ζ n

(20a)

(20b)

where Hn ¼

bn α2n1 þ s2 λn ζ n Dn L αn1 θn

1 sinhbn αn1  sin bn αn2 λ δn ¼ n ζ n coshbn αn1 þ cos bn αn2 θn ¼ 

λn sinhbn αn1 þ sin bn αn2 1 coshbn αn1 þ cos bn αn2 ζn

  

1=2 o1=2 1 n  αn1 ¼ pffiffiffi  r 2 þ s2 þ r 2  s2 þ 4=b2n αn2 2

(21a)

(21b) (21c)

(21d)

For the case of [(r2  s2)2 þ 4/b2]1/2 < (r2 + s2), the nth-order natural frequency ωn of the Timoshenko beam is determined from [18] h  i 2 bð r 2 þ s 2 Þ 0 2 þ b2 r 2  s2 þ 2 cos bα01 cos bα2   1=2 sin bα1 sin bα2 ¼ 0 b2 r 2 s2  1 (22) ^ n and φ ^ n for the cantilever Timoshenko beam are The nth normal modes w ~ n ¼ D n f w n ðξÞ w

¼ Dn cos bn α0n1 ξ þ λ0n ζ n ηn sin bn α0n1 ξ  cos bn αn2 ξ þ ηn sin bn αn2 ξ

(23a)

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e n ¼ Dn f φn ðξÞ φ   μ ¼ H n cos bn α0n1 ξ  0 n sin bn α0n1 ξ  cos bn αn2 ξ þ μn sin bn αn2 ξ λn ζ n

429

(23b)

where Hn ¼ 

bn α2n1 þ s2 λ0n ζ n Dn L αn1 μn

1 sin bn α0n1  sin bn αn2 λ0n ηn ¼ ζ n cos bn α0n1 þ cos bn αn2 μn ¼

λ0n sin bn α0n1  sin bn αn2 1 cos bn α0n1 þ cos bn αn2 ζn

  

1=2 o1=2 1 n α0n1 ¼ pffiffiffi r 2 þ s2  r 2  s2 þ 4=b2n 2

(24a)

(24b) (24c)

(24d)

In Eqs. (19)–(24), one has ζ n ¼ ðαn1 2 þ r 2 Þ=ðαn1 2 þ s2 Þ ¼ ðαn2 2  s2 Þ=ðαn2 2  r 2 Þ ¼ ðαn1 2 þ r 2 Þ=ðαn2 2  r 2 Þ ¼ ðαn2 2  s2 Þ=ðαn1 2 þ s2 Þ

(25a)

λn ¼ αn1 =αn2 ¼ jλ0n

(25b)

Now, the study turns to the stochastic vibration of the Timoshenko beam from the viewpoint of energy analysis. The total energy En contained in the vibration mode n can be found by calculating the elastic energy at the instant of maximal deflection when the cantilever Timoshenko beam is momentarily stationary, i.e., ejωn t ¼ 1,     Ð L EI @φn 2 @wn 2 ¼ 0 þ βAG φn  dx 2 @x @x "  2  2 # Ð L EI @f φn @f wn þ βAG f φn  dx ¼ Dn 2 0 2 @x @x

Eelastic n

(26)

There is an average energy of kBT/2 per degree of freedom for all relevant lateral vibration modes. There are both elastic energy and kinetic energy in a vibration mode. Then, on average, hEni = kBT for each vibration mode [8]. From Eq. (26), it is easy to get

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"

  # @f wn 2 D n ¼ En þ βAG f φn  dx : 0 2 @x @x "ð L  2   # @f wn 2 EI @f φn ¼ kB T þ βAG f φn  dx : 2 @x @x 0 2

Ð L EI @f φn 2

(27)

Thus, it is straightforward to obtain the RMS amplitude of nth mode at x from Eq. (6) and the RMS amplitude of the CNT at x from Eq. (7).

2.3

Comparison Between Different Beam Models

To predict the thermal vibration of a SWCNT from the analytical results in Sect. 2.2, it is necessary to know the Young’s modulus E and the shear modulus G or the Poisson’s ratio μ in advance. The previous studies based on the REBO potential gave a great variety of the Young’s moduli and the Poisson’s ratios of the SWCNTs [17, 19]. For example, the MD simulation of pure bending was carried out to obtain material parameters. The Young’s modulus is E = 0.740 TPa, and the Poisson’s ratio are μ = 0.254 for the (5, 5) armchair SWCNTs, while E = 0.798 TPa and μ = 0.317 hold for the (10, 10) armchair SWCNTs when the thickness of the SWCNTs was chosen as 0.34 nm. The mass density of the CNTs is ρ = 2237 kg/m3. Figure 2 shows the RMS amplitude of the thermal vibration of the first four modes of a (10, 10) armchair SWCNT of 24.6 nm long at 300 K. Here, the symbol EB represents the results predicted by using the model of an Euler beam, and the symbol TB represents results predicted by using the model of a Timoshenko beam. In Fig. 2, no difference between the results of Euler beam and Timoshenko beam can be identified for the RMS amplitude of the first-order mode of vibration. Little difference can be observed for the second-order mode of vibration. The difference of results between the two beam models becomes more and more obvious with an increase of the mode order. The RMS amplitude predicted by the model of Timoshenko beam is larger than that predicted by the model of Euler beam. Figure 3 illustrates the RMS amplitudes of the first four modes of the thermal vibration of a (10, 10) armchair CNT with 4.9 nm long. Here, the symbols EB and TB carry the same meaning as in Fig. 2. Figure 3a indicates that the difference between the results of models of the Euler beam the Timoshenko beam looks obvious even for the first-order mode of the thermal vibration. The rest subfigures show again that the difference for the beam models becomes more and more obvious with an increase of the mode order of vibration. The RMS amplitude predicted by the Timoshenko beam is also larger than that predicted by the Euler beam. Figure 4 presents the RMS amplitudes of thermal vibration of armchair SWCNTs with different sizes, where Fig. 4a is for a (10, 10) armchair CNT of 14.5 nm long, Fig. 4c is for a (10, 10) armchair CNT of 9.6 nm long, Fig. 4e is for a (5, 5) armchair CNT of 9.6 nm long, and Fig. 4b, d, and f are the zoomed views of Fig. 4a, c, and e,

15

a

Thermal Vibration of Carbon Nanostructures

b

0.3

RMS amplitude (nm)

RMS Amplitude(nm)

TB EB 0.2

0.1

0.0

431

0.06

TB EB 0.04

0.02

0.00 0

5

10

15

20

25

0

5

10

0.020

d

TB EB

RMS amplitude (nm)

RMS amplitude (nm)

c

0.015

0.010

0.005

0.000

5

10

15

X Coordinate (nm)

20

25

20

25

0.010

20

25

TB EB

0.008 0.006 0.004 0.002 0.000

0

15

X Coordinate (nm)

X Coordinate (nm)

0

5

10

15

X Coordinate (nm)

Fig. 2 The RMS amplitude of thermal vibration of a (10, 10) armchair CNT of 24.6 nm long at 300 K. (a) The first mode, (b) the second mode, (c) the third mode, (d) the fourth mode

respectively. Here, the symbols EB and TB represent the same meaning as before. For shorter SWCNTs, the difference between the results of the models of Timoshenko beam and Euler beam becomes very obvious.

3

Thermal Vibration of a SWCNT with Quantum Effects

On the other hand, it has been realized that quantum effect may become dominant for some kinds of mechanics problems in nanostructure. For example, Hone et al. observed the quantum size effects on the nanotube phonon spectrum [20]. O’Connell et al. showed that a mechanical mode can be cooled to its quantum ground state by using a microwave-frequency mechanical oscillator coupled to a quantum bit, which was used to measure the quantum state of the resonator [21]. The law of energy equipartition predicts with high accuracy when hv

Rð2Þ  Rð1Þ > > : 0

r < Rð1Þ Rð1Þ  r  Rð2Þ r > Rð2Þ

(4)

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Mechanics of Carbon Nanotubes and Their Composites

487

are a smooth cutoff function with R(1) = 0.17 nm and R(2) = 0.2 nm. For bond lengths beyond 0.2 nm, the above cutoff function suggests vanishing bond energy. The multi-body coupling term Bij in Eq. 1 depends on atoms other than i and j: 2 Bij ¼ 41 þ

X k ð6¼i, jÞ






312

G φijk f c ðrik Þ5 ,

(5)

where rik is the distance between atoms i and k (k 6¼ i, j), φijk is the angle between bonds i-j and i-k, and the function G(φijk) is GðφÞ ¼ 0:27186 þ 0:48922 cos φ  0:43286 cos2 φ  0:56140 cos3 φ þ 1:2711 cos4 φ  0:037931 cos5 φ for

0o  φ  109:47o

¼ 0:69669 þ 5:5444 cos φ þ 23:432 cos2 φ þ 55:948 cos3 φ þ 69:876 cos4 φ þ 35:312 cos5 φ for

109:47o < φ < 120o

(6)

¼ 0:00260  1:0980 cos φ  4:3460 cos2 φ  6:8300 cos3 φ  4:9280 cos4 φ  1:3424 cos5 φ for

120o  φ  180o

The second-generation interatomic potential [32] is used to predict the mechanics properties of graphene and carbon nanotubes in this section.

1.3

A Plate Theory for Graphene Based on the Interatomic Potential

Huang et al. [34] developed a plate theory for graphene based on the interatomic potential. The bond angle is 120o for the initial state of graphene without deformation. The equilibrium bond length r0 is obtained by minimizing the potential V at  o @V  ¼ 0. For the second-generation Brenner the bond angle 120 , i.e., @rij  rij ¼r0 , φijk ¼120o potential given in Eqs. 1, 2, 3, 4, 5, and 6, r0 = 0.142 nm. For the graphene subject to an infinitesimal in-plane membrane strain «, the changes of bond length rij  r0 and bond angle φijk  120o are also infinitesimal. pffiffi The Taylor series expansion of the cosine of bond angle is cos φijk ¼  12  23  h i

2
3 φijk  120o þ 14 φijk  120o þ O φijk  120o such that cos φijk þ 12 is infinitesimal. The interatomic potential in Eq. 1 can then be expanded in a Taylor series expansion as

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!  2 1 1 @2V
V ¼ V0 þ cos φijk þ rij  r0 þ 2 2 2 @rij k6¼i, j 0 0 !   X
 @2V 1 þ rij  r0 cos φijk þ @rij @ cos φijk 2 k6¼i, j 0 !     1 X @2V 1 1 þ cos φijk þ cos φijl þ , 2 k , l6¼i, j @ cos φijk @ cos φijl 2 2 0 X

@V @ cos φijk

! 

(7)

where the subscript “0” denotes the initial, equilibrium state (rij = r0 and φijk = 120o); the terms higher than the second order are neglected; and the first-order derivative  @V vanishes. The above Taylor series is expanded with respect to cos φijk (instead @rij 0

of φijk) because the interatomic potentials in Sect. 1.2 depend on the bond angle via cos φijk (see Eq. 6). Equation 7 involves five constants, the first-order  namely,    @V @2V @2V derivative , @ cos φijk 0 and second-order derivatives @rij @ cos φijk 0 , @r2ij 0   @2 V @2 V and ð k ¼ 6 l Þ, which can be obtained analytically. , @ cos φ @ cos φ @ cos φ @ cos φ ijk

ijk

0

ijk

ijl

0

For example, for the second-generation Brenner potential given in Eqs. 1, 2, 3, 4, 5, and 6:     @V @2 V 2 @2V ¼ 1:592eV, ¼ 4356eV=nm , 2 @ cos φijk 0 @rij @ cos φijk 0 ¼ 59:14eV=nm, @rij 0  @2 V @2 V @ cos φijk @ cos φijk 0 ¼ 3:099eV, and @ cos φijk @ cos φijl 0 ðk 6¼ l Þ ¼ 0:3673eV. Here the  @V reflects multi-body atomistic interactions. first-order derivative @ cos φ ijk

0

The Cauchy-Born rule [20] equates the energy stored in atomic bonds to strain energy at the continuum level. Atoms move according to a single mapping F = @x/@X from the initial to the current (deformed) configurations of the Bravais lattice, where X and x denote positions of the atom in these two configurations, respectively. A bond between a pair of atoms i and j in the initial ð0Þ configuration is described by the vector rij ¼ r0 n, where r0 is the initial bond length and n is the unit vector along the initial bond direction. For a simple ð0Þ Bravais lattice, h the deformed bond is riij ¼ F  rij , which has the length rij ¼ pffiffiffiffiffiffiffiffiffiffiffiffi rij  rij ¼ r0 1 þ n  «  n  12 ðn  «  nÞ2 for infinitesimal in-plane membrane strain «. The single mapping F ensures equilibrium of atoms for a simple Bravais lattice. Graphene, however, is a Bravais multi-lattice, for which a single mapping F cannot ensure the equilibrium of atoms anymore [17]. For uniform in-plane normal membrane strains e11 and e22 and shear strain e12 (Fig. 2a), graphene can be decomposed to two simple Bravais sub-lattices [17, 18], as shown in Fig. 3 by the open and solid circles, respectively. Each sub-lattice follows the single mapping F, but the two sub-lattices may have a shift ζ to ensure the equilibrium of atoms. The deformed bond (between atoms from two different sub-lattices) then becomes

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489

(a)

(b) Fig. 2 Schematic diagram of a graphene sheet subject to (a) in-plane membrane strain and (b) curvature


 ð0Þ rij ¼ F  rij þ ζ, which has the length rij given by r2ij ¼ r20 δαβ þ 2eαβ ðnα þ xα Þ
 nβ þ xβ , where δαβ is the second-order identity tensor and x ¼ r10 F 1  ζ is to be determined analytically via energy minimization (note the summation over Greek subscripts).

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Fig. 3 A shift vector ζ between two sub-lattices is introduced to ensure the equilibrium of atoms

Fig. 4 A straight segment of length r0 bent to a circular arc of radius R without stretch

r0 r R

For the graphene subject to the combined in-plane membrane strain eαβ(α, β = 1, 2) and curvature κ 11, κ 22 and κ12 (Fig. 2b), which stretch and bend the graphene, respectively, the distance between two nearest-neighbor atoms (from different sub-lattices) becomes [34]

 r4


 2 r2ij ¼ r20 δαβ þ 2eαβ ðnα þ xα Þ nβ þ xβ  0 κ αβ ðnα þ xα Þ nβ þ xβ , 12

(8)

where the shift vector x is to be determined analytically via energy minimization; the first term on the right-hand side is the leading term in bond length due to in-plane membrane strain, and the second term represents the leading term in bond length due to the curvature. Figure 4 illustrates this second term via a straight segment of length r0 bent to a circular arc of radius R without stretch. The arc length is r0, and the

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Mechanics of Carbon Nanotubes and Their Composites

491


r0  . Its straight length r between the two end pointshof the segment is r ¼i 2R sin 2R

r 0 5 r0 1 r0 3 Taylor expansion for r0 > 3 n þ x þ x n > > β γ γ β > = < 6   7> r20 6 7 ð2Þ ð2Þ  δαλ þ 2eαλ þ καβ κγλ 6 2 nβ þ xβ nγ þ xγ 7 , > 4  12  5> > > > > ð1Þ : 2 nβ þ xβ nðγ1Þ þ xγ ;

cos φijk ¼

(9)

where n(1) and n(2) represent the directions of i-j and i-k bonds prior to deformation, respectively. Substitution of Eqs. 8 and 9 into the Taylor expansion (7) gives the bond energy V as a quadratic function of membrane strain «, shift vector x, and curvature κ. The 3
 P energy associated with each atom is Φ ¼ 12 V rij ; cos φijk , k 6¼ i, j , with the j¼1

summation running over the three nearest-neighbor atoms and the factor one-half resulting from equipartition of bond energy. The shift vector is determined from @Φ energy minimization @x ¼ 0 as λ 3 X 2 nα nβ nλ , xλ ¼ Aeαβ 3 j¼1

(10)

which depends on the in-plane membrane strain « but not on the curvature κ (and therefore no shift for curvature only), and A is given in terms of the five derivatives and equilibrium bond length r0 by A¼

 1

8r20



@2V @r2ij

þ 12r0





@2V @rij @ cos φijk 0

0       : @V @2V @2V @2V 2 @2V 12 @ cos þ 4r þ 18  9 þ 12r 2 0 0 @r φ @ cos φ @ cos φ @ cos φ @ cos φ @rij @ cos φ ijk

0

ij

0

ijk

ijk

0

ijk

ijl

0

ijk

0

(11) The strain energy density (energy per unit area of graphene) is w ¼ SΦ0 , where S 0 pffiffi 3 3r2 ¼ 4 0 is the area per atom. It becomes the following quadratic function of « and κ once the shift vector x is substituted by Eq. 10:

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J. Wu et al.

! ! pffiffiffi 3 @V 1 @2V 2 w¼ ðκ11 þ κ22 Þ þ pffiffiffi ðe11 þ e22 Þ2 8 @ cos φijk @r2ij 4 3 0 0 i B h 2 2 þ pffiffiffi ðe11  e22 Þ þ 4e12 , 32 3

(12)

where B

" ! ! ! # 3 ð 1  AÞ 2 @V @2V @2V ¼ 4 þ6 3 @ cos φijk @ cos φijk @ cos φijk @ cos φijk @ cos φijl r20 0 0 0 !
   2 2 1  A @ V 2 þ 4ð1 þ AÞ2 @@rV2  12 : ij r0 @rij @ cos φijk 0 0

(13) The strain energy density w depends on the curvature κ only via κ11 + κ22 and is independent of κ11  κ22 and κ 12. For equi-biaxial curvature κ 11 = κ 22 = κ (and no membrane strain, which gives a vanishing shift vector in Eq. 10),
both1 the bond length in Eq. 8 and bond angle in Eq. 9 change according to r2ij ¼ r20 1  12 κ 2 r20 and cos φijk ¼  12 þ 38 κ2 r20 . For terms up to κ 2, the bond length change does not contribute to the interatomic potential V in Eq. 7 since (rij  r0)2 is on the order O (κ4). The bond however, contributes to V via !  angle change,  X @V 1 cos φijk þ . For κ 11 = κ and κ22 =  κ, i.e., saddle shape, @ cos φijk 2 k6¼i, j 0

it can be shown that the k-summation over the three bonds at each atom vanishes and therefore does not contribute to the strain energy density. The membrane stress (force per unit length) is the work conjugate of membrane strain, @e@wαβ , and is given by ! 1 @2V ðσ 11 þ σ 22 Þt ¼ pffiffiffi ðe11 þ e22 Þ, 3 @r2ij 0



B ðσ 11  σ 22 Þt e11  e22 ¼ pffiffiffi , σ 12 t e12 8 3

(14)

It is noted that the membrane stress naturally emerges as the product of “conventional membrane stress” σ αβ (force per unit area) and “thickness” t of the graphene because the strain energy density is the energy per unit area (instead of volume). The shear rigidity (ratio of σ 12t to e12) is proportional to B,  and the  biaxial tension rigidity (ratio of (σ 11 + σ 22)t to (e11 + e22)) is proportional to

@2 V @r2ij

. 0

The moment (bending/twisting moment per unit length) is the work conjugate of curvature, @κ@wαβ , and is given by

16

Mechanics of Carbon Nanotubes and Their Composites

m11 þ m22

493

! pffiffiffi 3 @V ¼ ðκ11 þ κ 22 Þ, 2 @ cos φijk 0

m11  m22 ¼ 0: m12

(15)

Its torsion rigidity vanishes at infinitesimal strain (since m12 = 0), andthe biaxial @V bending rigidity (ratio of (m11 + m22) to (κ 11 + κ 22)) is proportional to @ cos , φ ijk

0

which reflects multi-body atomistic interactions. The above moment-curvature relation is different from the classical linear elastic shell theory in the following aspects because the strain energy density in Eq. 12 depends on the curvature only via κ11 + κ 22: (i) the torque vanishes, m12 = 0, and the normal components of the moment are always equal, m11 = m22; (ii) the moment is proportional to the sum of principal curvatures κ11 + κ22 and is independent of deviatoric components of curvatures. For example, curvatures κ 11 =  κ22 give vanishing moments. The membrane energy density and bending energy density can be obtained h C from the classical linear elastic thin shells [35] as 2ð1v ð e þ e22 Þ2 2ð1  vÞ 2Þ 11 h i

  e11 e22  e212 , D2 ðκ 11 þ κ 22 Þ2  2ð1  vÞ κ 11 κ22  κ212 , where C and D are the tension and bending rigidities, respectively, and v is Poisson’s ratio. A comparison of strain energy density in Eq. 12 with  the  above membrane energy density [35] C ¼ p1ffiffi3 suggests the identifications 1ν

@2 V @r2ij

 @2 V @r2ij

B C ¼ pffiffiffi   4 3 @2 V @r2ij

0

 0

B þ 8 0

C and 1þν ¼ 8pB ffiffi3, which give



,



B 8 v ¼  0 : B 2 @ V þ @r2ij 8 0 @2 V @r2ij



(16)

However, the comparison of strain energy density in Eq. 12 with the above pffiffi  3 @V and D(1  ν) = 0, bending energy density [35] suggests Dð1 þ νÞ ¼ 2 @ cos φ ijk

0

which give ! pffiffiffi 3 @V , D¼ 4 @ cos φijk

v ¼ 1:

(17)

0

Poisson’s ratios in Eqs. 16 and 17 are contradictory. This is because in a classical linear elastic shell, the bending rigidity results from the tension and compression on opposite sides of the neutral plane. Consequently, its bending, tension, torsion, and shear rigidities satisfy bending rigidity torsion rigidity ¼ : tension rigidity shear rigidity

(18)

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In graphene, the bending rigidity results from atomistic interactions,  multi-body @V as reflected from the bond angle dependence @ cos of the interatomic potential φ ijk

0

in Eq. 17 and is not directly related to the tension rigidity such that bending rigidity torsion rigidity 6¼ : tension rigidity shear rigidity

(19)

In fact, the right-hand side in Eq. 19 is zero due to the vanishing torsion rigidity at infinitesimal strain in Eq. 15. The above inequality makes it impossible to define a single Poisson’s ratio for both tension and bending. Equation 18 is the basis for defining the shell thickness in the prior continuum t2 shell modeling of CNTs [3] since the ratio of bending to tension rigidities equals 12 in the classical linear elastic shell theory. The effective thickness of graphene (or CNTs) has been defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 bending rigidity : (20) h¼ tension rigidity For the proportional biaxial stretching e22 = λe11 and curvature κ22 = λκ 11, the thickness given by Eq. 20 becomes vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  u u 2 @V u @ cos θijk 0 , h ¼ 3u t @ 2 V 1  λB þ @rij 2 0 1þλ8

(21)

which clearly depends on the type of loading represented by λ. Figure 5 shows the graphene thickness h versus λ for the second-generation Brenner potential. The effective thickness reaches the maximum at equi-biaxial stretching (λ = 1). However, even Eq. 21 cannot represent the general loading since it requires the curvature ratio κ 22/κ 11 and strain ratio e22/e11 to be the same. For the graphene subject to uniaxial tension (σ 22 = 0), λ = e22/e11 = κ22/ κ11 =  ν, the effective thickness becomes very simple:

huniaxial tension

v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u @V u u @ cos θijk ¼ 3t  2 0 : @ V @rij 2 0

(22)

For uniaxial stretching and curvature, λ = e22/e11 = κ 22/κ 11 = 0, the effective thickness of graphene is

huniaxial stretching

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  u 2 @V u @ cos θijk 0 : ¼ 3u t 2 B @ V þ 2 @rij 0 8

(23)

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495

Fig. 5 The effective thickness h of graphene versus the loading type λ (Reproduced with permission from Ref. [34])

For equi-biaxial stretching and curvature, λ = e22/e11 = κ 22/κ 11 = 1, the effective thickness becomes

hequi-biaxial stretching

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  @V u2 u @ cos θijk ¼ 3t  2 0 , @ V @rij 2 0

(24)

while for equi-biaxial tension/compression, λ = e22/e11 = κ22/κ11 =  1, the effective thickness is zero, and the graphene loses its bending rigidity. It should be emphasized that the tension and bending rigidities in Eqs. 14 and 15 are well defined and are independent of the type of loading. The dependence on the type of loading appears only when one wants to define the thickness and elastic modulus of graphene.

1.4

A Finite-Deformation Shell Theory for Carbon Nanotubes Based on the Interatomic Potential

1.4.1 General Description of a Curved Surface A point P on a curved surface can be represented by
 P ¼ P ξ1 , ξ2 ,

(25)

where (ξ1, ξ2) are curvilinear coordinates on the surface. The covariant base vectors @P Aα ¼ @ξ α ðα ¼ 1, 2Þ are in the tangent plane of the surface. The covariant base vectors give the coefficients of the first fundamental form as

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Aαβ ¼ Aα  Aβ :

(26)

2 The unit normal to the surface is N ¼ jAA11 A A2 j . The coefficients of the second fundamental form, which characterize the curvatures of the surface, are

Bαβ ¼ N 

@2P : @ξα @ξβ

(27)

For two points on the surface, P(ξ1, ξ2) and P + ΔP with coordinates ξα and ξ + Δξα, the Taylor series expansion around the point P gives α

ΔP ¼

@P α 1 @ 2 P 1 @3P α β Δξ Δξ þ Δξα Δξβ Δξγ α Δξ þ @ξ 2! @ξα @ξβ 3! @ξα @ξβ @ξγ  þ O jΔξj4 ,

(28)

@P and the second- and third-order derivatives of P involve the where @ξ α ¼ Aα coefficients Aαβ, Bαβ, and their derivatives [36]. For P and P + ΔP representing two neighbor atoms on the surface, any continuum theory that represents the collective behavior of atoms holds only when the characteristic length L of deformation is much larger than the atomic Δ, as schematically illustrated in h
spacing, i Δ Δ 3 Fig. 6. For the order of error O L , R , the Taylor series expansion (28) then

becomes [27, 28]. ΔP ¼

  1 1 Δξα  Bβλ Bγμ Aαμ Δξβ Δξγ Δξλ Aα þ Bαβ Δξα Δξβ N, 6 2

where Aαμ is the inverse of Aαμ. The length of ΔP is given by

Fig. 6 Schematic diagram of the characteristic length L of deformation pattern and the atomic spacing Δ. (a) L  Δ such that the collective behavior of atoms can be represented by a continuum. (b) L Δ such that a continuum description is not applicable

L D

(a)

D (b)

(29)

16

Mechanics of Carbon Nanotubes and Their Composites

jΔP j2 ¼ Aαβ Δξα Δξβ 

2 1
Bαβ Δξα Δξβ : 12

497

(30)

The angle φ between two vectors ΔP(1) and ΔP(2), which is needed in the multiΔP  ΔP body interatomic potential for carbon, is given by cos φ ¼ ΔP ð1Þ ΔPð2Þ , where j ð1Þ jj ð2Þ j ΔP ð1Þ  ΔP ð2Þ ¼ Aαβ Δξαð1Þ Δξβð2Þ  1 þ Bαβ Bγλ Δξαð1Þ Δξλð2Þ 3Δξβð1Þ Δξγð2Þ  2Δξβð2Þ Δξγð2Þ  2Δξβð1Þ Δξγð1Þ : 12 (31)

1.4.2 Membrane Strains and Curvatures of a Deformed Surface The surface P(ξ1, ξ2) deforms to a new surface p(ξ1, ξ2). The base vectors for the @p a1 a2 deformed surface are aα ¼ @ξ α ; the unit normal is n ¼ ja1 a2 j; and the coefficients of

p the first and second fundamental forms are aαβ = aα  aβ and bαβ ¼ n  @ξ@α @ξ β , 2

respectively. Two points P and P + ΔP on the surface deform to p and p + Δp, where Δp can be obtained from Eq. 29 by replacing Aα, N, Aαβ, and Bαβ with aα, n, aαβ, and bαβ, respectively. Its length and angle with neighbor bond can be similarly obtained from Eqs. 30 and 31. The components of the Green strain are the difference between the coefficients of the first fundamental form before and after the deformation, i.e., Eαβ ¼

 1
aαβ  Aαβ : 2

(32)

Similarly, the curvatures are the difference between the coefficients of the second fundamental form, K αβ ¼ bαβ  Bαβ :

(33)

The length of Δp and its angle with the neighbor bond are then related to the components of Green strain and curvatures by
  2 1
Bαβ þ K αβ Δξα Δξβ , jΔpj2 ¼ Aαβ þ 2E αβ Δξα Δξβ  12 cos φ ¼

(34)

Δp  Δp

ð 1Þ ð 2Þ , and jΔpð1Þ jjΔpð2Þ j

Δpð1Þ  Δpð2Þ ¼




 
 1
Bαβ þ K αβ Bγλ þ K γλ Aαβ þ 2E αβ Δξαð1Þ Δξβð2Þ þ 12  (35) Δξαð1Þ Δξλð2Þ 3Δξβð1Þ Δξγð2Þ  2Δξβð2Þ Δξγð2Þ  2Δξβð1Þ Δξγð1Þ :

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1.4.3 Deformation of the Single-Wall Carbon Nanotube As discussed in Sect. 1.3, carbon nanotubes do not have simple Bravais lattice since their hexagonal lattice structure can be decomposed to two triangular sub-lattices marked by open and solid circles in Fig. 3. The two sub-lattices undergo a shift vector η in order to minimize the energy and ensure the equilibrium of atoms. The vector ΔP between two atoms from different sub-lattices has coordinates Δξα on the CNT prior to deformation. It becomes Δp on the deformed CNT and has the new coordinates α

Δξ ¼ Δξα þ ηα ,

(36)

where ηα are to be determined. The bond length and angle in Eqs. 34 and 35 are α obtained by replacing Δξα by Δξ ,
 α β  α β i2 1 h
Bαβ þ K αβ Δξ Δξ , jΔpj2 ¼ Aαβ þ 2E αβ Δξ Δξ  12 cos φ ¼

(37)

Δp  Δp

ð 1Þ ð 2Þ , and jΔpð1Þ jjΔpð2Þ j

Δpð1Þ  Δpð2Þ ¼




 α 
 1
β α λ Bαβ þ K αβ Bγλ þ K γλ  Δξð1Þ Δξð2Þ Aαβ þ 2E αβ Δξð1Þ Δξð2Þ þ 12  β γ β γ β γ  3Δξð1Þ Δξð2Þ  2Δξð2Þ Δξð2Þ  2Δξð1Þ Δξð1Þ Þ: (38)

The bond length and angle, and also the interatomic potential, depend on the components of Green strain Eαβ, curvatures Kαβ, and shift vector ηλ,


 rij ¼ rij E αβ , K αβ , ηλ , φijk ¼ φijk E αβ , K αβ , ηλ ; V ¼ V rij , φijk ; k 6¼ i, j : (39) The strain energy density w is the energy per initial unit area and is given by X


 V rij , φijk ; k 6¼ i, j


 1 1j3 w E αβ , K αβ , ηλ ¼ , (40) 2 S0 ffi Ð Ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A11 A22  A212 dξ1 dξ2 is the average area per atom. The shift vector, where S 0 ¼

ηλ, is determined by minimizing energy to ensure the equilibrium of atoms, i.e., @w ¼ 0, @ηλ

(41)

which gives ηλ in terms of Eαβ and Kαβ,
 ηλ ¼ ηλ E αβ , K αβ :

(42)

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The strain energy density in Eq. 40 then becomes the function of Green strain Eαβ and curvature Kαβ:


 w ¼ w E αβ , K αβ , ηλ E αβ , K αβ :

(43)

1.4.4 Membrane Stresses and Moments in a Carbon Nanotube The membrane strain tensor, whose components are Eαβ, is E = EαβAαAβ, where Aα is the contravariant base vector related to the covariant base vector Aβ by Aα = AαβAβ. The reciprocal pair of base vectors satisfy Aα  Aα = 1 (no sum over α) and Aα  Aβ = 0 (α 6¼ β). Similarly, the curvature tensor is K = KαβAαAβ. The second Piola-Kirchhoff membrane stress tensor T is the work conjugate of the Green strain tensor E: T¼

Dw @w @w @η @w ¼ þ  ¼ , DE @E @η @E @E

(44)

where @w @η ¼ 0 in Eq. 41 has been used. The components of Eq. 44 are T αβ ¼

@w , @Eαβ

(45)

where Tαβ are the contravariant components of T, i.e., T = TαβAαAβ. It is important to distinguish the covariant and contravariant components and base vectors for curvilinear coordinates. The moment tensor M is the work conjugate of the curvature tensor K: M¼

Dw @w @w @η @w ¼ þ  ¼ : DK @K @η @K @K

(46)

Its contravariant components are M αβ ¼

@w , @K αβ

(47)

with M = MαβAαAβ. Equations 44 and 46 give the constitutive relation for CNTs. Their incremental form is given by _ T_ ¼ L : E_ þ H : K, _ M_ ¼ H T : E_ þ S : K,

(48)

where L = LT and S = ST are the symmetric tension and bending rigidities given by

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   2 1 D @w @2w @2w @ w @2w   , L¼  ¼ DE @E @E@E @E@η @η@η @η@E     1 D @w @2w @2w @2w @2w S¼   ,  ¼ DK @K @K@K @K@η @η@η @η@K

(49)

T and @w @η ¼ 0 in Eq. 41 has been used: H and H are the coupled tension/bending rigidities given by

   2 1 D @w @2w @2w @ w @2w   ,  ¼ DK @E @E@K @E@η @η@η @η@K    2 1 D @w @2w @2w @ w @2w T H ¼   ,  ¼ DE @K @K@E @K@η @η@η @η@E H¼

(50)

which represent the coupling between membrane stresses and curvatures (or between moments and membrane strains) for a curved surface.

1.4.5 Equilibrium Equations for Membrane Stresses and Moments The Cauchy membrane stress tensor t, defined for the deformed configuration, is related to the second Piola-Kirchhoff membrane stress T by 1 t ¼ F  T  FT, J

(51)

F ¼ a1 A1 þ a2 A2

(52)

where

is the deformation gradient tensor which maps an infinitesimal element dP = dξ1A1 + dξ2A2 of the CNT prior to deformation to dP = FdP = dξ1a1 + dξ2a2 of the deformed CNT, a1 and a2 are the co-deformed base vectors for the deformed qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi CNT, and FT = A1a1 + A2a2 and J ¼ a11 a22  ða12 Þ2 are the transpose and Jacobian of F, respectively.

The Cauchy membrane stress is then  obtained from Eqs. 51 and 52 as t ¼ J1 T 11 a1 a1 þ T 22 a2 a2 þ T 12 ða1 a2 þ a2 a1 Þ , which has the same components Tαβ as the second Piola-Kirchhoff membrane stress (except the factor J1 ) but on different base vectors, and Jt is called the contravariant “push forward” of T. Similarly, the moment tensor m for the deformed configuration is 1 m ¼ F  M  FT, J

(53)

 which gives m ¼ J1 M 11 a1 a1 þ M 22 a2 a2 þ M 12 ða1 a2 þ a2 a1 Þ and has the same components as the moment tensor M (except the factor J1) but different base vectors,

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Jmαβ = Mαβ, and Jm is called the contravariant “push forward” of M. Wu et al. [37] discussed the relations among t, m, and asymmetric membrane stress and moment tensors in most shell theories. In the current, deformed configuration, the constitutive relation (48) can be expressed via the Oldroyd rate as d
αβ  Jt ¼ L αβλγ d λγ þ H αβλγ κ λγ , ðJ t ÞOldr ¼ L : d þ H : κ, dt or d
 ðJ mÞOldr ¼ H T : d þ S : κ, Jmαβ ¼ H λγαβ d λγ þ S αβλγ κλγ , dt

(54)

where d and κ are related to the rates of Green strain E_ and curvature tensor K_ by
 d ¼ F T  E_  F 1 ¼ E_ 11 a1 a1 þ E_ 22 a2 a2 þ E_ 12 a1 a2 þ a2 a1 ,
 κ ¼ F T  K_  F 1 ¼ K_ 11 a1 a1 þ K_ 22 a2 a2 þ K_ 12 a1 a2 þ a2 a1 ,

(55) (56)

which have the same components as E_ αβ and K_ αβ but different base vectors, i.e., they are the covariant “push forward” of E_ and K_ , respectively. Their components are related to the covariant and contravariant components, vα and vα (α = 1, 2), of the velocity v and the component v3 = v3 normal to the CNT surface by [31].

bλ:β

 1
d αβ ¼ E_ αβ ¼ vα;β þ vβ;α  v3 bαβ , 2

(57)

καβ ¼ K_ αβ ¼ v3;αβ þ vλ;β bλα þ vλ;α bλβ þ vλ bλα;β  v3 bαλ bλ:β ,

(58)

λα



λα

a11 a21

a12 a22



where ¼ a bαβ , a is the inverse of aλα and is related to aλα by  1 a11 a12 ¼ , and “;” denotes the covariant derivatives. a21 a22 The tension, bending, and coupling rigidities in the current, deformed configuration in Eq. 54 are related to their counterparts in the initial configuration by   L ¼ ðFFFF Þ  L, S ¼ ðFFFF Þ  S,    H ¼ ðFFFF Þ  H, H 

T

 ¼ ðFFFF Þ  H T , 

(59)

(60)

and they have the same contravariant components, L αβλγ ¼ Lαβλγ, S αβλγ ¼ S αβλγ, and H αβλγ ¼ H αβλγ but again on different base vectors (the quad-dot product between FFFF and any fourth-order tensor simply change the base vectors from the initial configuration to the current, deformed configuration).

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The equilibrium equations in the finite-deformation shell theory can be written in terms of t and m as [31] t αβ ;β þ 2bαβ mβλ

;λ

α

þ bαλ;β mλβ þ X ¼ 0

bαβ t αβ þ bαβ bβλ mαλ  mβλ

;βλ

ðα ¼ 1, 2Þ,

þ X 3 ¼ 0,

(61) (62)

α

where X and X 3 are the effective in-surface and out-of-surface area forces (applied force per current unit area), respectively, and tαβ are the contravariant components of t. The traction boundary conditions are [31]:


 λ t αβ þ 2bαλ mλβ υβ ¼ t α þ bαλ m mαβ ;α υβ þ

ðα ¼ 1, 2Þ,

 T @
αβ @m m υα sβ ¼ t 3 þ , @s @s

(63) (64)

B , mαβ υα υβ ¼ m

(65)

where t α and t 3 are the prescribed effective in-surface and out-of-surface tractions, B and m T are the prescribed bending moment and torque, s is the unit vector along m the boundary, υ = s  n is the in-surface unit normal, @/@s is the directional λ is the component of boundary moment derivative along the direction of s, and m 1 2 ¼m  a1 þ m  a2 ¼ m T s þ m B υ. m The incremental equilibrium equations are α d  αβ d  α βλ d  α λβ d X t ;β þ 2 bβ m ;λ þ bλ;β m ¼0 þ dt dt dt dt

d  d d
bαβ t αβ þ bαβ bβλ mαλ  mβλ dt dt dt

1.5

;λβ

þ

ðα ¼ 1, 2Þ,

d X 3 ¼ 0: dt

(66)

(67)

The Rigidity and Uniform Deformation of the Single-Wall Carbon Nanotubes

1.5.1 Bending Rigidity of Graphene Yakobson et al. [3] calculated the bending rigidity of graphene in order to determine its thickness. The difference in energy (per atom), ΔU, between single-wall CNTs pffiffi S0 3 3 2 and graphene is related to the CNT radius by ΔU ¼ EI 2R2, where S 0 ¼ 4 r0 is the area of graphene per atom and EI is the bending rigidity of graphene. Yakobson et al.

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[3] then obtained the bending rigidity EI by fitting the molecular dynamics results to S0 the ΔU ¼ EI 2R 2 relation. We study the rolling of a graphene sheet into a CNT in order to validate the above 1 S0 expression ΔU ¼ EI 2R and ξ2 be the Cartesian coordinates on the 2 . Let ξ (undeformed) graphene, which is rolled to a CNT along ξ1 (and therefore ξ2 is along the CNT axis). The coefficients of the first and second fundamental forms are Aαβ = δαβ and Bαβ = 0. Let e1 and e2 denote the engineering strains to be determined in the circumferential and axial directions (ξ1 and ξ2) during rolling of the graphene to the single-wall carbon nanotube. Only armchair and zigzag CNTs are studied such that there is no shear. The point P(ξ1, ξ2) on the graphene becomes p = ReR + (1 + e2)ξ2eZ on the CNT, where R is the CNT radius and eR and eZ are 1 the unit vectors along the radial and axial directions. The polar angle is Θ ¼ ð1þeR1 Þξ . @p @p ¼ ð1 þ e2 ÞeZ , where eΘ is The base vectors are a1 ¼ @ξ 1 ¼ ð1 þ e1 ÞeΘ and a2 ¼ @ξ2 the unit vector in the circumferential direction. The nonvanishing coefficients of the first and second fundamental forms become a11 = (1 + e1)2, a22 = (1 + e2)2, and 2 b11 ¼  ð1þeR 1 Þ , which give the nonvanishing components of the Green strain and 2

curvature tensors E 11 ¼ e1 þ 12 e21 , E22 ¼ e2 þ 12 e22 , and K 11 ¼  ð1þeR 1 Þ . The strain energy density w becomes a function of e1 and e2 (and shift vector η1 and η2). Since the CNT is not subject to any externally applied strain during the rolling from the graphene, e1 and e2 are determined by minimizing the strain energy, @w @w i.e., @e ¼ @e ¼ 0. 1 2 Figure 7 shows the energy (per atom) versus the CNT radius R for armchair S0 and zigzag CNTs. The relation ΔU ¼ EI 2R 2 is also shown, with the bending rigidity

S0 Fig. 7 Dependence of the energy according to the present theory and of EI 2R 2 on the carbon nanotube radius R (Reproduced with permission from Ref. [27])

504

EI ¼ Mκ1111 ¼

J. Wu et al.

pffiffi  3 @V 4 @ cos φijk 0

for graphene under uniaxial curvature κ11 (κ 22 = 0). The

S0 present analysis agrees very well with the relation ΔU ¼ EI 2R 2.

1.5.2 Tension of Carbon Nanotubes An armchair or zigzag CNT is subject to simple tension with the (engineering) axial strain eaxial. The strain elateral in the circumferential direction due to lateral contraction is to be determined. Let ξ1 = RΘ and ξ2 = Z, where Θ and Z are the cylindrical coordinates and the initial radius R of the CNT prior to tension is used to scale Θ such that ξ1 has the length dimension. The nonvanishing coefficients of the first and second fundamental forms are A11 = 1, A22 = 1, and B11 ¼  R1 . The point P = ReR + ZeZ on the CNT prior to tension becomes p = R(1 + elateral)eR + Z @p (1 + eaxial)eZ on the deformed CNT. The base vectors a1 ¼ @ξ 1 ¼ ð1 þ elateral ÞeΘ and @p a2 ¼ @ξ2 ¼ ð1 þ eaxial ÞeZ give the nonvanishing coefficients of the first and second fundamental forms a11 = (1 + elateral)2, a22 = (1 + eaxial)2 and b11 ¼  1þeRlateral. The nonvanishing components of the Green strain and curvature tensors are E 11 ¼ elateral þ 12 e2lateral , E22 ¼ eaxial þ 12 e2axial , and K 11 ¼  elateral R . The nonvanishing components of the second Piola-Kirchhoff membrane stress and moment tensors are obtained from Eqs. 45 and 47. The equilibrium equations become T 11  M 11 Rð1þelateral Þ ¼ 0 , which gives the strain elateral in the circumferential direction in terms of the axial strain eaxial, i.e., elateral = elateral(eaxial). Figure 8 shows the axial force, normalized by the CNT circumference 2πR prior to tension, versus the axial strain for several armchair and zigzag CNTs. The curves

Fig. 8 The axial force, normalized by the carbon nanotube circumference 2πR prior to deformation, versus the engineering strain eaxial for several armchair and zigzag carbon nanotubes in simple tension (Reproduced with permission from Ref. [28])

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Fig. 9 The force/strain ratio at the vanishing strain versus the radius of armchair and zigzag carbon nanotubes (Reproduced with permission from Ref. [28])

do not depend strongly on the CNT radius R, but the chirality has a strong effect. At the limit of strain approaching zero, the force/strain ratio is given analytically as 2π [R(L2222  v12L2211) + v12H2211], where Lαβλγ , Sαβλγ , and Hαβλγ are the components of the tension, bending, and coupled tension/bending rigidity tensors on the R2 L1122 RH 2211 base vectors, eΘ and eΖ, of the CNT prior to tension, v12 ¼ 2 ð0Þ is R L1111 2RH 1111 þS 1111 þRM 11

Poisson ratio, and the superscript “(0)” denotes the initial residual membrane stress and moment in the CNT. Figure 9 shows the force/strain ratio (for the infinitesimal strain), normalized by 2πR, versus the CNT radius for armchair and zigzag CNTs. The force/strain ratio depends weakly on the CNT chirality since the curves for armchair and zigzag are rather close. Furthermore, the force/strain ratio normalized by 2πR becomes a constant once the CNT radius exceeds 1 nm.

1.5.3 Pressure of Carbon Nanotubes For an armchair or zigzag CNT of radius R subject to internal pressure Pint (force per initial unit area), the deformation is uniform and axisymmetric prior to bifurcation. The analysis in Sect. 1.5.2 still holds except that the equilibrium equation becomes M 11 Pint R T 11  Rð1þe ¼ 1þe when finite-deformation effects are taken into account. The lateral Þ lateral traction-free condition at the end of CNT requires T22 = 0. These two equations give the strains eaxial and elateral in the axial and circumferential directions in terms of the internal pressure Pint. Figure 10 shows the internal pressure, multiplied by the initial radius R of the CNT, versus the lateral strain for several armchair and zigzag CNTs. Similar to Fig. 8, the curves do not depend strongly on the CNT radius R, but the chirality (armchair vs. zigzag) has a strong effect. At the limit of internal pressure approaching zero, the pressure/strain ratio is given analytically as

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Fig. 10 The internal pressure, multiplied by the initial radius R, versus the engineering strain elateral for several armchair and zigzag carbon nanotubes in internal pressure (Reproduced with permission from Ref. [28])

Fig. 11 The internal pressure/strain ratio at small strains, multiplied by the initial radius R, versus the radius of armchair and zigzag carbon nanotubes (Reproduced with permission from Ref. [28])

 ð0Þ R3 R2 L1111  v21 R2 L1122  2RH 1111 þ v21 RH 2211 þ S 1111 þ RM 11 , where v21 ¼ RL2211 H 2211 RL2222

is the corresponding Poisson’s ratio. Figure 11 shows the pressure/strain ratio (at the infinitesimal strain), multiplied by R, versus the CNT radius for armchair and zigzag CNTs. The pressure/strain ratio depends weakly on the chirality since the

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curves for armchair and zigzag CNTs are rather close. The pressure/strain ratio, multiplied by R, becomes a constant once the CNT radius exceeds 2 nm.

1.5.4 Torsion of Carbon Nanotubes An armchair or zigzag CNT is subject to torsion, and the twist (rotation per unit initial length) is denoted by κ. For ξ1 = RΘ and ξ2 = Z, the nonvanishing coefficients of the first and second fundamental forms are A11 = 1, A22 = 1, and B11 ¼  R1 , where R is the CNT radius prior to torsion. The point P = ReR + ZeZ prior to torsion moves to p = R(1 + elateral)er + Z(1 + eaxial)eZ, where the initial unit vectors eR and eΘ distort during torsion to become er = eR cos κZ + eΘ sin κZ and eθ =  eR sin κZ + eΘ cos κZ since the polar angle Θ becomes θ = Θ + κZ. The strains elateral and eaxial in the circumferential and axial directions due to finite torsion @p are to be determined. The base vectors after the torsion are a1 ¼ @ξ 1 ¼ ð1 þ elateral Þeθ @p and a2 ¼ @ξ2 ¼ ð1 þ elateral ÞRκeθ þ ð1 þ eaxial ÞeZ , which give the coefficients of first and second fundamental forms 

a11 a21

 ð1 þ elateral Þ2 ð1 þ elateral Þ2 Rκ ¼ , ð1 þ elateral Þ2 Rκ ð1 þ elateral Þ2 R2 κ2 þ ð1 þ eaxial Þ2 #   " 1 þ elateral b11 b12 ð1 þ elateral Þκ  : ¼ R b21 b22 ð1 þ elateral Þκ ð1 þ elateral ÞRκ2

a12 a22





(68)

(69)

The components of Green strain and curvature tensors are 

E 11 E 21

E12 E22





2

3 1 ð1 þ elateral Þ2 Rκ 7 2 5, 1 2 1 2 2 2 eaxial þ eaxial þ ð1 þ elateral Þ R κ 2 2 (70) " #  elateral ð1 þ elateral Þκ  K 12 : (71) ¼ R K 22 ð1 þ elateral Þκ ð1 þ elateral ÞRκ2

1 2 6 elateral þ 2 elateral ¼ 41 ð1 þ elateral Þ2 Rκ 2

K 11 K 21

The components of the second Piola-Kirchhoff membrane stress and moment are obtained from Eqs. 45 and 47. The equilibrium equations become 2 2 12 þκ R M 22 T 11 þ 2κRT 12  M 11 þ2κRM ¼ 0. The boundary conditions for pure torsion Rð1þelateral Þ require T22 = 0. These two equations give the strains eaxial and elateral in the axial and circumferential directions in terms of the twist κ. Figure 12 shows the torque, normalized by 2πR2, versus the normalized twist κR for several armchair and zigzag CNTs. Similar to Figs. 8 and 10, the CNT radius has little effect, but the chirality has a strong influence since the curves for armchair CNTs are far above the zigzag CNTs. At the limit of twist approaching zero, the
ð0Þ torque/twist ratio is given analytically as 2πR R2 L1212  4RH 1212 þ 4S 1212 2RM 22 Þ.

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Fig. 12 The torque, normalized by 2πR2 prior to deformation, versus the normalized twist κR for several armchair and zigzag carbon nanotubes in pure torsion (Reproduced with permission from Ref. [28])

Fig. 13 The torque/twist ratio at vanishing twist, normalized by 2πR2 prior to deformation, versus the nanotube radius (Reproduced with permission from Ref. [28])

Figure 13 shows the torque/twist ratio (at infinitesimal twist), normalized by 2πR3, versus the CNT radius for armchair and zigzag CNTs, which once again depends weakly on the CNT chirality. The normalized torque/twist ratio also becomes a constant once the CNT radius exceeds 2 nm.

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Instability of Carbon Nanotubes

For a CNT subject to tension/compression, internal/external pressure, and torsion, the deformation is uniform prior to instability, as given in Sect. 1.5. The increment of deformation becomes nonuniform after bifurcation takes place [38].

1.6.1 Instability of Carbon Nanotubes in Tension The increment (rate) of deformation becomes nonuniform at the onset of bifurcation. Let v ¼ U_ denote the velocity with the covariant components vΘ = v1, vZ = v2, and vR = v3. For a CNT of length L subject to the axial force but no shear at two ends Z = 0 and L, the velocity (prior to and) at the onset of bifurcation satisfies vZ ¼

@vΘ @vR ¼ ¼0 @Z @Z

Z ¼ 0, L:

at

(72)

We first consider axisymmetric bifurcation. The axisymmetric velocity field, satisfying the incremental equilibrium Eqs. 66 and 67 and boundary condition Eq. 72, is given by vΘ ¼ 0, vZ ¼ V Z0 sin

mπZ mπZ , vR ¼ V R0 cos , L L

(73)

where m = 1, 2, 3,    is the eigenmode number and (VZ0, VR0) is the corresponding eigenvector satisfying the following homogeneous algebraic equation: 

QZZ QRZ

QZR QRR



V Z0 V R0

 ¼ 0:

(74)

The coefficients QIJ are given in terms of the tension; bending and coupled tension/bending rigidity tensors, L, S, and H; and the membrane stress and moment tensors, T and M, by " QZZ ¼  L2222 þ QZR QRR

#

T 22

mπ 2

, L ð1 þ eaxial Þ " #   mπ 3 1 þ elateral H 2211 mπ M 22 þ H 2222 þ L2211  ¼ QRZ ¼  , 2 L R Rð1 þ elateral Þ L ð1 þ eaxial Þ " # ð1 þ elateral Þ2 2H 1111 S 1111 M 11 þ 2 ¼ L1111  þ 2 2 R ð 1 þ e Þ R lateral R ð1 þ elateral Þ Rð1 þ elateral Þ3 " # mπ 2 mπ 4 2ð1 þ elateral Þ 2 1 þ elateral H 1122  2 S 1122  T 22  þ M  S 2222 : 11 2 R L L R Rð1 þ eaxial Þ 2

(75)

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Fig. 14 The critical strain for bifurcation in tension based on the second-generation interatomic potential versus mR/L for the (8,8) armchair carbon nanotube (Reproduced with permission from Ref. [28])

In order to have a nontrivial solution, the determinant of the 2  2 matrix in Eq. 74 must vanish. The smallest axial strain eaxial that satisfies this condition is the critical strain for bifurcation in tension and is denoted by etension . cr For non-axisymmetric bifurcation, the velocity field, satisfying the incremental equilibrium Eqs. 66 and 67 and boundary condition Eq. 72, is given by mπZ mπZ , vZ ¼ V Z0 cos nΘ sin , L L mπZ , vR ¼ V R0 cos nΘ cos L

vΘ ¼ V Θ0 sin nΘ cos

(76)

where n = 1, 2, 3,   , n = 0 degenerates to the axisymmetric bifurcation Eq. 73, and the eigenvector (VΘ0, VZ0, VR0) satisfies a set of homogeneous algebraic equations. The vanishing of the determinant of coefficient matrix gives the condition for non-axisymmetric bifurcation. Figure 14 shows the critical strain etension for bifurcacr tion in tension based on the second-generation potential [32]. The critical strain for axisymmetric bifurcation (n = 0) is approximately a constant, 0.299. The critical strain for non-axisymmetric bifurcation (n = 1, 2, 3,   ) is smaller than the axisymmetric bifurcation strain (n = 0) such that non-axisymmetric bifurcation prevails.

1.6.2 Instability of Carbon Nanotubes in Compression We use the finite-deformation shell theory formulated in Sect. 1.4 to study the instability of CNTs in compression. The deformation is uniform prior to bifurcation and is the same as that in Sect. 1.5.2 except for the axial strain eaxial < 0. Once the

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Fig. 15 The critical strain for bifurcation in compression based on the second-generation interatomic potential versus L/(mR) for the (8,8) armchair carbon nanotube (Reproduced with permission from Ref. [28])

compressive strain reaches a critical value, ecompression , bifurcation occurs. In the cr following ecompression is called the buckling strain. cr The onset of bifurcation is governed by the analysis in Sect. 1.6.1, except that based on the secondeaxial < 0. Figure 15 shows the buckling strain ecompression cr generation interatomic potential [32] versus L/(mR) for the (8,8) armchair CNT in compression, where m(=1, 2, 3,   ) is the eigenmode number. Both axisymmetric (n = 0) and non-axisymmetric bifurcations (n = 1, 2, 3,   ) are considered, where the velocity field for n = 1 is identical to that for the Euler beam buckling. For long CNTs L/(mR) > 8.2, n = 1 (Euler beam buckling) gives the smallest buckling strain ecompression . For relatively short CNTs L/(mR) < 8.2, n = 2 gives the smallest cr . Axisymmetric bifurcation (n = 0) never occurs in compression. ecompression cr

1.6.3 Instability of Carbon Nanotubes Subject to Internal Pressure Similar to Sect. 1.5.3, the deformation is still uniform at the onset of bifurcation, but its increment is not and is given by Eq. 76. The incremental equilibrium, Eqs. 66 and 67, then gives a homogeneous algebraic equation for (VΘ0, VZ0, VR0) as 0

QΘΘ @ QZΘ QRΘ

QΘZ QZZ QRZ

10 1 QΘR V Θ0 QZR A@ V Z0 A ¼ 0, QRR V R0

(77)

where, similar to Eq. 75, the coefficients QIJ are given analytically in terms of the rigidity tensors L, S, and H and the membrane stress and moment tensors T and M. In order to have a nontrivial solution, the determinant of the 3  3 matrix in Eq. 77 must

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Fig. 16 The critical internal pressure for bifurcation based on the second-generation interatomic potential versus L/(mR) for the (8,8), (12,12), and (16,16) armchair carbon nanotubes. The bifurcation corresponds to n = 1 (non-axisymmetric mode) (Reproduced with permission from Ref. [28])

vanish, which gives the critical condition for bifurcation of CNTs subject to internal pressure. Figure 16 shows the critical internal pressure for bifurcation, Pint cr , based on the L second-generation interatomic potential [32] versus mR for (8,8), (12,12), and (16,16) armchair CNTs, where m (=1, 2, 3  ) is the eigenmode number and R and L are the CNT radius and length, respectively. The bifurcation mode always corresponds to n = 1 (the same mode as the Euler beam buckling). The CNT radius has little effect on the critical internal pressure for bifurcation since the curves in Fig. 16 are close. This is similar to the bifurcation in tension (Sect. 1.6.1) since the membrane stress state is also in tension for the CNT subject to the internal pressure.

1.6.4 Instability of Carbon Nanotubes Subject to External Pressure For an armchair or zigzag CNT of radius R subject to external pressure Pext, the deformation is uniform and axisymmetric prior to bifurcation. The analysis in Sect. 1.6.3 still holds except that the internal pressure Pint is replaced by Pext. Equation 77 also gives the critical condition for bifurcation into the modes according to Eq. 76 under external pressure (if Pint is replaced by Pext). Figure 17 shows the critical external pressure for bifurcation, Pext cr , based on the second-generation interatomic potential [32] versus L/(mR) for the (8,8) armchair CNT, where m (=1, 2, 3  ) is the eigenmode number and R and L are the CNT radius and length, respectively. Both the axisymmetric (n = 0) and non-axisymmetric bifurcations (n = 1, 2, 3,   ) are considered. For CNT lengths L/(mR) > 3.8, n = 2 gives the lowest critical external pressure Pext cr . For very short

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Fig. 17 The critical external pressure for bifurcation based on the second-generation interatomic potential versus L/(mR) for the (8,8) armchair carbon nanotube (Reproduced with permission from Ref. [28])

CNTs L/(mR) < 3.8, other bifurcation modes (n = 3, 4,   ) appear, but the axisymmetric bifurcation (n = 0) or non-axisymmetric bifurcation with n = 1 never occurs. The bifurcation modes (n = 2, 3, 4,   ) for the external pressure are completely different from that for the internal pressure (n = 1) in Sect. 1.6.3.

1.6.5 Instability of Carbon Nanotubes in Torsion The deformation is still uniform at the onset of bifurcation, but its increment is not. Let v ¼ U_ , with components vR, vΘ, and νZ, denote the velocity away from the homogeneous solution. The velocity field in the CNT can be expressed as vΘ ¼ V Θ0 cos

      mπZ mπZ mπZ  nΘ , vZ ¼ V Z0 cos  nΘ , vR ¼ V R0 sin  nΘ , L L L (78)

where n = 0, 1, 2, 3,   , m (=1, 2, 3  ) are the eigenmode numbers, and (VΘ0, VZ0, VR0) is the corresponding eigenvector. Substitution of Eq. 78 into the incremental equilibrium Eqs. 66 and 67 yields a set of homogeneous algebraic equations for (VΘ0, VZ0, VR0). The vanishing of determinant of the coefficient matrix gives the condition for the critical twist κcr at which the torsional bifurcation takes place. Figure 18 shows the critical twist for bifurcation, κ crR, based on the secondgeneration interatomic potential [32] versus L/(mR) for the (8,8), (12,12), and (16,16) armchair CNTs, where m (=1, 2, 3  ) is the eigenmode number and R and L are the CNT radius and length, respectively. For long CNTs (large L/(mR)), the bifurcation mode is always n = 2. The corresponding critical twist for bifurcation κ cr

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Fig. 18 The critical twist for bifurcation based on the second-generation interatomic potential versus L/(mR) for the (8,8), (12,12), and (16,16) armchair carbon nanotubes (Reproduced with permission from Ref. [28])

approaches a constant value for each CNT as L/(mR) increases. For relatively short CNTs, other bifurcation modes n = 3, 4,    appear, but n = 0 and n = 1 never occur.

1.7

Concluding Remarks

A finite-deformation shell theory for single-wall carbon nanotubes (CNT) is established directly from the atomic structure of CNT and the interatomic potential. It is different from all existing shell theories because of the coupling in the constitutive model between the membrane stress and curvature and between moment and membrane strain. It is also different from the classical Cauchy-Born rule linking atomistic models to the three-dimensional continuum theory since it accounts for the important effect of moment and curvature (of CNTs) [39]. The atomistic-based finite-deformation shell theory is used to study the rigidity and instability of single-wall CNTs subject to tension, compression, internal and external pressure, and torsion.

2

Mechanics of Carbon Nanotubes-Reinforced Composites

2.1

Introduction

Due to their superior mechanical properties, carbon nanotubes (CNTs) have been used as reinforcements in polymer matrix composites [40–45]. Carbon nanotubes usually do not bond well to polymers [46–48] such that their interaction is the van

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der Waals force [49–53], which is much weaker than covalent bonds. This leads to sliding of CNT in polymer matrix when subjected to loading. Continuum models have been developed for CNT-reinforced polymer matrix composites [51, 54–58]. As compared to atomistic simulations such as molecular dynamics, continuum models are not constrained on the length and time scales. This gives the suitability for the study of nanocomposites. However, modeling of CNT/ polymer interfaces has always been a challenge because of the difficulties to account for the van der Waals force in continuum models. Cohesive zone models have been widely used in the continuum study of interface debonding and sliding in composites [59–61]. A cohesive zone model assumes a relation between the normal (and shear) traction(s) and the opening (and sliding) displacement(s). When implemented in the finite element method, the cohesive zone model is capable of simulating interface debonding and sliding [62–71]. However, these cohesive models are all phenomenological due to the difficulties in measuring directly the cohesive laws for interfaces. There are also some experimental studies of microscale cohesive laws [71–77] but none on nanoscale cohesive laws such as the CNT/polymer interfaces. Jiang et al. [78] developed a cohesive zone model for interfaces between singlewall carbon nanotubes (SWCNTs) and polymer matrix directly from the van der Waals interactions. It focuses on the van der Waals force and does not consider the possible chemical bonding even though the latter may contribute to CNT/polymer interactions [79, 80]. Jiang et al.’s model gives the relation between the normal (and shear) traction(s) and the opening (and sliding) displacement(s) across the interface in terms of parameters for the van der Waals interactions. Such a cohesive law provides the link between the nanoscale cohesive law and macroscopic properties of composite materials. For example, it has been applied to study the effect of CNT/polymer interfaces on the macroscopic behavior of CNT-reinforced composite materials [68, 81]. Pantano et al. [82, 83] used nonlinear springs to represent the van der Waals force in their finite element analysis of multi-wall carbon nanotubes (MWCNTs). Lu et al. [84, 85] extended Jiang et al.’s [78] cohesive law to interfaces between MWCNTs and polymer matrix since MWCNTs are one of the widely used reinforcements in nanocomposites. Tan et al. [86] incorporated Jiang et al.’s [78] nonlinear cohesive law for CNT/polymer interfaces in the study of the macroscopic behavior of CNT-reinforced composites. In Jiang et al. [78] and Lu et al.’s [84, 85] cohesive laws, the energy between two atoms of distance r due to the van der Waals force is represented by the LennardJones 6–12 potential: V ðrÞ ¼ 4e

 12  σ σ6  , r12 r6

(79)

pffiffiffi where V is the energy between a pair of atoms of distance r, 6 2σ is the equilibrium distance between the atoms, and e is the bond energy at the equilibrium distance. For a pair of carbon atoms, the bond energy and equilibrium distance are

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eC  C = 0.002390 eV and σ C  C = 0.3415 nm [87]. For a carbon atom and a –CH2– unit in polymer, eC  C = 0.004656 eV and σ C  C = 0.3825 nm [88], respectively. The purpose of Sect. 2 is to review the cohesive laws between CNTs and a polymer matrix, which link the macroscopic behavior of CNT-reinforced nanocomposites and the nanoscale behavior of interfaces. Section 2.2 describes the cohesive law for interfaces between single-wall carbon nanotubes and polymers due to the van der Waals force [78]. Section 2.3 describes the cohesive law for interfaces between multi-wall carbon nanotubes and polymers due to the van der Waals force. A micromechanics model is provided in Sect. 2.4 to predict the nonlinear behavior of CNT-reinforced composite in terms of the properties of CNT/matrix interfaces, as well as those of CNTs and matrix.

2.2

A Cohesive Law for Interfaces between Single-Wall Carbon Nanotubes and Polymers Due to the van der Waals Force

2.2.1

The Cohesive Law for a Single-Wall Carbon Nanotube in an Infinite Polymer Matrix Firstly, the effect of CNT radius is neglected, and the interaction between a graphene (i.e., an infinite plane of carbon atoms) and polymer is studied. The graphene is parallel to the polymer surface, and h denotes their equilibrium distance (Fig. 19a). In order to establish a continuum cohesive law, carbon atoms on the graphene are homogenized to a plane with an area density ρc, where ρc is
related pffiffiffi to the equilibrium bond length l0 of graphene prior to deformation by ρc ¼ 4= 3 3l 0 2 . The number of carbon atoms over an area dA on the graphene is ρcdA. Similarly, ρp denotes the volume density of polymer molecules and the number of polymer molecules over a volume dV is ρpdV. The distance between a point (0,0) p onffiffiffiffiffiffiffiffiffiffiffiffiffiffi the graphene and a point (x, z)(x   h, z 0) in the polymer (Fig. 19a) is r ¼ x2 þ z2. The energy due to the van der Waals force is given by V(r) in Eq. 79. For an infinitesimal area dA, the energy stored due to the van der Waals force is

Fig. 19 A schematic diagram of a graphene parallel to the surface of an infinite polymer: (a) the distance between the graphene and polymer surface is h; (b) the graphene is subjected to the opening and sliding displacements (Reproduced with permission from Ref. [78])

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ð ρC dA

V ðrÞρp dV polymer ¼ 2πρp ρC dA

517

ð h

V polymer

ð1 dx

1

V ðrÞzdz:

(80)

0

The cohesive energy Φ is the energy per unit area and is given by Φ ¼ 2πρp ρC

ð h

ð1 dx

1

V ðrÞzdz ¼

0

 9  2π 2σ σ3 ρp ρC eσ 3  , 3 15h9 h3

(81)

pffiffiffi where 6 2σ is the equilibrium distance in the van der Waals force and e is the corresponding bond energy. The equilibrium distance h is determined by minimizing the energy, @Φ/@h = 0, as h¼

 1=6 2 σ ¼ 0:858σ: 5

(82)

For the opening displacement v and sliding displacement u beyond the equilibrium distanceh, as shown in Fig. 19b, the cohesive energy can be similarly obtained as ! 9 3 2π 2σ σ Φ ¼ ρp ρC eσ 3  , 3 15ðh þ vÞ9 ðh þ vÞ3

(83)

which can also be obtained from Eq. 81 by simply replacing h with h + v. Equation 83 is independent of the sliding displacement u because sliding does not change the van der Waals force for infinite graphene and polymer. This leads to a vanishing shear cohesive stress: τcohesive ¼

@Φ ¼ 0: @u

(84)

The tensile cohesive stress is obtained from Eq. 84 as

σ cohesive

" # @Φ σ4 2σ 10 2 ¼ 2πρp ρC eσ : ¼  @v ðh þ vÞ4 5ðh þ vÞ10

(85)

Equations 84 and 85 give the cohesive law for graphene/polymer based on the van der Waals force. Such a cohesive law gives the following cohesive properties:
1=6 (1) Initial slope ðdσ cohesive =dv at v ¼ 0Þ ¼ 30 25 πρp ρC eσ 2 (2) Cohesive strength (maximum cohesive stress) σ max ¼ 6π 5 ρp ρC eσ h
21=6 i (3) Critical separation δ0 ¼ σ  h ¼ 1  5 σ

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(4) Total Φtotal

cohesive energy qffiffi 5 3 ¼ 4π 9 2ρp ρC eσ

(area

underneath

the

σ cohesive~v

curve)

The cohesive law in Eqs. 83 and 85 can be rewritten in terms of the cohesive strength σ max and total cohesive energy Φtotal as

Φ¼

9 > =

8 >
2 > σ max ; : 1 þ 0:682 σmax v 1 þ 0:682 v Φtotal Φtotal 8 >
=

1 1 σ cohesive ¼ 3:07σ max h i4  h i10 : > > σ max ; : 1 þ 0:682 σ max v 1 þ 0:682 v Φtotal Φtotal

(87)

The initial slope and critical separation can also be expressed in terms of σ max and Φtotal by 12:6σ 2max =Φtotal and 0.242Φtotal/σ max, respectively. For a CNT embedded in an infinite polymer matrix, let h denote the equilibrium distance between the CNT and polymer surface (Fig. 20). Carbon atoms are homogenized on the CNT and represented by an area density ρC (which may be slightly different from that of graphene due to the effect of CNT radius). The volume density of polymer molecules is still denoted by ρp. The cylindrical coordinates (R, θ, z) are used with z as the central axis of the CNT. Without losing generality, a point (R, 0, 0)

Fig. 20 A schematic diagram of a single-wall carbon nanotube (SWCNT) in a polymer matrix. The SWCNT radius is R, and the distance between the SWCNT and polymer surface is h (Reproduced with permission from Ref. [78])

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519

on the CNT and a point (ζ, θ, z)(ζ R + h) in the polymer are picked, where R is the CNTpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi radius. The distance between these two points is ffi r ¼ ζ 2  2Rζ cos θ þ R2 þ z2 . The energy due to the van der Waals force is still given by V(r) in Eq. 79. For a section of CNT and polymer with height dz, the energy stored in this section due to the van der Waals force is ð ρC 2πRdz

V ðrÞρp dV polymer ¼ 2πρp ρC Rdz

ð1 Rþh

V polymer

ζdζ

ð 2π

ð1 dθ

0

1

V ðrÞdz0 , (88)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ ζ 2  2Rζ cos θ þ R2 þ z0 2 . The cohesive energy Φ is the energy per unit area, and is given by Ð 2π Ð 1 ζdζ 0 dθ 1 V ðrÞdz0   Φ¼ h 2π R þ dz 2 Ð1 Ð 2π Ð 1 ρp ρC R Rþh ζdζ 0 dθ 1 V ðrÞdz0 , ¼ h Rþ 2 2πρp ρC Rdz

Ð1

Rþh

(89)

where 2π(R + h/2)dz is the average of CNT area 2πRdz and polymer surface area 2π (R + h)dz. Figure 21 shows theqcohesive energy, normalized by the total cohesive energy for ffiffi graphene, Φtotal ¼ 4π 9

5 3 2ρp ρC eσ

, versus the distance h between the CNT and

polymer surface for several CNT radii, where h is normalized by its equilibrium
1=6 value 25 σ in Eq. 82 for graphene and σ = 0.3825 nm is the characteristic length in the van der Waals force. The curves for different CNTs are very close, and they are Fig. 21 The cohesive energy, normalized by the total cohesive energy for graphene, versus the distance h between the single-wall carbon nanotube (SWCNT) and polymer surface for several SWCNT radii and graphene, where h is normalized by its equilibrium value (2/5)1/6σ for graphene and σ = 0.3285 nm is the characteristic length in the van der Waals force (Reproduced with permission from Ref. [78])

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Fig. 22 The tensile cohesive stress σ cohesive normalized by the cohesive strength σ max versus the normalized opening displacement v/δ0, where δ0 = 0.242Φtotal/σ max is the critical separation at which the cohesive strength is reached and δ0 = 0.0542 nm for SWCNTs/polyethylene (Reproduced with permission from Ref. [78])

all close to that for graphene. This suggests that the CNT radius has little effect on the cohesive energy. Each curve has a minimum, which occurs at the equilibrium distance between the CNT and polymer surface, and this distance is very close to
21=6 σ in Eq. 82. 5 Similar to graphene, the cohesive energy for a CNT in an infinite polymer matrix is independent of the sliding displacement u such that the shear cohesive stress vanishes. Since the effect of CNT radius is small as shown in Fig. 21, the cohesive energy and tensile cohesive stress are still given approximately by Eqs. 83 and 85, respectively, and the cohesive law also holds. Figure 22 shows the tensile cohesive stress σ cohesive, normalized by the cohesive strength σ max, versus the normalized opening displacement v/δ0, where δ0 = 0.242Φtotal/σ max is the critical separation at which the cohesive strength is reached and δ0 = 0.0542 nm for CNTs/polyethylene. The cohesive stress increases rapidly at small opening displacement and gradually decreases after the cohesive strength is reached.

2.2.2

The Cohesive Law for a Single-Wall Carbon Nanotube in a Finite Polymer Matrix The interaction is firstly studied between a graphene and polymer that overlaps over a length L, as shown in Fig. 23. The distance between the graphene and polymer surface is h. The area density of the graphene and volume density of polymer molecules are ρC and ρp, respectively. Figure 23 shows the Cartesian coordinates (x, y, z), where x is the direction normal to the graphene (and polymer surface) and y is along the graphene free edge. The graphene-free edge is represented by (0, y, 0), while the free edge of polymer surface is (h, y, L ). Without losing generality, a point (0, 0, zC)(zC 0) on the graphene and a point (x, y, zp)(x   h, zC  L ) in the polymer are picked

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Fig. 23 A schematic diagram of a graphene parallel to the surface of a semi-infinite polymer; the graphene/ polymer overlap length is L, and the distance between the graphene and polymer surface is h (Reproduced with permission from Ref. [78])

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 (Fig. 23). The distance between these two points is r ¼ x2 þ y2 þ zp  zC . Unlike the analysis in Sect. 2.2.1, the deformation for finite graphene and polymer is nonuniform. The energy per unit thickness Φline (along the free edge) stored due to the van der Waals forces is defined as Φline ¼

ð1

ð ρc dzC

0

¼ ρp ρC

ð1

V ðrÞρp dV polymer V polymer

0

ð1

ð h

dzC

dx 1

ðL dy

1

1

V ðrÞdzp ,

(90)

which can be integrated analytically to give Φline

8  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 39 2  h9 5h2 3h4 h6 h8 > > 2 2 > > þ h L 1 þ þ L 1 þ  þ  > > > 9 2 4 6 8 7> 6 > > 9 6 8L 128L 64L 16L 8L > > 7 σ > > > 6 7> > > 2 4 6 = < 96 7 69h 15h 9h 2π 15h 4 5 pffiffiffiffiffiffiffiffiffiffiffi  q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   : ¼ ρp ρC eσ 3
  3 5 128 h þ L 2 2 > > 3 128 h þ L2 128 h þ L2 > > > > > > > >    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > > > > σ3 h3 h2 > > 2 2 > > ; :  3 L 1þ 3 þ h þL 1 2 2L 2L 2h

(91) For the overlap length L in Fig. 23 larger than h, the Taylor expansion of Eq. 91 with respect to h/L gives  9    3  2π σ3 h 3 2σ  3 L 1þO 3 , Φline ðL, hÞ ¼ ρp ρC eσ (92) 9 3 L 15h h  σ3 3 2σ 9 where 2π ρ ρ eσ  is identical to Eq. 81 for infinite graphene and polymer 9 3 3 p C 15h h matrix; the terms neglected are on the order of h3/L3 and therefore are very small. The equilibrium distance h determined from @Φline/@h = 0 is

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  3  h h ¼ 0:858σ 1 þ O 3 , L

(93)

which is essentially the same as Eq. 82 for infinite graphene and polymer. It is shown that the energy Φline given in Eq. 92 is an excellent approximation of the exact solution in Eq. 91 for L > h. For the opening displacement v and sliding displacement u beyond the equilibrium distance h, Φline is obtained from Eq. 91 by replacing L and h with L  u and h + v, respectively, i.e., Φline ¼ Φline ðL  u, h þ vÞ:

(94)

For L  u > > h + v, the Taylor expansion in Eq. 94 becomes " # 2π 2σ 9 σ3 3  Φline ðL, hÞ ¼ ρp ρC eσ ðL  uÞ, 3 15ð0:858σ þ vÞ9 ð0:858σ þ vÞ3

(95)

where h has been replaced by 0.858σ. The tangential and normal forces (per unit thickness along the graphene free edge) are therefore " # @Φline 2π σ3 2σ 9 3  ρp ρC eσ F tangent ¼  , (96) 3 @u ð0:858σ þ vÞ3 15ð0:858σ þ vÞ9 " # @Φline σ4 2σ 10 2  2πρp ρC eσ  ðL  uÞ: (97) F normal ¼ @v ð0:858σ þ vÞ4 5ð0:858σ þ vÞ10 The average shear and tensile cohesive stresses over the current overlap length L  u can be defined as " # F tangent 2π σ3 2σ 9 σ 2 , (98)  ρp ρC eσ τcohesive ¼  3 9 Lu 3 Lu ð0:858σ þ vÞ 15ð0:858σ þ vÞ " # F normal σ4 2σ 10 2 σ cohesive ¼  2πρp ρC eσ  , (99) Lu ð0:858σ þ vÞ4 5ð0:858σ þ vÞ10 where the terms neglected are on the order of (σ/(L  u))3. Equations 98 and 99 give the cohesive law for a finite graphene/polymer based on the van der Waals force. The average tensile cohesive stress σ cohesive is identical to Eq. 85 for an infinite graphene/polymer and is independent of the sliding displacement u. Therefore the tensile cohesive law in Eq. 87, which is given in terms of the cohesive strength σ max and total cohesive energy Φtotal, still holds. The average shear cohesive stress τcohesive in Eq. 98 is much smaller than σ cohesive because L  u > > σ, and it depends on both sliding and opening displacements, u and v, i.e., tension/shear

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523

Fig. 24 The average shear cohesive stress τcohesive, normalized by the current overlap length L  u and total cohesive energy Φtotal, versus the normalized opening displacement v/δ0, where δ0 = 0.242Φtotal/σ max is the critical separation at which the tensile cohesive strength is reached and δ0 = 0.0542 nm for SWCNTs/polyethylene (Reproduced with permission from Ref. [78])

coupling in the cohesive law. The shear cohesive law in Eq. 98 can be rewritten in terms of the cohesive strength σ max and total cohesive energy Φtotal as

τcohesive ¼

8 >
:

3 1 þ 0:682 Φσmax v total

i3  h

1 1 þ 0:682 Φσmax v total

i9

9 > =

Φtotal : (100) > ; σ max ðL  uÞ

Figure 24 shows the average shear cohesive stress τcohesive normalized by the current overlap length L  u and total cohesive energy Φtotal, (L  u)τcohesive/Φtotal, versus the normalized opening displacement v/δ0, where δ0 = 0.242Φtotal/σ max is the critical separation at which the tensile cohesive strength is reached and δ0 = 0.0542 nm for CNTs/polyethylene. The average shear cohesive stress is not zero even at v = 0 due to the asymmetry of the two sides of the overlap length L; therefore the tangential force does not vanish. In fact, at v = 0 τcohesive has a zero slope and a maximum Φtotal/(L  u), which is much smaller than σ max, and τcohesive decreases monotonically as v increases. Similar to Sect. 2.2.1, the effect of CNT radius is small such that it is negligible, i.e., the analytical expressions of the cohesive law for graphene/polymer can be effectively used for CNT/polymer.

2.2.3 Polymer Surface Roughness The polymer surface is assumed to be flat in Sects. 2.2.1 and 2.2.2, in which the polymer surface roughness is neglected. In fact, the polymer of chain molecules exhibits irregular surface structure and roughness [89–91]. A simple model shown in Fig. 25 can be used to estimate the effect of polymer surface roughness on the cohesive law. The polymer surface is wavy in z direction and has an amplitude Δ and wavelength λ. The average distance between the polymer surface and graphene is

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Fig. 25 A schematic diagram of a graphene on a wavy polymer surface that has the amplitude Δ and wavelength λ. The average distance between the graphene and polymer surface is denoted by h. The graphene is subjected to the opening and sliding displacements (Reproduced with permission from Ref. [78])

Fig. 26 The cohesive law for SWCNT/polymer matrix with different polymer surface waviness Δ, where h is the equilibrium spacing, σ max is the cohesive strength and δ0 is the critical separation at which the cohesive strength is reached (Reproduced with permission from Ref. [94])

still denoted by h. Let x be the coordinate along the normal direction of the graphene. The distance between a point (0, 0, zC) on the graphene and a point (x, y, zp) in the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
2 2πz 2 polymer (Fig. 26) is r ¼ x þ y2 þ zp  zC , where x  h  Δ cos λ p . The average cohesive energy for an arbitrary pair potential V(r) is given by

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ð ð 1 λ=2 dzC V ðrÞdV polymer λ λ=2 V polymer ð1 ð hΔ cos 2πzp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ð1 λ
2 ρp ρC λ=2 ¼ dzC dy dzp V x2 þ y2 þ zp  zC dx, λ 1 λ=2 1 1

Φ ¼ ρp ρC

(101) Ð λ=2 where 1λ λ=2 dzC represents the average over the wavelength λ. For the opening displacement v and sliding displacement u (along the wavy direction x) as shown in Fig. 25, the cohesive energy can be similarly obtained by replacing x with x  v and zp  zC with zp  zC  u, respectively, Φ¼

ρp ρC λ ð1 ð hvΔ cos 2πðz0 þzC þuÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð λ=2 ð1 λ  dzC dy dz0 V x0 2 þ y2 þ z0 2 dx0 , λ=2

1

1

1

(102) where the integration variables have been changed to z0 = zp  zC  u and x0 = x  v. Ð1 Ð1 ρp ρC Ð λ=2 The shear cohesive stress is given by τcohesive ¼ @Φ @u ¼ 2πΔ λ2 λ=2 dzC 1 ffidy 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 0 2π ðz0 þzC þuÞ C þuÞ 0 0 V ðr0 Þ sin 2π ðz þz dz , where r ¼ h þ v þ Δcos þ y2 þ z0 2 . This λ λ integral can be evaluated analytically to give τcohesive ¼ 0,

(103)

i.e., the shear cohesive stress vanishes for a wavy polymer surface and flat graphene. In fact, such a conclusion of vanishing shear cohesive stress also holds for an arbitrary periodic polymer surface and flat graphene and an arbitrary polymer surface and wavy graphene that have the same period. Ð1 ρ ρ Ð λ=2 The tensile cohesive stresses is given by σ cohesive ¼ @Φ ¼  pλ C λ=2 dzC 1 dy @v Ð1 0 0 0 1 V ðr Þdz , where r is given above. This integral can also be evaluated analytically to give σ cohesive ¼ 2πρp ρC

ð 1

1 V ðrÞrdr  π hþvΔ

ð hþvþΔ hþvΔ

V ðrÞr cos

1

 rhv dr : Δ (104)

Since the amplitude Δ of polymer wavy surface is assumed to be less than h + v (otherwise the polymer surface penetrates the graphene), the Taylor expansion of Eq. 104 with respect to Δ gives

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σ cohesive ¼ 2πρp ρC

ð1

V ðrÞrdr

hþv


 π þ ρp ρC ½V ðh þ vÞ þ ðh þ vÞV 0 ðh þ vÞΔ2 þ O Δ3 : 2

(105)

Figure 26 shows the tensile cohesive stress given by Eq. 104 versus the normalized opening displacement v/δ0 for several waviness amplitudes Δ = 0, 0.2, 0.4, 0.6 and 0.8h, where Δ = 0 corresponds to no waviness and h is equilibrium spacing. The tensile cohesive stress σ cohesive is normalized by the cohesive strength σ max without waviness. As the waviness Δ increases, both the cohesive strength (peak value of the curve) and total cohesive energy (area underneath the curve) decrease. For example, the cohesive strength drops from σ max for Δ = 0 to 0.81σ max for Δ = 0.2h, 0.60σ max for Δ = 0.4h, 0.48σ max for Δ = 0.6h, and 0.41σ max for Δ = 0.8h, respectively [78]. The average shear cohesive stress also vanishes τcohesive = @Φ/@u = 0. This represents shear stress averaged over the period of wavy polymer surface. However, the local shear stress may not be zero.

2.3

A Cohesive Law for Interfaces between Multi-Wall Carbon Nanotubes and Polymers Due to the van der Waals Force

2.3.1

The Cohesive Law for an Infinitely Long Multi-Wall Carbon Nanotube Firstly the effect of CNT radius is neglected, and the interaction is studied between two parallel graphenes (i.e., two infinite planes of carbon atoms). The equilibrium distance between two graphenes (Fig. 27a) is denoted by h. In order to establish a continuum cohesive law, carbon atoms on each are homogenized and are ffiffiffi graphene
p  represented by an area density, ρC ¼ 4= 3 3l 0 2 , where l0 is the equilibrium bond length of graphene prior to deformation. The number of carbon atoms over an area dA on the graphene is ρCdA. The distance between two points on two graphenes is Fig. 27 A schematic diagram of two parallel graphenes: (a) the distance between two graphenes is h; (b) the graphene is subjected to the opening and sliding displacements (Reproduced with permission from Ref. [84])

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527

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ h2 þ z2 , where z is the projected distance between the two points on each graphene (Fig. 29a). The energy due to the van der Waals force is given by V(r) in Eq. 79. For an infinitesimal area dA, the energy in the graphene due to the van der Waals force is ð ρC dA V ðrÞρC dA ¼ A

2πρ2C dA

ð1

V ðrÞzdz,

(106)

0

where the integration is over the infinite area A of the graphene. The cohesive energy Φ is the energy per unit area and is given by  10  ð 2σ σ4 Φ ¼ 2πρ2C V ðrÞzdz ¼ 2πρ2C eσ 2  , 5h10 h4 0

(107)

h ¼ σ,

(108)

pffiffiffi where 6 2σ is the equilibrium distance between two carbon atoms interacting via the van der Waals force and e is the corresponding bond energy. The equilibrium distance h is determined by minimizing the energy, @Φ/@u = 0, as

i.e., the equilibrium distance h between two graphenes is the same as the characteristic length σ in the van der Waals force. For the opening displacement v and sliding displacement u beyond the equilibrium distance as shown in Fig. 29b, the cohesive energy can be similarly obtained as " Φðu, vÞ ¼ 2πρ2C eσ 2

2σ 10 5ðσ þ vÞ10



σ4 ðσ þ vÞ4

# ,

(109)

which can also be obtained from Eq. 107 by simply replacing h with σ + v. Equation 109 is independent of the sliding displacement u because sliding does not change the van der Waals force for two infinite graphenes; therefore the shear cohesive stress vanishes, i.e., τcohesive ¼

@Φ ¼ 0: @u

(110)

The tensile cohesive stress is obtained from Eq. 109 as σ cohesive

" # @Φ σ5 σ 11 2 ¼ 8πρC eσ ¼  : @v ðσ þ vÞ5 ðσ þ vÞ11

(111)

Equations 109 and 111 give the cohesive law for two graphenes based on the van der Waals force. Such a cohesive law gives the following cohesive properties:

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(i) Initial slope (initial value of dσ cohesive/dv at v = 0) 48πρ2C e (ii) Cohesive strength (maximum cohesive stress) σ max ¼ ð48π=11Þð5=11Þ5=6 ρ2C eσ (iii) Critical separation δ0 = [(11/5)1/6  1]σ at which the cohesive strength is reached (iv) Total cohesive energy (area underneath the cohesive–v curve) Φtotal ¼ 6πρ2C eσ 2 =5
pffiffiffi  For e = 0.00239 eV, σ = 0.3415 nm [87], ρC ¼ 4= 3 3l 20 , and l0 = 0.142 nm for graphene, the total cohesive energy Φtotal = 0.245 Jm2, which is rather small and reflects the weak van der Waals interactions. It is on the same order as the CNT adhesion energy reported in experiments and atomistic simulations [92]. The cohesive strength, however, is very large, σ max = 1.35 GPa. The cohesive law in Eqs. 109 and 111 can be rewritten in terms of the cohesive strength σ max and total cohesive energy Φtotal as Φtotal Φðu, vÞ ¼ 3

(

2 ½1 þ 0:530ðσ max =Φtotal Þv10 (

σ cohesive ¼ 3:54σ max



1 ½1 þ 0:530ðσ max =Φtotal Þv

5

)

5

,

½1 þ 0:530ðσ max =Φtotal Þv4



1

(112) )

½1 þ 0:530ðσ max =Φtotal Þv11

:

(113) The initial slope and critical separation are given in terms of σ max and Φtotal by 11:3σ 2max =Φtotal and 0.265Φtotal/σ max, respectively. The interactions between multilayer graphenes and polymer are then studied. The graphenes are parallel to the polymer surface, and their distances are denoted by hi, with i = 1 corresponding to the layer next to the polymer and i = n the outmost layer (Fig. 28). The cohesive energy between the ith and jth graphenes is 2πρ2C eCC σ 2CC   2σ 10 σ 4CC CC ði < jÞ. Therefore, the cohesive energy among all graphenes is 10  4 5ðhj hi Þ ðhj hi Þ   P 2σ 10 σ 4CC 2 2 CC Φgraphenegraphene ¼ 2πρC eCC σ CC 1i > > 10 = X < 5½0:858σ þ ð i  1 Þσ þ v  CCH2 CC 1 :  σ 4CCH2 > > > 1in > > >  ; : ½0:858σ CCH2 þ ði  1Þσ CC þ v1 4

σ cohesive ¼

(117)

It degenerates to the solutions for the polymer with a single graphene. Then the effect of CNT radius is considered and two concentric CNTs are studied. Let h denote the equilibrium distance between two CNT walls (Fig. 30). Carbon atoms are homogenized on each CNT and represented by an area density ρc (which may be slightly different from the area density of graphene due to the effect of CNT radius). The cylindrical coordinates (R, θ, z) are used with z denoting the central axis of CNTs. Without losing generality, a point (R, 0, 0) on the inner CNT wall and a point (R + h, θ, z) on the outer CNT wall are picked, where R is the radius of inner

Fig. 30 A schematic diagram of two carbon nanotube (CNT) walls. The radius of inner CNT is R, and the distance between two CNTs is h (Reproduced with permission from Ref. [84])

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CNT.qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The distance between ffi these two points is 2 2 2 r ¼ ðR þ hÞ  2RðR þ hÞ cos θ þ R þ z . The energy due to the van der Waals force is still given by V(r) in Eq. 79. For a section of CNTs with height dz, the energy stored in this section of CNTs due to the van der Waals force is ð ð 2π ð þ1 ρC 2πRdz V ðrÞρC dA ¼ 2πρ2C RðR þ hÞdz dθ V ðrÞdz0 , A

0

(118)

1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ ðR þ hÞ2  2RðR þ hÞ cos θ þ R2 þ z0 2 . The cohesive energy Φ is the energy per unit area and is given by Ð 2π Ð þ1 2πρ2C RðR þ hÞdz 0 dθ 1 V ðrÞdz0 Φ¼ 2π ðR þ h=2Þdz Ð 2π Ð þ1 ρ2 RðR þ hÞ 0 dθ 1 V ðrÞdz0 , ¼ C ðR þ h=2Þ

(119)

where 2π(R + h/2)dz is the average of inner CNT area, 2πRdz, and outer CNT area, 2π(R + h)dz. The above integral can be expressed in terms of the elliptic integral. Figure 31 shows the cohesive energy, normalized by the total cohesive energy Φtotal ¼ 6πρ2C eσ 2 =5 which equals 0.245 Jm2 for graphene, versus the distance h between two CNT walls for several radii of the inner CNT, where h is normalized by the characteristic length σ = 0.3415 nm in the van der Waals force. The curves for different CNTs are very close, and they are all close to that for graphene. This suggests that the CNT radius has little effect on the cohesive energy. Each curve has

Fig. 31 The cohesive energy, normalized by the total cohesive energy Φtotal for graphene, versus the distance h between two carbon nanotubes for several CNT radii and graphene, where h is normalized by the characteristic length σ = 0.3415 nm in the van der Waals force (Reproduced with permission from Ref. [84])

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Fig. 32 A schematic diagram of multi-wall carbon nanotubes (MWCNTs) in a polymer. The radii of nanotube walls are R1, R2, . . ., where R1 is the outermost wall, Rn is the innermost wall, and n is the number of nanotube walls. The spacing between the multi-wall carbon nanotube and polymer is h (Reproduced with permission from Ref. [85])

a minimum value which occurs at the equilibrium distance between two CNT walls, and this distance is very close to σ. For a MWCNT embedded in an infinite polymer matrix as shown in Fig. 32, the radii of the CNT walls are denoted by Ri, with i = 1 being the outermost CNT wall (next to polymer) and i = n the innermost CNT wall. The spacing between the outermost CNT wall and the polymer surface is denoted by h. The cohesive energy (per unit length) between the ith and jth CNT walls is obtained as 2πρ2C eCC σ 2CC  
 2σ 10 σ 4CC CC  π Ri þ Rj ði < jÞ . Therefore, the cohesive energy among 10  4 5ðRi Rj Þ ðRi Rj Þ all CNTs is ΦCNTCNT ¼

2π 2 ρ2C eCC σ 2CC
  Ri þ Rj :

"

X 1i > σ 8CCH2 σ 2CCH2 3 > > > > > >  ; : þ 2 σ CCH2 10ðR1  Ri þ hÞ8 ðR1  Ri þ hÞ2  It can be shown that, within the order of error

h Rþh

(121)

2 , the spacing between the

outermost CNT and polymer is h ¼ 0:858σ CCH2 and the spacing between CNTs are Ri  Ri + 1 = σ C  C (i = 1, 2, . . ., n  1). The total cohesive energy is the sum of ΦCNT  CNT and ΦCNT  polymer. For the opening displacement v1 between the outermost CNT and polymer beyond the equilibrium distance, h, the cohesive energy can be obtained by replacing h with h + v1. The derivative of total cohesive energy with respect to v1 gives the tensile cohesive stress. For a SWCNT and polymer, the cohesive stress is given by σ cohesive

 1þ

v1 ¼ 2R1 þ 0:858σ CCH " # 2 4 2σ 10 σ CCH2 CCH2   , 5½0:858σ CCH2 þ v1 10 ½0:858σ CCH2 þ v1 4 2πρCH2 ρC eCCH2 σ 2CCH2

 (122)

where the average area π ð2R1 þ 0:858σ CCH2 Þ between the outermost CNT wall and polymer is used in the cohesive stress calculation. For a double- or triple-walled CNT and polymer, the cohesive stress is  X  v1  ði  1Þσ CCH2 1þ σ cohesive ¼ 2πρCH2 ρC eCCH2 σ 2CCH2  2R1 þ 0:858σ CCH2 1in 9 8 10 2σ CCH2 > > > > > > < 10 = 5½0:858σ CCH2 þ ði  1Þσ CC þ v1  ,  σ 4CCH2 > > > > > >  ; : ½0:858σ CCH2 þ ði  1Þσ CC þ v1 4

(123)

where n = 2 or 3 for double- or triple-walled CNTs, respectively. For the polymer with more than three-walled CNTs, only the summation for i = 1, 2, and 3 are needed. Figure 33a shows the cohesive stress σ cohesive versus the opening displacement v1 for a SWCNT (n = 1) and polyethylene. As the CNT radius decreases, the cohesive stress increases, but the increase is small. For double- walled (n = 2) and triplewalled (n = 3) CNTs shown in Fig. 33b and c, respectively, the cohesive stress

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Mechanics of Carbon Nanotubes and Their Composites

535

Fig. 33 The cohesive stress σ cohesive versus the displacement v1 for multi-wall carbon nanotubes (MWCNTs) and polymer: (a) single-wall carbon nanotubes (SWCNTs) (n = 1), (b) double-walled carbon nanotubes (DWCNTs) (n = 2), and (c) triple-walled carbon nanotubes (TWCNTs) (n = 3) (Reproduced with permission from Ref. [85])

becomes essentially independent of the CNT radii and is therefore well approximated by the corresponding results for multilayer graphenes given in Eq. 117.

2.3.2 The Cohesive Law for Two Finite Multi-Wall Carbon Nanotubes For two graphenes that overlap over a length L as shown in Fig. 34, the distance between them is denoted by h. The area density of a graphene is still denoted by ρC. The Cartesian coordinates (x, y, z) are used with x along the direction normal to the graphenes and y along the graphene-free edge. The free edge of one graphene is represented by (0, y, 0), while the free edge of the other is (h, y, L). Without losing
generality, 
a point  (0, 0, zC)(zC 0) on one graphene and a point h, y, z0C z0C  L on the otherqgraphene are picked for study. The distance ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 between these two points is r ¼ h2 þ y2 þ z0C  zC . The energy due to the van der Waals force is given by V(r) in Eq. 79. Unlike the analysis in Sect. 2.3.1, the deformation for finite graphenes is nonuniform. The energy per unit thickness Φline (along the free edge) stored between two graphenes due to the van der Waals forces is obtained by

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J. Wu et al.

Fig. 34 A schematic diagram of two parallel graphenes with overlap length L; the distance between two graphenes is h (Reproduced with permission from Ref. [84])

Φline ¼

ð1 0

ð ρC dzC A

V ðrÞρC dA ¼ ρ2C

ð1

ð1 dzC

0

ðL dy

1

1

V ðrÞdz0C :

(124)

Here the upper limit of integration for dzC is taken as 1 since the van der Waals force decreases very fast with the distance. Similarly, the limits of integration for dy and the lower limit of integration for dz0 are taken as 1 and 1, respectively. After integration, Eq. 124 then becomes Φline ðL, hÞ

8 2 39 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > h > > 2 2 > > > 6 L þ h þ L  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7> > > 2 2 > > 6 7 10 > > 2 h þL σ 6 > > 7 > > > > 6 7 4 6 8 > h h 5h < 5h10 6 = 7> qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    4 5 2 2
2 

 ¼ 2πρC eσ : 5 7 3 > > 8 h þ L2 16 h2 þ L2 128 h2 þ L2 > > > > > > " # > > > > 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > σ4 > > > h > > 2 2 > > p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  L þ h þ L  > > : 2h4 ; 2 2 2 h þL

(125) For the overlap length L larger than h, the Taylor expansion of Eq. 125 with respect to h/L gives Φline ðL, hÞ ¼ 2πρ2C eσ 2

 10    4  2σ σ4 h  L 1 þ O , L4 5h10 h4

(126)

in which the terms neglected are on the order of h4/L4 and therefore are very small. The equilibrium distance h determined from @Φline/@h = 0 is   4  h h¼σ 1þO 4 : L

(127)

It is shown that for length L > h, Eq. 126 is an excellent approximation of the exact solution (Eq. 125).

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Mechanics of Carbon Nanotubes and Their Composites

537

For the opening displacement v and sliding displacement u beyond the equilibrium distance, the energy per unit thickness Φline (along the free edge) is obtained from Eq. 125 by replacing L and h with L  u and h + v, respectively, that is, Φline ¼ Φline ðL  u, h þ vÞ:

(128)

It depends on the sliding and opening displacements through the current overlap length L  u and distance h + v between two graphenes. For L  u > > h + v, the Taylor expansion in Eq. 126 becomes " Φline 

2πρ2c eσ 2

2σ 10 5ðσ þ vÞ10



σ4 ðσ þ vÞ4

# ðL  uÞ,

(129)

where h has been replaced by σ. The tangential and normal forces (per unit thickness along the graphene free edge) are F tangent

F normal

" # @Φline σ4 2σ 10 2 2  2πρC eσ ¼  , @u ðσ þ vÞ4 5ðσ þ vÞ10

" # @Φline σ5 σ 11 2  8πρC eσ ¼  ðL  uÞ: @v ðσ þ vÞ5 ðσ þ vÞ11

(130)

(131)

We define the average shear and tensile cohesive stresses over the current overlap length L  u as " # F tangent σ4 2σ 10 σ 2 ,  2πρC eσ τcohesive ¼  Lu ðσ þ vÞ4 5ðσ þ vÞ10 L  u " # F normal σ5 σ 11 2 σ cohesive ¼  8πρC eσ  , Lu ðσ þ vÞ5 ðσ þ vÞ11

(132)

(133)

where the terms neglected are on the order of σ 4/(L  u)4. Equations 132 and 133 give the cohesive law for finite graphenes based on the van der Waals force. The average tensile cohesive stress σ cohesive is identical to that for infinite graphenes and independent of the sliding displacement u. Therefore, the tensile cohesive law, given in terms of the cohesive strength σ max and total cohesive energy Φtotal, is still valid. The average shear cohesive stress τcohesive in Eq. 132 is much smaller than σ cohesive because L  u > > σ, and it depends on both sliding and opening displacements, u and v, i.e. tension/shear coupling in the cohesive law. The shear cohesive law, Eq. 132, can be rewritten in terms of the maximum cohesive strength and total cohesive energy as

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J. Wu et al.

Fig. 35 The average shear cohesive stress τcohesive, normalized by the current overlap length L  u and total cohesive energy Φtotal, versus the normalized opening displacement v/δ0, where δ0 = 0.265(Φtotal/σ max) is the critical separation at which the tensile cohesive strength is reached and δ0 = 0.0480 nm for carbon nanotubes (Reproduced with permission from Ref. [84])

τcohesive ¼

8 > >
> =

5σ max ½1 þ 0:530ðσ max =Φtotal Þv4 Φtotal : 2 > 3 > σ > max ðL  uÞ > : 10 ; 5½1 þ 0:530ðσ max =Φtotal Þv

(134)

Figure 35 shows the average shear cohesive stress, τcohesive, normalized by the current overlap length L  u and total cohesive energy total, Φtotal, (L  u)(τcohesive/ Φtotal), versus the normalized opening displacement v/δ0, where δ0 = 0.265(Φtotal/ σ max) is the critical separation at which the tensile cohesive strength is reached and δ0 = 0.0480 nm for two graphenes. The average shear cohesive stress is not zero even at v = 0 because the two sides of the overlap length L are not symmetric such that the tangential force does not vanish. In fact, at v = 0, τcohesive has a zero slope and a maximum Φtotal/(L  u), which is much smaller than σ max, and τcohesive decreases monotonically as v increases.

2.4 2.4.1

The Effect of the van der Waals-Based Interface Cohesive Law on Carbon Nanotubes-Reinforced Composite Materials

A Micromechanics Model for Nanocomposites with Nonlinear Interface Cohesive Law For a composite consisting of a matrix and long fibers since CNTs have very large aspect ratio (e.g., 105), the fiber volume fraction is denoted by f. The macroscopic stress σ and strain e of the nanocomposite represent the collective, homogenized behavior of fibers and matrix. They are distinguished from the microscopic stress σ and strain e in fibers and matrix since the latter satisfies the constitutive laws in the corresponding constituents and are nonuniform due to the material inhomogeneities. The macroscopic strain is given by [70, 95]

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Mechanics of Carbon Nanotubes and Their Composites

    e ¼ M m : σ þ f ðM f  M m Þ : σ f þ eint ,

539

(135)

where the matrix and fibers are linear elastic with the elastic compliance tensor Mm for the matrix and Mf for fibers; hσ fi is the average stress in fibers; and heinti, which represents the contribution from the fiber/matrix interface, is given by 

 1 eint ¼ 2Ωf

ð ð½u n þ n ½uÞdA:

(136)

S int

Here Ωf is the total fiber volume, and Sint represents all fiber/ matrix interfaces. For perfectly bonded interfaces, heinti = 0. For not-well-bonded interfaces, as in CNT-reinforced composites, heinti and hσ fi are to be determined from the interface cohesive law. For a simple case of an isotropic matrix containing randomly distributed, long, and isotropic fibers subjected to hydrostatic tension σI, where I is the second-order identity tensor, the macroscopic strain of the composite is eI and e is obtained from Eq. 135 as 1 f e¼ σ þ 3K m 3



  1 1  f   int   σ kk þ ekk , 3K f 3K m

(137)

where  Km and Kf are the elastic bulk moduli of the matrix and fibers, respectively, and eint kk is related to the displacement jump [u] in the normal direction across the fiber/matrix interface by  int  1 ekk ¼ Ωf

ð ½udA:

(138)

S int

model, such as the dilute solution, is needed to determine  fA micromechanics   in terms of σ (or e). Equation 137 then gives the nonlinear stress-strain σ kk and eint kk relation. Since the CNT volume fraction is small, the dilute solution which neglects the interactions among fibers may be used. For a long fiber of radius R in an infinite matrix subjected to remote hydrostatic tension, let σ and e denote the macroscopic stress and strain of the composite and σ int the normal cohesive stress at the fiber/ matrix interface. The stresses in the fiber are uniform, σ frr ¼ σ fθθ ¼ σ int ¼ constant, and σ fzz ¼ constant, where (r, θ, z) are the cylindrical coordinates. The strains in the


 fiber are efrr ¼ efθθ ¼ ð1  υf Þσ int  υf σ fzz =E f and efzz ¼ σ fzz  2υf σ int =Ef , where Ef and υf are Young’s modulus and Poisson’s ratio of  the fiber. The radial displacement at the fiber boundary is R ð1  υf Þσ int  υf σ fzz =Ef . The matrix stress in the fiber direction is the same as macroscopic stress σ. The radial displacement in the matrix is represented by ur = Ar + B/r [96], where the coefficients A and B are to be determined by the remote boundary condition σ m rr jr!1 int at the fiber/matrix ¼ σ and the interface condition σ m rr jr¼R ¼ σ . The displacement  σ  ð1 þ υm Þσ int =E m, where Em and υm interface r = R is then given by R ð2  υm Þ are Young’s modulus and Poisson’s ratio of the matrix, respectively.

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The opening displacement is the displacement jump across the fiber/matrix interface, i.e., ½ u ¼

 R

 R

ð2  υm Þσ  ð1 þ υm Þσ int  ð1  υf Þσ int  υf σ fzz : Em Ef

(139)

It is recalled that the normal cohesive stress σ int = σ int([u]) is related to the opening displacement [u] via the nonlinear cohesive law in Eq. 139 then gives the macroscopic stress σ in terms of [u] and σ fzz, and σ fzz is to be determined by the Reuss approximation in the following. The macroscopic strain e can be obtained from Eq. 137. For very long fibers, Reuss approximation [97] gives the fiber stress σ fzz in the fiber direction to be the same as the macroscopic stress, i.e., σ fzz ¼ σ:

(140)

The Reuss approximation, however, gives a lower bound solution and  under estimates the reinforcing effect of fibers. The average stress in fiber is σ fkk ¼ 2 σ int þ σ. The additional strain eint kk due to interface opening is 2[u]/R. Equation 139 gives the macroscopic stress in terms of the interface opening displacement [u] as   ½u 1 þ υm 1  υf int þ þ σ ð½uÞ R Em Ef σ ¼ : 2  υm υf þ Em Ef

(141)

The macroscopic strain is obtained in terms of [u] from Eq. 137 as e¼

1 f σ þ 3K m 3



  1 1 int ½ u  : 2σ ð½uÞ þ σ þ 2 3K f 3K m R

(142)

Equations 141 and 142 give the macroscopic stress σ and strain e in terms of [u] based on the Reuss approximation and therefore provide the nonlinear stress-strain relation of the CNT-reinforced composite.

2.4.2 Stress-Strain Curve of CNT-Reinforced Composite Materials CNT Young’s modulus Ef is on the order of 1 TPa [98], which is several orders of magnitude higher than the modulus Em of polymer matrix (e.g., 0.9 GPa for polyethylene). (The CNT elastic modulus in the CNT radial direction may be smaller than that in the axial direction but is still much larger than the polymer modulus.) For Ef > > Em, the stress-strain relation in Eqs. 141 and 142 is simplified to

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Mechanics of Carbon Nanotubes and Their Composites

σ ¼

E m ½u 2  υm R 1 þ υm þ 3:07 σ max 2  υm

"

σ max 1 þ 0:682 ½u Φtotal

4

541

 10 # σ max  1 þ 0:682 ½u , Φtotal (143)



e¼



1 þ υ m þ f ½ u f 1  2υm σ max þ 3:07 1 þ υm  ð5  υm Þ 2  υ R 3 2  υm E m m  4  10 σ max  1 þ 0:682 Φσmax ½ u   1 þ 0:682 ½ u  : Φ total total

(144)

Figure 36 shows the stress-strain curve for a CNT-reinforced polyethylene composite subjected to hydrostatic tension as well as the straight line for the linear elastic polyethylene matrix for comparison. Young’s modulus and Poisson’s ratio of polyethylene are Em = 0.9 GPa and υm = 0.3, respectively. The nanotube radius is R = 1.25 nm for the armchair (18,18) CNTs. The CNT volume fraction is f = 10%. (It is well known that the dilute solution holds only for the small reinforcement volume fraction since it neglects the reinforcement interactions.) Tan et al. [70] has shown that the dilute solution is different from the Mori-Tanaka method, which accurately accounts for the reinforcement interactions, by a factor of 1  f. For CNT volume fraction f = 10%, this factor is 90%. For strain less than 13.9%, the stressstrain curve is almost (but not exactly) a straight line that is above the straight line for the matrix and reflects the CNT reinforcing effect. The curve for the composite is not significantly higher than the straight line for the polyethylene matrix because the Reuss approximation gives a lower bound solution and therefore underestimates the CNT reinforcing effect. Once the strain reaches 13.9% (and the stress reaches the Fig. 36 The macroscopic stress-strain relation of a carbon nanotube-reinforced polyethylene matrix composite subjected to hydrostatic tension. The volume fraction of armchair (18,18) carbon nanotubes is 10%. The linear elastic stressstrain relation of the polyethylene matrix is also shown (Reproduced with permission from Ref. [86])

542

J. Wu et al.

Fig. 37 The macroscopic stress-strain relation of a carbon nanotube-reinforced polyethylene matrix composite. The volume fraction of armchair (18,18) carbon nanotubes is 10%, 5%, and 1% (Reproduced with permission from Ref. [86])

peak strength 406 MPa), both stress and strain decrease because of the interface softening behavior displayed in the cohesive law. The stress and strain start to increase again as the strain drops to 9.1%, but the curve becomes lower than the straight line for the matrix because the completely debonded CNTs are equivalent to voids and therefore weaken the matrix. For the displacement-controlled experiment, the stress-strain curve would drop vertically from 406 MPa to 289 MPa at the 13.9% strain, as illustrated by the arrow in Fig. 36, and then gradually increase. Figure 37 shows the stress-strain curves for the polyethylene matrix with armchair (18,18) CNTs at three volume fractions, f = 1%, 5%, and 10%. The peak strengths for different f are exactly the same, 406 MPa, which suggests that the CNT volume fraction has no effect on the composite strength. The critical strain at the peak strength changes slightly, e.g., 13.9% for f = 10%, 14.6% for f = 5%, and 15.2% for f = 1%. The CNT volume fraction f = 10% gives a higher initial modulus than f = 5% (or 1%), but it also gives a lower incremental modulus at large strain (after CNTs are completely debonded). For the displacement-controlled experiment, the vertical stress drop is 117 MPa for f = 10% and 68 MPa and 16 MPa for f = 5% and 1%, respectively. Therefore, CNTs may reinforce the composite at the small strain but weaken it at the large strain. Figure 38 shows the effect of CNT size on the stress-strain curve. The CNT volume fraction is fixed at f = 10%. Three different armchair CNTs, (18,18), (12,12), and (6,6), are reexamined, and the corresponding radii are R = 1.25, 0.83 and 0.42 nm, respectively. The peak strength increases from 406 MPa to 419 and 462 MPa as the CNT radius decreases, from (18,18) to (12,12) and (6,6). The critical strain for the peak strength also increases, from 13.9% to 14.6% and 16.8%. Small CNTs clearly give stronger reinforcing effect than large CNTs because, at a fixed CNT volume fraction, there are more small CNTs than large ones and therefore more

16

Mechanics of Carbon Nanotubes and Their Composites

543

Fig. 38 The macroscopic stress-strain relation of a carbon nanotube-reinforced polyethylene matrix composite. The volume fraction is 10% for three different armchair carbon nanotubes (18,18), (12,12), and (6,6) (Reproduced with permission from Ref. [86])

Fig. 39 The macroscopic stress-strain relation of a carbon nanotube-reinforced polyethylene matrix composite. The volume fraction is 10% for armchair (18,18) carbon nanotubes. Three values of the total cohesive energy are taken, 0.107, 0.214, and 0.429 Jm2. The cohesive strength is fixed at 479 MPa (Reproduced with permission from Ref. [86])

interfaces. This observation of strong reinforcing effect for small CNTs also holds after CNTs are debonded from the matrix. The stress drop is 117 MPa, 106 MPa, and 41 MPa for the (18,18), (12,12), and (6,6) CNTs. Therefore, small CNTs are more effective than large CNTs. In order to investigate whether the interface bonding has a strong influence on the macroscopic behavior of CNT-reinforced composites, the total cohesive energy are doubled and quadrupled to Φtotal = 0.214 Jm2 and 0.429 Jm2 while maintaining the same cohesive strength σ max = 479 MPa. Figure 39 shows the stress-strain curve

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of the composite with f = 10% volume fraction of armchair (18,18) CNTs. The peak strength increases from 406 MPa to 432 and 496 MPa when the total cohesive energy changes from 0.107 Jm2 to 0.214 and 0.429 Jm2, respectively. The corresponding strain for the peak strength also increases from 13.9% to 15.3% and 18.9%. The overall stress-strain curve for large Φtotal (0.429 Jm2) becomes much higher than that for low Φtotal (0.107 Jm2). Therefore, strong adhesion between CNTs and polymer matrix may significantly improve the mechanical behavior of CNT-reinforced composites.

2.5

Concluding Remarks

A cohesive law is established for interfaces between carbon nanotubes and polymers due to the van der Waals force. The cohesive law, which links the macroscopic behavior of CNT-reinforced nanocomposites and the nanoscale behavior of interfaces, is given in terms of the area density of carbon nanotube and volume density of polymer, as well as the parameters in the van der Waals force. A micromechanics model for CNT-reinforced nanocomposites with the incorporation of the nonlinear cohesive law is established to investigate the mechanical behavior of nanocomposites.

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Flexoelectric Effect at the Nanoscale

17

Lele L. Ma, Weijin J. Chen, and Yue Zheng

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Theoretical Descriptions of Flexoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Thermodynamic Model of Ferroelectrics with Flexoelectricity . . . . . . . . . . . . . . . . . . . . . . 2.2 Microscopic Theory of Flexoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Theoretical Calculations of Flexoelectric Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Size Effect of Flexoelectric Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Experimental Characterization of Flexoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Experimental Determination of Flexoelectric Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Flexoelectricity-Induced Novel Phenomena in Nano-ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . 4.1 Modifications on Dielectric and Mechanical Response of Nanoscale Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Imprint Behaviors in Ferroelectric Nanofilms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Influence of Flexoelectric Effect on Ferroelectric Nanodomains . . . . . . . . . . . . . . . . . . . . 4.4 Novel Domain Wall Properties Resulted by Flexoelectric Effect . . . . . . . . . . . . . . . . . . . . 5 Applications of Flexoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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L. L. Ma · Y. Zheng (*) State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou, China Micro&Nano Physics and Mechanics Research Laboratory, School of Physics, Sun Yat-sen University, Guangzhou, China School of Engineering, Sun Yat-sen University, Guangzhou, China e-mail: [email protected] W. J. Chen State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou, China Micro&Nano Physics and Mechanics Research Laboratory, School of Physics, Sun Yat-sen University, Guangzhou, China e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_18

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Abstract

Flexoelectric effect is a universal electromechanical coupling effect in solids. Nevertheless, it has been ignored for a long time, due to the usual small strain gradients in materials. The case is different for nanomaterials, where strain gradients can usually reach to the order of 106 m
1. Moreover, as the flexoelectric effect scales with dielectric susceptibility, it can be much more apparent in materials with higher dielectric constants, such as ferroelectrics. Due to this reason, increasing attention has been attracted on the flexoelectric effect in nanomaterials (with ferroelectrics as the representatives) during the past few years. The large flexoelectric effect in nanomaterials not only makes them promising candidates to design novel electromechanical nanodevices but also strongly modifies many material properties such as polarization switching, ferroelectric domains and domain walls, polarization-mediated electronic transport effects, etc. Focusing on flexoelectric effect at the nanoscale and with ferroelectrics as the representatives, this chapter aims to provide an overview of this rapidly growing field, including the theoretical models and experimental characterization methods of flexoelectric effect, the recent important progress of flexoelectric effect, as well as the potential applications. Keywords

Flexoelectricity · Ferroelectric · Nanoscale · Strain gradient · Polarization

1

Introduction

Flexoelectric effect, also called flexoelectricity or nonlocal piezoelectric effect, describes the linear electromechanical coupling between electrical polarization and strain gradients in materials. Concretely speaking, it characterizes the induced polarization by inhomogeneous strain or the strain resulted from the polarization inhomogeneity, where the former corresponds to the direct flexoelectric effect and the latter corresponds to the converse flexoelectric effect. Flexoelectricity manifests itself in a variety of materials, such as solid dielectrics, liquid crystals and biological systems, etc. The physical mechanisms in different kinds of materials can be quite different. Here the discussions are mainly confined in the flexoelectricity of solids. When we talk about electromechanical couplings, it is inevitable to mention piezoelectricity, which characterizes the linear coupling between polarization and homogeneous strain. A big difference between piezoelectricity and flexoelectricity is their different requirements on the symmetry of materials. Piezoelectricity can only be found in 20 point groups which are non-centrosymmetric, for homogeneous strain cannot change the symmetry of materials. In contrast, with inhomogeneous strain or strain gradients, the symmetry of materials will be broken, which results in the

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Fig. 1 Schematic of the effect of homogeneous and inhomogeneous strain on the symmetry of centrosymmetric lattice. It can be seen that the homogeneous strain has no effect on the lattice symmetry and introduces no polarization, but the inhomogeneous strain or strain gradient does break the lattice symmetry and induces net polarization in the centrosymmetric lattice

generation of polarization and enables the universality of flexoelectricity in the whole 32 point groups (see Fig. 1). So compared with piezoelectricity, flexoelectricity is more general and it broadens the scope of candidate materials possessing desired electromechanical behaviors. The flexoelectric effect in solids was predicted theoretically in the 1950s [1]. In 1964, Kogan [2] proposed a phenomenological description for this effect firstly and estimated the order of magnitude of the flexoelectric coefficients. In 1968, Mindlin [3] first introduced polarization gradient term into the continuum theory. At the same year, Bursian [4] experimentally observed the bending of a thin BaTiO3 (BTO) film due to polarization, which is the beginning of investigating converse flexoelectric effect, i.e., the linear coupling between polarization gradient and strain. In 1974, Bursian [5] used the term “nonlocal piezoelectric effect,” and the terminology was substituted by “flexoelectric effect” in 1981 by Indenbom [6]. Then Tagantsev [7] presented a phenomenological description systematically on flexoelectricity and made the first attempt to investigate this effect microscopically. The derived results in his work showed that the flexoelectric effect could be divided into several contributions and the bulk flexoelectric coefficients are scaling with the dielectric susceptibility proportionally. Toward applications, Fousek et al. [8] proposed the idea of “flexoelectricity-based piezoelectric composites” which have no piezoelectric constituents in 1999. Though these achievements have been made, there have been only a few interests in this area until at the beginning of the new century.

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The situation changed after a series of experimental works conducted by Ma and Cross et al. [9]. They determined the flexoelectric coefficients of several ferroelectric ceramics and confirmed the susceptibility dependence of the flexoelectric coefficients, but the obtained values of the coefficients in some materials were much larger than the preceding theoretical estimations. Then Zubko et al. [10] experimentally investigated the flexoelectricity of SrTiO3 (STO) single crystals and obtained its all three flexoelectric coefficients for the first time. Some theoretical works also began to take flexoelectric effect into consideration, and flexoelectricity was drawing more and more attentions. During the past decade, flexoelectric effect has aroused great interests, and the investigations have been broadened to several aspects. These include (a) the theoretical calculation of flexoelectric coefficients both phenomenologically and microscopically as well as those by the first-principle methods, (b) the experimental determination of flexoelectric coefficients, (c) the manifestation of flexoelectric effect in the physical properties of materials, (d) the explanation of some confusing but novel phenomena by taking this effect into consideration, (e) the innovative applications and device design based on flexoelectricity and (f) the construction of continuum theories involving the flexoelectric term, etc. The ignorance of flexoelectricity in its early years was mainly due to the relative weakness of the effect in bulk materials with common scales, for the order of magnitude of the flexoelectric coefficients are estimated as ~e/a (10–9C/m), where e is the electron charge and a is the lattice constant. Accompanying the progress in material science and technology, however, the dimensions of materials scale down, and the strain gradients obtained can be much larger. Especially, with the dimensions of materials down to nanoscale, the flexoelectric effect can be quite significant and even plays a dominant role. Moreover, as the flexoelectric coefficients scale with dielectric susceptibility, the effect can be much more apparent in materials with higher dielectric constants. In this chapter, a review on the flexoelectric effect at the nanoscale, with ferroelectrics as the representatives, will be presented. Ferroelectrics are typical materials exhibiting high dielectric constants; thus, they are promising prototypes for the investigations of nontrivial flexoelectricity. They also exhibit a wide range of fascinating properties such as polarization switching, ferroelectric domains and domain walls, polarization-mediated electronic transport effects, etc. Importantly, many properties of ferroelectrics show strong size-dependence, due to the collective feature of the ferroelectricity. In combination with the flexoelectric effect and size effects, nanoscale ferroelectrics could present abundant phenomena and introduce interesting applications. In the following, Section 2 builds up a thermodynamic model involving the bulk static flexoelectricity in nano-ferroelectrics and introduces Tagantsev’s microscopic rigid ion model. Then the theoretical works about the calculations of the flexoelectric coefficients are briefly retrospected, and the size effects of flexoelectricity are discussed. The experimental characterization of flexoelectricity is introduced in Sect. 3. Section 4 involves the influence of flexoelectricity on material properties and its effects on domains and domain walls. Section 5 introduces the applications of flexoelectricity, and a conclusion is made in Sect. 6.

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2

Theoretical Descriptions of Flexoelectric Effect

2.1

Thermodynamic Model of Ferroelectrics with Flexoelectricity

The flexoelectricity-induced polarization can be written as Pl ¼ μijkl

@eij @xk

(1)

where eij is the strain tensor and μijkl is the flexoelectric tensor. Einstein summation convention is assumed hereafter with the dummy index running from 1 to 3. The flexoelectric tensor is a fourth-rank tensor which guarantees the existence of flexoelectricity in all crystals with any symmetry mathematically. It has 54 independent components for most general cases. For cubic crystals, it has only three independent components known as the longitudinal coefficient μ11, transverse coefficient μ12, and shear coefficients μ44. To characterize flexoelectricity (including both the direct and converse effect), the total free energy density of a ferroelectric system should be supplemented by adding the linear coupling term between the polarization and strain gradient and the term between the strain and polarization gradient. This term is denoted as fflexo, which represents the flexoelectric coupling energy density @eij @Pk ð2Þ
f ijkl eij @xl @xl  ð1Þ ð2Þ ð1Þ ð2Þ
f ijkl þ f ijkl @ðPk eij Þ f ijkl
f ijkl @eij @Pk Pk ¼


eij @xl 2 2 @xl @xl ð1Þ

f flexo ðPi , eij , ∇Pi , ∇eij Þ ¼
f ijkl Pk

ð1Þ

(2)

ð2Þ

where f ijkl and f ijkl are the coupling coefficients. The volume integral of the first term on the right-hand side can be transformed to a surface integral, and it makes no contribution in the energy minimization process. Therefore, the flexoelectric coupling energy density can be written as the Lifshitz invariant 

f flexo Pi , eij , ∇Pi , ∇eij ð1Þ




 @eij 1 @Pk
Pk ¼ f ijkl eij 2 @xl @xl

(3)

ð2Þ

where f ijkl ¼ f ijkl
f ijkl is the bulk flexoelectric coupling coefficients and it has the same symmetry with μijkl. The total free energy of ferroelectric system can be expressed as a function of polarization P, electric filed E, and strain eij, F ¼ FLand þ Felas þ Fgrad þ Felec þ Fflexo þ Fsurf ð ð  ¼ f Land þ f elas þ f grad þ f elec þ f flexo dV þ f surf dS

(4)

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where FLand, Felas, Fgrad, Felec, Fflexo and Fsurf are the Landau free energy, elastic energy, gradient energy, electric energy, flexoelectric coupling energy, and surface energy, respectively, and fLand, felas, fgrad, felec, fflexo and fsurf are the corresponding energy densities. For a typical perovskite ferroelectric material such as BaTiO3, the Landau free energy density can be written as an eighth-order polynomial:     f Land ðPi Þ ¼ a1 P1 2 þ P2 2 þ P3 2 þ a11 P1 4 þ P2 4 þ P3 4     þ a12 P1 2 P2 2 þ P1 2 P3 2 þ P2 2 P3 2 þ a111 P1 6 þ P2 6 þ P3 6        þ a112 P1 2 P2 4 þ P3 4 þ P2 2 P1 4 þ P3 4 þ P3 2 P1 4 þ P2 4   (5) þ a123 P1 2 P2 2 P3 2 þ a1111 P1 8 þ P2 8 þ P3 8  6 2   2   2  2 6 2 6 2 þ a1112 P1 P2 þ P3 þ P2 P1 þ P3 þ P3 P1 þ P2   þ a1122 P1 4 P2 4 þ P1 4 P3 4 þ P2 4 P3 4   þ a1123 P1 4 P2 2 P3 2 þ P2 4 P1 2 P3 2 þ P3 4 P1 2 P2 2 where a1, a11, a12, a111, a112, a123, a1111, a1112, a1122 and a1123 are the phenomenological coefficients. With the existence of eigenstrain eij0 caused by electrostriction, the elastic energy density of the system can be written as   1    f elas Pi , eij ¼ Cijkl eij
eij 0 ekl
ekl 0 2   1 ¼ C11 e11 2 þ e22 2 þ e33 2 þ C12 ðe11 e22 þ e11 e33 þ e22 e33 Þ 2     þ 2C44 e12 2 þ e13 2 þ e23 2 þ β11 P1 4 þ P2 4 þ P3 4      þ β12 P1 2 P2 2 þ P2 2 P3 2 þ P3 2 P1 2
e11 q11 P1 2 þ q12 P2 2 þ P3 2      
e22 q11 P2 2 þ q12 P1 2 þ P3 2
e33 q11 P3 2 þ q12 P1 2 þ P2 2
2q44 ðe12 P1 P2 þ e13 P1 P3 þ e23 P2 P3 Þ (6) with

β12

  1 β11 ¼ C11 Q11 2 þ 2Q12 2 þ C12 Q12 ð2Q11 þ Q12 Þ 2   ¼ C11 Q12 ð2Q11 þ Q12 Þ þ C12 Q11 2 þ 3Q12 2 þ 2Q11 Q12 þ 2C44 Q44 2 q11 ¼ C11 Q11 þ 2C12 Q12 q12 ¼ C11 Q12 þ C12 Q11 þ C12 Q12 q44 ¼ 2C44 Q44

where C11, C12 and C44 are the elastic moduli and Q11, Q12 and Q44 are the electrostrictive coefficients.

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Flexoelectric Effect at the Nanoscale

555

The gradient energy density can be written as   1 1 f grad ∇j Pi , ∇k eij ¼ gijkl Pi, j Pk, l þ wijklmn eij, k elm, n 2 2

(7)

where gijkl and wijklmn are the gradient coefficients. The second term on the righthand side of Eq. (7) appears to guarantee a smooth distribution of the order parameter in the presence of flexoelectric coupling term. Under the condition f 2klmn < gijkl Cijmn, this term can be neglected, and the gradient energy can be simplified as 

f grad ∇j Pi , ∇k eij



"


 # @P1 2 @P2 2 @P3 2 þ þ @x1 @x2 @x3
 @P1 @P2 @P1 @P3 @P2 @P3 þ G12 þ þ @x1 @x3 @x2 @x3 1 @x2 "@x
2

 # 1 @P1 @P2 @P1 @P3 2 @P2 @P3 2 þ G44 þ þ þ þ þ 2 @x2 @x1 @x3 @x1 @x3 @x2 "


 # 1 @P1 @P2 2 @P1 @P3 2 @P2 @P3 2 0 þ G44
þ
þ
2 @x2 @x1 @x3 @x1 @x3 @x2

1 ¼ G11 2

(8) The electric energy density is 1 1 f elec ¼
Pi Ei
ϵ b Ei Ei ¼
ðP1 E1 þ P2 E2 þ P3 E3 Þ
ϵ b ðE21 þ E22 þ E23 Þ (9) 2 2 where ϵ b is the background dielectric coefficient. The flexoelectric coupling energy density is
   1 @eij @Pk
Pk f flexo Pi , eij , ∇Pi , ∇eij ¼ f ijkl eij 2  @x l @xl

 1 1 @P1 1 1 @P2 ¼ f 11 e11 þ f 12 ðe22 þ e33 Þ þ f 11 e22 þ f 12 ðe11 þ e33 Þ 2 2 2 2 @x @x2 1

 1 1 @P3 þ f 11 e33 þ f 12 ðe11 þ e22 Þ 2
2 
@x3 
 @P1 @P2 @P1 @P3 @P2 @P3 þ f 44 e12 þ þ þ þ e13 þ e23 @x2 @x
@x @x1
@x3 @x2 1 3

1 @e11 1 @e22 @ϵ 33 @e12 @e13
f 11 þ f 12 þ þ þ f 44 P1 @x1 2
@x1 @x1  @x3 

2
@x2 1 @e22 1 @e11 @e33 @e12 @e23
f 11 þ f 12 þ þ þ f 44 P2 @x2 2
@x2 @x2  @x3 

2
@x1 1 @e33 1 @e11 @e22 @e13 @e23
f 11 þ f þ þ þ f 44 P3 2 @x3 2 12 @x3 @x3 @x1 @x2 where f11, f12 and f44 are the flexoelectric coupling coefficients.

(10)

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The surface energy density can be written as f surf ðPi Þ ¼

D1 s P1 2 D2 s P2 2 D3 s P3 2 þ þ 2δ1 eff 2δ2 eff 2δ3 eff

(11)

with δieff being the so-called extrapolation length and Dis being the material coefficients related to the gradient energy coefficients and surface orientations. To minimize the total free energy F, apply the Euler-Lagrange equations with respect to polarization Pi and strain eij, respectively, @F @
@Pi @xj

@F   @ @Pi =@xj

@F @
@eij @xk

@F   @ @eij =@xk

! ¼0

(12a)

¼ σ ij

(12b)

!

For simplicity, if just consider the second order of P in the Landau energy density, one can have
 @ekl Pi ¼ ϵ 0 χ ij Ej þ f jmkl @xm

(13)

  @Pk σ ij ¼ Cijkl ekl
ekl 0 þ f ijkl @xl

(14)

These are the constitutive equations of ferroelectrics considering the flexoelectric effect. kl From Eq. (13), it can be seen that the term f jmkl @ϵ @xm behaves just like the electric field Ej, so it is defined as the flexoelectric field Efj ¼ f jmkl

@ekl @xm

(15)

Furthermore, rewrite Eq. (13) by defining μijkl = ϵ 0χ imfmjkl, and rewrite Eq. (14) in @ 2 ϵ kl terms of E by neglecting the term related to the higher-order @x , i @xj Pi ¼ ϵ 0 χ ij Ej þ μijkl

@ekl @xj

  @Ek σ ij ¼ Cijkl ekl
ekl 0 þ μijkl @xl

(16)

(17)

where μijkl is the so-called flexoelectric coefficients appearing in Eq. (1). It should be noted that Eq. (17) is only limited to the cases of small strain gradients; for large strain gradients such as in domain walls and interfaces, it is not rigorous because the

17

Flexoelectric Effect at the Nanoscale

neglect of the term

@ 2 ϵ kl @xi @xj

557

is not reasonable in such cases. Moreover, with the

piezoelectric coupling term involved in the total free energy, the constitutive equations of “general piezoelectricity” can be derived as Pi ¼ ϵ 0 χ ij Ej þ eijk ϵ jk þ μijkl

@ϵ kl @xj

(18)

  @Ek σ ij ¼ Cijkl ekl
ekl 0 þ eijk Ek þ μijkl @xl

(19)

with eijk being the piezoelectric constants. To investigate the domain structure evolution of ferroelectric system and study the system’s responses and physical properties under different boundary conditions, the time-dependent Ginzburg-Landau (TDGL) equations @Pi δF ¼
L δPi @t

(20)

are commonly employed to describe the evolution of order parameters. For a given ferroelectric system, Eq. (20) should be solved under a set of boundary conditions, i. e., polarization boundary condition, mechanical boundary condition, and electrical boundary condition. The polarization boundary condition can be obtained by varying the total free energy with respect to Pi and equating the variation δF to zero, that is, @f surf @Pi

¼0

þ nl



@f bulk @ ð@Pi =@xl Þ

 @f @f elas @f elec @f flexo þ @ ð@Pgrad þ þ þ =@x Þ @ ð @P =@x Þ @ ð @P =@x Þ @ ð @P =@x Þ i l i l i l i l

s

(21)

To obtain the mechanical boundary conditionin the presence of flexoelectric effect, first substitute the strain eij by 12 ui, j þ uj, i , then vary the total free energy with respect to ui and equating the variation δF to zero,
 @f grad @f surf @f bulk @f elas @f elec @f flexo ¼0 þ þ þ þ þ nl @ui @ ð@ui =@xl Þ @ ð@ui =@xl Þ @ ð@ui =@xl Þ @ ð@ui =@xl Þ @ ð@ui =@xl Þ s

(22) Take a mechanical-free, open-circuit ultrathin ferroelectric film as an example, the equations of boundary conditions at the top and bottom surfaces are


 f lmij Dsi @Pk e P þ n g þ n j ijkl j lm ¼ 0 s i δi @xl 2 s
 f ij3l @Pl f ¼0 σ i3
l33l þ 2 2 @xj s ∇  Djs ¼ 0

(23)

(24) (25)

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with Eqs. (23), (24), and (25) being the polarization, mechanical, and electrical boundary conditions, respectively. It should be noted that the consideration of flexoelectric effect not only contributes to the TDGL equation but also contributes to the boundary conditions. Nevertheless, in many theoretical works, the modifications of flexoelectric effect on the boundary conditions were ignored. It is still indistinct how much the flexoelectric boundary conditions can influence the domain structure and physical properties of ferroelectric systems. But it is expected that in nanoscale materials, the effect of the flexoelectric boundary conditions should not be neglected not only because of the significance of flexoelectricity but also because of the importance of the surfaces at the nanoscale. It is noteworthy to point out that the flexoelectricity discussed above is the bulk static flexoelectricity which is somewhat analogous to piezoelectricity. According to Tagantsev’s work [7], the dynamic bulk flexoelectric response, for example, the polarization caused by the strain gradients induced by an acoustic wave, can also be derived in a similar way by considering the kinetic energy and can be written as Pi ¼ ϵ 0 χ ij Ej þ ðμijkl þ μdijkl Þ

@ekl @xj

(26)

where μdijkl is the dynamic flexoelectric coefficients which is also proportional to the susceptibility χ ij and is approximately of the same order of magnitude with the bulk static fleoxelectric coefficients. For a system in quasi-static states, the dynamic flexoelectric effect can be ignored. Moreover, a complete phenomenological description of flexoelectricity for finite samples must include the effects of surfaces or interfaces. A typical example of surface effects that can mimic bulk flexoelectric response is the surface piezoelectricity, where thin piezoelectric layers near the surfaces cause a built-in field across the sample, and the field depends on the strain state of the surfaces. Despite a lack of clear understanding of the surface flexoelectricity, it is believed that it makes an important contribution to the large difference between the experimental flexocoupling coefficients and the theoretical bulk ones. In general, properties near the surface (e.g., bonding and screening environment, surface polarization, surface work function, interface dipole, etc.) should change in response to local strain. As a result, a long-range surface-specific flexoelectric field may arise across the sample due to the strain difference between the surfaces. In this context, besides solving the bulk flexoelectricity, one can include a surface-specific flexoelectric field to the system, for example, for a nanofilm as Eiflexo, S ¼ f Sijkl ΔeSkl =h

(27)

where ΔeSkl are the components of strain difference between the top and bottom surfaces of the nanofilm, h is the film thickness, and f Sijkl are the surface-specific flexocoupling coefficients. It is also noteworthy that recent first-principle calculation has shown that bending strain can induce antiferrodistortive (AFD) displacement associated with the rotation

17

Flexoelectric Effect at the Nanoscale

559

of oxygen octahedral in ultrathin ferroelectric nanofilms. [11] Interestingly, the bending nanofilm possesses spontaneous polarization (which is absent in flat film) due to the coupling between ferroelectricity and the AFD phase transition. In other words, polarization is induced by strain gradient via the triggering of AFD phase transition. This actually indicates flexoelectricity can be originated from more complex mechanisms. However, a complete phenomenological description of flexoelectricity in nanomaterials taking into account such mechanisms is still lacking.

2.2

Microscopic Theory of Flexoelectric Effect

Though the phenomenological description sheds light on the flexoelectric response of materials and obtains the flexoelectric constitutive equations, it cannot give the magnitude of the flexoelectric coefficients. To estimate the flexoelectric coefficients and uncover the microscopic mechanism of flexoelectricity, various theoretical works have been conducted. In 1964, Kogan [2] first gave the order of magnitude estimation of the flexoelectric coefficients. Following his method, consider a cubic unit cell with lattice constant a, if the strain variation over the interatomic distance equals one, then the strain gradient is 1a. With the polarization being of the order of ea ea
2 e
2 a3 ¼ ea , the flexoelectric coefficient can be obtained as  1=a ¼ a, where e is the absolute value of the electron charge. After Kogan, Tagantsev [7] developed a microscopic description of flexoelectricity based on the “rigid ion” model, which focused on the ionic flexoelectricity and neglected the electronic contribution. Here a brief introduction to this model is given; more details can be found in Reference [7]. Consider a crystal lattice sketched in Fig. 2. Atom B is the nth atom of the lattice, and its jth coordinate is denoted by Rn, j before deformation. When the lattice is subjected to an inhomogeneous deformation, one can get two kinds of values of the ith components of the displacement of atom B. The first value is an approximation 0 based on the continuum theory and is called the external displacement wext n, i , set xj the coordinates of an immobile reference point, then wext n, i

¼

ð R n, j x0j

@ui dy @xj j

(28)

i where @u @xj is the so-called external strain. The other value is the real displacement of

the atom B according to the discrete nature of the lattice, wn,i. The inhomogeneous deformations break the centrosymmetry of the lattice, so wext n, i is not coincident ext with wn,i. The difference between wn,i and wn, i is the so-called internal strain wint n, i, and it is proportional to the strain gradient jkl ext wint n, i ¼ wn, i
wn, i ¼ N n, i

@ejk @xl

(29)

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Fig. 2 Schematic of a lattice before a and after b inhomogeneous deformation

where N jkl n, i can be calculated by the theory of lattice dynamics. If the nth atom has the charge Qn, the variation of the average polarization induced by the inhomogeneous deformation can be written as δPi ¼ V
1 def

X

X   Qn Rn, i þ wn, i
V
1 Q n Rn , i

n

(30)

n

where V and V def are the volume of the sample before and after deformation, respectively. Substituting Eq. (29) into Eq. (30), one can find the strain gradient induced polarization variation in a unit cell with volume v, δPi ¼ v
1

X n

Qn N jkl n, i

@ejk @xl

(31)

So the flexoelectric coefficients can be obtained as μijkl ¼

X δPi ¼ v
1 Qn N jkl n, i @ejk =@xl n

(32)

Based on this rigid ion model, Tagantsev [7] divided the flexoelectric effect into four contributions, i.e., static bulk, dynamic bulk, surface piezoelectricity, and

17

Flexoelectric Effect at the Nanoscale

561

surface flexoelectricity, and the four contributions are generally of the same order of magnitude. As mentioned before, surface piezoelectricity shows its existence in finite samples, the surfaces of which break the symmetry of the system and result in effective piezoelectric layers in vicinity of the surfaces. This contribution is also susceptibility dependent and is independent of the surface/volume ratio. The last contribution, i.e., surface flexoelectricity, is not dependent on the dielectric constant thus relatively insignificant. It should be noted that the surface contribution to the flexoelectric response was challenged by Resta [12], who developed a microscopic description of the electronic contribution to flexoelectricity based on the classical piezoelectric theory.

2.3

The Theoretical Calculations of Flexoelectric Coefficients

To quantify the significance of flexoelectric response, the relatively accurate values of flexoelectric coefficients are needed, not just the order of magnitude estimations. The theoretical calculations of flexoelectric coefficients were conducted just in recent years though the microscopic theories were put forward much earlier. Based on Tagantsev’s microscopic model, Maranganti et al. [13] first implemented the calculation of bulk static flexoelectricity. Later, Hong et al. [14] performed the firstprinciple calculations of longitudinal flexoelectricity using a direct approach. They directly calculated the induced polarization by a given strain profile and extracted the flexoelectric coefficients. The direct methods have the merits of including the extrinsic contributions, such as surface and defect effects. Then Hong and Vanderbilt [15] extended Resta’s theory to yield the transverse flexoelectric coefficients and obtained the longitudinal flexoelectric coefficients for a number of materials such as BTO, PbTiO3 (PTO), and STO. Later they developed a more general first-principle theory which included both the electronic and lattice contributions in the context of density-functional calculations [16]. They obtained the relationship of flexoelectric coefficients between the fixed electric field (μEijkl ) and fixed electric displacement ðμD mjkl ) boundary conditions: μEijkl ¼ ðδim þ χ im ÞμD mjkl

(33)

where δim is Kronecker delta. So μD mjkl could be seen as a “ground-state bulk property” E to calculate the μijkl at finite temperature by scaling with the dielectric constant. Ponomareva et al. [17] developed an effective Hamiltonian method to study the flexoelectric effect in ferroelectric thin films at finite temperatures and investigated the temperature and film thickness dependence of the flexoelectric coefficients. Recently, Stengel [18] built up a new continuum thermodynamic functional and constructed the relationship between the continuum description and ab initio phono dispertion curves, which was used to calculated the flexoelectric coefficients. Besides of calculating the exact values of the flexoelectric coefficients, the upper limits for the magnitude of the static bulk flexoelectric coefficients in ferroelectrics

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were determined by Yudin et al. [19]. By studying the requirement of stability of the parent phase, the upper bounds for the flexoelectric coefficients were derived as

2.4

f 244 < c44 g44

(34a)

ðf 11
f 12 Þ2 < ðC11
C12 Þðg11
g12 Þ

(34b)

Size Effect of Flexoelectric Response

The significance of the flexoelectric response is not only related to the magnitude of the flexoelectric coefficients but also to the strain gradients. The flexoelectric response is expected to be largely size-dependent and tends to be much stronger when the structural size scales down to nanometers because the strain gradients are closely linked with structural feature size. The size scale of strain gradients can be shown in a simple way as discussed by Majdoub et al. [20]. As shown in Fig. 3, consider two triangular inclusions with different length scales but the same aspect ratio, when they were subjected to the same stress, the strain distributions of the two inclusions would be identical. So the strain gradients scaled inversely with ai, which was the distance between two points inside the inclusion. Majdoub et al. [20] also investigated the flexoelectric size effects of piezoelectric and non-piezoelectric cantilevers by using both the flexoelectric continuum theory and the atomistic simulations. As illustrated in Fig. 4, when the thickness of the cantilever was below 10 nm, an apparent piezoelectric response can be found in nonpiezoelectric paraelectric BTO which was attributed to the flexoelectricity. With the decrease of thickness of the cantilever, the piezoelectric response of paraelectric BTO became much stronger, and its piezoelectric constant could be as high as five times of the piezoelectric BTO constant when the thickness of the cantilever was s¥ ∂ε ∂x

~

s¥

Dε

∂ε

a1

∂x

~

Self-similar

Dε a2

a2

a1 scaling »mm

» nm

s¥

s¥

Fig. 3 The illustration of size effects of flexoelectricity by considering two triangular inclusions with the same aspect ratio, subjected to the same stress but with different length scales. Reprinted with permission from [20]. Copyright (2008) American Physical Society

17

Flexoelectric Effect at the Nanoscale

563

Normalized effective piezoelectric constant

6 PZT BT(paraelectric) Bulk PZT

5 4 3 2 1 0 0.4

5

10

15

20 25 30 Thickness h (nm)

35

40

45

50

Fig. 4 The enhancement of piezoelectric response of piezoelectric PZT (dashed blue) and paraelectric BTO (solid red) with the decrease of cantilever thickness. The piezoelectric constants are normalized with respect to the bulk piezoelectric PZT (solid blue) (dPZT =
274pC/N ) and piezoelectric BTO (dBT =
78pC/N ), respectively. Reprinted with permission from [20]. Copyright (2008) American Physical Society

smaller enough. The enhancement of piezoelectric response of PZT (lead zirconate titanate) was relatively moderate compared with that of BTO, but the piezoelectric constant was still doubled at 15 nm. Inspired by the size effect of flexoelectricity, Majdoub et al. [21] also found a size-dependent effect of the harvested power of a PZT cantilever due to flexoelectricity. Their results indicated that the energy harvesting of piezoelectric nanostructures could be dramatically enhanced in a definite small range of sizes by taking the flexoelectricity into consideration. Due to the size effects and its significance in smaller size, flexoelectric response is quite strong in systems with nanoscale sizes and comparable to the piezoelectric response or even plays a dominant role. To clarify the influence of flexoelectricity and its size effects in nanostructures, there are also more and more interests focused on the flexoelectric continuum theory, which will not be covered here, and are referred to, e.g., Reference [22].

3

Experimental Characterization of Flexoelectric Effect

3.1

Experimental Determination of Flexoelectric Coefficients

The measurement techniques of flexoelectric coefficients include direct methods and the converse ones. The direct methods measure the polarization of the samples induced by the external inhomogeneous deformations. In contrast, the converse methods utilize the converse flexoelectric effect, measure the strains of the sample under a nonuniform

564

L. L. Ma et al.

electric field, and extract the converse flexoelectric coefficients. It has been shown that the direct and converse flexoelectric coefficients are equivalent. The direct methods are more commonly used and include various designs such as cantilever bending, threepoint bending, four-point bending and pyramid compression, etc. The typical experimental setups of cantilever bending, three-point bending, fourpoint bending and pyramid compression are illustrated in Fig. 5a, b, c and d, respectively. To illustrate the measurements, take the three-point bending measurement conducted by Zubko et al. [10] as an example. As shown in Fig. 5b, oscillatory bending strain can be induced by the probe on single crystal STO, with the strain gradient being

3
 @e11 L L
x1 , ¼ 3z0 2 2 @x3

(35)

where L is the distance between the knife edges under the sample, z0 is the displacement of the probe, and xi is the distance between the center of the sample and the interested position. The induced average out-of-plane polarization can be measured as P3 ¼

I ωA

(36)

where ω is the angular frequency of the oscillatory mechanical bending, A is the area of the electrode and I is the measured ac current. In addition, the flexoelectricityinduced average polarization can also be obtained as P3 ¼ μ

@e11 12z0 ¼ μ 3 ð L
aÞ @x3 L

(37)

where μ is the effective flexoeletric coefficient and a is the half length of the electrodes. From Eqs. (36) and (37), it has μ¼

IL3 12ωAz0 ðL
aÞ

(38)

It should be noted that μ is the effective flexoelectric coefficient, which is a combination of the flexoelectric coefficients defined before. To obtain the full three flexoelectric coefficients of STO, the measurement should be repeated for samples with different orientations. It is also necessary to use a different method to get another independent equation because the pure bending method can only provide two independent equations. Then the flexoelectric coefficients can be extracted by solving the independent equations of different μ from different measurements. Moreover, the existence of the heater and liquid N2 bath in Fig. 5b enables the controllability of temperature; thus, the dependence of the flexoelectric coefficients can also be obtained.

(250mm)

Movable stage

Actuator (max. disp.: 37μm)

(85mm)

Contact cylinder pin (diameter: 8mm)

Power Amplifier

Displacement sensor

Specimen

Loudspeaker

Sample

Displacement Transducer

40mm

Lock-in Amplifier

Temperature Controller

d

b

Au

3

4

7

5

9

10

x2

8

x3

6

Charge Amplifier

A.C. driving force

Load Cell

Liquid N2

L

x1

SrTiO3

lock-in

Lock-in Amplifier

INSTRON 5866

heater

Flexoelectric Effect at the Nanoscale

Fig. 5 The experimental setups for different direct measurement methods of flexoelectric coefficients. (a) Cantilever bending (Reprinted with permission from [9]. Copyright (2001) AIP Publishing LLC.), (b) three-point bending (Reprinted with permission from [10]. Copyright (2007) American Physical Society.), (c) four-point bending (Reprinted from [23]. Copyright (2011) The Japan Society of Mechanical Engineers.) and (d) pyramid compression. (Reprinted with permission from [24]. Copyright (2006) Springer Nature.)

XY-table

c

a

80mm (110mm)

17 565

566

a

L. L. Ma et al.

0.07

b 1.6

x1/L = 0.18 x1/L = 0.67

1.4 2

0.05 0.04 0.03

Polarization (μC/m )

2

Polarization (μC/m )

0.06

–6

m12 = 4.1 x 10

C/m

0.02

m12 = 3.4 x 10

–6

C/m

0.01

1.2 1.0 0.8 0.6

m12 = 100 mC/m

0.4

0.00 0.2 –0.01 –0.004 0.000 0.004 0.008 0.012 0.016 0.020 –1 Strain gradient (m )

0.0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 –1 Strain gradient (m )

c

Polarization (μC/m2)

1.6

1.2

0.8

0.4 0.5 μC/m 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

–1

Strain gradient (m )

Fig. 6 The relationship between the flexoelectricity-induced polarization and strain gradient in (a) relaxor PMN (Reprinted with permission from [9]. Copyright (2001) AIP Publishing LLC.), (b) paraelectric BST (Reprinted with permission from [25]. Copyright (2002) AIP Publishing LLC.) and (c) unpoled ferroelectric PZT (Reprinted with permission from [26]. Copyright (2003) AIP Publishing LLC.)

By using the developed experimental techniques, the flexoelectric coefficients of a number of ferroelectric and relaxor ceramics have been measured. The results of the direct and converse methods show a good agreement. The relationships between the flexoelectricity-induced polarization and strain gradient of relaxor PMN (lead magnesium niobate), paraelectric BST (barium strontium titanate), and unpoled ferroelectric PZT are shown in Fig. 6. It can be seen that for paraelectric ceramics, the flexoelectric polarization scales linearly with strain gradient as expected. In contrast, for unpoled ferroelectric one, the flexoelectric behavior is nonlinear, which is associated with the ferroelastic domain switching under large mechanical loading. According to Tagantsev [7], the relationship between flexoelectric coefficients and dielectric permittivity can be described as μij ¼ γχ ij

e a

(39)

where χ ij is the dielectric susceptibility and γ is a constant of the value close to 1.

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Flexoelectric Effect at the Nanoscale

567

Fig. 7 (a) Relationship between flexoelectric coefficient μ12 and dielectric permittivity and (b) the temperature dependence of μ11 and χ of BST. Reprinted with permission from [24]. Copyright (2006) Springer Nature

The experimental results characterizing the permittivity dependence of μ12 for PMN and BST are shown in Fig. 7a. The temperature dependence of μ11 and χ of BST are shown in Fig. 7b. It is clear that the flexoelectric coefficients scale with the dielectric permittivity. The enhancement of μ12 in BST above ϵ r~11000 is perhaps associated with the existence of ferroelectric domains above the general transition temperature. However, it can be seen that γ of PMN and BST are not as expected and are also much different with each other, with γ PMN~0.65 and γ BST~9.3. Moreover, for unpoled PZT and BTO ceramics, the scale factors have been determined as γ PZT~0.57 and γ BT~11.4, respectively. Compared with Pb-based PZT and PMN, the Ba-based BST and BTO have stronger flexoelectric effects and larger values of γ. Thus the chemical makeup might be the source of the difference of the flexoelectricity in these two perovskite structures. The dielectric permittivity dependence of the flexoelectric coefficients indicate a large flexoelectric response at the temperature close to the paraelectric-ferroelectric transition temperature of ferroelectrics, where the ϵ r is much higher. The measured flexoelectric coefficients of BST, PMNT (lead magnesium niobate-lead titanate), and PST (lead strontium titanate) under such conditions are indeed particularly high (~10
100μC/m). But for relaxor ferroelectric PMN, the large μ12 is attributed to not only the large permittivity but also the reorientation of the polar nanodomains under strain gradients. The flexoelectric coefficients have also been measured for some kinds of single crystals. The temperature dependence of the flexoelectric response of STO and BTO are shown in Fig. 8a and b, respectively. The inset of Fig. 8a shows the linear proportionality between flexoelectricity-induced polarization and strain gradients, as expected. For STO, the flexoelectric response increases qualitatively following the dielectric permittivity with a decreasing temperature. However, there is an anomaly of the flexoelectric response when the temperature is below 105K, which is the cubic to tetragonal phase transition temperature of STO. The anomaly can be explained as follows: ferroelastic domains appear below 105K and tend to release the elastic energy by readjustment of domain walls under mechanical deformation, thus leading

568

6x10

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4x10–10 (111) 2x10

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–10

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 –1 Strain gradient (m )

30

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2.0 1.5

20

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–50 –150 –100 Temperature (°C)

0

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|P3| (C/m2)

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μ13eff (μC/m)

(101)

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50

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L. L. Ma et al.

BTO (111)

10 1 0.1

0.01

50

100 150 200 Temperature (°C)

250

300

Fig. 8 Temperature dependence of the flexoelectric response of (a) single crystal STO (Reprinted with permission from [10]. Copyright (2007) American Physical Society.) and (b) single crystal BTO. The inset of a shows the relationship between the polarization induced by flexoelectric effect and strain gradient. The red and blue curves in b are the measured values on heating and cooling, respectively (Reprinted with permission from [27]. Copyright (2015) American Physical Society.)

Table 1 Flexoelectric coefficients for BTO and STO from theoretical calculations and experiment. Reprinted with permission from [13]. Copyright (2009) American Physical Society.

STO BTO

μ11(10
13C/m) Ab initio Experiment
26.4 20 15.0

μ12(10
13C/m) Ab initio Experiment
374.7 700
546.3 106

μ44(10
13C/m) Ab initio Experiment
357.9 300
190.4

to the relaxation of strain gradients and consequently the reduction of the flexoelectric response. For BTO single crystal, a sharp decrease of flexoelectric coefficients from 10–100 μC/m to 1–10 μC/m can be seen above Tc, which corresponds to the ferroelectric-paraelectric phase transition. Since the reported values of BTO ceramics in paraelectric phase are 5–50 μC/m, it indicates that the grain boundaries play a role of enhancing the flexoelectric response. To compare the flexoelectric coefficients obtained through theoretical calculations and experimental measurements, BTO and STO are taken as examples as displayed in Table 1. It can be seen that the order of magnitude of flexoelectric constants for STO show good agreement between the ab initio calculations and the experiments, while the measured transverse flexoelectric coefficient of BTO are three to four orders higher than the calculated one. This discrepancy is partly attributed to the large temperature dependence of the dielectric permittivity of BTO, since the temperature of the calculation is 0K, while the experiment is conducted at room temperature. Taking this effect into consideration, the gap can be narrowed to about one to two orders of magnitude. Other sources of this gap involve the surface piezoelectricity and the flexoelectricity-induced alignment of

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precursor polarization existing in the paraelectric phase of BTO. Furthermore, since the experimental data for BTO used here is from BTO ceramics, the grain boundary enhancement of flexoelectricity is also a contribution to this gap.

4

Flexoelectricity-Induced Novel Phenomena in Nano-ferroelectrics

When the size of ferroelectric system decreases to nanoscale, various triggering phenomena arise. Many of the phenomena can be explained without taking into account the flexoelectric effect, while some of the anomalies cannot. In this section, we will discuss the triggering phenomena and properties induced by flexoelectric effect in nanoscale ferroelectrics, from relatively global characteristics such as dielectric responses and imprint behaviors to some local characteristics, e.g., the profile and conductivity of domain walls.

4.1

Modifications on Dielectric and Mechanical Response of Nanoscale Ferroelectrics

4.1.1 Modified Dielectric Response It has been reported that for ferroelectric thin films close to the transition temperature, the permittivity is depressed rather than the sharp peak of bulk ferroelectrics. With decreasing film thicknesses, the depression tends to be more pronounced. The smearing of the temperature-dependent permittivity for BST thin films with different thicknesses is shown in Fig. 9a. By taking the flexoelectric effect into consideration, Catalan et al. [29] calculated the ferroelectric properties in terms of the LGD theory in the assumption of an exponentially relaxed strain profile. The results show a good agreement with the experimental data, as displayed in Fig. 9b. Unlike bulk ferroelectrics, thin films are subjected to misfit strains by the underlying substrates. The misfit strains relax across the direction of film thickness and generate a strain gradient in the film. Due to the flexoelectric effect, this strain gradient will induce a built-in flexoelectric field, which influence the dielectric responses. With decreasing film thickness, the flexoelectric response becomes more apparent and leads to a larger degradation of the dielectric permittivity. Besides the smearing of the dielectric permittivity, flexoelectricity can also modify the critical thickness of the ferroelectric films. The reported vanishing critical thickness of free standing PT (lead titanate) films has been demonstrated to be caused by flexoelectricity. And it is proposed that the critical thickness of epitaxial BTO films can be increased by the flexoelectric effect. Furthermore, the flexoelectricity-induced modifications on Curie temperature and pyroelectric coefficients of ferroelectric thin films have also been reported. These phenomena indicate that thin films with desirable dielectric and ferroelectric properties could be designed by introducing or avoiding specific strain gradients.

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b 2000 1500 1000 0.1 500 0

100

0.05

150

200 250 300 350 Temperature / K

0 400

calculated

2000

1000 t

0

400

measured

2000 t

350 Tm(K)

1440 nm 950 nm 660 nm 340 nm 280 nm 220 nm

Loss Tangent relative permitivity

Dielectric Constant

a

Calculated 300

1000 250 Measured

0

250 500 750 t(nm)

200

300

400

500

600

T(K)

Fig. 9 (a) Experimental data of relative permittivity for BST grown on SRO substrate (Reprinted with permission from [28]. Copyright (2002) AIP Publishing LLC.) and (b) The calculated temperature dependence of permittivity and the comparison with the measured values (Reprinted with permission from [29]. Copyright (2005) American Physical Society.)

4.1.2 Modified Mechanical Response In indentation tests of ferroelectrics, it is found that the indentation hardness was enhanced when the indenter radius decreases. This indentation size effect of hardness is demonstrated to be induced by the flexoelectric effect by Gharbi et al. [30], both theoretically and experimentally. Barium titanate and quartz were both investigated to testify their theory. The indentation hardness with different indenter radius is shown in Fig. 10a. It can be seen that the flexoelectric model is in excellent agreement with experiments. For macroscale indentation, the strain field is relatively uniform, and the strain gradient is small; therefore, the flexoelectricity-induced enhancement can be neglected. For nanoscale indentation, there existed an extremely large strain gradient so that the enhancement is apparent for BTO which has large flexoelectric coefficients. While quarts shows no enhancement even for nanoindentation because of its weakness of flexoelectric response. This enhancement phenomenon also provides us a method to extract the flexoelectric coefficients by indentations. For BTO in the present case, the extracted flexoelectric coefficients is μ12 = 4μC/m, which is in good agreement with the value measured by cantilever bending. In fact, besides the indentation hardness, various mechanical properties are modified by flexoelectricity and thus present a size effect according to the developed flexoelectric continuum theory. As an example, the effective flexural stiffness of PMN nanofilm is shown in Fig. 10b. It can be seen that correction of flexural stiffness due to flexoelectricity is fairly obvious when the film thickness decreases to several nanometers and cannot be neglected in the design and fabrication of nanodevices.

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Flexoelectric Effect at the Nanoscale

Contact stiffness/Contact radius (N/m2)

a 4.5

571

x 1011 Piezoelectric model for Barium Titanate Flexoelectric model for Barium Titanate Experimental model for Barium Titanate Experimental data for Quartz Linear fitting for Quartz

4 3.5 3 2.5 2 10-8

10-7

10-6 Contact radius a(m)

10-5

400

800

10-4

b Effective Flexural Stiffness (1E-9)

101

C11 + E11 C11

100

10-1

10-2

10-3

200

600

1000

Thickness (nm) Fig. 10 (a) Size effect of the indentation hardness of BTO and quartz (Reprinted with permission from [30]. Copyright (2009) AIP Publishing LLC.). (b) Effective flexural stiffness of PMN nanofilms. The red line is based on the flexoelectric theory, and the blue curve represents the classical piezoelectric theory (Reproduced from [31]. # IOP Publishing Ltd. All rights reserved.)

4.2

Imprint Behaviors in Ferroelectric Nanofilms

The phenomenon that the polarization of ferroelectrics has a preferential direction and asymmetric hysteresis loops is known as imprint. That is, one polarization state is more stable than the other. This asymmetric polarization bistability on the one hand will degrade the performance of devices such as nonvolatile memory in ferroelectric thin film; on the other hand, it can also lead to an interesting

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phenomenon, such as smearing of phase transition. Ferroelectric nanocapacitors are typical examples to exhibit imprint behavior. For these systems, first-principle calculations have showed that the different bonding environment at the top and bottom electrode/ferroelectric interfaces can lead to asymmetric stability of polarization. (See reference [32] and references therein.) Meanwhile, phenomenological models usually attribute the intrinsic source of this asymmetric polarization stability to the so-called non-switchable dead layer at the film/electrode interface. This layer may be caused by various reasons such as a secondary low-permittivity phase at the film surface, the presence of misfit dislocations, nearby-surface variation of polarization and electric field penetration into the electrodes, etc. Importantly, it has been demonstrated that the dominant contribution to the dead layer is the flexoelectric effect trigged by the large strain gradient near the interface. The misfit strain between the film and electrode relaxes by the appearance of dislocations. Full strain relaxation will generate a strain gradient as large as ~106 m
1 with a relaxation length of 10 nm. The flexoelectric filed is estimated as Ef = f ∇e~107 V/m by using the typical value of the flexocoupling coefficients f~10 V. Although this estimated value is smaller than value of the coercive field, it is thought to result in the non-switchable dead layer at the region of interface, as shown in Fig. 11. Assuming the direction of flexoelectricity-induced polarization in the dead layer is upward, as illustrated, the head-to-head polarization configuration of the capacitor will result in a larger electrostatic energy compared with the head-to-tail configuration and is less energetically favorable. This leads to a preferential polarization oriented upward in the rest of the film; thus, imprint behavior occurs. This flexoelectricity-induced imprint behavior reminds us that large strain gradients need to be avoided to prevent the degradation of device performance in ferroelectric nanofilms. Imprint behavior can also be introduced by the application of external strain gradients. Gruverman et al. [34] demonstrated that the PZT thin films could be imprinted through flexoelectric effect by applying bending stress. The PFM

Fig. 11 Schematics of non-switchable dead layer at the region of interface of a ferroelectric capacitor. Reprinted with permission from [33]. Copyright (2006) AIP Publishing LLC

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Fig. 12 Imprint behavior generated by mechanical bending in PZT thin film. (a) and (b) are the PFM amplitude and phase images for the as-grown film before bending. (c, d) and (e, f) are the corresponding images after tensile and compressive bending, respectively. Reprinted with permission from [34]. Copyright (2003) AIP Publishing LLC

magnitude and phase images of the as-grown film are shown in Fig. 12a and b. Corresponding images after tensile and compressive bending are shown in Fig. 12c, d, e and f, respectively. The dark and light regions in phase images represent the upward and downward polarization, respectively. The dark lines in the amplitude image represent the domain boundaries. It can be seen that before bending, the polarization has no preferential direction. After the application of tensile bending stress, the as-grown capacitor is switched into a single domain with upward polarization, while the application of compressive bending stress results in a single domain with downward polarization. The different preferential polarization directions are due to the opposite sign of the strain gradient for tensile and compressive bending. This kind of imprint behavior suggests that the domain configurations of nano-ferroelectrics can be influenced and controlled by external strain gradients, which will be discussed in detail in the next section.

4.3

Influence of Flexoelectric Effect on Ferroelectric Nanodomains

4.3.1 Mechanical Switching of Polarization in Ferroelectric Thin Films Since the flexoelectric field induced by strain gradients is analogous to electric field, it can also be used to switch the polarization of ferroelectrics mechanically. The experimental demonstration of the mechanically induced domain switching was conducted by Lu et al. [35]. BTO ultrathin film grown on STO substrate was used in their investigation. The compressive misfit strain of the film resulted in the out-ofplane polarization; thus, only 180 polarization switching was allowed. The film was first written into a bipolar domain state electrically, the PFM images of which are shown in Fig. 13a and b. Then it was scanned in the center area by an AFM tip with an increasing loading force from 150 to 1500 nN. The PFM images of the scanned film are shown in Fig. 13c and d. It can be seen that with increasing tip force, the upward polarization is switched downward. This mechanical switching behavior is

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b

e

c

d

free energy, J/m3

4 108

no stress homogeneous stress strain gradient

3 108

2 108

1 108 -0.8

-0.4

0

0.4

0.8

polarization, C/m2

Fig. 13 Mechanically induced reversal of ferroelectric polarization. (a) and (b) are the PFM phase and amplitude images of the film in bipolar states. Dark areas in the phase image represent the domain with upward polarization. (c) and (d) are the PFM phase and amplitude images after scan. (e) Free energy calculations for the epitaxially clamped BTO ultrathin film subjected to no stress (blue curve), homogeneous compressive (red curve), and tip-induced strain gradient (green curve). From [35]. Reprinted with permission from AAAS

attributed to the flexoelectric filed induced by the local stress inhomogeneity under the AFM tip. The estimated flexoelectric field is as strong as 2 MV/cm, which is comparable to the coercive field. This switching behavior can also be explained in perspective of free energy. Figure 13e shows the calculated free energy for the epitaxially clamped BTO ultrathin film under no stress, homogeneous compressive stress, and tip-induced strain gradient. Under homogeneous stress, the double well is still symmetric though the barrier decreased. While under strain gradient, the double well is skewed, and the downward polarization state is more energetically favorable, thus driving the upward polarization to switch downward. The effects of epitaxial strain on mechanical switching were investigated, and it was found that the mechanical threshold for polarization switching increased with increasing in-plane compressive strain, which was resulted from the increase of coercivity and tetragonality. The loading mode of AFM tip was also reported to influence mechanical switching process and resulting domain. Sliding contact loading mode requires smaller load to realize the switching, and the written domains were much more stable, compared with the perpendicular loading mode. This was attributed to the different strain distributions under the AFM tip in different loading modes. The nonvolatile ferroelectric memory written mechanically and read electrically were recently demonstrated in ferroelectric polymer. The force needed to record electronic signals on the polymer film was less than 100 nN and the mechanically written domains were stable and electrical erasable. Note that the flexoelectric field induced by the AFM tip is localized near the tip, which makes it suitable to write local nanodomains but not suitable to erase them. From theoretical point of view, a nonlocal flexoelectric field can be also induced by other kinds of strain gradients such as when a nanofilm is subjected to cylindrical

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bending or wavy bending. Simulation has showed that the nonlocal flexoelectric fields associated with these two kinds of mechanical bending can largely affect the stability of the nanodomains in ferroelectric nanofilm and lead to effective erasing of the information stored by the nanodomains [36]. Take the wavy bending as an example, as illustrated in Fig. 14. To study the erasing effect of wavy bending, the stability of pre-written 180 cylindrical domains with different sizes are investigated. The strain profile of a nanofilm under wavy bending can be approximately descriped as a cosine form (Fig. 14a). The flexoelectric effect is considered with a flexocoupling f12 = 10V. Fig. 14b and c depict the distributions of in-plane strain and the flexoelectric field and as a function of the wave amplitude. Firstly, it can be seen that wavy bending has significant impact on the stability of cylindrical domains, which can lead to the instability of domains and thus lead to the erasing effect. As shown in Figure. 14d, where the flexoelectric field is switched off, the critical size of stable cylindrical domain changes from 11 nm to 30 nm when the bending amplitude increases from 0.2 nm to 1.0 nm. When the flexoelectric field is switched on, significant difference can be found in the controllability on domain stability of the bending loads (Fig. 14e). Flexoelectric field can both enhance and depress the stability of the 180 cylindrical domain, depending on the direction of the domain polarization and the flexoelectric field. Due to the modulated flexoelectric field, large change in domain shape not only happens at region near domain instability but also happens when the domain size is comparable to the wave length, as indicated by the curves as shown in Fig. 14g. By influencing the stability of domains, the nonlocal wavy bending and flexoelectricity can be used to erase information, and furthermore, control the domain shapes. Compared with electrical switching, mechanical switching can avoid the associated breakdown and leakage problem. As the critical mechanical load to induce polarization switching is generally small, the nanofilm cannot be damaged in the mechanical switching process. It is therefore promising for application in nonvolatile ferroelectric memory with ultrahigh data storage density and less power consumption. However, mechanical switching has only been reported to switch the upward polarization to downward, not vice versa. The reversal switching may be realized by strain gradient engineering or by changing the surface screening condition of the film, which so far has not yet been investigated.

4.3.2 Impacts on Domain Patterns of Nanoscale Ferroelectrics The properties of ferroelectrics such as dielectric and piezoelectric response can be largely modified by the domain structure. Since flexoelectricity is significant at nanoscale, its effect on the domain configurations should be considered for applications of ferroelectrics. The influence of flexoelectricity on the domain patterns of HoMnO3 epitaxial thin films has been reported by Lee et al. [37]. To investigate the role flexoelectricity, films with different large strain gradients were grown. The strain gradients were modulated by controlling the oxygen partial pressures PO2 during the growth process. Figure 15a shows the strains for films deposited at high and low PO2. It can be seen that strains for both high and low PO2 were relaxed exponentially from the film/substrate interface to the film surface, and the strain variation for high PO2 is

Fig. 14 (continued)

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larger than low PO2. Thus the films deposited at higher PO2 had larger strain gradient. It was estimated that the flexoelectric field Es for film grown with PO2 = 10 mTorr and PO2 = 350 mTorr was ~0.7MV/m and ~5MV/m, respectively. The estimated values of Es were comparable to the coercive filed of the film at temperature close to Tc so the flexoelectric effect was giant. The domain configurations for films deposited at different PO2 are shown in Fig. 15b. For the film deposited at low PO2 with small Es, a mixed multidomain pattern was formed due to the depolarization energy. Meanwhile, for the film deposited at high PO2 near Tc, Es was large enough to induce the formation of monodomain. The effect of Es on the domain configurations was further investigated by a periodic two-dimensional model illustrated in Fig. 15c, where α and β are the domain width of the upward and downward domains. The results are shown in Fig. 15d. It can be clearly seen that the flexoelectric field had a large effect on the domain configurations, which indicated that the domain configurations could be tuned by flexoelectricity. Another experimental evidence suggesting that the flexoelectricity had effect on the domain configurations was reported by Catalan et al. [38]. They noticed that there existed large unexpected horizontal strain gradients in the c-domains of PTO thin film with a-c twins. The strain gradients were attributed to the attachment of the film with ferroelastic domains onto the flat surface of the substrate. According to flexoelectricity, this strain gradient would lead to a horizontal contribution to the polarization of c-domain and cause a strong polarization rotation. They measured the lattice distortion (Fig. 16a) and mapped polarization of the twins in the film (Fig. 16b). Strong polarization rotation of c-domains was observed, which was in agreement with the flexoelectric theory. The influence of flexoelectricity on domain configurations was also theoretically investigated by Ahluwalia et al. [39]. They predicted the domain patterns under different strength of the flexoelectric coupling by using two-dimensional phase field simulations. The roles of different flexoelectric components were also analyzed under mechanically constrained and stress-free boundary conditions. It was found that a fine structure in domain patterns formed when the strength of flexoelectric coupling exceeded a critical value, as shown in Fig. 17a. The polarization (Py in Fig. 17b) in these fine domains had a nearly sinusoidal modulation, indicating the formation of the so-called “incommensurate” or modulated phase. The finding that flexoelectricity could induce incommensurate phase formation in ferroelectrics was in agreement with previous investigations. Two-dimensional modulated domain ä Fig. 14 Control of domain stability by wavy bending on 128 nm  128 nm  8 nm simulation cells at room temperature. The cells are initially written with cylindrical domains with size r from 1 nm to 64 nm. (a) Schematics of a cell under wavy bending, with λ = 128 nm. Distributions of (b) strain and (c) flexoelectric field as a function of Ab in the x-z plane of a cell under wavy bending. Phase diagrams of equilibrium domain pattern in cells under wavy bending with flexoelectric field (d) switched off and (e) on. (f) and (g) the average polarization of the equilibrium domain patterns in z-direction, i.e., , in the initial cylindrical domain region, for the two bending cases. Reprinted with permission from Macmillan Publishers Ltd. Nature Scientific Reports [36], Copyright (2014)

(

Surface (z = 0)

0

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asubstrate

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) : +PFE(–PFE) domain

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Misfit strain

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Up (down) domain with +PFE (–PFE)

4

Es (MV m-1)

2

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6

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100

Upward (downward) Ddefect

AFE or PE

polydomain

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iii

Medium Medium Es PO2

ii

i

Flexoelectric effect Alignment near TC of defect dipoles

(α/β)2

Fig. 15 Flexoelectricity tuned domain configurations. (a) Strain profiles for films deposited at high and low PO2. (b) Schematic diagram showing how Es can affect domain configuration, which depends on PO2 during the growth process. (c) A simple model of two-dimensional periodic square-like 180 domains. (d) Temperature and Es dependence of the domain configuration. Reprinted with permission from [37]. Copyright (2011) American Physical Society

Es

c

Tensile strain u

b Lattice constant a Temperature (K)

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Fig. 16 Direct observations of the polarization rotation. (a) Positions of atoms with zoom-ins showing the a-domain (I) and c-domain (II). (b) Out-of-plane strain (color map) and electric polarization (vector map) with the line scan showing the rotations angles from normal in the c-domains. Reprinted with permission from Macmillan Publishers Ltd. Nature Materials [38], Copyright (2011)

patterns, such as domain patterns with alternating vortices (Fig. 17c) and with coexisting phases of the stripes and vortices (Fig. 17d), had also been observed under specific flexoelectric coupling strengths and mechanical boundary conditions. It was proposed by the authors that more flexoelectricity-induced domain configurations might occur in the real 3D crystals. It is noteworthy that effect of mechanical strain/stress on the vortex transformation has been revealed [40], which showed quite distinct characteristics from that of conventional electric field. Considering the similarity between flexoelectric field and electric field, mechanical strain/stress gradients might act like the role of electric field to induce transformation of ferroelectric vortices. However, it is still not clear now. Furthermore, the controllability of domain evolution by mechanical loads and flexoelectricity has been reported by Chen et al. [41]. They investigated the stability and evolution paths of domain patterns under uniform strain and cylindrical bending strain gradient in ferroelectric nanofilms. The role of both the bulk and surface contributions to the flexoelectricity was studied for different initial domain patterns. According to their results, the domain patterns were largely affected by strain gradient even in the absence of flexoelectricity. The presence of bulk flexoelectricity made the a/c and b/c domain variants more favorable and thus modified the domain patterns. Due to the weakness of the bulk flexoelectricity, this modification is small. The contributions of bulk flexoelectricity to the mechanical and polarization boundary conditions also had effects on the domain patterns. The domain pattern evolution was further investigated with different strength of flexoelectric fields, which involves both the bulk and surface contributions of flexoelectric effect. It was demonstrated

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Px

1

Py

0.5 0 -0.5 -1 -1.5

0

20

40

60

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120

Rescaled Distance

c

d

Fig. 17 (a) Domain pattern with f44 = 15 V under stress-free boundary conditions. (b) Polarization profile of the domain pattern in a along the x direction. (c) Domain pattern with f11 = 15 V under stress-free boundary conditions. (d) Domain pattern with f11 = 12 V under mechanically constrained boundary conditions. Reprinted with permission from [39]. Copyright (2014) American Physical Society

that with large flexoelectric field, the domain patterns were switched into the monodomain state, which indicating the erasing effect. The domain evolution of a nanofilm written with “SYSU” under various bending loads was investigated to show the effect of mechanical loads on the carried information in the film, as shown in Fig. 18. One can see that with the combined effect of strain gradient and flexoelectric field, the information was erased under a large downward bending. While under upward bending, the written information would be unstable and evolved to a lager domain, which could be used to assist writing domains.

4.4

Novel Domain Wall Properties Resulted by Flexoelectric Effect

Compared with domains, domain walls have been demonstrated to own attracting properties such as high electronic conductivity, chirality, and oxygen vacancy

17

a

Flexoelectric Effect at the Nanoscale

initial state

final state

bending

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evolution

initial state

final state

bending

c

evolution

initial state

final state

bending

d

581

evolution

initial state

final state

bending

P

c +domain

evolution P c domain

Eflexo

up bending

down Eflexo bending

Fig. 18 Evolution of a written domain pattern “SYSU” under various cylindrical bending condition. (a) etop =
etop =
0.005, (b) etop =
etop =
0.01, (c) etop =
etop = 0.005, and (d) etop =
etop = 0.01. Reprinted from [41], Copyright (2015), with permission from Elsevier

segregation in ferroelectrics. It is becoming a trend to regard domain walls as promising alternative engineering elements in smart materials. Due to the sharp change of order parameter normal to the ferroelectric domain wall, the strain gradients in the wall can be very large as a result of electrostriction. Therefore the flexoelectric effect is very pronounced and has important effects on the domain wall properties.

4.4.1 Domain Wall Profile The 180 domain wall separating domains with antiparallel polarizations is very common in ferroelectrics. The profile of this kind of domain wall has been considered as Ising-like for a long time. However, this viewpoint should be renewed by considering flexoelectric effect because the flexoelectricity-induced polarizations can result in more complicated domain wall profiles. The role of flexoelectricity on domain wall profiles has been reported recently. As an example, we discuss the domain walls of tetragonal BTO. Following the work of Gu et al. [42], consider a 180 domain wall with spontaneous polarization along x3, as illustrated in Fig. 19a. The calculated domain wall by phase field simulation at wall orientation θ = 0 is shown in Fig. 19b. Nonzero sinusoidal P1 is observed, while P2 is exactly zero, which corresponds to a Néel-like feature. Considering the dependence of flexoelectric and electrostrictive coefficients on domain wall orientation, the effect of flexoelectricity on domain wall profile is anisotropic. The calculated maximum absolute values of P1 and P2 in different oriented domain wall are shown in Fig. 19c.

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Domain Wall Plane

Polarization (normalized)

b

a

1.0 P1/0.044μC-m-2 0.5 0.0 -0.5 P3/25.56μC-m-2 -1.0 -4

-3

-2

-1

0

1

2

3

4

Distance (nm)

10 8 6 150 4 2 0 180 2 4 6 210 8 10

120

P1

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240 270

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500 400 300 200 100 0 -100 -200 -300 -400 -500

E f1 E d1 E f2

Electric field (kV/cm)

90 Maximum polarization (10-2 μC/cm2)

c

-2

-1 0 1 Distance (nm)

2

Fig. 19 (a) 180 ferroelectric domain walls in tetragonal BTO. x1 is normal to the domain wall plane, while x2 and x3 are parallel to the domain wall plane. θ indicates the angle between domain wall and crystallographic direction [100]C. (b) Domain wall profile of phase field simulation at θ = 0. (c) Maximum absolute values of P1 and P2 vs. domain wall orientation θ. (d) Flexoelectric filed Efi and depolarization field Edi around the domain wall with θ = 5π/12. Reprinted with permission from [42]. Copyright (2014) American Physical Society

As expected, a strong angular dependence of P1 and P2 is observed. The P1 component is nonzero for all orientation θ, while P2 vanishes when θ = nπ/4, with n being an integer, indicating that the pure ferroelectric domain wall is Ising-Neel type when θ = nπ/4 and Ising-Bloch-Neel type for all other orientations. In addition, since the strain resulted from the electrostrictive effect is symmetric with respect to the center of the domain wall, the flexoelectric filed proportional to strain gradient is along opposite direction on opposite side of the wall center, as shown in Fig. 18d, thus leading to the chiral profiles of P1 and P2 in tetragonal BTO domain walls. It should be noted that the chirality of P1 induces a strong depolarization filed (see Fig. 19d), which would further influence the conductivity of the domain walls, as will be discuss later. The flexoelectricity-induced non-Ising characteristics of domain wall profiles are in agreement with previous first-principle investigations, where the Bloch-like and Ising-like characters were reported but not explained. Furthermore, it should be noted that the polarization component normal to the domain walls (P1 here) was also

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found to be nonzero in other kinds of domain walls [43]. In incipient ferroelectrics STO, for example, polar ferroelastic twin walls have been predicted theoretically and confirmed experimentally. The polarization of its domain walls was attributed to flexoelectricity and rotostriction (or the so-called flexo-roto field), analogous to the mechanism in ferroelectrics, i.e., flexoelectricity and electrostriction.

4.4.2 Conductivity of Domain Walls With different rotation angles with respect to the spontaneous polarization, ferroelectric domain walls can be nominally charged or not, as illustrated in Fig. 20a. Compared with the domain regions far from the domain walls, charged domain walls possess enhanced conductivity due to the accumulation of free charge carriers by screening the bound charges in the wall. Recently, the nominally uncharged ferroelectric domain walls and ferroelastic twin walls are also reported be conductive or with enhanced conductivity. It has been demonstrated that the conductivity of uncharged domain walls is closely related to flexoelectricity. Due to flexoelectricity, the polarization components and the domain wall profile are modified, as discussed in the previous section. The presence of polarization component perpendicular to the domain wall plane changes the bound charge distribution and results in a strong electric field at the wall. The accumulation of free screening charge carries is thus affected, and the modification on the domain wall conductivity arises. It should be noted that the domain wall conductivity was influenced by inhomogeneous a

b 300

a

Ps

Bound charge free charge

Holes density p/p 0

100

-Ps

10

1

0.1

0

100

200

300

Angle α (grad) Fig. 20 (a) Schematic of a charged domain wall (Reproduced from [44]. # IOP Publishing Ltd. All rights reserved.) (b) Relative hole density at different rotation angles in uncharged 180 domain walls in multiferroic BiFeO3. Black-dotted curves are calculated without deformation potential Ξpij ¼ 0 and flexoelectric coupling fij = 0. Solid curves are calculated for different coefficients: f11 =
1.38  10
11C
1m
3, f12 = 0.67  10
11C
1m
3, f44 = 0.85  10
11C
1m
3, and Ξpij ¼ 0 (red solid curves); 2fij and Ξpij ¼ 21 eV (magenta solid curves); 3fij and Ξpij ¼ 21 eV (blue solid curves) (Reprinted with permission from [45]. Copyright (2012) American Physical Society)

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deformations through two mechanisms, i.e., the so-called deformation potential and the flexoelectric coupling (flexoelectric field in ferroelectric domain walls and flexoroto field in twin walls). Figure 20b shows the influence of flexoelectric coupling on the hole density of the uncharged 180 domain wall in BiFeO3, with different flexoelectric coefficients and rotation angles. One can see that the variations of hole density was about one order of magnitude, depending on the strength of flexoelectric coupling. The anisotropy of the hole density was attributed to the angle-dependent electrostriction. Furthermore, the conductivity of charged domain walls has also been demonstrated to be modified by flexoelectricity.

5

Applications of Flexoelectric Effect

Being a commonly existing electromechanical coupling effect, flexoelectric effect provides us more possibilities to meet the urging need for new design and fabrication of nanodevices. The first potential application of flexoelectric effect resides in the socalled pseudo-piezoelectric composites. As discussed before, flexoelectricity is a universal mechanism in materials with any symmetry, so it can still induce polarizations in non-piezoelectric materials by inhomogeneous deformation. This effect largely broadens the candidate materials for piezoelectric devices, which can be leadfree, biocompatible, and even flexible. Fousek [8] first proposed the idea to design piezoelectric composites with non-piezoelectric components. This flexoelectric-type piezoelectric composites were realized by a simple design shown in Fig. 21a. The composites contain an active flexoelectric phase with truncated pyramid shape and a second dielectric phase (air here). When the composites are sandwiched by electrodes and compressed homogeneously, the induced strains in the pyramids are inhomogeneous and result in polarizations by flexoelectricity. Thus the composites exhibit piezoelectric responses.

Fig. 21 Schematics of pseudo-piezoelectric composites. (a) The design with truncated pyramid array (Reprinted from [46], Copyright (2013), with permission from Elsevier.). (b) The design employing transverse flexoelectricity (Reprinted with permission from [47]. Copyright (2009) AIP Publishing LLC.

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An imaginative design to employ the large transverse flexoelectricity of BST is illustrated in Fig. 21b. The homogeneous compression leads to the bending of the paraelectric BST layers and results in charge separations by flexoelectricity. Metallic stripes are placed at positions with maximum strain gradients to collect the induced charges. An effective piezoelectric coefficients is reported to be higher than 4300 pC/ N for a six-unit three-layer composite. It should be noted that the converse piezoelectric effect in this structure is weak due to the relative homogeneity of electric field between the parallel strips. This separate control of direct and converse piezoelectric effect is favorable for applications requiring sensing but not actuating and vice versa. Theoretical analysis on pseudo-piezoelectric composites has been conducted. Various applications of the pseudo-piezoelectric composites, such as sensors, actuators, transducers, energy harvesters, and nanogenerators, were also reported. The interaction between flexoelectricity and piezoelectricity can also lead to a novel nanoscale ferroelectric device, i.e., strain diode, as illustrated in Fig. 22. The cantilever is a buffered sandwich structure with a (110)-oriented PZT active layer as the core, STO as electrodes, and yttria-stabilized zirconia (YSZ) as the bottom buffer. When the top surface is exerted to a voltage, negative for example, the bending curvature of the cantilever will be large or small, depending on the ferroelectric polarization directions in the PZT layer. If the polarization is upward, the negative voltage of the top surface will increase the polarization, which leads to an out-of-plane (parallel to the polarization direction) expansion and an in-plane contraction due to electrostriction. Thus an upward bend of the cantilever occurs by the constraint of the buffer. Meanwhile, for downward polarization, the voltage will cause an out-of-plane contraction and in-plane expansion, which leads to a downward bend of the cantilever. On the contrary, the voltage-induced bending by flexoelectricity is independent with the polarization directions, which is always upward because of the negative flexoelectric coefficients for (110)-oriented perovskites. Therefore, for upward polarization, both piezoelectricity and flexoelectricity

Fig. 22 Schematic illustration of the strain diode, which is attributed to the enhancement and suppression of flexoelectricity on piezoelectric response. Reproduced from [48] with permission of The Royal Society of Chemistry

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Fig. 23 (a) Schematic of strain relaxation and the associated flexoelectric field in an epitaxial HoMnO3 film grown on Pt(111)/Al2O3(0001) substrate. (b) The I-V curve of such a film which shows a rectifying diode effect. Reprinted with permission from [49]. Copyright (2012) American Chemical Society

favor the upward bending and cause a large curvature. In contrast, for downward polarization, the direction of the bending caused by piezoelectricity and flexoelectricity is opposite, which leads to a smaller curvature. A two-state strain diode is thus obtained. Besides strain diode, flexoelectricity can also lead to an electrical diode, as reported by Lee et al. [49]. For epitaxial HoMnO3 film grown on Pt(111)/ Al2O3(0001) substrate, as shown in Fig. 23a, the misfit strain is as high as þ3.5%. The flexoelectric field due to strain relaxation is estimated to be Es  10 MVm
1, which is comparable to the electric field in conventional p-n junctions. The flexoelectric field-induced polarization causes the positive bound charge at top of the HoMnO3/Pt interface and negative bound charge at bottom of the interface. These bound charge would result in a high Schottky barrier at the interface. Due to the small barrier height at surface, electrons are more likely to be injected there and result in a rectifying diode effect, as the I-V curve shown in Fig. 23b. Another promising application of flexoelectricity is the nonvolatile ferroelectric memories with the ferroelectric domains written and erased through mechanical loads. The mechanism has been discussed in Sect 4.3.1. The typical mechanical writing and electrical reading process using an AFM tip is illustrated in Fig. 24. Without applying an external electric field, mechanical writing/erasing of ferroelectric domain can avoid the associated dielectric breakdown and current leakage problem, so it is promising for application in nonvolatile ferroelectric memory with ultrahigh data storage density and less power consumption. One of the challenges of such an application is that mechanical switching reported in ferroelectric film so far is unidirectional, as the strain gradient is fixed in direction.

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Mechanical Writing

V

Electrical Reading Fig. 24 Schematic illustration of the mechanical writing and electrical reading on ferroelectric thin film by AFM tip. Reprinted with permission from [50]. Copyright (2015) AIP Publishing LLC

6

Summary

In this chapter, we have presented a review on the recent progress of flexoelectricity at the nanoscale, with emphasis on a special class of materials, namely, ferroelectrics. Flexoelectricity is a universal electromechanical coupling effect between polarization and strain gradients and between strain and polarization gradients. Due to its size-dependent behavior, flexoelectricity plays a significant role at the nanoscale. It can influence the dielectric and electromechanical responses of materials, generate imprint behaviors in ferroelectric nanofilms, and affect the global and local characteristics of nanodomains and domain walls. It is promising to utilize flexoelectric effect in potential applications on micro-nano electromechanical systems, such as pseudo-piezoelectric composites, diodes, and nonvolatile ferroelectric memories. It is noteworthy that there still exist some confusable issues in this rapid growing field. Flexoelectric coefficients, which characterize the significance of flexoelectricity, exhibit large discrepancies in their magnitudes between theoretical calculations and experimental measurements. Meanwhile, the available database of flexoelectric coefficients is also limited. Consequently, there are difficulties in assessing the role of flexoelectricity in novel phenomena, in predicting undiscovered behaviors of flexoelectricity by theoretical modeling and simulations, and in precise design of nanodevices based on flexoelectricity. To solve this problem, both the experimental methods and theories of flexoelectricity need to be further developed. The techniques of employing flexoelectricity, such as strain gradient engineering, are also awaiting to be further explored. Though challenging, it is believed that the progress of flexoelectricity in the near future will bring us more possibilities.

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28. Sinnamon LJ, Bowman RM, Gregg JM. Thickness-induced stabilization of ferroelectricity in SrRuO3/Ba0. 5Sr0. 5TiO3/Au thin film capacitors. Appl. Phys. Lett. 2002;81(5):889–91. 29. Catalan G, Noheda B, McAneney J, Sinnamon LJ, Gregg JM. Strain gradients in epitaxial ferroelectrics. Phys. Rev. B. 2005;72(2):020102. 30. Gharbi M, Sun ZH, Sharma P, White K. The origins of electromechanical indentation size effect in ferroelectrics. Appl. Phys. Lett. 2009;95(14):142901. 31. Liang X, Yang W, Hu S, Shen S. Buckling and vibration of flexoelectric nanofilms subjected to mechanical loads. J. Phys. D. Appl. Phys. 2016;49(11):115307. 32. Chen WJ, Zheng Y, Luo X, Wang B, Woo CH. Ab initio study on the size effect of symmetric and asymmetric ferroelectric tunnel junctions: A comprehensive picture with regard to the details of electrode/ferroelectric interfaces. J. Appl. Phys. 2013;114(6):064105. 33. Tagantsev AK, Gerra G. Interface-induced phenomena in polarization response of ferroelectric thin films. J. Appl. Phys. 2006;100(5):051607. 34. Gruverman A, Rodriguez BJ, Kingon AI, Nemanich RJ, Tagantsev AK, Cross JS, et al. Mechanical stress effect on imprint behavior of integrated ferroelectric capacitors. Appl. Phys. Lett. 2003;83(4):728–30. 35. Lu H, Bark CW, De Los Ojos DE, Alcala J, Eom CB, Catalan G, et al. Mechanical writing of ferroelectric polarization. Science. 2012;336(6077):59–61. 36. Chen WJ, Zheng Y, Xiong WM, Feng X, Wang B, Wang Y. Effect of mechanical loads on stability of nanodomains in ferroelectric ultrathin films: Towards flexible erasing of the nonvolatile memories. Sci. Rep. 2014;4 37. Lee D, Yoon A, Jang SY, Yoon JG, Chung JS, Kim M, et al. Giant flexoelectric effect in ferroelectric epitaxial thin films. Phys. Rev. Lett. 2011;107(5):057602. 38. Catalan G, Lubk A, Vlooswijk AHG, Snoeck E, Magen C, Janssens A, et al. Flexoelectric rotation of polarization in ferroelectric thin films. Nat. Mater. 2011;10(12):963–7. 39. Ahluwalia R, Tagantsev AK, Yudin P, Setter N, Ng N, Srolovitz DJ. Influence of flexoelectric coupling on domain patterns in ferroelectrics. Phys. Rev. B. 2014;89(17):174105. 40. Chen WJ, Zheng Y, Wang B. Vortex domain structure in ferroelectric nanoplatelets and control of its transformation by mechanical load. Sci. Rep. 2012;2:796. 41. Chen W, Zheng Y, Feng X, Wang B. Utilizing mechanical loads and flexoelectricity to induce and control complicated evolution of domain patterns in ferroelectric nanofilms. J. Mech. Phys Solids. 2015;79:108–33. 42. Gu Y, Li M, Morozovska AN, Wang Y, Eliseev EA, Gopalan V, Chen LQ. Flexoelectricity and ferroelectric domain wall structures: Phase-field modeling and DFT calculations. Phys. Rev. B. 2014;89(17):174111. 43. Morozovska AN, Eliseev EA, Glinchuk MD, Chen LQ, Gopalan V. Interfacial polarization and pyroelectricity in antiferrodistortive structures induced by a flexoelectric effect and rotostriction. Phys. Rev. B. 2012;85(9):094107. 44. Yudin PV, Tagantsev AK. Fundamentals of flexoelectricity in solids. Nanotechnology. 2013;24 (43):432001. 45. Morozovska AN, Vasudevan RK, Maksymovych P, Kalinin SV, Eliseev EA. Anisotropic conductivity of uncharged domain walls in BiFeO3. Phys. Rev. B. 2012;86(8):085315. 46. Jiang X, Huang W, Zhang S. Flexoelectric nano-generator: Materials, structures and devices. Nano Energy. 2013;2(6):1079–92. 47. Chu B, Zhu W, Li N, Cross LE. Flexure mode flexoelectric piezoelectric composites. J. Appl. Phys. 2009;106(10):104109. 48. Bhaskar UK, Banerjee N, Abdollahi A, Solanas E, Rijnders G, Catalan G, Flexoelectric MEMS: towards an electromechanical strain diode. Nanoscale. 2016;8(3):1293–8. 49. Lee D, Yang SM, Yoon JG, Noh TW. Flexoelectric rectification of charge transport in straingraded dielectrics. Nano Lett. 2012;12(12):6436–40. 50. Chen X, Tang X, Chen XZ, Chen YL, Guo X, Ge HX, et al. Nonvolatile data storage using mechanical force-induced polarization switching in ferroelectric polymer. Appl. Phys. Lett. 2015;106(4):042903.

Mechanical Properties of Nanostructured Metals: Molecular Dynamics Studies

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Haofei Zhou and Shaoxing Qu

Contents 1 Strengthening by Nanoscale TBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Threading Dislocation in Nanotwinned Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Jogged Dislocation in Nanotwinned Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Twin-Size Induced Transition in Dislocation Mechanism and Continuous Strengthening in Nanotwinned Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Nanotwinned Metals Under Nanoindentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Toughening by Nanoscale TBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Crack Propagation in Nanotwinned Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Toughening Mechanisms Governed by Nanoscale TBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Delocalized Deformation in Nanotwinned Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Torsion of Nanotwinned Nanorods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Physical Origin of the Torsional Detwinning Domino in Nanotwinned Nanorods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Plastic Deformation in Metallic Glass Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Non-localized Deformation in Metallic Alloys with Amorphous Structure . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

592 593 595 597 598 599 600 601 602 602 604 607 610 612 612

Abstract

This chapter overviews our recent work on MD simulations of the mechanical properties of nanostructured metals with an emphasis on revealing the controlling deformation mechanisms, interpreting the experimental data, and guiding further research in structural optimization and processing.

H. Zhou School of Engineering, Brown University, Providence, RI, USA e-mail: [email protected] S. Qu (*) Department of Engineering Mechanics, Zhejiang University, Hangzhou, China e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_19

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To achieve stronger and tougher metals by architecting their microstructures into the nanometer scale has been an enduring pursuit in materials science and engineering. During the past few decades, nanostructured metals have emerged as a new class of materials with tunable distribution of microstructures. Due to their unique microstructures, nanostructured metals could be prepared to exhibit an unusual combination of mechanical properties, such as ultrahigh strength, good tensile ductility, enhanced strain hardening, superior fracture toughness, and fatigue resistance [1–4]. However, the current lack of understanding of the underlying mechanisms that control the mechanical properties of these materials severely limits our ability to tailor or optimize their properties for specific applications. Recent advances in computational modeling provide unprecedented opportunities to advance the frontier of knowledge in our understanding of mechanical behaviors of materials at nanoscales. With the development of teraflop computing power and sophisticated empirical atomic interaction potentials, large-scale molecular dynamics (MD) simulations have acted as an increasingly important role in complementing theories and experiments in tackling complex problems in computational materials science [5–8]. In this chapter, we will overview our recent work on MD simulations of the mechanical properties of nanostructured metals with an emphasis on revealing the controlling deformation mechanisms, interpreting the experimental data, and guiding further research in structural optimization and processing. This chapter is organized as follows: Sections 1, 2, and 3 will cover a range of mechanical behaviors of nanotwinned metals along with several unique deformation mechanisms governed by nanoscale twin boundaries (TBs). Sections 4 and 5 will focus on the improvement of mechanical behaviors of metallic glasses (MGs) by introducing structural inhomogeneity into the glass matrix.

1

Strengthening by Nanoscale TBs

Nanoscale TBs are planar defects where atom arrangements on one side are mirror reflections of those on the other side. Ultrahigh strength and hardness have been reported in nanotwinned metals, ceramics, diamond, and biomaterials [9–19]. In response to the external loading, nanoscale TBs are known to hinder the movement of lattice dislocations that accommodate plastic deformation, thereby strengthen the polycrystals similar to traditional grain boundaries (GBs) in polycrystalline metals. Recently, it has been reported that there exists a transition in deformation mechanism from the conventional Hall-Petch strengthening mechanism to a dislocation-nucleation-governed softening mechanism that involves the nucleation and motion of twinning partial dislocations along the twin planes below a critical twin thickness [20, 21]. This transition in deformation mechanism has been demonstrated by experiments and simulations in nanotwinned polycrystalline metals with equiaxed grains where TBs and GBs are randomly oriented, as well as nanotwinned nanopillars and nanowires with only TBs but no other general GBs [22].

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More recently, nanotwinned metals with columnar grains have been fabricated by direct current electrodeposition [23–25]. Such samples have a strong {111} out-ofplane texture and preferentially oriented nanoscale twins with TBs nearly perpendicular to the axial directions of the columnar grains. Columnar-grained nanotwinned Cu exhibits a remarkable plastic anisotropy and excellent fatigue properties, as their deformation behaviors are strongly dependent on the TB spacing and orientation. These phenomena provide a promising foundation to optimize the mechanical properties of nanotwinned materials by tailoring their twin microstructures. In the columnar-grained nanotwinned metals, when the loading direction is oriented in parallel to the twin planes, both dislocation pile-ups due to TB impediment and easy glide of partial dislocations along the twin planes are to a large extent suppressed, leading to the activation and glide of threading dislocations confined between neighboring TBs. A similar mechanism has been previously reported in nanolayered metallic composites where the plastic deformation is strongly associated with the glide of threading dislocation loops confined in the nanolayers [26, 27]. In these lamellar materials, the strength is enhanced as the layer thickness is reduced, which can be described by the so-called confined layer slip (CLS) model of threading dislocations [26]. This chapter will discuss the mechanical behavior of nanotwinned metals under uniaxial tension [28] and nanoindentation [29] loading conditions. It will be shown that the strengthening of nanotwinned metals is closely related to the deformation mechanisms governed by nanoscale TBs.

1.1

Threading Dislocation in Nanotwinned Metals

MD simulations were performed on fully three-dimensional (3D) Cu polycrystals. Each sample contains four columnar grains, where the coherent TBs are perpendicular to the axial direction of the grains (i.e., the thickness direction of the sample), exhibiting a strong {111} texture similar to the experiments [23–25]. The in-plane misorientations between neighboring grains are larger than 30
to ensure high-angle GBs. Uniaxial tension with a constant strain rate of 2  108 s1 was applied parallel to the twin planes. Figure 1a, b shows dislocation structures in two representative grains in the sample with grain size d = 50 nm and λ = 5.01 nm at 6.18% strain. Most of the dislocation lines in Fig. 1a, b exhibit a hairpin-like shape. This kind of dislocation loops is generally referred to as threading or channeling dislocations. Figure 1c shows the atomic configuration of dislocation loops confined in twin lamellae. A typical threading dislocation has two misfit segments on the neighboring twin planes and one threading segment spanning across the twin lamella. During the simulation, threading dislocation loops are found to initiate from GBs, glide through grain interior, and become finally absorbed at the opposite GBs. Such deformation mechanism prevails in the samples with TB spacing larger than 3.75 nm. In thin films [30, 31] or nanolayered metallic composites [23–25], the nucleation and motion of threading dislocation are known to be controlling mechanisms responsible for plastic deformation, resulting in enhanced strength with decreasing

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Fig. 1 (a, b) Dislocation structures in two representative grains of a simulated nanotwinned sample with d = 50 nm and λ = 5.01 nm at 6.18% train. (c) Atomic structures of threading dislocations in one twin lamella via position-based coloring, where the colors represent the distance of atoms from the center of a simulated grain. It is observed that two misfit segments lie on the twin planes and the threading segment is confined in the twin lamellae [28]

lamellar thickness. In the CLS model suitable for describing such size strengthening effect in thin films, the yield strength is expressed as [30, 31]      μb sin φ 4  v αλ ¼M ln þ τ0 8πλ 1v b sin φ 

σ cls

(1)

where λ is the film thickness, μ the shear modulus, v the Poisson’s ratio, τ0 the resistance to dislocation glide from lattice and other obstacles, M the Taylor factor, φ the angle between slip plane and the layer interface, b the Burgers vector of dislocation, and α a coefficient representing the extent of dislocation core. For a nanolayered metallic composite, the CLS model is modified as [23–25] σ 0cls ¼ M

    μb sin φ 4  v αλ γ μb ln  þ 8πλ 1v b sin φ λ lð1  vÞ

(2)

after the influence of interfaces is accounted for. The first term on the right-hand side of Eq. 2, similar to that of Eq. 1, represents the stress required to drag the misfit segments of a threading dislocation. The second term represents the prestress associated with the interfaces, with γ being the interfacial energy (J/m2). The third

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term reflects the interaction between a glide threading dislocation and a pre-existing array of misfit dislocations on the interfaces, with l denoting the mean dislocation spacing. For a twin lamellae with thickness on the order of several nanometers, the interfacial energy of TB being γ T = 24.0 mJ/m2 for Cu [32], the contribution from the second term in Eq. 2 to strength is negligible. Therefore, regarding the last two terms in Eq. 2 as a constant σ d independent of TB spacing, one may rewrite Eq. 2 as [28] σ TB cls ¼ M

    μb sin φ 4  v αλ ln þ σd 8πλ 1v b sin φ

(3)

In subsequent analysis, we will use Eq. 3 to account for the strengthening due to threading dislocations confined in twin lamellae.

1.2

Jogged Dislocation in Nanotwinned Metals

Figure 2 shows deformation patterns in two representative grains in the sample with d = 50 nm and λ = 0.83 nm at 10.30% strain, which is very distinct from those shown in Fig. 1. It is observed that the plastic deformation is governed by activities of necklace-like dislocations extending over multiple twin planes, which we refer to as jogged dislocations. Each of these jogged dislocations is nucleated as a whole from a GB, rather than through assembly or interactions of separately emitted dislocations. The nucleation process should be assisted by diffusion and coalescence of vacancies.

Fig. 2 (a, b) Dislocation structures in two representative grains of a simulated nanotwinned sample with d = 50 nm and λ = 0.83 nm at 10.30% strain. Multiple extended dislocations connect with each other, constituting a jogged dislocation. Atoms are painted via the position-based coloring method [28]

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Fig. 3 (a–c) Detailed atomic structures of a jogged dislocation (circled in Fig. 2a) from different views. The jogged dislocation exhibits a zigzag shape from a certain perspective. (d–e) Magnification of the segments within the dashed line frames in (a) and (b). The extended and constricted jogs of one segment are indicated by the red arrows. (f) Double Thompson tetrahedron representation in the twin-matrix system. The upper tetrahedron represents the slip systems for the matrix, while the lower one for the twin. Colors are assigned to atoms based on their local crystal orders measured by the common neighbor analysis (CNA) method: gray stands for perfect atoms, red for atoms in stacking faults or TBs, green for atoms in dislocations or GBs, blue for atoms near vacancies, and yellow for fully disordered atoms [28]

Figure 3a, b shows the detailed atomic structures of a single-jogged dislocation spanning across multiple twin planes, while Fig. 3c, d shows the magnification of three consecutive segments of the jogged dislocation. The dislocation pattern consists of multiple extended dislocations, each bounded by two twin planes. Every extended dislocation contains a pair of leading and trailing partial dislocations separated by a stacking fault. Because of the geometrical constraints, each extended dislocation is terminated at a twin plane and joined with another dislocation in the neighboring twin through a common segment in the twin plane. Based on the double Thompson tetrahedron representation in Fig. 3e, the leading and trailing partials in matrix have Burgers vectors of Aβ and βC, while those in the twin plane have Burgers vectors of Aβ0 and β0 C. Thus, the common segment in the twin plane has

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the Burgers vector of a stair-rod dislocation β0 β. Such stair-rod segment possesses the characteristic of a unit jog and is pinned at the twin plane. Depending on its Burgers vector and the corresponding line tension, the unit jog may be either of extended or constricted type, as illustrated in Fig. 3d. Plastic deformation governed by jogged dislocations prevails for TB spacings below 3.75 nm. Interestingly, the motion of jogged dislocation involves the collective movements of multiple extended dislocations which are linked by numerous unit jogs pinned at twin planes. As the extended part of a jogged dislocation is driven forward by the applied stress, the unit jogs undergo de-pinning at the twin planes, which is assisted by diffusion of point defects.

1.3

Twin-Size Induced Transition in Dislocation Mechanism and Continuous Strengthening in Nanotwinned Metals

Comparison between Figs. 1 and 2 demonstrates that there exists a distinct transition from threading dislocation to jogged dislocation governed deformation mechanisms as the twin thickness is reduced. Figure 4a shows typical stress-strain curves of the samples with d = 20 and 30 nm. If we take the average of stresses after yielding as the flow stress, it is observed from Fig. 4b that the flow stress gradually increases with deceasing TB spacing. This means that the presence of TBs always strengthens the columnar-grained nanotwinned metals if the external loading is parallel to the TBs, in contrast to the softening phenomenon previously revealed in the equiaxed nanotwinned metals in the similar range of twin thickness [20, 21]. We find that the CLS model in Eq. 3 fails to correlate the size dependence of materials strength in Fig. 4b. In fact, the prediction from Eq. 3 is found to decrease when TB spacing λ falls below a certain value, as shown in Fig. 4b. As the layer thickness becomes comparable to the cutoff radius of dislocation core, the first term in Eqs. 1 and 3 becomes negative due to the logarithmic component, causing a stress drop which is

Fig. 4 (a) Stress-strain curves of simulated samples with d = 20 and 30 nm and a range of TB spacings from 0.63 nm to 12.50 nm. (b) The twin-size dependence of material strength. There exists a transition from threading dislocation to jogged dislocation governed deformation regimes with reduced TB spacing [28]

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Table 1 Parameters used in Eqs. (3) and (5) for Cu [28] μ (GPa) 48 M 3

ν 0.35 σ d (GPa) 0.60

b (nm) 0.147 τ0 (GPa) 0.59

φ 70.5
bf (nm) 0.255

α 0.13, 0.16 fpin (pN) 31.48

contradictory to our simulation results. Relevant parameters used in Eq. 3 are summarized in Table 1. The deformation patterns shown in Fig. 2 imply that the strengthening of a columnar-grained nanotwinned metal, as the TB spacing is reduced to around 1 nm (comparable to the cutoff radius of dislocation core), must be associated with the collective motion of jogged dislocations. Consequently, the CLS model is no longer valid at sufficiently small TB spacings (see Fig. 4b) due to a transition in deformation mechanism. To explain the above TB strengthening behavior, especially in the small twin-thickness range, we propose a simple model based on the motion of jogged dislocation and de-pinning of unit jogs on twin planes. For every constituent loop in the jogged dislocation, the pinning force fpin exerted by TBs on the unit jog during its motion is compensated by the associated Peach-Koehler force acting on the dislocation. Accordingly, the equilibrium equation of a dislocation should be given by [28] f pin ¼ ðτ  τ0 Þbf λ

(4)

where τ is the resolved shear stress, τ0 the friction stress, and bf the magnitude of Burgers vector of dislocation. Here the subscript “f” in b is used to emphasize the full-dislocation nature of the jogged dislocation. Solving Eq. 4 and considering the Taylor factor M, we obtain the following steady-state flow stress σ jog due to the de-pinning process [28] σ jog ¼ M

  f pin þ τ0 bf λ

(5)

As shown in Fig. 4b, as the TB spacing falls below 3.75 nm, the strength enhancement can be well described by Eq. 5 with fpin = 31.48 pN and τ0 = 0.59 GPa for the case of Cu. Based on our simulations and the theoretical predictions in Fig. 4b, the transition from threading to jogged dislocations occurs around 2–5 nm. Within this regime, the threading and jogged dislocations could coexist during plastic deformation. As the TB spacing is reduced across this size range, the dominant mechanism of plastic deformation gradually transits from a threading dislocation-dominated regime to one that is fully dominated by jogged dislocations throughout the grain interiors.

1.4

Nanotwinned Metals Under Nanoindentation

Strengthening effect governed by nanoscale TBs is also demonstrated in nanoindentation simulations on quasi-3D polycrystalline Cu samples with an idealized

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Fig. 5 (a) Simulated load-displacement curves of nanotwinned and twin-free samples under nanoindentation. (b–c) Deformation patterns of the samples at the moment indicated by the arrow in (a). Atoms are painted by the CNA method [29]

{110} texture. Each sample consists of 6 randomly oriented grains with a mean grain size of 20 nm, and the sample size is 60  40  2 nm3. During nanoindentation, a cylindrical indenter was pushed toward the top sample surface at a speed of 0.01 nm/ ps until reaching a maximum indentation depth of 4 nm. Each Cu atom in the sample interacts with the rigid indenter via a repulsive potential V(r) = Aθ(R-r)(R-r)3, where A = 2.67 nN/Å2 is a constant, θ(R-r) is the standard step function, R = 4 nm is the indenter radius, and r is the distance between the atom to the center of the cylindrical indenter. The grain underneath the indenter is oriented such that its (111) planes are tangential to the indenter surface before contact. The vertical motion of ten layers of atoms at the bottom of the sample is fixed. Figure 5a shows the load-displacement curves of two simulated samples. Sample “Cu-20 nt-5” has a TB spacing of 5 nm, while sample “Cu-20 nc” is twin-free. It is seen that the nanotwinned sample has a larger resistance to nanoindentation than the twin-free sample. Figures 5b, c show deformation patterns captured at the moment indicated by the arrow in Fig. 5a. It is seen that the dislocations in the nanotwinned grain in direct contact with the indenter are severely blocked by the nanoscale TBs (Fig. 5b). In the twin-free case, however, the dislocations can easily traverse the grain and interact with the GB on the other side of the grain (Fig. 5c).

2

Toughening by Nanoscale TBs

Experiments have reported that a high and controlled density of nanoscale TBs could serve to enhance the ductility and fracture toughness of nanocrystalline materials [33–35]. In this section, attention is focused on exploring the atomistic toughening mechanisms governed by nanoscale TBs in Ni nanocrystals by MD simulations [36].

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Crack Propagation in Nanotwinned Metals

The fracture process of nanocrystalline Ni samples with and without nanoscale TBs is examined by MD simulations. Figure 6a shows a quasi-3D nanocrystalline Ni sample containing nanoscale TBs. The mean grain size is 20 nm, and the sample size is 80  120  1.99 nm3. An initial edge crack with a length of 13 nm was introduced into the sample. At the beginning of each loading step, a displacement field acquired from the mode-I crack tip solution was applied to the boundary atoms within 1 nm from the edges of the sample. The increment in the stress intensity factor is 0.01 MPam1/2. Afterward, the boundary atoms were fixed, while the other atoms were allowed to relax for 1 ps. It is observed that a high density of TBs with inclined TB orientations can greatly enhance the fracture toughness of polycrystalline Ni (Fig. 6b, c). The detailed atomistic toughening mechanisms governed by nanoscale TBs will be discussed in Sect. 2.2.

a y z

x

Stress intensity factor (MPa √m)

b

7 6 5 4 3 2 GS–20 nt–2 GS–20 nt–5 GS–20 nc–20

1 0

0

100

200

300

400

500

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Crack tip propagation (Å) Stress intensity factor (MPa √m)

c

7 6.5 6 5.5 5 4.5 nt nc–20

4 0

45

90

135

180

TB orientation (°)

Fig. 6 (a) Quasi-3D nanotwinned Ni polycrystals with a TB spacing of 2 nm. The blue arrows indicate the direction of the externally applied stress. (b, c) Stress intensity factor versus crack tip propagation distance for samples with different TB densities and orientations [36]

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Toughening Mechanisms Governed by Nanoscale TBs

Dislocation motion along TBs is found to be an effective way to release the stress concentration at the crack tip. Figure 7a shows that plastic deformation is largely suppressed in nanocrystalline Ni samples without nanoscale TBs. In nanotwinned Ni samples, however, numerous partial dislocations are emitted from the TB-GB intersections and glide along the nanoscale TBs, leading to crack blunting as shown in Fig. 7b. The failure mechanism in nanocrystalline Ni samples switches from intergranular fracture to intragranular fracture as the TB density increases (Fig. 7). In twin-free nanocrystalline Ni sample, the crack propagates along the GBs (Fig. 7a), while in nanotwinned samples with high density of TBs, the crack tends to propagate along or through the nanoscale TBs within the grains (Fig. 7b). The GB energy of Ni predicted by the EAM potential is 1572 mJ/m2, much higher than its TB energy of 63 mJ/m2. The substantial difference between these boundary energies suggests that the crack extends along an intergranular path. However, a high density of TBs near the crack tip is able to promote local nucleation and motion of dislocations, hindering the intergranular fracture mechanism in nanotwinned Ni samples. The dislocation motion along the TBs, on the other hand, may disturb the coherency of the TBs and trigger intergranular fracture during crack blunting. Another toughening mechanism revealed in nanotwinned Ni samples is the nucleation and growth of an incipient daughter crack in the grain ahead of the mother crack (Fig. 7b). The formation of the daughter crack is associated with a sequence of partial dislocations gliding along a TB, squeezing into a TB-GB intersection, and creating a local wedge-shaped crack embryo. As the daughter crack grows, the neighboring TBs become curved, and numerous partial dislocations

Fig. 7 (a) Twin-free nanocrystalline Ni showing limited plastic deformation. (b) Nanotwinned Ni exhibiting crack blunting and intragranular fracture with numerous dislocations gliding along TBs. TBs become curved (indicated by the arrow) due to the growth of a daughter crack ahead of the main crack [36]

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are deposited along these TBs (indicated by the arrow in Fig. 7b), forming geometrically necessarily dislocations. This provides an effective toughening mechanism that locally accommodates a large amount of plastic deformation.

3

Delocalized Deformation in Nanotwinned Metals

Localized deformation is one of the most frequent causes of material failure in metallic components and structures. How to maintain sustained deformation in nanostructures without localized failure is an important question for many applications of nanotechnology. MD simulations demonstrated that nanoscale TBs in onedimensional nanostructures can lead to giant rotational deformation without localized failure through a novel torsional detwinning domino mechanism [37].

3.1

Torsion of Nanotwinned Nanorods

Figure 8a shows the typical surface morphology of a twisted nanotwinned nanorod with diameter D = 20 nm. TBs are embedded in the nanorod with uniform spacing of λ = 1.05 nm and aligned nearly perpendicular to the axis of the cylindrical nanorod with a tilt angle less than 5
. A constant surface shear strain rate of 3.33  107 s1 was adopted to twist the nanorod to a total surface shear strain of 420%. It is seen that the surface becomes rough with numerous slip steps. The average depth of the slip

Fig. 8 Deformation patterns of nanotwinned (a, b) and single-crystalline (c, d) nanorods after being loaded by torsion to 420% of surface shear strain [37]

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steps are on the order of 1 nm. The roughness is associated with the depletion of TBs (i.e., detwinning) during torsion, as indicated by Fig. 8b. It is clear that all twin planes underneath the rough surface have been dissolved, leaving an “infected” region filled with various types of lattice defects. Similar plastic deformation patterns were observed in all simulated nanotwinned nanorods. For comparison, the deformation patterns of a twisted single-crystalline twin-free counterpart were also displayed in Fig. 8c. The surface of the twin-free nanorod after torsion is featured by a few slip ledges, which have much larger scales (about 5 nm in depth) and are distributed much more narrowly than those observed in Fig. 8a. This implies that the plastic deformation is localized to several atomic planes in the twin-free samples. The “infected” region is filled with a few stacking fault ribbons and dislocation lines, as shown in Fig. 8d. Figure 9a shows the surface shear stress versus strain relations of nanotwinned and single-crystalline nanorods during torsion. The shear stress and shear strain at the surface are given by τ = 2 T/πR3 and e = κR, respectively, where T is the torque, κ is the twist per unit length, and R is the radius of the nanorod. It is observed from Fig. 9a that both samples responded cyclically to the external load with a period of about 17.5% strain, corresponding to a twist angle of 60
. In each cycle of the stress-strain curve of the twin-free samples, a number of dislocations nucleated from the surface and then reacted with each other, leading to dislocation multiplication and formation of complex dislocation networks on the (111) plane. In contrast to the twin-free samples, it is seen that each cycle on the stress-strain curve of the nanotwinned samples corresponds to the depletion of one TB, as evidenced by Fig. 8b. Figure 9b–g shows the shear strain contours on the axial sections of the twisted nanorods, where atoms are colored according to their local von Mises strain values. The twistinduced strain gradient along the radial direction is clearly identified. For the nanotwinned nanorod, detwinning started at a TB near the middle of the nanorod, which was transformed into a twist boundary to accommodate a certain amount of plastic strain. Further deformation caused the twisted boundary to disappear, and the original TB was dissolved. The same detwinning process then spread on after the other into the neighboring TBs under increasing torsion, as shown in Fig. 9b–d. Plastic deformation was accumulated and delocalized through the consecutive depletion of TBs in the “infected” region outlined by a dashed box in Fig. 9b. This is contrast to the highly localized plastic deformation in the single-crystalline nanorod, as shown in Fig. 9e–g. More interestingly, only one TB is depleted in the nanotwinned nanorod for every 60
of torsion, which may be referred to as a single detwinning event. Every detwinning event sets off a subsequent detwinning event on a neighboring TB, leading to a detwinning domino propagating along the nanorod. Each peak in the stress-strain curves shown in Fig. 9a corresponds to dislocation nucleation on a TB. A significant difference between the two curves in Fig. 9a is that the peak stress remains at about 2.0 GPa throughout the torsion for the nanotwinned nanorod, while it decreases gradually from as high as 3.2 GPa in the first cycle to less than 1.3 GPa at the end for the single-crystalline nanorod. The reduction in the peak stresses of the twin-free sample implies that dislocations there can nucleate and multiply with ease due to progressive strain localization and accumulation of defects.

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Fig. 9 (a) Stress-strain relations of nanotwinned (top) and single-crystalline (bottom) nanorods under torsion. (bd) Atomic shear strain contours of nanotwinned nanorods during twisting. (eg) Atomic shear strain contours of single-crystalline nanorods during twisting [37]

However, the peak stresses of the nanotwinned sample remain at a nearly constant stress level associated with the onset of individual detwinning event.

3.2

Physical Origin of the Torsional Detwinning Domino in Nanotwinned Nanorods

Figure 10 illustrates the physical nature of the twist-induced detwinning process in nanotwinned nanorod under torsion. First imagine a typical nanotwinned structure with a stacking sequence of CABCBAC. Figure 10a shows a diagram of three

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Fig. 10 Physical mechanism of twist-induced detwinning. (a) Three adjacent (111) atomic layers arranged in a twinned structure. (b) The same stack of atomic layers arranged in a perfect fcc structure after a twist of 60
in imposed on the layers above the twin plane C [37]

consecutive (111) layers. If all atomic layers above the twin plane C are rigidly twisted by an angle of 60
, the resulting configuration is shown in Fig. 10b. It is seen that the mirror symmetry across the twin plane C has been removed. The final stacking sequence is CABCABC, recovering the perfect fcc structure. Thus, the detwinning domino essentially originates from the rotational symmetry of {111} planes in fcc metals. Figure 11 shows a series of snapshots that capture the structural evolutions of a TB at various strain levels. Initially, a partial dislocation nucleates from the free surface and glides along the TB, as shown in Fig. 11a. This dislocation is then trapped at the center due to the stress gradient in the radial direction. Meanwhile, another partial dislocation nucleates from the free surface, as shown in Fig. 11b. More partial dislocations emit from the free surface and react with each other, leading to the formation of dislocation junctions indicated in Fig. 11c. Each arm of the junction is an extended dislocation consisting of leading and trailing partial dislocations separated by a stacking fault. As more dislocations interact with the preexisting dislocation structures, a complex dislocation network becomes visible in Fig. 11d. Crystallographically, such dislocation network corresponds to a high-angle twist boundary. The partial dislocations in the network are geometrically necessary to accommodate the lattice mismatch between the original TB and the adjacent (111) plane. A twist from 0
to 30
gradually dismantles the coherency of the TB, which results in a fully disordered configuration shown in Fig. 11e. Interestingly, when the twist angle increases from 30
to 60
, the dislocation network gradually coarsens as dislocations progressively escape from the free surface, as shown in Figs. 11f–h. Such phenomenon is accompanied by a significant drop in the applied shear stress after yielding (Fig. 9a). The final configuration after twisted by 60
is just a perfect fcc structure. It is noted that each detwinning event is mediated by the formation and annihilation of a twist boundary and intense dislocation interactions occur during the detwinning process, leading to the formation of various types of defects, such as vacancies, stacking fault tetrahedrons (SFTs), and even extrinsic stacking faults (ESFs), as shown in Fig. 11i. To understand why detwinning occurs sequentially, rather than randomly at any twin planes, the distribution of shear stress σ θz in the twisted nanotwinned nanorod is plotted in Fig. 12. Significant stress concentration is found at the intersections between TBs and the free surface. More importantly, the maximum σ θz = 3.56 GPa

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Fig. 11 Structural evolution of a twin plane in a twist-induced detwinning event. (a, b) Nucleation of partial dislocations from the free surface. (c, d) Formation of dislocation junctions due to dislocation interactions. (e) Atomic configuration of TB after twisting by ~30
. (f, g) Coarsening of dislocation network as dislocations escape from the surface. (h) Depletion of TB after twisting by 60
. (i) Abundant residual lattice defects in the “infected” region after a series of detwinning events [37]

occurs at the outermost TB of the nanotwinned region, about 0.5 GPa higher than that on the rest of the TBs. This induces detwinning always at the edge of the nanotwinned region, leading to the observed sequential detwinning. The torsional detwinning domino in nanotwinned nanorods departs from all previously reported detwinning mechanisms [20, 38, 39] in that it does not require TB migration. Instead, detwinning is realized through sequential transformation (into a twist boundary) and annihilation of TBs with a rotational period of 60
, which contributes to the delay of shear localization in the nanostructure. The sequential depletion of TBs leads to a sustained rotational plastic deformation. Take a nanorod containing a total of 60 TBs and with diameter/length = 1/3, for instance; the complete depletion of these coherent interfaces via the domino detwinning mechanism gives rise to ~1000% of plastic strain. Thus, the operation of a detwinning domino could lead to practically unlimited rotational deformation in 1D nanostructures.

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Fig. 12 Distribution of virial shear stress σ θz in a twisted nanotwinned nanorod right before the next detwinning event occurs at the twin plane indicated by the black arrows. (a) Color map of σ θz revealing stress concentration at the intersections between TBs and the free surface. The maximum σ θz occurs at the outermost TB of the nanotwinned region indicated by the black arrow. The topright schematic depicts the stress component σ θz that is calculated to draw the stress contour. (b) Sliced nanotwinned nanorod exposing the intersections between the TBs and the free surface [37]

4

Plastic Deformation in Metallic Glass Matrix Composites

Metallic glasses (MGs) have attracted tremendous attention in the past few decades due to their superior mechanical properties such as high elastic limit, remarkable hardness, high fracture toughness, and good corrosion resistance [40–43]. MGs in their bulk form suffer from a strong tendency to plastic strain localization in a narrow region called shear band and exhibit macroscopically catastrophic failure under tension at room temperature. A great deal of effort has been made in recent years to improve the plasticity of MGs. Some monolithic MGs exhibiting considerable compressive plasticity were developed, possibly due to a high Poisson’s ratio with the relative ease of shearing over dilatation [43], in situ nanocrystallization during deformation [44], and more free volume by faster cooling [45] or minor alloying

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b

W

h

[010]

a

Y X [100]

Fig. 13 Schematic of a MG-matrix composite model containing second-phase nanocrystals with dimensions of 78.5  84.2  5.6 nm3. The orientation of the nanocrystals is indicated as [100] along the X-axis and [010] along the Y-axis. The nanocrystal pattern can be adjusted by varying the geometry parameters a, b, w, and h. Uniaxial tension is applied along the Y-axis. Free surface condition is applied along the X-axis [50]

[46]. However, MGs still show limited plastic strain under tension. In order to overcome this issue, MG/crystal composites have recently been developed by virtue of the interaction between the glass matrix and crystalline particles. Remarkable tensile ductility of these composites has been reported [47–49]. This section will be focused on the microstructural evolution and deformation mechanisms of a crystalreinforced MG composite model with discontinuous Cu particles inserted into a Cu64Zr36 glass matrix (Fig. 13). By controlling the size, spacing, and orientation of the nanocrystals, various patterns of MG-matrix composites can be generated for simulations [50]. The stress-strain relations of the MG-matrix composites with three different shapes of b/a = 6.5, 5.7, and 3.7 are shown in Fig. 14a, while the normalized number of atoms in shear transformation zones (STZs) in the glass matrix is plotted against the applied strain in Fig. 14b. At the initial elastic deformation range, the tensile stresses of the composites increase linearly with the applied strain. At about 7% strain, local STZs start to nucleate in the glass matrix, as shown in Fig. 14c. The global yielding of the sample is a result of STZ localization and the formation of embryo shear bands at the free surface and glass-crystal interfaces, as shown in Fig. 14d. The thickness of these embryo shear bands is ~10 nm. It is seen that the normalized number of STZ atoms increases radically at this time (see Fig. 14b). In the stress-strain curves, this is registered as an abrupt stress drop (see Fig. 14a). Interestingly, the stress drops of the MG-matrix composites are noticeably smaller than that of the monolithic MG sample (~2 GPa). The underlying deformation mechanisms are illustrated in Fig. 14e, f. It is seen that, as the embryo shear bands evolve in the glass matrix, the nanocrystals act as barriers to shear band propagation. Some shear bands cease their propagation due to the suppression of nanocrystals, while others deviate from

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Mechanical Properties of Nanostructured Metals: Molecular Dynamics Studies

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5

100

4

90

Stress (GPa)

3.5 3 2.5 2 1.5 1

Composite I Composite II Composite III Monolithic MG

0.5 0 0

c

5

10 15 Strain (%)

d

20

609

110

4.5 Normalized Number of STZ Atoms

18

Composite I Composite II Composite III Monolithic MG

80 70 60 50 40 30 20 10 0 0

25

e

deflection

5

10 15 Strain (%)

f

20

25

penetration

Fig. 14 (a) Typical stress-strain curves of MG-matrix composites with various shapes of nanocrystals. (b) The relations between the normalized number of STZ atoms in the glass matrix and the applied strain. (c–f) Atomistic configurations of a MG-matrix composite showing suppression and deflection of shear bands by nanocrystals [50]

their original shear path and deflect along the glass-crystal interfaces. In this way, shear localization might be effectively suppressed in the glass matrix, improving the global plasticity of the composites. As the volume fraction of the nanocrystals increases to ~50%, the glass matrix becomes limited for the formation of embryo shear bands. Figure 15a shows that, at this relatively large volume fraction, STZs are uniformly distributed in the glass matrix at 13% strain and no localized deformation in the form of shear bands is observed, in distinct contrast to the case with a lower volume fraction of 14.5% shown in Fig. 14d. The blue arrow in Fig. 15a indicates that the operation of STZs near the glass-crystal interfaces can trigger the emission of dislocations from the interface. Further plastic deformation of this composite is thus governed by the cooperative operations of dislocations in the nanocrystals and STZs in the glass matrix, as shown in Fig. 15b.

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Fig. 15 (a, b) Atomic configurations of a composite with a large volume fraction of nanocrystals at two different applied strains. The arrow indicates the activation of lattice dislocations at the glass-crystal interfaces [50]

5

Non-localized Deformation in Metallic Alloys with Amorphous Structure

It has been argued that atomic-, nano-, and microscale spatial fluctuations in material density and strength may play important roles in the mechanical behavior of MGs [51–53]. Inspired by the concept of heterogeneous microstructures, pre-plastic deformation by compression or cold-rolling was performed to improve the plasticity of MGs during subsequent bending and compression tests, which is attributed to the introduction of pre-existing shear bands during pre-deformation [54, 55]. The coldrolled MGs even exhibited an obvious plastic deformation during tension [56, 57]. The underlying mechanism of how the pre-existing shear bands affect the deformation behavior of MGs, however, is still unclear. This section is focused on understanding the deformation mechanisms of MGs with tailor-designed pre-existing shear bands. The simulation details can be found in Ref. [58] Stress-strain curves for MGs with a composition of Cu64Zr36 and various atomic percents of pre-existing shear bands that are aligned parallel to the loading axis are shown in Fig. 16a. It is seen that the macroscopic yield stress decreases as the atomic percent of pre-existing shear bands increases from 0 at.% to 7.38 at.%. It is interesting that the flow stress after the stress overshoot first increases with increasing amount of pre-existing shear bands in the range of 0 at.% to 3.27 at.% and then reaches a plateau as the atomic percent further increases from 3.43 at.% to 7.38 at.%. The contrast is also observed in Fig. 16b, where the normalized number of atoms in STZs is plotted against the applied strain. It is clear that after sample yielding, the number of STZ atoms increases rapidly for samples with less than 3.43 at.% preexisting shear bands, while it remains on a plateau for samples with more preexisting shear bands. Deformation patterns of samples with 3.27 at.% and 3.42 at.% pre-existing shear bands are shown in Fig. 17. Atoms in operating STZs are colored in green, while those in the pre-existing shear bands are painted in red. Figure 17a shows that a shear band of ~10 nm in width is formed in the sample with 3.27 at.% pre-existing shear bands at a total strain of 12%. The sample features two shear offsets on the free surfaces. The shear localization first appears in the glass matrix near the surface and

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Fig. 16 Simulated stress-strain relations (a) and normalized number of atoms in STZs as a function of strain (b) for MGs with various contents of pre-existing shear bands [58] Fig. 17 Deformation patterns of two typical MGs embedded with pre-existing shear bands. (a) Shear band occurs in the sample with 3.27 at.% pre-existing shear bands. (b) Uniform STZs are observed in the sample with 3.43 at.% pre-existing shear bands [58]

then extends across the entire sample. In contrast, in the sample with a slight larger amount (3.42 at.%) of pre-existing shear bands, plasticity is dominated by nonlocalized deformation with slight necking at strains up to 20%, as shown in Fig. 17b. The activation of STZs is uniformly distributed, and highly localized shear banding

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is suppressed during tension simulation. As a result, there exists a transition in tensile deformation of MGs at a critical atomic percent of pre-existing shear bands between 3.27 at.% and 3.43 at.%. Similar transition in plastic deformation is also observed for different orientations of pre-existing shear bands, albeit at slightly different critical atomic percents.

6

Summary

In this chapter, enhanced mechanical properties (strength, toughness, and ductility) of metallic materials by introducing structural heterogeneities, such as nanoscale twins, second-phase particles, or pre-deformed materials, have been demonstrated by MD simulations under various loading conditions (tension, indentation, and torsion). The knowledge about the underlying unique deformation mechanisms gained from these simulations not only extends the fundamental understanding of the relationship between nanoscale structures and macroscopic properties but also provides insights for future design of nanostructured metals with superior mechanical behaviors.

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Processes in Nano-Length-Scale Copper Crystal Under Dynamic Loads: A Molecular Dynamics Study

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Contents 1 2 3 4 5 6 7 8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Research of Energy Absorption by the Nanostructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Research of Crystal Structure Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Numerical Method of Calculation of Rotary Field Characteristics . . . . . . . . . . . . . . . . . . The Study of Rotary Fields in Nanostructures at the Compressive Actions . . . . . . . . . . . . . The Research of the Rotary Fields in the Nanostructures during Stretching at a Constant Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Nanostructures Cross-Size Influence on the Rotary Field Characteristics . . . . . . . . . . . 10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

616 617 618 620 624 629 632 635 642 649 653

Abstract

This chapter is devoted to the research of rotary field formation in a nano-lengthscale metal crystal under acting of different kinds of mechanical loads by molecular dynamics method. It was considered two sorts of loads: compressive dynamic load and stretching at a constant deformation velocity. The simulation technique of such vortex structures in solid was developed. It is shown that there exists a critical energy flux at which the system experiences an avalanche change both in the time and load dependence of energy absorption and in the type of wave processes in its structure. It was revealed that this process is a type of nanostructure self-organization in response to an external energy flux with subsequent development of a strong rotational field. The critical role of a rotary wave in the process of material fracture was defined, as the rotary wave energy exceeds I. F. Golovnev · E. I. Golovneva (*) Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, Russia e-mail: [email protected]; [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_20

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30% of the total internal energy of the structure at the strain rate greater than 200 m/s. The interpretation of vortex structure formation and spread in solids is proposed from the point of view of structure self-organization. The authors studied the structure size influence on rotary field formation, and it was revealed their appearance is not a result of nanoscale smallness of the sample. At the same time, the influence of a nanostructure’s cross size on the rotary field energy is being researched. Keywords

Molecular dynamics method · Metal nanostructures · Dynamic load · Rotary field · Size influence · Mesoanalysis

1

Introduction

In this paper, a study of rotary fields in the nanostructure, occurring during such external loads as acting of compressive stress impulses and stretching at a constant deformation velocity, was conducted. The relevance of such studies is due to the development of the cluster structure and rotary field during the expansion of nonlinear wave flow of local structural transformations in a nanoscale single crystal under the influence of the external applied stresses. In the work [1], it is shown that these structures radically affect all processes in solids. The study of the issue showed that not only were rotary fields not studied, but also they were not even mentioned. There is no any reference to rotary fields even in the works of venerable scientists [2–6], devoted to the study of deformations, damage, and fracture of solids and to the study of the crack development. At the same time, the analytical theory of nonlinear waves of structural transformations was developed in the works [7–9]. According to the conclusions of the work [1], the control parameter of nonlinear waves of local structural transformations is the local curvature in a highly nonequilibrium structure. In other words, the expansion of nonlinear waves of localized plastic flow of nondislocational nature is a translational and rotary process. Experimental verification [10] of this statement during the uniaxial stretching of flat samples of metal materials with nanostructured surface layers confirmed predictions of the nonlinear wave theory. In this regard, in the works [11–13], numerical microanalysis of these processes for the case of the external compression stress influencing the nanostructure within a certain period was conducted. Moreover, it was found that for a certain critical value of the external stress, the leading front of defects’ wave is formed, which is a complex of interfaces with different radii of curvature, beyond which the rotary field is formed. All this made it necessary to study the rotary field formation and their properties manifested under the influence of external stresses that lead to fracture. In this regard, the molecular dynamics study of such processes for the case of uniaxial stretching at a constant deformation velocity and at the compressive loads external pulses was conducted.

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It should be emphasized that the method of molecular dynamics is a numerical method of studying the processes and calculating the characteristics of the system, which is seen as a set of interacting atoms and molecules. With the help of the numerical solution of the equations of classical mechanics movement, the phase trajectories of all the atoms are calculated, i.e., the maximum information about the system is determined. Then, by averaging over the atom system, all the necessary characteristics of the system are obtained.

2

Physical Model

The study of the formation and development of the wave processes in nanostructures was carried out using the example of a copper cluster of cuboid shape with dimensions of 50*5*5 crystal cells along the X, Y, and Z axes, respectively. The crystal orientation (1, 0, 0) was selected. To describe the interatomic interaction, widely approved Voter’s potential [14] for metals was used. It was obtained with the help of the method of interstitial atom. Atoms were initially placed in sites of the perfect crystal fcc lattice. Then, using the method of artificial viscosity [15], the coordinates and momenta of the system in the state of minimum potential energy were found. Finally, the obtained values of the coordinates and momenta were used as the initial data. To simulate the compressive stress pulses, the following method was used. The external disturbance was generated by the impulse force in the form of the Heaviside function. The impulse value was specified by external constant compressive stress σ 0 acting on the left lateral face lying in the plane X = 0 at the initial point of time. Since in the molecular dynamics model forces fa acting on individual atoms are specified as the reference quantity, they are calculated in the program by the Cauchy model: f ax ¼ σ 0 S=nf where S is the area of the left face of the crystal and nf is the number of atoms on this face. Since the area of the face may change under compression, it was recalculated on each time step, as well as external forces acting on atoms, given that σ 0= const. Pulse duration Tσ was also a controlled external parameter, like the value of external stress σ 0. Simulation of the uniaxial stretching at a constant deformation velocity was conducted as follows. Atoms
of the fixed clamp  left extreme plane simulating the  0 0 0 ! k 0 2 were placed in the potential V r i ¼ 4 xi  xi . Here and below, xi , yi , zi are coordinates of the i-th atom after relaxation of the entire system to the thermodynamic equilibrium after cooling. Atoms of the right extreme plane simulating the moving with velocity v0 were placed in the potential
 clamp   2 ! V r i ¼ 4k xi  x0i þ v0 t .

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Numerical Simulation

The numerical scheme used in the paper for all kinds of external actions was a widely known modified Verlet velocity scheme of the second order of accuracy with the time step 1016 s. Below the illustration of numerical error is presented for external compressive stress. After removal of the external stress at the time point Tσ, the system becomes closed and isolated. From this time on, the integration error was calculated: err ¼

Et  ETσ  100%: ETσ

Here, Et and ETσ are the energy of the system at the time points t and Tσ , respectively. As authors earlier demonstrated [11], with the external stress growth from 0.1 to 4 GPa, the error increases by 9 orders from 1010 to 101 [11]. In the calculations the Verlet list radius was larger by 0.1 nm than the cutoff radius of the Voter’s potential, and the update step was 1014 s (or 100 time steps). Later investigations, however, revealed that the external stress growth gives rise to highfrequency oscillations in the volume with a period less than 1014 s. The numerical scheme was therefore improved by significantly increasing the Verlet radius until it exceeded the potential cutoff radius by 0.3 nm. All calculation results below are given for this case. Figure 1 exemplifies the error behavior within the range up to 50 ps for three external stress values. The energy error within the interval of 50 ps (500,000 time steps) did not exceed 105%. The mesoanalysis was used to calculate the spatial distribution of thermomechanical characteristics. It was realized by the next way. The crystal orientation (1, 0, 0) provides the most simple orientation of atomic planes in space, i.e., perpendicularly to the X axis. This allows the entire crystal to be divided into mesovolumes shaped as rectangular parallelepipeds that contain one atomic plane. The size of a mesovolume along the X axis is determined by the quantity ΔLi ¼

Fig. 1 Time dependence of the logarithm of energy error: external stress 0.1 (1), 2 (2), 4 GPa (3)

xiþ1 þ xi xi þ xi1  , 2 2

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where xi is the average value of the x atomic coordinates of the i-th plane. Knowing a set of mesovolume lengths at the initial and an arbitrary point of time, we determine the spatial distribution of local relative strains ex at any point of time. The lateral dimensions of a mesovolume are calculated based on the average values of the y and z coordinates of outermost atoms lying in the i-th plane. This detailed approach allowed us to calculate density, center-of-mass velocity, temperature in mesovolumes, and cohesive energy of atomic planes at required points of time for a certain range of external stresses at which the atomic planes do not break and under which no crystal structure defects are generated. For this purpose, an additional mesoanalysis of the state of nanostructures was developed. The main distinctive feature of this approach is that atoms are not assigned fixed atomic plane numbers. The entire space is divided by fixed planes perpendicular to the X axis along which the wave front propagates. The spacing between the planes is a/2, where a = 0.3615 nm is the lattice cell size of copper. With this size, there was only one atomic plane in a mesocell in an undisturbed state (at the initial time point) (Fig. 2). The crystal length is 18.075 nm (50 lattice cells along the X axis). We used 1010 cells (182.5575 nm) to account for the displacement of the entire crystal in space. Then, the position of each atom of the structure was analyzed at a particular point of time, and the atom was assigned the number of the mesocell it occupied. Using this procedure, the following mesocharacteristics were determined: the number of atoms in a mesocell, center-of-mass velocity components of the atomic subsystem in each mesocell, kinetic energy of the chaotic motion of atoms or mesocell temperature, potential interaction energy of atoms in a mesocell, potential binding energy of atoms in neighboring mesocells, and atomic interaction force for neighboring mesocells. (The x component of the force acting on mesocell atoms from atoms of the right-hand mesocell was calculated). Fig. 2 Schematic of mesovolume formation

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The Research of Energy Absorption by the Nanostructure

The example of absorption energy by the nanostructure at the compressive action is below. The most fundamental quantity is the total energy (Fig. 3) absorbed by the system under externally applied load. One can use the thermodynamic definition of the internal energy of structure as the total energy in the center-of-mass system. Therefore its dependence on the external stress has the form represented in Fig. 4. As one can see, the same three external stress intervals can be distinguished with respect to all energy characteristics: 1. σ 0  2.6 GPa. The dependence of all energy components on the applied stress is close to quadratic, which is indicative of the elastic compression of the crystal. 2. 2.6  σ 0  2.8 GPa. The function jumps to the third interval. 3. 2.8  σ 0  4 GPa. In the third interval, the total energy grows almost linearly with external stress increase. It is interesting to analyze energy redistribution after external stress removal. In Fig. 5 an illustration is given for the case of the external stress σ 0 = 2 GPa and

Fig. 3 Increment of total energy vs external stress

Fig. 4 Increment of the internal energy of structure vs external stress

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Fig. 5 Partial fractions of energy versus time: total energy (1), total kinetic energy (2), kinetic energy of the chaotic motion of atoms (3), variation of the potential energy of atoms (4), internal energy of the system (5)

Fig. 6 Partial fractions of energy versus time: total energy (1), total kinetic energy (2), variation of the potential energy of atoms (4), internal energy of the system (5)

Tσ = 6 ps, which corresponds to the first interval of the elastic response of the system. It is seen that not only the total but also the internal energy of the system remains constant. The potential and kinetic energies of the chaotic motion of atoms are redistributed in an oscillatory manner, which corresponds to wave processes in the structure. The total kinetic energy repeats oscillations of the chaotic component and is shifted up by the value of the constant energy of the center of mass of the system. The mode of energy redistribution in the interval up to 10 ps (the wave reflection from the free face ends) for the third interval (σ 0 = 4GPa, Tσ = 6 ps) is illustrated in Fig. 6. Total and internal energies remain constant, too, but the complication took place because of the abnormal behavior of the potential energy. Within the interval of 4 ps, it is almost constant up to 10 ps, and then the system gets into the state with the deeper minimum of the potential energy. Up to the moment of external action being off (6 ps), the external energy flux is consumed to the increment of the kinetic energy of the chaotic component of the kinetic energy dEkin and on the kinetic energy of the system mass center (it should be accented that dEkin contains also the kinetic energy

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of the wave motion which has not yet transformed into the thermal energy at this stage). Further reflection of the wave from end faces results in the following: by the instant of 20 ps, the transfer into the state with even lower energy takes place, which in turn causes further increment of dEkin of the structure temperature. Then, the transfer of the wave part of the internal energy is observed between the potential and kinetic components. Analysis of the irreversible changes process as a property of structure selfassembly is of special interest. In the paper the irreversibility process is considered to mean two constituents. The first one is the transition of “structured” external mechanical force to “unstructured” thermal energy. The second corresponds to irreversible changes in the crystalline structure of the system. An essential characteristic of the first constituent of irreversible changes is the ratio of the fraction of the total energy going to internal energy to the total energy of the system or dissipation of external mechanical force: Diss ¼

Ein  100%: Et

Figure 7 shows the dependence of this coefficient on the externally applied stress. It should be noted that the internal energy found above contains energy components related to wave motion. However, as is known from practice, the energy of wave motion also dissipates into thermal energy in a certain time interval of relaxation. As this graph shows, the dependence of the dissipation coefficient on the applied stress is absolutely different in above pointed intervals of external stresses. Hence the first external stress interval now has an additional interval up to σ 0  0.5 GPa, though the general division into intervals with respect to physical properties is the same for the dissipation coefficient as well. To analyze the plotted dependence of the given characteristic, the following numerical procedure was carried out. The coordinates and momenta of atoms at the point of external stress removal (the impulse duration Tσ = 6 ps) were used as Fig. 7 Ratio of the increment of the internal energy of structure to its total energy vs external stress

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initial values. Then, we subtracted the center-of-mass velocity of the structure as a whole from atomic velocities, and the system was cooled down with the use of the artificial viscosity method [15]. Thus, we defined the finite potential energy of the system related to the irreversible deformation of the crystal structure or, in conventional terms, to plastic strains arising under externally applied compressive stress. Figure 8 demonstrates how the potential energy varies with respect to the initial energy of the structure dUfr = Ufr(Tσ )  Ufr(0). It is seen that the first interval corresponds to an elastic response of the nanostructure to external load. For a more detailed analysis of the dissipation coefficient jump, Fig. 9 illustrates the behavior of this quantity for the first interval only. Under the external compressive stress σ  0.4 GPa, the system recovers its initial state after cooling – this is the region of perfectly elastic response. With stress growth to 0.5 GPa partial irreversibility arises from residual displacements of atoms in outermost atomic planes. As an example, Fig. 10 demonstrates the arrangement of atoms of the cooled system in the first atomic plane (side view in the XZ plane) for two external stresses σ 0 = 0.2 and 0.5 GPa. It is seen that the atoms on parallelepiped ribs shifted. Evident that, as the size rose, the contribution into the total energy of these atoms becomes ignorable, and the dependence of the dissipation coefficient in the external action area of 0.5 GPa becomes smooth.

Fig. 8 Variation of the energy of plastic deformation of the crystal versus external compressive stress

Fig. 9 Variation of the energy of plastic deformation of the crystal versus external compressive stress (first interval)

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Fig. 10 Arrangement of atoms in the first atomic plane. XZ plane (black circles). External stress 0.2 and 0.5 GPa (gray circles)

Fig. 11 Arrangement of atoms in the structure in the XZ plane

Thus, the study of structural changes in the crystal allows the phenomenon to be finally analyzed from the physical point of view.

5

The Research of Crystal Structure Evolution

It seems appropriate to start analyzing the interval of irreversible deformations (σ 0  2.8 GPa) with considering the arrangement of atoms in space. Figure 11 illustrates the arrangement of atoms in the XZ plane at the point of external stress removal (the case of σ 0 = 3.6 GPa, the impulse duration Tσ = 6 ps). The time is chosen so that the elastic wave reaches the right face of the structure.

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Fig. 12 Temperature distribution in the crystal

Fig. 13 Distribution of the center-of-mass velocities of mesovolumes in the crystal

The elastic precursor is followed by a wave propagating through the crystal in which irreversible changes of the crystal structure take place. Below are the distributions of temperature in space (Fig. 12) and center-of-mass velocities for mesovolumes (Fig. 13). At the wave front, temperature increases abruptly up to 450 K, then decreases down to 100 K, and remains almost invariable along the entire length of the crystal. The distribution of the center-of-mass velocities of mesovolumes demonstrates how the wave of irreversible deformation is generated. A region of size 30 nm is formed ahead the front in which the velocity is negative and equal to about 150 m/s. Further, matter is strongly accelerated in the wave front; the velocity becomes positive and equal to about 400 m/s. This pattern of velocity fields leads us to the conclusion that the rotation fields must be studied. The final detailed analysis of the processes of irreversible deformations in the crystal permits performing the research of the crystal structure in different mesovolumes. Figure 14 illustrates the atomic arrangement of the cooled system (gray circles) in the XZ plane and Fig. 15 in the YZ plane. To greater clearness the scales of axes are different.

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Fig. 14 Spatial arrangement of atoms in the XZ plane. Gray circles, atomic arrangement at the point of wave arrival; black circles, atoms of the cooled system

Fig. 15 Spatial arrangement of atoms in the YZ plane. Black circle, atoms of the initial structure during the wave passage; gray circles, atoms of the cooled structure

One can clearly see the formation of numerous defects in the region behind the front. To study the crystalline structure of matter, we use a pair distribution function which is calculated in each mesocell in the time moment 6 ps: gð r Þ ¼

V 1 ΔN ,  N 2 4πr 2 Δr

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Here, V is the mesocell volume, and N is the number of atoms in the mesocell. Judging from Fig. 14, of particular interest are the following domains: (1) 3 nm – the crystal face to which external stress was applied, (2) 6 nm – the central part of the wave of irreversible deformation, (3) 9.5 nm – the leading edge of the plastic wave at which a series of defects were generated after system cooling, and (4) 16 nm – the elastic wave region. During propagation of the wave of irreversible deformation, short-range ordering is preserved for the first coordination sphere in the mesocell on the crystal face (Fig. 16) while for the second and third very slight deviations from the amorphous state. For higher orders the function is close to the structure of a liquid. The pair function for the same region of the crystal after cooling is represented in Fig. 17. The picture is seen to remain unchanged, though the center of the second peak is slightly shifted to lower values. The pair distribution function in the region of the developed wave of irreversible deformation is given in Fig. 18. There is only the first peak left (short-range ordering), which means that matter is in a liquid state. After system cooling the pair distribution function has the form shown in Fig. 19. As a result of cooling, the crystalline structure of the fcc lattice is recovered, with the appearance of double peaks near the third, fourth, and sixth peaks, which indirectly points to the generation of structural (cluster) defects.

Fig. 16 Pair distribution function. Black line, perfect crystal; gray line, in the time moment of wave propagation in the crystal. Mesocell with coordinate 2.801 nm

Fig. 17 Pair distribution function. Solid line, perfect crystal; gray line, cooled crystal after the wave passage. Mesocell with coordinate 2.801 nm

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Fig. 18 Pair distribution function. Black line, perfect crystal; gray line, in the time moment of wave propagation in the crystal. Mesocell with coordinate 6.055 nm

Fig. 19 Pair distribution function. Solid line, perfect crystal; gray line, cooled crystal after the wave passage. Mesocell with coordinate 6.055 nm

Fig. 20 Pair distribution function. Black line, perfect crystal; gray line, in the time moment of wave propagation in the crystal. Mesocell with coordinate 9.67 nm

The distribution of atoms in the wave front is demonstrated in Fig. 20. One can clearly see the fcc structure with slightly broadened lines. The pair function after cooling of deformed crystal is given in Fig. 21. The lines of the first three coordination spheres are partially retained. Further behavior of the pair function rather corresponds to the liquid (amorphous) state of matter. In the both cases, the radial distribution functions coincide with perfect structure (the mesocell with coordinate 15.99 nm). Consequently, there are no irreversible changes of the crystal structure in this region and the wave propagates elastically.

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Fig. 21 Pair distribution function. Solid line, perfect crystal; gray line, cooled crystal after the wave passage. Mesocell with coordinate 9.67 nm

Fig. 22 Scheme of the elementary mesovolume in order to calculate the angular momentum of atomic subsystem

6

The Numerical Method of Calculation of Rotary Field Characteristics

For the assigned objective of the study of wave rotary properties, it was carried out [13] an additional partition of the space into calculated volumes of cuboid shape (Fig. 22) with the base rib length equal to the half of the crystal cell size. To determine the moment value in the cell along a certain axis, the side ribs of these cells were set parallel to these axes. Note that the taken lengths of the computational cells along the Y and Z axes were much bigger than the sizes of the nanostructure along these axes (namely, three times bigger). It permitted automatically “not losing” the atoms as the structure expands in the plane perpendicular to the compression axis. In this case, it is the X axis. In the case the moments along the X axis were calculated, the side ribs length was equal to five crystal cells. Figure 23 presents the example of calculation cell orientation. To study the rotation along the X axis, all space was divided into mesocells of the same volume, according to the scheme of Fig. 23a. Mesovolumes shown in Fig. 23 were additionally divided into 15 cuboids with generators parallel to the Y axis (see Fig. 23b) and to the Z axis (see Fig. 23c). This made it possible to calculate the distribution of the components of the angular momenta relative to the axes passing through the centers of mesovolumes with the

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Fig. 23 The construction of mesovolumes to researching the rotary movement along axes X (a), Y(b), Z(c)

coordinates (xc, yc, zc). The mesovolume axis in Fig. 23a crosses the plane YZ in the points (yc, zc), the mesovolume in Fig. 23b crosses the plane XZ in the points (xc, zc), and the mesovolume in Fig. 23c crosses the plane XY in the points (xc, yc). It should be noted that these axes are parallel to the correspondent coordinate axes. The calculation is carried out using the ordinary formulas:

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McX ¼ McY ¼ McZ ¼

X

ðyi  yc Þpzi  ðzi  zc Þpyi

631



i

X

ððzi  zc Þpxi  ðxi  xc Þpzi Þ

i

X

ðxi  xc Þpyi  ðyi  yc Þpxi



i

Here, summing up is carried out by the atoms in the computational cell. The vectors (xi, yi, zi) and ( pxi, pyi, pzi, ) are the coordinates and components of momentum of the i-atom, and (xc, yc, zc) are the coordinates of the crossing symmetry axis of the computational cell passing through the parallelepiped base center and any plane (XY), (XZ), or (YZ). The following expressions of analytical mechanics for the material point system were used to find out the energy of atom revolution motion in the computation cell:  X
 2   EcRX ¼ McX = 2  I cX ; I cX ¼ m ðyi  yc Þ2 þ ðzi  zc Þ2 ic

EcRY

 X
 2   ¼ McY = 2  I cY ; I cY ¼ m ðx i  x c Þ2 þ ðzi  zc Þ2 ic

 X
 2   EcRZ ¼ McZ = 2  I cZ ; I cZ ¼ m ðx i  x c Þ2 þ ðy i  y c Þ2 ic

To find out the total rotation energy, the energies of individual computational cells were summed up: ERX ¼

X c

EcRX ; ERY ¼

X c

EcRY ; ERZ ¼

X c

EcRZ

ERt ¼ ERX þ ERY þ ERZ Further physical analysis requires such a characteristic as the energy per one atom. Averaging by an individual computational cell is taken as a base. For this cell, each energy component and the total amount of atoms in the cell Nc are known in each time instant. Then, one can introduce such a characteristic as the specific rotational energy of the cell correlating with a specific component of the impulse moment per one atom: EcRXa ¼ EcRX =N c ; EcRYa ¼ EcRY =N c ; EcRZa ¼ EcRZ =N c ; ERta ¼ ERXa þ ERYa þ ERZa :

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Fig. 24 Distribution of rotational moments over crystal mesocells along the X axis. Gray circles – moments MZ, black circles – moments MY, t = 6 ps, σ = 4 GPa, Т σ = 6 ps (the point of removal of external stress)

7

The Study of Rotary Fields in Nanostructures at the Compressive Actions

The next step was to analyze the moments of momenta in the above-described mesovolumes. First of all, it was necessary to define the relationship between the values of moments McY , McZ in perpendicular planes. The moment distribution along the X axis is given in Fig. 24. The values of moments McY , McZ in the X mesovolumes coincide. The leading edge of the wave of defects coincides with the front of significant rotational motion. In this respect, the term “rotational wave” will be used below. The distribution of these moments at a fixed x-coordinate value is illustrated for two X mesovolumes with coordinates 4.07 and 9.5 nm. They correspond to the middle part and front of the rotational wave (Figs. 25 and 26). The moment distribution in the planes perpendicular to the X axis is seen to coincide. Additional investigation showed that the sums over each X mesovolume are equal to zero: X c

McY ¼ 0,

X c

McZ ¼ 0

Hence, illustrations below will be given only for moments MZ. The distribution of moments MX is demonstrated in Fig. 27. Notice that they are much lower in magnitude than the moments McY , McZ . It is interesting to consider the rotational wave evolution in the time range up to 6 ps, when the external energy flux is different from zero. Figures 28, 29, 30, and 31

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Fig. 25 Distribution of rotational moments over crystal mesocells along the Y and Z axes at x = 4.07 nm. Gray circles, moments MZ; black circles, moments MY, t = 6 ps. σ = 4 GPa, Т σ = 6 ps

Fig. 26 Distribution of rotational moments over crystal mesocells along the Y and Z axes at x = 9.50 nm. Gray circles, moments MZ; black circles, moments MY, t = 6 ps. σ = 4 GPa, Т σ = 6 ps

give a dynamic picture of the rotational wave development in the crystal at the time points 2 and 5 ps. Figures 28 and 30 demonstrate the spatial arrangement of atoms in the XZ plane. Figures 29 and 31 show the distribution of the moments of momenta MZ over the X mesovolumes along the X axis.

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Fig. 27 Distribution of rotational moments MX over crystal mesocells along the X axis, t = 6 ps, σ = 4 GPa, Т σ = 6 ps

Fig. 28 Arrangement of structure atoms in the XZ plane, t = 2 ps, σ = 4 GPa, Т σ = 6 ps

Fig. 29 Distribution of rotational moments over crystal mesocells along the X axis, t = 2 ps, σ = 4 GPa, Т σ = 6 ps

The most convenient characteristics for the determination of the rotational wave velocity are the distribution of temperature and center-of-mass velocity of atoms in mesocells along the X axis over the crystal. Knowing the coordinate of a particular wave point at a particular point of time, we can determine the wave velocity under the action of external stress. The average wave velocity hVTi determined by the temperature peak within the range from 1 to 6 ps is 1.3 km/s.

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635

Fig. 30 Arrangement of structure atoms in the XZ plane, t = 5 ps, σ = 4 GPa, Т σ = 6 ps

Fig. 31 Distribution of rotational moments over crystal mesocells along the X axis, t = 5 ps, σ = 4 GPa, Т σ = 6 ps

8

The Research of the Rotary Fields in the Nanostructures during Stretching at a Constant Strain Rate

The important points of time and movement velocity values of the free boundary, at which the fracture in the same nanostructure as in the one that is studied in this paper takes place, were found earlier by the authors, as described in the article [16]. In this connection, the results that are presented below were obtained considering the information of the work [16]. It was shown that for the free boundary movement velocity of 300 m/s at the point of time of 6.94 ps, the reflection of the wave front from the fixed edge of the crystal begins. The force exerted from the side of the fixed clamp reaches a maximum value in its modulo; and crack appears in the layer of the surface atoms. After the point of time of 9.14 ps, plastic deformation followed by fracture occurs. At 12.5 ps in the ninth mesocell, “crack” appeared in the bulk part as well. At 13.2 ps the complete fracture of the nanocrystal comes: a solid crack grows and reaches three mesocells. This is the critical size for the interatomic interaction potential – atoms do not interact at this distance. Figure 32a–c shows the views of crystal in the different time moments to better visual presentation of structure deformation.

636 Fig. 32 View of the crystal at the points of time of 9.14 ps (a), 12.5 ps (b), 13.2 ps (c); the free boundary movement velocity is 300 m/s

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Fig. 33 Distribution of the angular momenta in the crystal at the point of time of 3 ps; the free boundary movement velocity is 300 m/s. Black dots, MY; gray ones, MZ

Fig. 34 View of the crystal in the XZ plane at the point of time of 3 ps; the free boundary movement velocity is 300 m/s

The following is the results obtained on the basis of the method of the calculation of rotary fields for the free boundary movement velocity of 300 m/s, described in Sect. 6. During the first stage, the analysis at the time interval up to 5 ps was conducted, until the wave front had not reached the fixed clamp. In order to illustrate, Fig. 33 shows the distribution of the value of the momenta MY and MZ (SI unit: 1034kg  m2 s) in the space along the X axis at the point of time of 3 ps. The view of the structure in the XZ plane is presented nearby for convenience (Fig. 34). It can be seen that the components of momenta along the axes perpendicular to the wave expansion direction coincide with high accuracy. A more detailed analysis showed that the cells with the same x-coordinate have the symmetry of the components of the momenta MY, MZ: when the cells are turned on the angle of 180 around the X axis, their values coincide in the correspondent points of space (Fig. 35). At the same time, the total sum of the components of the momenta MY, MZ in cells with the same x-coordinate equals to zero. The diagram does not include the values of the components of momenta along the X axis, because they are negligible at this time interval. To illustrate this statement,

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Fig. 35 Distribution of the angular momenta in the crystal along the Y and Z axes, respectively. The point of time of 3 ps; the free boundary movement velocity is 300 m/s. Black dots, MY; gray ones, MZ

Fig. 36 Dependence of the angular momenta MX, MY, MZ on the time. The free boundary movement velocity is 300 m/s. MX, black dashed line; MY, black solid line; MZ, gray line

Fig. 37 Dependence of components of the angular velocity of the Y and Z axes in symmetrical cells with the coordinate x ¼ 159:96 A. The free boundary movement velocity is 300 m/s

Fig. 36 shows the dependence of three components of summarized momenta MX, MY, MZ at the time interval up to 15 ps for the moving clamp velocity of 300 m/s. As it can be seen, in the process of fracture, the symmetry of the rotary field begins to break, and there occurred the considerable angular momentum MX of vortices, directed along the X axis. The dependence of angular velocity of the identified vortices in conventional units (rad/s) helps to understand their physical meaning more than the momentum values shown in the diagrams above. Figure 37 shows the dependence of

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639

Fig. 38 Distribution of the angular momenta in the crystal at the point of time of 5 ps; the free boundary movement velocity is 300 m/s. Black dots, MY; gray ones, MZ

Fig. 39 View of the crystal in the XZ plane at the point of time of 5 ps; the free boundary movement velocity is 300 m/s

components of the angular velocity along the Y and Z axes in symmetrical cells with  the coordinate x ¼ 159:96 A . Complete synchronization of these characteristics in time and the symmetry of their absolute values are visible. The dynamics of the rotary field development is of interest. As an example, the diagrams for the velocity of 300 m/s for the same characteristics that were given earlier are presented below (Figs. 38 and 39), but for the time moment 5 ps. At the point of time of 5 ps, when the wave front reaches the fixed clamp, the symmetry of momenta is observed almost within the whole space (Fig. 38). It began to break only near the moving clamp, as it holds an active generation of transformational defects (Fig. 39). By 9.14 ps, when the wave front began to be reflected from the fixed clamp, values of the rotary field momenta sharply increased in the area of crack formation (Fig. 40). It can be seen in Fig. 41. At the same time, the rotary field began to spread from the moving clamp in the form of oscillations (see. Fig. 41). By 13.2 ps, the nanostructure collapsed near the fixed clamp, as evidenced by the main (solid) crack, which can be seen in Fig. 42. Moreover, Fig. 42 shows that near

640 Fig. 40 Distribution of the angular momenta in the crystal at the point of time of 9.14 ps; the free boundary movement velocity is 300 m/s. Black dots, MY; gray ones, MZ

Fig. 41 View of the crystal in the XZ plane at the point of time of 9.14 ps; the free boundary movement velocity is 300 m/s

Fig. 42 Distribution of the angular momenta in the crystal at the point of time of 13.2 ps; the free boundary movement velocity is 300 m/s. Black dots, MY, gray ones, MZ

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Fig. 43 View of the crystal in the XZ plane at the point of time of 13.2 ps; the free boundary movement velocity is 300 m/s

Fig. 44 The rotary field energy from time for different velocities of the moving clamp. The velocity value is given on the diagram

Fig. 45 Dependence of the rotary field energy at the point of time of 4 ps on the moving clamp velocity (gray line); the approximation obtained by the least square method (black line)

the moving clamp, a crosspiece was distinctly formed (right side of the crystal), which is accompanied by oscillations of the rotary field (Fig. 43). The dependence of total energy of the rotary field both on time and velocity of the moving clamp is of interest (see Fig. 44). In due course, the energy grows approximately linearly (especially in the range up to 4 ps), as well as the tilt angle with the increasing velocity of the moving clamp. Figure 45 shows the dependence of the rotary field energy at the point of time of 4 ps on the moving clamp velocity. The approximation obtained by the least square method is as follows:

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Fig. 46 The portion of the rotary field energy from the nanostructure internal energy, from time for different velocities of the moving clamp. The moving clamp velocity value in km/s is given near the diagrams

ERotS ¼ 87422:64  V  15214:164 The energies obtained above say little about the contribution of rotary fields in the process of damage and fracture of the structure. For this, it is more convenient to use such characteristic as portion of the rotary field energy from the nanostructure internal energy (see Fig. 46): δrot ¼ Erot

 Ein

 100%:

As can be seen, for velocities over 200 m/s at the time of arrival of the wave front to the fixed clamp (5 ps), the rotary field energy is about 40%.

9

The Nanostructures Cross-Size Influence on the Rotary Field Characteristics

As it is known [17], nanostructures make a special class of objects because at this scale level, their size and shape influence their physical and mechanical properties. Thus it is of interest to determine the effect of the size on rotary field characteristics. There are a number of works [18–29] wherein more or less attention is paid on the analysis of system size effect on the destruction (the value of the stress critical value, cracking, pore formation). It should however be noted that none of these works deal with the phenomenon of rotary fields occurring in a solid body. The present section illustrates the effect of the transversal size of the nanostructure on the parameters of the rotary fields for the case of the external compressing stress acting on the nanostructure within a certain time. The Sect. 7 presents the results for the properties of the rotary fields occurring in the nanostructures shaped as rectangular parallelepipeds under shock loadings. In the transversal cross section (YZ plane), the sizes were 5*5 crystal cells. The effect of the size is illustrated by the nanostructures with the transversal size of 10*10 and 15*15 crystal cells. The crystal structure of these nanosystems is of primary interest.

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Fig. 47 Position of atoms in the XZ plane. Time instant, 6 ps; external stress, 4 GPa; transversal size, 5 crystal cells

Fig. 48 Position of atoms in the YZ plane. Time instant, 6 ps; the external stress, 4 GPa; transversal size, 5 crystal cells

For these sizes, the share of surface atoms was 34%, 19%, and 14%, respectively. The compressing stress was applied to the left end face; the sides and right face were free. Initial temperature of the nanostructure was almost zero (T  106 K ). Initial data for the coordinates and impulses of the system atoms were found by the artificial viscosity method [15]. The illustration below is given for the case of the external stress of 4 GPa acting within 6 ps. Below there are the patterns of atom positions in the planes XZ and YZ at the time instant of 6 ps, when the elastic wave reaches the free boundary (Figs. 47, 48, 49, 50, 51, and 52). Every graph corresponds to the case of the external stress of 4 GPa acting within 6 ps. Evident that for the nanostructure of 10 and 15 transversal cells, the leading front of the transformation wave is noticeably behind the case with the transversal size of 5 cells. Thus, extra researches were carried out for the time instant

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Fig. 49 Position of atoms in the XZ plane. Time instant, 6 ps; external stress, 4 GPa; transversal size, 10 crystal cells

Fig. 50 Position of atoms in the YZ plane. Time instant, 6 ps; external stress, 4 GPa; transversal size, 10 crystal cells

10 ps (Fig. 53), when the reflection from the free boundary had already started but the reflected wave did not interact yet with the transformation defect wave. The pattern of atom position in the XZ plane demonstrated that the stable barrel-like shape for 10 and 15 cells had formed by this time. Atom positions in the transversal cross section are in qualitative agreement for every size. It is necessary to stress the high symmetry of the atom positions in every case. Let us compare the characteristics which should not depend on the transversal size. The sizes of 5, 10, and 15 cells will be illustrated within the time interval up to 6 ps, when the reflection from the free surface has not started yet. Comparison of the mass center velocities of the first and last atomic planes, as well as the mass center of the whole structure gives rather important information (Figs. 54 and 55).

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Fig. 51 Position of atoms in the XZ plane. Time instant, 6 ps; external stress, 4 GPa; transversal size, 15 crystal cells

Fig. 52 Position of atoms in the YZ plane. Time instant, 6 ps; external stress, 4 GPa; transversal size, 15 crystal cells

As it is seen, the structure size highly influences the process of wave reflection from the free surface as well as the motion of the end faces after the action end. The share of the total energy transforming into the internal energy of the system is of interest, too (Fig. 56). Evident that the thermal energy share of the mass center decreases and the kinetic energy share of the same rises as the size increases. For the analysis of distribution of the thermomechanical characteristics over the space, the system of immovable computational cells described above is the most convenient. Above it was shown that the velocities of the atom center masses in the cells and their temperature are the most important characteristics leading to the conclusion of the rotary field existence. Below, the characteristics are given for 5, 10, and 15 cells in time instants of 6 and 10 ps (Figs. 57 and 58).

646 Fig. 53 Position of atoms in the XZ plane. Time instant, 10 ps; external stress, 4 GPa; transversal size, 15 crystal cells

Fig. 54 Mass center velocity versus the structure with the transversal size of 5 crystal cells. 1, black line, the last atomic plane; 2, black dashed line, the mass center of the whole structure; 3, gray dashed line, the 1st atomic plane

Fig. 55 Mass center velocity versus the structure with the transversal size of 15 crystal cells. 1, black line, the last atomic plane; 2, black dashed line, the mass center of the whole structure; 3 gray dashed line, the 1st atomic plane

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Fig. 56 The share of the total energy transforming into the internal energy versus time. Transversal sizes: 1, black line, 5 cells; 2, gray line, 10 cells; 3, black, dashed line, 15 cells

Fig. 57 Mass center velocity in computational cells versus the coordinate. Time instant 6 ps. Transversal sizes: 1 black line, 5 cells; 2 gray line, 10 cells; 3 black dashed line, 15 cells

Fig. 58 Mass center velocity in computational cells versus the coordinate. Time instant10 ps. Transversal sizes: 1 black line, 5 cells; 2 gray line, 10 cells; 3 black dashed line, 15 cells

At the time instant 6 ps, velocity distributions for the transversal sizes of 5 and 10 crystal cells are very close to each other. There is the formed area with the constant mass velocity and accuracy to fluctuations. The front of the transformation defect wave is pronounced. Before the front, the velocity distribution is of non-monotony character and has an evident minimum. The difference is that for the structure of five cells, this minimum is negative, i.e., the substance in this point moves toward the wave, whereas for the size of 10 cells, the velocity in the minimum

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Fig. 59 Temperature in computational cells versus coordinates. Time instant 6 ps. Transversal sizes: 1 black line, 5 cells; 2 gray line, 10 cells; 3 black dashed line, 15 cells

point is positive. For the structure of 15 cells, the velocity distribution from the 1st atomic plane to the last one decreases approximately linearly. At the time instant of 10 ps, the transformation defect wave has formed for every size. It evident that the mass velocity behind the wave front decreases as the transversal size of the structure rises. At this moment, the reflection from the free boundary takes place. The reflected wave however does not yet interact with the transformation defect wave, which permits making a number of conclusions about the wave itself. Temperature distribution in the space (Fig. 59) indicates the forming front of transformation defects for the structures of all sizes. However, the front becomes flatter as its size rises. Thus, the distribution of the thermomechanical characteristics in the space has the similar character for the studied sizes of the nanostructures; it indicates the presence of the rotary fields in them. Below there are the results of vortices distribution along the compression axis, with the impulse moments parallel to the Y (black bubbles) and . 2 Z (gray dots) axes. The measurement unit of the impulse moment is 1034 êãì . en Evident that for the structure with the size of 5 crystal cells (Figs. 60 and 61), the values of rotary moments by the Y and Z axes precisely coincide both before the elastic wave arrival to the free boundary and during the reflection from it. Moreover, within the whole range under consideration, there is the symmetry about the X axis; in other words, the sum of moments in the computational cells at the fixed coordinate x is zero. In the structure of bigger size (10 and 15 crystal cells), these properties are preserved just up to the elastic wave arrival to the free boundary (the time instant 6 ps). Then, insignificant deviation of the moment values from the exact symmetry is observed along the Y and Z axes (from the time instant 10 ps). The distribution of the impulse moments parallel to the X axis before the reflection start of the elastic wave from the free boundary has the same features of symmetry about the X axis (the sum of moments in the computational cells at the fixed x-coordinate is zero). Then this symmetry is slightly perverted.

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Fig. 60 Impulse moment distribution MY,Z in the space. MY, black bubbles; MZ, gray dots. Structure size 5 cells, time instant 6 ps

It is interesting to analyze the dependence of the total energy of the rotary motion of atoms on time within the range from the start of the external action to the start of interaction of the wave reflected from the free boundary (Fig. 62). By the moment of elastic wave arrival to the free boundary, certain-fold growth of the transversal size results in the same growth of the rotary field total energy (twoand three-time growth). In the bigger structures, however, more atoms participate in the rotary motion. In this context, comparison of the rotary energy values per one atom is of interest (Figs. 63 and 64). Evident that the specific values of the rotary energies for the rotations around the Y and Z axes are very close to each other and weakly depend on the structure sizes (Fig. 64). The opposite effect is observed for the rotary fields with the impulse moment direction along the X axis (Fig. 63). The energy of such vortex fields rises dramatically together with the growth of the system transversal size. The graphs clearly show that the 2–3 time growth of the structure transversal size causes the growth of the rotary wave energy ERXa for one atom by 12 and 50 times, respectively. Let us compare the specific values of the total rotary energy (Fig. 65). Similarly to the rotary energies for the rotations around the Y and Z axes, they are very close to each other and weakly depend on the structure size. It means that the specific total rotary energy may be the general characteristics of the nano-size crystal (shape, substance, etc.).

10

Conclusion

A synergetic mechanism of how a solid nanosystem responds to an external energy flux is disclosed. There exists an ultimate external load at which the mechanism of energy absorption in a solid nanosystem with perfect lattice changes abruptly. Below the critical load, the absorbed energy is expended in two ways: (1) to maintain the elastic wave motion and (2) to increase the kinetic energy of the center of mass of the system. Above the critical external load, the total energy flux exceeds

Fig. 61 Impulse moment distribution MY,Z in the space. MY, black bubbles; MZ, gray dots. Размер structure 5 cells, time instant 10 ps

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Fig. 62 Total energy of the rotary wave versus time. Transversal sizes: 1 black line, 5 cells; 2 gray line, 10 cells; 3 black dashed line, 15 cells.

Fig. 63 Energy of the rotary wave per one atom with the impulse moments parallel to the X axis on time. Transversal sizes: 1 black line, 5 cells; 2 gray line, 10 cells; 3 black dashed line, 15 cells

Fig. 64 Energy of the rotary wave per one atom with the impulse moments parallel to the Y axis on time. Transversal sizes: 1 black line, 5 cells; 2 gray line, 10 cells; 3 black dashed line, 15 cells

651

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Fig. 65 Total energy of the rotary wave per one atom versus time. Transversal sizes: 1 black line, 5 cells; 2 gray line, 10 cells; 3 black dashed line, 15 cells

the fluxes expended in these ways. A wave of irreversible structural changes is generated in which the perfect lattice transforms into an amorphous state. In this case, the rotational component of deformation induced by the wave propagation arises. Much more energy is needed to maintain this wave; in so doing, a stationary though thermodynamically nonequilibrium process takes place. The structure of matter in the wave of irreversible changes corresponds to a liquid phase. It is more correct to use the term “amorphous phase” for this state, because the temperature of matter behind the wave front is about 100 K. By cooling the system down to cryogenic temperatures, we revealed the generation of numerous structural defects. The formation of these defects is a synergetic rearrangement of structure under the action of an external energy flux. It revealed the formation of a rotational wave in which cluster defects of different atomic configurations are most intensively generated. The leading edge of the rotational wave is a complex formed by numerous interfaces with different curvature radii. The nanostructure after self-organization forms a large number of interfaces with different curvature radii. The interfaces are generators of the defects that are metastable and decrease the thermodynamic nonequilibrium of the nanocrystal. The velocity of the rotational wave front for copper nanostructure with cross size 5  5 crystal cells was calculated to be about 1300 m/s. The rotational wave front is equal to about 7 nm and has a complex structure: the mass velocity is negative at the leading edge of the front (matter moves against the wave motion direction due to cluster structure formation) and then it becomes positive (400 m/s), being almost two times higher in amplitude than the velocity at the leading edge (200 m/s). A temperature peak of up to 300 K is formed above the front with negative velocity, which is about 3.5 times higher than behind the front. Analysis of the transversal size influence has demonstrated that it influences the shock wave processes caused by the uniaxial compression action. The growth of the structure size results in the increasing intensity of rotary field formation and in the relative increase of the rotary field energy with the moments along the X axis. This can be interpreted as the formation of the rotational freedom

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653

degree along the nanostructure compression axis. Before the arrival of the elastic wave to the free boundary, the rotary moments in the computational cells by the Y and Z axes which are perpendicular to the shock compression direction precisely coincide. Moreover, there is the symmetry for all impulse moment components in the computational cells about the X axis (or the sum of the moments in the computational cells at the fixed coordinate x is zero). At the same time, the total specific energy of the rotary motion (the energy per one atom) weakly depends on the sizes and is the general characteristic for this substance. Analysis of the phenomena in the metal nanostructure for the uniaxial stretch of the metal nanostructure with the constant deformation rate has shown that the movement of the wave front from the moving clamp is accompanied by the emergence of a rotary field. The difference from the shock action is that the rotary wave movement velocity for this type of load coincides with the elastic wave velocity. At the movement of the wave front till the moment of reflection from the fixed clamp, the momentum value in calculated cells increases with time. Similarly to the shock action in cells with the same x-coordinate, the symmetry of the components of the momenta MY, MZ is observed, when cells are turned on the angle of 180 around the X axis and the total sum of the components of the momenta MY, MZ in cells with the same x-coordinate equals to zero. During the formation of translational defects on the moving clamp, this symmetry disappears; and an oscillation picture of the rotary field begins to form in the space, which is completed by the time of fracture. When the wave moves toward the fixed clamp, the rotary field energy is about 20% of the internal energy for the movable clamp velocities up to 200 m/s. For higher speeds, proportion rises to 40% of the internal energy. Thus, it can be concluded that the occurrence of rotary fields significantly contributes to fracture of the structure at speeds higher than 200 m/s. This chapter is based on the following papers of the authors [11, 13, 30, 31] with the permission of journals editors.

References 1. Panin VE, Egorushkin VE. Phys Mesomech. 2013;16(3):5–6. 2. Eidel B, Hartmaier A, Gumbsch P. Multiscale Model Plast Fract Means Dislocat Mech CISM Int Centre Mech Sci. 2010;522:1–57. 3. Zhang L-Y, Li Y, Cao Y-P, Feng X-Q, Gao H. J Appl Mech. 2013;80(6). https://doi.org/10.1115/ 1.4023977061018. 4. http://www.engin.brown.edu/Faculty/gao/gaogroup/publications.html 5. D. Huang, M. Wang, G. Lu, J. Nanomater. 2014;7. https://doi.org/10.1155/2014/ 732434732434. 6. Ravi-Chandar K. Dynamic fracture. Oxford: Elsevier; 2004. p. 219. ISBN: 0–08–044352-4 7. Egorushkin VE, Panin VE, Savushkin EV, Khon YA. Russ Phys J. 1987;30(1):5–16. 8. Korteveg DJ, de Vries G. Phil Mag. 1895;39(5):422–43. 9. Egorushkin VE. Russ Phys J. 1992;35(4):316–34. 10. Panin VE, Egorushkin VE, Panin AV. Physics-Uspekhi. 2012;55(12):1260–7.

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11. Golovnev IF, Golovneva EI, Merzhievsky LA, Fomin VM. Phys Mesomech. 2013;16 (4):294–302. 12. Golovnev IF, Golovneva EI, Merzhievsky LA. News Altai State Univ. 2014;1(81):40–2. https:// doi.org/10.14258/izvasu(2014)1.1-08. 13. Golovnev IF, Golovneva EI, Merzhievsky LA, Fomin VM, Panin VE. Phys Mesomech. 2015;18(3):179–86. 14. A.F. Voter. Embedded atom method potentials for seven FCC metals: Ni, Pd, Pt, Cu, Ag, Au, and Al, Los Alamos unclassified technical report # LA-UR 93-3901. 1993. 15. Golovnev IF, Golovneva EI, Fomin VM. Phys Mesomech. 2003;6(5–6):41–5. 16. Golovneva EI, Golovnev IF, Fomin VM. Comput Mater Sci. 2015;97:109–15. 17. Roco MC, Williams RS, Alivisatos P, editors. Nanotechnology research directions: IWGN workshop report. Vision for nanotechnology R&D in the next decade. Dordrecht/ Boston/London: Kluwer Academic; 2000. 18. Zhuo XR, Beom HG. Materials science and engineering A-structural materials properties microstructure and processing. 2015;636:470–75. 19. Sadeghian H, Goosen JFL, Bossche A, Thijsse BJ, van Keulen F. Effects of size and surface on the elasticity of silicon nanoplates: molecular dynamics and semi-continuum approaches. Thin Solid Films. 2011;520(I.1):391–9. https://doi.org/10.1016/j.tsf.2011.06.049. 20. Huang D, Qiao P. Mechanical behavior and size sensitivity of Nanocrystalline nickel wires using molecular dynamics simulation. J Aerosp Eng. 2011;24(I.2):147–53. https://doi.org/ 10.1061/(ASCE)AS.1943-5525.0000006. 21. Tang T, Kim S, Horstemeyer MF, Wang P. Atomistic modeling of crack growth in magnesium single crystal. Eng Fract Mech. 2011;78(I.1):191–2011. https://doi.org/10.1016/j.engfracmech.2010.11.009. 22. Tang T, Kim S, Horstemeyer M F, Wang P. A molecular dynamics study of fracture behavior in magnesium single crystal. In: Magnesium technology. Book Series: Magnesium Technology Series. Conference: Conference on Magnesium Technology held during TMS 140th Annual Meeting and Exhibition; 2011. p. 349–55. 23. Tang T, Kim S, Horstemeyer MF. Molecular dynamics simulations of void growth and coalescence in single crystal magnesium. Acta Mater. 2010;58(I.14):4742–59. https://doi.org/ 10.1016/j.actamat.2010.05.011. 24. Sadeghian H, Yang Chung-Kai, Goosen, JFL, Bossche A, Staufer U, French PJ, van Keulen F. Effects of size and defects on the elasticity of silicon nanocantilevers. J Micromech Microeng. Conference: 20th Micromechanics Europe Workshop (MME 09). 2010;20 I.6:064012. https:// doi.org/10.1088/0960-1317/20/6/064012. 25. Sadeghian H, Goosen JFL, Bossche A, Thijsse BJ, van Keulen F. Size-dependent elastic behavior of silicon nanofilms: molecular dynamics study. In: IMECE 2009: Proceedings of the ASME International Mechanical Engineering Congress and Exposition. 2010;12 parts A and B:151–6. 26. I-Ling C, Yu-Chiao C. Is the molecular statics method suitable for the study of nanomaterials? A study case of nanowires. Nanotechnology. 18(I.31):315701. https://doi.org/10.1088/09574484/18/31/315701. 27. Golovnev IF, Golovneva EI, Fomin VM. The influence of a nanocrystal size on the results of molecular-dynamics modeling. Comput Mater Sci. 2006;36:176–9. https://doi.org/10.1016/j. commatsci.2004.12.082. 28. Potirniche GP, Horstemeyer MF, Wagner GJ, Gullett PM. A molecular dynamics study of void growth and coalescence in single crystal nickel. Int J Plast. 2006;22(I.2):257–78. https://doi.org/ 10.1016/j.ijplas.2005.02.001. 29. Wu HA, Soh AK, Wang XX, Sun ZH. Strength and fracture of single crystal metal nanowire. In: Kishimoto K et al., editor. Advances in fracture and failure prevention. Conference. Book Series: Key engineering materials. 2004;p. 261–263. Part 1&2:33–8. 30. Golovnev IF, Golovneva EI, Fomin VM. Comput Mater Sci. 2015;110:302–7. 31. Golovnev IF, Golovneva EI, Merzhievsky LA. Phys Mesomech. 2016;19(5):66–72. (in Russian).

Understanding Fracture and Fatigue at the Chemical Bond Scale: Potential of Raman Spectroscopy

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Contents 1 Introduction: Why Vibrational Spectroscopy Probes Mechanics Well at the Subnanometer Scale: Anharmonicity of the Chemical Bond? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Choice of Probe(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A Few Examples: Elastic and Viscoelastic Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Understanding Fracture Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Understanding Fatigue Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Coupled mechanical and Raman analysis of a material under tension or compression provides much information on the material’s (nano)structure. Raman extensometry can be applied to synthetic and natural polymer fibers (e.g., polyamides [polyamide 66], polyethyleneterephthalate, polypropylene, poly[paraphenylene benzobisoxazole], keratin/hair, and silkworm and spider silks). The technique allows differentiation between crystalline and amorphous macromolecules. Bonding is similar in the two cases, but each exhibits different Raman signatures, especially at low wavenumbers, and a broader distribution of conformations is observed for amorphous macromolecules. These conclusions are used to discuss modifications induced by the application of a tensile or compressive stress up to the point of fracture – in particular the effects of fatigue.

P. Colomban (*) De la Molécule aux Nano-Objets: Réactivité, Interactions Spectroscopies, MONARIS, Sorbonne Université, CNRS, Paris, France e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_23

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Keywords

Mechanics · Fiber · Raman spectroscopy · Infrared · Nanostructure

1

Introduction: Why Vibrational Spectroscopy Probes Mechanics Well at the Subnanometer Scale: Anharmonicity of the Chemical Bond?

As illustrated in Fig. 1a, matter consists of atoms that interact through chemical bonds. Atoms are usually considered to be hard spheres, and the bonds are represented by springs. A chemical bond is well characterized by its potential (Fig. 1b). The harmonic oscillator approximation is the simplest model that describes atomic/molecular motions. Bond stiffness is a constant (k); if Δl = (l
l0) is the shift in atomic position, with l0 being the interatomic distance at rest (i.e., the bond length), a single diatomic oscillator would have the following restoring force: F =
k  Δl/l0. The corresponding potential well is symmetric (V = ½ k [l
l0]2), and the corresponding fundamental vibration is ν = 1/2π (√[k/μ]), where μ represents the reduced mass of the oscillator (see, e.g., [5] and the references included). With this harmonic approximation, the potential well (Fig. 1b), lattice spacing, and vibrational properties are independent of temperature, pressure, and stress. Anharmonicity should be taken into account with, for example, a Morse potential whose shape describes the behavior of a “hard sphere” under compression (l < l0; the left side is steeper than a parabolic shape, and the right side describes the dissipation mechanism up to a continuum, with a shape less steep than the harmonic one). The potential can be expressed as [5] Vanharmonic ¼ k=2ðl
l0 Þ2 þ k0 ðl
l0 Þ3 : High-order developments are necessary to precisely account for the different origins of the anharmonicity (e.g., mechanical, electrical). Consequently, any

Fig. 1 (a) Schematic showing the perturbation of the interatomic bond of a stretched solid. (b) Comparison between harmonic (black parabola) and anharmonic (Morse curve in red) potential wells. Note the modification of the vibrational levels. The dotted line is a guide showing the mean position (thermal expansion)

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modification of the chemical bond length changes the force constant and results in a vibration wavenumber shift. This is the principle of vibrational extensometry [5, 40]. The empirical relationship between the strain-/stress-/temperature-induced wavenumber shift (Δν) is linear. For example, for strain Δe, Δν ¼ S  Δe where S is the Raman factor (cm
1 is the unit used in spectroscopy, cannot be replaced by centimeter per percentage strain). As illustrated in Fig. 1b, the difference between the parabolic (harmonic) and Morse (anharmonic) potential wells increases from the bottom to the top (in other words, when the temperature increases) and hence the S parameter will be larger for second-order vibrations, third-order vibrations, and so on. The S value also depends on the nature of the chemical bond; it is maximal for “soft,” anharmonic bonds such as those found in carbons, compounds containing an aromatic cycle, and H-bonded compounds. Historically, Raman extensometry was first applied to carbon fibers [1, 15, 24, 25, 40, 44, 49], taking advantage of resonance Raman Effect. Typically, S measures ~9 cm
1/% strain for fundamental carbon modes (around 1350 or 1600 cm
1) and reaches around 29 cm
1/% strain if the second harmonics bands are used (~4300 cm
1) [24]. S values are much lower for oxides (2–3 cm
1/%), and generally only fundamental modes are observed (nonresonance Raman spectroscopy) [7, 24, 47]. Raman scattering is preferred to infrared (IR) spectroscopy because Raman bands are much narrower than IR ones (a Raman probe is local, and electric coupling does not affect the bond polarizability much; on the contrary, an IR probe is an instant electric dipole coupled to its environment, and therefore it is greatly broadened by neighboring defects). Furthermore, carbon materials exhibit very strong resonance in the Raman spectrum (second and even third overtones can be measured!) and dissipate heat well [5, 24]. It is important to keep in mind that the wavenumber shift induced by hundred of degrees Celsius can be an order of magnitude larger than that induced by a stress close to the fracture. Thus perfect control of the sample temperature under the laser beam (see Sect. 3) is mandatory. Raman extensometry was first used to probe the compressive or tensile effect of a composite matrix on fiber reinforcement [1, 15, 24, 25, 40, 44, 47]. Its main objective was to determine whether fibres reinforcing a matrix were in tension or in compression and what were the applied stress values. The technique allows the ineffective length and load distribution to be measured [2, 15, 24, 44]. No information regarding the mechanical behavior of the matter or associated structural characteristics was researched. Intrinsic information about the mechanical behavior of the matter itself could be obtained [7]; this is the subject of this chapter. Example case studies provided here concern fibers.

2

Choice of Probe(s)

Figure 2 shows typical Raman spectra of a polymer fiber (for instance, polyamide 66 [PA66], which is named as such because 6 carbon atoms are located between the amide group [-HN-C=O]) recorded with the electric vector of the laser beam parallel

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Fig. 2 Example of Raman spectra recorded for a single polyamide (PA66) fiber with the electric vector of the laser beam parallel and perpendicular to the fiber axis. Arrows indicate characteristic modes to be used as specific probes of mechanical behavior

or perpendicular to the fiber axis. The differences arise from the axial character (structural anisotropy) of fibers. However, the scattered Raman light depends on the sample symmetry (second-rank tensor) and reveals the symmetry of the illuminated matter. Vibrational spectroscopy analyzes both the atom mechanics and electric transfer (conductivity) [24]. Consequently, vibrational modes involving light species are observed at a high wavenumber and those of heavy ones are observed at a low wavenumber. Strongly bonded species contribute to high wavenumber modes and weak-bonded ones, to low wavenumbers. Raman intensity depends on the charge transfer when an atom moves; in other words, it will be strong for covalent bonds (with an electron “cloud” between atoms) and almost null for ionic bonds (no common electron cloud between atoms). Bands between approximately 2500 and 3800 cm
1 correspond to vibrational modes involving hydrogen, a light atom, namely C-H and N-H modes for polymers [7, 33]. Bands below 200 cm
1 involve the motion of whole polymer chains and reflect the long-distance structure (translational/lattice modes). Modes in between correspond to stretching and bending vibrations of atoms with intermediate mass/bond strength. Hydrogen atoms of X-H bonds could establish couplings with their environment; consequently, X-H modes probe the closest environment well. Lattice modes, on the other hand, best probe the long-range structure [7]. Each vibrational mode informs a specific “view” of the matter. As mentioned above, expected wavenumber shifts should be small, and a compromise should be made between band intensity and narrowness (for easy measurement), wavenumber range, and so on.

3

Procedures

Figure 3a shows a fiber under tension located below a microscope objective with a long working distance in a Raman spectrometer. Fiber diameter typically ranges between a few microns (carbon and silk fibers) and hundred(s) of microns (hair, fiber precursors,

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Fig. 3 (a) View of a fiber in tension, analyzed by Raman microspectrometry; the fibre is in water. (b) Detail of the focused laser beam on a fiber under tension. (c) Comparison of the Raman wavenumber shift measured as a function of illumination power for an SiC fiber, freestanding in air or embedded in a ceramic (oxide) or a metal matrix. Carbon (~1345 cm
1, solid line) and SiC (~ 780 cm
1, dash line) probes are used (After [26]). A constant value guarantees the absence of heating

etc.). Selection of the microscope objective fixes the probed volume, from about 5  5 (xy plan)  15 (z direction) μm3 for a standard 50 objective to less than 0.5  0.5  2 μm3 with a 200 objective. Precise measurement of the stress and structural order/disorder profile across the fiber diameter is therefore possible [5, 6, 7, 14]. A tensile machine should compensate for fiber (visco)elasticity [28, 33] by moving one grip and hence applying a definite stress or strain. Temperature should be perfectly controlled [26] (see Fig. 3b and further discussion). Because of the focusing of the light and the weak Raman contribution of water, measurement can be made inside water [36] (Fig. 3a). Applying a compressive stress is much more complicated, and the main technique is the use of a diamond (or more rarely alumina) single-crystal anvil cell (DAC) [6, 7, 9, 13, 37]. The DAC volume is small: typically, the diameter is 200 μm and the height 100 μm. Boucle and curved supports/matrices have also been used [29, 42, 43]. Hydrostatic pressure in a DAC is measured from the Cr3+ ion fluorescence spectrum of a small ruby ball. In the case of a boucle of curved support, the pressure should be calculated based on the substrate/ sample geometry.

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As noted in previous paragraph, the wavenumber shift induced by strain, stress, or pressure generally reaches a few centimeters, no more, because the “hard” spherical characteristic of atoms – that is, the steep slope of the bottom of the potential well (Fig. 1b). High-resolution (resolution 3 GPa), polyethyleneterephthalate [PET], polyamide 66 [PA66], and isotactic polypropylene [i-PP]) and natural (silk, keratin) fibers. (b) An example of Weibull distributions for silks from different silkworms (wild Gonometa postica [Gp], wild Gonometa rufobrunea [Gr], degummed single fiber from Bombyx mori [Bm-D], and a single fiber from yarn of silk from Bombyx mori [Bm-yarn]) and spiders (Nephila madagascariensis [Neph]). See the text for the meaning of the acronyms (After [18]). Plots show the rather slow decrease of the Young modulus (c) and the strong decrease of the ultimate tensile stress as a function of fiber diameter (d) for different silks (Bm, BmO and BMO*, representing domestic Bombyx mori silkworm from different provenances; Anth: wild Antheraea silkworm) (After [10] and [19])

Ganometa postica harvested in South Africa, the distribution is narrow and mechanical properties are poor, with ultimate strength less than 150 MPa [17]. The mechanical strength of Nephila madagascariensis spider silk fibers, however, can reach 1 GPa, but the Weibull parameter (the slope) is less than 2 because of the huge distribution; Weibull parameter values up to 8–10 are measured for silk from well selected, domestic Bombyx mori silkworms (a degummed single fiber and a single fiber extracted from a yarn). The ultimate strength also depends from the fiber diameter. The best values are measured for small-diameter fibers (Fig. 4b). Extrapolation of the fiber diameter decrease will lead to approach the strength of the ultimate mechanical brick, the chemical bond! (Fig. 1a) On the contrary, the evolution of the Young modulus measured as the slope of the initial stress/strain curve versus fiber diameter is weak; this parameter reflects more the characteristics of the matter [19]. It is thus important to study the relationship between the structure of the material and its mechanical properties.

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Fig. 5 Schematic of the different model structures of polymers: a Prevorsek model (a0 ), where a polymer chain belongs to both ordered and disordered/amorphous regions; the region between the ordered and disordered regions is called the isotropic amorphous domain; an Oudet model (b0 ), where crystals form islands in the amorphous matrix (After [5]). Polarized (perpendicular and parallel) low-wavenumber Raman spectra of synthetic (polyamide 66 [PA66], a and b) and natural polyamide fibers (Nephila spider silk and Bombyx mori silk shown in c and d, respectively). In (d), the spectra have been recorded with a perpendicular setting in two different spots along the fiber (After [5, 7, 45])

Polymer materials are made of atoms (C, N, O) with a similar mass connected by strong covalent bonds. Bond lengths and angles are almost constant, but dihedral rotation is easy [12]; this can lead to orientation defects and hence to materials buckling easily. This is the origin of the poor crystallinity of these compounds. Ordered regions form within the amorphous matrix (crystalline “islands” of the Oudet model; Fig. 5b0 ) of some polymer materials, whereas in others the same polymeric chain belongs both to ordered (or even crystalline) and to amorphous regions (the Prevorsek model, Fig. 5a0 ). An intermediate state is called the amorphous “oriented” region [7, 39]. These different types of local structures contribute differently to the vibrational signature. The Raman bands of a crystalline material have a Lorentzian shape, that is, a narrow peak with elongated wings [7, 5] (Fig. 5a). Amorphous structures are characterized by a variety of short-range conformations – namely, a distribution of bond rotation angles. The vibrational spectrum reflects all the configurations, and bands get a Gaussian (or more complex) shape (see the dashed component in Fig. 5a). Most of the literature differentiates the signature of “ordered” and “disordered” parts by checking the band width and the shape of the internal (stretching and bending) modes of the backbone chains measured in the 800- to 2000-cm
1 range [3, 21, 23,

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Fig. 6 (a) Comparison of the infrared spectra recorded on a polyamide 66 [PA66] fiber, as received, saturated in water and D2O, and then dried to get an isotopic dilution of hydrogen among deuterated species (After [7]). (b) Evolution of the center of gravity of the X-H stretching mode for X=O and X=N as a function of the X (donor)–Y (acceptor) bond length (After [11] and [35])

31, 32, 38, 41, 48]. The main reason for this is instrumental: this spectral window can be easily measured with standard instruments. We showed that two other regions are much more informative: the low wavenumber range (3000 cm
1), where modes involving H-bonded vibrators are observed [7, 9] (Fig. 6). The latter modes are stronger in IR than in Raman spectroscopy. However, because of the small mass of protons, instant dipole couplings are important. Consequently, the band broadens, and a complex shape is common [4, 35]. The best technique to suppress coupling-induced broadening is isotopic dilution: replacing more than 80% of H atoms with deuterium (or tritium) atoms, which are two (or three) times heavier, suppresses most couplings, thereby drastically narrowing the bandwidth [6, 9]. The resulting bandwidth depends only on a short-range structural (temperature independent) or dynamic (temperature dependent) disorder. The case of PA66 fibre is shown as an example in Fig. 6. It became possible to distinguish contributions of ordered/crystalline (narrow component) and disordered/amorphous (broader component) regions of the material. The donor-acceptor bond length can be calculated from the band wavenumber using Novak or Gruger relationships [11, 35]. Except when X-H vibrators are present, especially when isotopic dilution can occur, Raman microspectroscopy is much more efficient than IR spectroscopy to study structural modifications induced by pressure, stress, or strain. Although some internal modes are sensitive to the crystallinity of polymers (in which broader

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features are observed for the signature of the amorphous part), the most spectacular effects are observed with low wavenumbers. Figure 5a–d compares lowwavenumber Raman spectra recorded for synthetic (PA 66) and natural (silkworm and spider silk) polyamides. The backbone chain of polyamides consists of an amide (–O=C–CH–N–H–) unit (separated by C–C bonds in PA66). Note that silk can be considered as one of the most “simple” polypeptide fibers and is a good model compound with which to study the mechanical properties of more complex fibrous polypeptides/polyamides such as ligaments and arterial walls [10]. A semicrystalline polymer consists of ordered and disordered macromolecular chains. Ordered and disordered/amorphous areas cannot always be separated because orientational disorder does not involve the formation of defined domains; the X-ray diffraction signature is similar for a mixture of ordered and disordered phases and for continuous paracrystals [27, 33]. Nevertheless, at the chemical bond scale – the scale of vibrational spectroscopy – contributions of chemical bonds with a rather well-defined environment and those with a disordered one are different and distinguishable. It is well established that some internal bonds peaking between 800 and 1600 cm
1 – for example, those involving the ring deformation of PET (approximately 995, 1097, 1175, and 1185 cm
1, the 1175 cm
1 peak of been associated to “amorphous” material) [3, 14, 23, 31, 38, 41, 48] – give information about ordered and disordered macromolecular chains. Aromatic rings serve as a short-range probe because of their strong Raman signatures. Similar features are encountered for other polymers [32], but the best information can be extracted from those with a low wavenumber [7, 21]. Synchrotron radiation and X-ray diffractions are alternative methods (albeit more expensive, with information limited to crystalline molecular chains [22]) that obtain similar results, as shown for hair [30] and single PBO fibers [20]. As explain earlier, the external/lattice modes of the vibrational unit are observed at low wavenumbers and depend on the long-range order. For crystalline materials, narrow modes are observed, and their number depends on the vibrational unit (generally identical to the crystallographic unit/cell) symmetry. The lower the symmetry class, the higher the number of modes/bands. Only one crystalline mode is observed at ~100 cm
1 for synthetic PA66 [7, 14] and at ~143 cm
1 for silks [8, 10, 19, 45]. More bands are observed for PET, PBO, Kevlar, and polypropylene fibres [7, 14]. The amorphous character of polymer materials results from orientational disorder. At a short range, the structure is not very different from that of a crystalline homologue, but because of the defects, the bond density by unit volume is lower; consequently, the mean distance between moving chains is a little larger and the corresponding translational modes less energetic. The amorphous equivalent of the crystalline peak appears at a lower wavenumber (Fig. 5). The elastic Rayleigh scattering depends on local heterogeneity, and strong Rayleigh wings are formed for amorphous phases. The low wavenumber spectrum of polyamide fibers is thus the superimposition of the components as a result of Rayleigh (Lorentzian) and Raman (amorphous [Gaussian] and crystalline [Lorenzian] components) scattering [7].

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Understanding Fracture Behavior

The evolution of different components as a function of applied perturbation (tensile stress/strain, hydrostatic pressure) informs the material’s nanostructure. Figure 7 compares the evolution of the low-wavenumber amorphous and ordered/“crystalline” component as a function of applied tensile strain/load for PET, i-PP, PA66, and fibers. In addition, the evolution of a νN-H vibrator is also given for silk and keratin fibers in Fig. 8 to show the large amount of information that can be extracted from Hbonded vibrators with very high wavenumbers. An example of behavior under hydrostatic pressure is given in Fig. 9. A comparison of macromechanical behavior (the stress vs. strain curve measured for a single fiber using a universal test machine [28]) with nanomechanical behavior (the shift of vibrational modes) shows the following results: – Behavior changes at macroscopic and nanoscopic levels are observed simultaneously. – Behaviors of amorphous/disordered and the crystalline/ordered signature are different and can be studied independently. Actually, it is important to note that the duration of vibrational and mechanical measurements differs. The strain-stress curve takes a few tens of seconds to measure, although the measurement of a Raman spectrum ranges between 15–30 min and 6 h, as a function of the fiber and spectrometer used (“modern” notches filters or “old” double monochromator instruments). The results presented here were established with different fiber number, between 2 (silk, Fig. 7d) and 6 (PA 66, Fig. 7b) fibers are measured, respectively. Although a similar behavior is observed for the crystalline and amorphous components in PA66 fibers, very different ones are measured for PET and i-PP fibers: no variation or a slight increase (Poisson effect resulting from the squeeze of the fiber diameter under tension) for the crystalline component of PET and i-PP polymers but a large wavenumber decrease (i.e., lengthening of the bonds) for the amorphous component. The different behaviors arise from the different nanostructures – Oudet and Prevorsek types, respectively (see Fig. 5). When the crystalline part forms islands, their bond behavior is affected very little or not at all; when macromolecular chains belong both to the crystalline and amorphous regions, however, as in a Prevorsek microstructure, the behaviors are similar. Comparison of PA66 from different producers demonstrated that most differences concern the amorphous part, and these differences explained large changes regarding fatigue behavior [39]. These results confirm that the amorphous part of semicrystalline polymer materials controls mechanical behavior. The lower density of bonds in an amorphous volume (i.e., more “voids” in amorphous regions) facilitates the relative motion of a chain with respect to others – a phenomenon at the origin of viscoelasticity. When such rearrangements are possible, the slope of the wavenumber shift increases drastically and the fracture limit arrives quickly.

Fig. 7 (continued)

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Silk and keratin fibers belong to polyamide fiber group but, contrary to synthetic polyamides (such as PA66), their side groups are variable and complex. In PA66 the side group is H (i.e., similar to a glycine residue); in silk and keratin, side groups are residues of amino acids (glycine, alanine, serine, tyrosine, phenylalanine, cystine, etc.). Short side chains allow a regular helix to form, and larger and rigid residues break the order; cystine residues lead to covalent S-S bridges between adjacent macromolecular chains [36], modifying the tensile mechanical behavior. Hence the S-S bridges couple the polyamide chains in keratin, and when a significant number of bridges break (Fig. 8a), the helix unrolls and lengthens (phase transition analog to Brill transition in synthetic polyamides) and a plateau is observed on the tensile stress/strain plot. Similar behavior is observed for silks, but the variability of the amino acid residue sequences along the fiber lead to a variety of mechanical behaviors (types 1 to 4), which are observed both at macro- and nanomechanical scales [8, 19, 45] (Fig. 8a, c). Indeed, similar behaviors are measured at macro- and nanoscales. To study small differences in behavior, H-bonded probes are the most informative because the large anharmonicity of the bond leads to a stronger wavenumber shift of the X-H vibrator (Fig. 6b). Figure 9 illustrates the potential of the technique to study compressive behavior. A small piece of fiber is cut and placed in a DAC chamber with a ruby ball (pressure probe). Helium gas is used as the pressure medium. The example is a PA66 fiber, but PET and PBO fibers have also been studied [16, 37]. Three vibrational probes are considered: (i) the crystalline component at ~100 cm
1 (the quality of spectra recorded through a DAC is not sufficient to allow extraction of the Rayleigh wing and the amorphous component), (ii) the amide I component at ~1634 cm
1, and (iii) the N-H modes at ~3300 cm
1 [13].

6

Understanding Fatigue Behavior

Figure 10 compares the shift of the amorphous low-wavenumber mode as a function of tensile stress; the data were measured postmortem at and close to the fatigue fracture tip of three fibers. Fibers that broke under the fatigue test exhibited a typical habit characterized by an elongated “tongue” (BC segment). As shown on the inset in Fig. 10, the residual (compressive/upward) stress shift reaches the maximum at point B (the fracture mirror) and decreases on both sides. If the maximal value is

ä Fig. 7 Comparison of the nano- and macromechanical behaviors of (a) polyethyleneterephthalate [PET], (c) isotactic polypropylene [i-PP], (c) polyamide 66 [PA66], and (d) silk fibers under uniaxial tension. On the right is shown the scale of stress applied to measure the macroscopic behavior (curve labeled “macro”); on the left is shown the wavenumber shift. The different behavioral domains are noted 1, 2, and so on (After [28, 39]). Comparison of the nanomechanical (Raman shift of ~143 cm
1 T0chain mode of crystalline islands) and macromechanical behavior for two Nephila spider silk fibers (d); the dashed line corresponds to a one-shot tensile measurement (of a few minutes’ duration), whereas the solid line is the stress/strain plot measured during the Raman measurement (30 min for each spectrum). Note the stress relaxation during the Raman counting (After [45])

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Fig. 8 Natural polyamide fibers. (a) Evolution of the intensity of an S-S peak normalized to that of a phenyl group (relative humidity of 40–50%), as a function of the applied strain for a Caucasian hair; S-S bond bridges adjacent polyamide chains). Comparison between the macroscopic stress/ strain curve and the wavenumber of the νN-H (a and c) and νAmide I probe (bottom curve in (a)) (After [36]). (b) Different types of mechanical macroscopic behavior observed for silk under uniaxial strain. (c) Similar behaviors are observed at the chemical bond scale; the νN-H bond was used as the probe (After [8])

plotted on the tensile plot, the position corresponds to the intersection between the y axis and the straight line issue of the slope measured after the plateau. In other words, at the fracture, the plateau that arises from the ability of the amorphous chains to rearrange under tension easily (i.e., without energy cost) disappears. This explains why different fiber products with similar ultimate tensile strengths and Young moduli but different Raman shifts versus tensile strain/stress behaviors have very different characteristics when fatigued [7, 28].

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Fig. 9 Wavenumber shift measured on a polyamide 66 fiber under compression (Diamond Anvil Cell) regarding νN-H (3300-3200 cm-1), νAmide I (~1634-1648 cm-1) anf νT' chain modes (~100-150 cm-1); solid and dash lines are guides for eyes; the lines correspond two data exctrated with different fittng and illustrate the error (After [13])

7

Conclusion

This miniature review illustrates the potential of vibrational studies of materials under tensile or compressive stress. The good spatial resolution of Raman spectroscopy (~0.5 μm), even with a microscope objective with a long working distance, allows precise measurements in various environments. The case studies reported here deal with fibers, but similar work can be conducted using films or membranes (e.g., Nafion polymer used as an electrolyte of fuel cells and electrolyzers). Load sensors up to 100–200 g are convenient to study single fibers [33, 34, 39], but sensors supporting a much higher load (500–3000 g) are required to study films [10] and machines supporting 10 kg or more are required to study composite coupons [10, 15] and a variety of composite materials. IR can be also used, but the spatial resolution is less (50–20 μm) and the broader bands make the measurements less reliable (Table 1).

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Fig. 10 Comparison of Raman shifts measured postmortem at the tip of three polyamide 66 [PA66] fibers fractured by fatigue (point B wavenumber residual shift; the inset shows the variation of the wavenumber of the amorphous component, ~ 100 cm
1), with the plot of the wavenumber position of the amorphous component at a different level of strain (After [7, 28]) Table 1 Raman S (cm
1 GPa
1) compression factor for different fibers PA66 [13]

PET [16]

PBO [37]

T0chain 7.5 4.5 νEthylene glycol

νN–H
17.5
7 δC–H

νAmide I 0 1.6 νC–C ring & C (O)–O

segment

0.3 νPhenylene ring þ δbenzene ring þ νC–C0 0.7 0.15

0.7 δphenylene ring

2.45

0.7 0.25

Remark 1 nm) vacancy clusters of h100i type was found to be significant. Indeed, the formation of loops and in particular the balance between the amount of 1/2h111i and h100i dislocation loops is known to depend on the irradiation conditions, temperature, and grain orientation. This has been extensively investigated experimentally in pure Fe and Fe ferritic alloys [75–77].

4.8

Role of Stress

Materials in-service conditions are far from being stress free, and the impact of stress and strain on the primary damage has also been investigated for some time. Studies have focused on the effect of hydrostatic, uniaxial, and bain strain on the formation and migration energies of point defects, the TDEs, and defect production in cascades. Twenty five years ago, Kirsanov et al. [79] showed that external stresses impact on the TDEs as well as on the RCS path and shape. As a consequence, they observed that the cascade volume size and the number of defects decreased and more compact vacancy clusters were obtained at the center of the cascade. Nonuniform stresses were shown to lead to more significant changes than uniform stresses. Strain impacts on the point defect formation energies and as a consequence on the number of point defect created during the cascade [80], the impact being smaller for uniaxial strain than hydrostatic strain [81, 82]. The strain field is also a key factor for SIA orientation [83], which is essential for its migration in Fe. Indeed, Fe is the only bcc metal whose stable SIA is oriented along a h110i direction because of magnetism. As a result, for an Fe SIA to move, it has to rotate to align along the h111i direction and the impact of strain on this mechanism is thus obvious. Zolnikov showed also that the occurrence of plastic deformation and more precisely twinning under irradiation depends on the applied load [84].

5

Primary Damage Evolution

The evolution of the point defects created by the PKAs, i.e., the annealing of the cascade, its interaction with other cascades, solute atoms and elements of the microstructure lead to the evolution of the microstructure and thus to the property changes of the materials. To extend the study up to the formation of experimentally resolvable damage features (point defect clusters, dislocation loops, solute precipitates. . .), one turns to computational tools based on KMC or on mean field rate theory (MFRT) [85]. KMC methods can be applied to objects (OKMC [86–89]), events (EKMC [90]), or atoms (AKMC [91]) in a specific volume. These techniques

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can be globally defined as coarse-grained models because atoms are not explicitly treated in either except in the atomistic (or lattice) KMC, i.e., the AKMC. Such simulations produce size distributions of experimentally resolvable defects (voids, loops, precipitates . . .), including defects with sizes lower than the resolution achievable experimentally. Another interest of these methods is to provide the complete evolution of the microstructure with time, whereas most of the time, for practical reasons only a few “snapshots” of the microstructure is obtainable during its evolution in experimental settings. In the worst case, one has only access to the final microstructure. The debris from individual cascades form the “source term” of larger scale models of the evolution of the microstructure under irradiation that must reach doses of several dpa to tackle fission and fusion relevant questions. In contrast with OKMC methods, in MFRT models, in order to take into account defect spatial correlations after the cascade, the cascade debris must be annealed before they are introduced and the annealing time is somewhat empirical. In [92], a space homogenization method based on the annealing of individual cascades using OKMC was derived that takes into account the amount of “pristine material” available before the introduction of the cascade debris. During the cooling down of the cascade, many recombinations take place, and the radiation-induced microstructural and compositional changes in solids are governed by the interaction between the fraction of defects that escape their nascent cascade and the material [93]. The amount of freely migrating defects has been estimated using OKMC [87, 93–95], and it was shown that the migration properties of point defects, i.e., the relative mobilities of point defects and point defect clusters of both types (vacancies and SIAs) and the dimensionability (1D versus 3D migration) played a crucial role. In particular, the migration properties of point defects and point defect clusters, clearly affect the fraction of interstitials that escape the cascades, the time required for clustering onset [87, 94, 96] and in general the long-term evolution of the microstructure. These results are confirmed in an indirect way in a very recent KMC simulation of loop formation in thin Fe films and in bulk Fe. Indeed, as mentioned above, the formation of h100i loops in Fe is an extensively studied problem. The formation of these loops is found inside MD cascades [51] in the vicinity of a surface but it can also be due to a 1/2h111i loop interaction mechanism as proposed in [78], and it appears to depend also on the presence of impurities that affect the defect migration [74]. Only long-term simulations of accumulation of cascades can reveal the sensitivity of the system to the properties of the displacement cascades (their spatial extent, morphology, and the spatial correlation of defects) in the evolution of point defect cluster size distributions. It has been shown that there is no simple answer and the sensitivity can be weakly or strongly dependent on other parameters such as the mobility of defects, the type of interactions, the presence of impurities, and temperature [97, 98]. Finally, one can add that the first stages of precipitation may occur during cascade annealing as shown in a work coupling OKMC and AKMC methods to study the evolution of the primary damage in FeCu alloys [99]. OKMC on isolated cascade debris provided the SIA-vacancy recombination ratio and the description of the interstitial clusters escaping from the cascade core and thus participating in the

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defect flux in the materials. AKMC gives then the interaction of the vacancies with the solute atoms and shows the formation of vacancy and copper clusters as well as vacancy-copper complexes in the cascade area. A single 20 keV cascade typically leads to the formation of more than one large object, probable nuclei of the precipitates, or the dilute atmospheres observed to form under irradiation. The number and size of these objects were shown to strongly depend on temperature. At low temperatures, small clusters cannot emit vacancies and only local rearrangements by vacancy jumps and slow diffusion can take place. One observed thus the formation of small mixed objects (Cu-V complexes or clusters). At intermediate temperatures (temperatures close to the vessel in-service temperature), vacancies from the smallest clusters or complexes are emitted and contribute to the growth of larger mixed objects. At even higher temperatures, vacancies do not bind with Cu atoms long enough to transport them far and no large Cu clusters did form.

6

Future Directions and Open Problems

In conclusion, the simulation of the radiation damage made a lot of progress, despite recurrent questions such as the impact of electronic effects, the accuracy of the cohesive models (equilibrium part, short-range interaction part), the effect of the changing microstructure, or the lack of statistics. The two techniques used to model the primary damage, BCA, and MD are complementary. The BCA allows to treat very high energy PKA whereas MD is more appropriate to determine point defect structures and the primary damage for lower PKA energies, but the space and timescales that it can cover are limited. One of the most important issues is the interatomic potential. The primary damage, i.e., the amount and structure of the defects produced, can be very sensitive to it as shown for Fe in [24]. This has been confirmed by other studies in Fe [55, 102], or in W [103]. The displacement (sub)cascades initiated by the PKA evolve too quickly, and their sizes are generally too small to be observed experimentally. The exception is the case of W which was investigated with the help of a field ion microscope [104]. It is thus almost impossible to compare the results obtained with experimental data, except in a few dedicated experiments. Furthermore, the primary damage depends on both the equilibrium part and the hardened part of the potential, with certainly a large contribution from the “connection” between the ZBL and the equilibrium part. In particular, the potential energy range from the FP formation energy (typically 10 eV in metals) up to few times the TDE (typically 100–200 eV in metals) has to be adjusted with caution, but it is not clear how to do it properly. Progress is done thanks to DFT results which can be used to harden the potential or to assess the hardening [106, 27]. The equilibrium part is also important as defect and defect clusters properties, or the shock and heat waves propagation properties, need to be predicted accurately. However due to the difficulty of adjustment on all the known properties of the material, one has to select those that seem essential, a notion that may evolve in time, as with the recent discovery of the C15 SIA clusters in Fe [107]. Furthermore, if reliable potentials for pure metals, i.e., ideal perfect

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materials, are not so easy to build, moving to real materials, i.e., dilute or concentrated alloys such as FeNiCr or high entropy alloys becomes an extremely complex task. Sensitivity studies are necessary in order to have more confidence on the quality of the results and also to determine the impact of each part of the potential on the primary damage, a fact still not fully understood. The solution to this problem is nowadays to assess the validity of the results obtained by using different interatomic potentials. Clarifying the effect of electrons and electron–phonon interactions on the primary damage in terms of number of defects and defect size distributions also made a lot of progress, thanks to theoretical models and DFT calculations. However, no straightforward method can easily be implemented in a MD code yet. The defect number produced as a function of the PKA energy at the end of the displacement cascade is rather easy to obtain with a reasonable accuracy, whereas the defect size distributions which have an impact on the evolution of the microstructure require large statistics and a large number of MD cascades (several hundreds of cascades per conditions typically). BCA calculations are much faster than MD, but unfortunately, these distributions cannot be obtained by BCA. To get good statistics, large computing resources are necessary and/or smart MD simulation of the cascades. Two very recent and innovative methods have been proposed to speed up the calculations. The first one is the cell molecular dynamics for cascades proposed by Crocombette [100]. In this method, the simulation box is divided into cells. During the simulation, classical MD is done only in “activated cells” where the activation criterion is based on the kinetic energy of the atoms of the neighboring cells. The speed-up permits the calculation of high energy cascades and the accumulation of significant statistics. In the second method, proposed by Ortiz [101], the idea is to use the BCA at the beginning of the cascade calculation where the time-step is small because of the high speed of the initial projectile and the energetical knock-onatoms. Indeed these atoms will only hit very few quasi static atoms, the collisions with which are well modeled by the scattering formula. The criterion of switching from BCA to MD is the energy of the knock-on-atom. Of course, the larger the BCA “window,” the larger the speed up but at the cost of accuracy, which is not so easy to quantify due to the stochastic nature of the displacement cascades. From the point of view of experiments, experimental validation/assessment of the primary damage could be revisited taking advantage of the progress achieved in the experimental techniques as well as the simulations possibilities. Very low temperature irradiation (ideally at liquid He temperature in order to minimize any microstructure evolution by diffusion) would allow to produce the primary damage, which could then be analyzed with different techniques, such as TEM to identify possible (TEM visible) loops formed, or X diffraction techniques to analyze small defects and defect clusters [105]. These observation techniques have to be combined with isochronal annealing experiments in order to better assess the different mechanisms taking place when a large population of point defects has been introduced in a material. Indeed, in metals with high point defect migration energy, conditions where point defect diffusion cannot take place can be found (e.g., Fe or Ni, where SIA migration barriers are larger than in W).

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Acknowledgements This work was partly supported within the European project SOTERIA (661913) and by CEA under the collaborative contract number V 3542.001 on fusion engineering issues. It contributes to the Joint Programme on Nuclear Materials (JPNM) of the European Energy Research Alliance (EERA).

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88. Soneda N, Ishino S, Takahashi A, Dohi K. Modeling the microstructural evolution in bcc-Fe during irradiation using kinetic Monte Carlo computer simulation. J Nucl Mater. 2003;323:169–80. 89. Martin-Bragado I, Rivera A, Valles G, Gomez-Selles JL, Caturla MJ. MMonCa: an object Kinetic Monte Carlo simulator for damage irradiation evolution and defect diffusion. Comput Phys Commun. 2013;184:2703–10. 90. Dalla Torre J, Bocquet J-L, Doan NV, Adam E, Barbu A. JERK, an event-based Kinetic Monte Carlo model to predict microstructure evolution of materials under irradiation. Philos Mag. 2005;85:549–58. 91. Becquart CS, Soisson F. Kinetic Monte Carlo simulations of precipitation under irradiation. This Book. 92. Jourdan T, Crocombette J-P. Rate theory cluster dynamics simulations including spatial correlations within displacement cascades. Phys Rev B. 2012. https://doi.org/10.1103/ PhysRevB.86.054113. 93. Soneda N, de la Rubia TD. Defect production, annealing kinetics and damage evolution in αFe: an atomic-scale computer simulation. Philos Mag A. 1998;78:995–1019. 94. Xu H, Osetsky YN, Stoller RE. Cascade annealing simulations of bcc iron using object kinetic Monte Carlo. J Nucl Mater. 2012;423:102–9. 95. Nandipati G, Setyawan W, Heinisch HL, Roche KJ, Kurtz RJ, Wirth BD. Displacement cascades and defect annealing in tungsten, Part III: the sensitivity of cascade annealing in tungsten to the values of kinetic parameters. J Nucl Mater. 2015;462:345–53. 96. Xu H, Stoller RE, Osetsky YN. Cascade defect evolution processes: comparison of atomistic methods. J Nucl Mater. 2013;443:66–70. 97. Souidi A, Hou M, Becquart CS, Malerba L, Domain C, Stoller RE. On the correlation between primary damage and long-term nanostructural evolution in iron under irradiation. J Nucl Mater. 2011;419:122–33. 98. Becquart CS, Souidi A, Domain C, Hou M, Malerba L, Stoller RE. Effect of displacement cascade structure and defect mobility on the growth of point defect clusters under irradiation. J Nucl Mater. 2006;351:39–46. 99. Domain C, Becquart CS, van Duysen JC. Kinetic Monte Carlo simulations of cascades in Fe alloys. MRS Online Proc Libr Arch. 2000. https://doi.org/10.1557/PROC-650-R3.25. 100. Crocombette J-P, Jourdan T. Cell Molecular Dynamics for Cascades (CMDC): a new tool for cascade simulation. Nucl Instrum Methods Phys Res Sect B Beam Interact Mater At. 2015;352:9–13. 101. Ortiz CJ. work in progress. 102. Terentyev D, Lagerstedt C, Olsson P, Nordlund K, Wallenius J, Becquart CS, Malerba L. Effect of the interatomic potential on the features of displacement cascades in α-Fe: a molecular dynamics study. J Nucl Mater. 2006;351:65–77. 103. Fikar J, Schäublin R. Molecular dynamics simulation of radiation damage in bcc tungsten. J Nucl Mater. 2009;386–388:97–101. 104. Seidman DN. The direct observation of point defects in irradiated or quenched metals by quantitative field ion microscopy. J Phys F Met Phys. 1973;3:393–421. 105. Ehrhart P. Investigation of radiation damage by X-ray diffraction. J Nucl Mater. 1994;216:170–98. 106. Olsson P, Becquart CS, Domain C. Ab initio threshold displacement energies in iron. Mater Res Lett. 2016;4:219–25. 107. Alexander R, Marinica M-C, Proville L, Willaime F, Arakawa K, Gilbert MR, Dudarev SL. Ab initio scaling laws for the formation energy of nanosized interstitial defect clusters in iron, tungsten, and vanadium. Phys Rev B. 2016. https://doi.org/10.1103/PhysRevB.94.024103.

Monte Carlo Simulations of Precipitation Under Irradiation

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Charlotte S. Becquart and Frédéric Soisson

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Master Equation: Dynamical Interpretation of Monte Carlo Simulations . . . . . . . . . . . 2.2 Cohesive and Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 AKMC Simulations of Precipitation Without Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 AKMC for Radiation Damage: Cascade Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Acceleration of Precipitation Versus Ballistic Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Flux Coupling: Radiation-Induced Segregation and Radiation-Induced Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Examples of Complex Systems: Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Beyond AKMC: Towards Larger Space and Timescales: OKMC/EKMC . . . . . . . . . . . . . . . . 5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Towards More Physical Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Reaching Realistic/Useful Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

705 706 706 708 712 712 714 715 717 719 720 723 723 725 726 727 727

C. S. Becquart University Lille, CNRS, INRA, ENSCL, UMR 8207, UMET, Unité Matériaux et Transformations, Lille, France EM2VM, Joint laboratory Study and Modeling of the Microstructure for Ageing of Materials, Lille and Moret sur Loing, France e-mail: [email protected] F. Soisson (*) DEN-Service de Recherches de Métallurgie Physique, CEA, Université Paris-Saclay, Gif-surYvette, France e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_24

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Abstract

Atomistic kinetic Monte Carlo (AKMC) is a powerful technique to study the microstructural and microchemical evolution of alloys controlled by diffusion processes. AKMC simulations are thus ideal tools to study precipitation, under irradiation and during thermal aging. In this chapter, we briefly present the method, underlining the different hypotheses usually made in the studies which have been done so far and the increasing contribution of density functional theory (DFT) calculations. We then proceed to present several simulations of the first stages of precipitation that can be quantitatively compared with experimental studies, in order to show the complexity introduced by the irradiation. We move to the mesoscale and introduce event kinetic Monte Carlo (EKMC) and object kinetic Monte Carlo (OKMC) methods which until now have mostly dealt with point defect cluster distributions in pure metals or “gray alloys” and were thus not really appropriate to study precipitation. However, they can be coupled with AKMC to speed up the calculations and recent developments take into account solute atoms more explicitly. We expose then recent advances that relieve some of the simplifying assumptions of standard AKMC models and conclude with a few challenging issues that we feel need to be addressed to predict correctly the behavior of alloys under irradiation but have been barely introduced in the models. Keywords

Kinetic Monte Carlo · Kinetics of precipitation · Irradiation effects · Atomistic simulations Acronyms

ABC AKMC ANN bac-MRM CRP DFT EKMC HSLA KMC NEB ODS OKMC PD RIP RIS RPV SIA

Autonomous basin climbing Atomic kinetic Monte Carlo Artificial neural network Basin auto-constructing mean rate method Copper-rich precipitates Density functional theory Event kinetic Monte Carlo High-strength low-alloy Kinetic Monte Carlo Nudged elastic band Oxide dispersion strengthened Object kinetic Monte Carlo Protective domain Radiation-induced precipitation Radiation-induced segregation Reactor pressure vessel Self-interstitial atom

22

1

Monte Carlo Simulations of Precipitation Under Irradiation

705

Introduction

Irradiation may affect precipitation, and more generally phase stability, in many different and often apparently contradictory ways. Depending on irradiation conditions, experimental observations have revealed cases of dissolution or fragmentation of stable precipitates, but also the accelerated precipitation of stable precipitates or the precipitation of unstable phases; the disordering of ordered precipitates or the accelerated ordering of precipitates, the amorphization of crystalline precipitates or the crystallization of amorphous phases, as well as complex interactions between the precipitates and the other elements of the microstructure such as grain boundaries and dislocations. Extensive reviews can be found in Refs. [1, 2]. Since mechanical properties of materials are frequently controlled by the microstructure of precipitation this is of great industrial importance (see ▶ Chap. 7, “Multiscale Modeling of Radiation Hardening”). Irradiated alloys are non-equilibrium systems, subjected to permanent fluxes of particles and energy and in general their evolution cannot be predicted by thermodynamics. A key difference with the usual situation is that not only the kinetic path but the final steady-state – if it exists – depends on the radiation flux and on the very details of atomic transport mechanisms. The latter are much more diverse under irradiation: in a typical substitutional alloy, in addition to the usual thermally activated vacancy jumps they include, for example, the athermal displacements of atoms within displacement cascades, usually referred to as ballistic mixing, the migration of self-interstitials of various configurations, the recombination of a vacancy and a self-interstitial, the elimination of point defects at sinks. Several approaches have been proposed to deal with the situation. Phenomenological models have been proposed for many experimental observations. On the opposite, a general approach has been proposed by Martin and coworkers to predict the steady-states of “driven alloys” [3] from effective potentials, playing a role similar to equilibrium thermodynamic potentials (but including some forcing and diffusion parameters). But so far analytical expressions have been obtained only for most simple systems and diffusion mechanisms (e.g., direct exchanges between atoms). Therefore, numerical simulations based on the atomic description of all the relevant diffusion mechanisms appear to be the most practical way to develop predictive models of precipitation under irradiation. Molecular dynamics simulations have given invaluable insights on radiation damage mechanisms, such as displacement cascades [4]. But with time steps of 10
15 s, they are limited to short-time evolutions (a few nanoseconds at best). Precipitation results from thermally activated jumps of point defects, with migration barriers between a few tenths and a few eV, and takes place over much larger times. Kinetic Monte Carlo simulations (see recent reviews in [5–7]) have shown to be especially useful to model systems of a few million atoms on the appropriate timescale because they deal with the jumps of atoms or point defects, not the lattice vibrations. One usually distinguishes atomistic kinetic Monte Carlo (AKMC) simulations where all the atoms are modeled and object- and event-kinetic Monte Carlo (OKMC and EKMC) simulations, where only special objects (solute atoms, point

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defects, and clusters) are considered. AKMC simulations are more precise but more time consuming. The basic problem is, for this approach, to give a realistic description of the relevant atomic displacement mechanisms: the displacements directly resulting from radiation damage, the point defect elimination mechanisms and the thermally activated jumps of point defects, including the way the migration barriers depend on the local atomic environment. In OKMC and EKMC simulations, the key point is to determine the effective parameters from atomic models as well as all the relevant mechanisms that can happen. This chapter reviews some recent AKMC simulations of precipitation in alloys under irradiation. We start with a brief introduction on Monte Carlo methods and algorithms (Sect. 2), using the Master Equation approach to explain the connection between the atomic jump frequencies and the kinetic path of the alloy, and especially why it evolves – or not – towards an equilibrium state (Sect. 2.1). We present the most common methods used to compute the jump frequencies in AKMC simulations in (Sect. 2.2), and stress out the recent contributions of ab initio calculations, and rigid lattice approximations – which are so far most commonly used to model precipitation under irradiation. Recent applications of KMC are presented in Sect. 3: first a brief review on precipitation during thermal aging – a necessary validation step, then simulations of alloys under irradiation. The main objective of this paragraph is to show how the puzzling diversity of behaviors experimentally observed can be rationalized from atomic mechanisms. In Sect. 4, we move to the mesoscale and introduce EKMC and OKMC methods which until now have mostly concentrated on point defect cluster distributions in pure metals or “gray alloys,” but which take into account solute atoms more explicitly in recent developments or can be coupled with AKMC to speed up the calculations. We conclude with a few challenging issues that we feel need to be addressed but have been barely introduced in the models so far.

2

Methods

2.1

Master Equation: Dynamical Interpretation of Monte Carlo Simulations

The master equation gives the evolution of the probabilities Pi(t) to find a physical system in a configuration i, with an internal energy Ei [8]:  dPi X  ¼
W ij Pi ðtÞ þ W ji Pj ðtÞ , dt j

(1)

Where Wij is the probability of transition per unit time between configurations i and j. For “Markovian processes,” the transitions rates are time independent and only depend on i and j. In the following, a configuration i will typically correspond to a given distribution of atoms and a few point defects on a perfect lattice. If the transition mechanism is ergodic and fulfills the detailed balance principle:

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Monte Carlo Simulations of Precipitation Under Irradiation

707

  eq W ij Pj E i
Ej ¼ eq ¼ exp , W ji Pi kB T

(2)

the system evolves towards its equilibrium state, i.e., the equilibrium (Boltzmann) distribution: Peq i

  1 Ei ¼ exp
Z kB T

(3)

where Z = i exp (
Ei/kBT) is the partition function. As each term in the sum of Eq. (1) cancels, dPeq i ðtÞ=dt ¼ 0: the equilibrium distribution is a stationary solution of the master equation. Monte Carlo simulations can be used to generate series of configurations according to these rules. To get equilibrium properties, one can choose any mechanism that fulfills the detailed balance condition (2). In practice, one can choose a mechanism that minimizes the computation cost (e.g., permutation between two atoms anywhere in the simulation box). But to simulate a precipitation kinetic path, the transition mechanisms i ! j and the transition rates Wij must be realistic. Typically in metallic alloys, the transitions occur by jumps of point defects and the transition rates are determined by their jump frequencies. Note that in alloys under irradiation (as in other driven systems) some transition mechanisms do not fulfill the detailed balance: the displacements and replacements occurring in displacement cascades, the recombination between vacancies and selfinterstitials, and the annihilation of point defects at sinks. Therefore, the system does no evolve towards an equilibrium state. In some cases, the system may reach steadystates – but in general the latter, contrary to the equilibrium states, will depend on the details of the transition mechanisms and transition rates. A detailed description of the point defect formation, migration, and elimination is thus especially important.

2.1.1 AKMC Algorithms In the standard equilibrium Metropolis algorithm, at each Monte Carlo step, one possible event i ! j is chosen and performed if:   ΔEij R < exp
kB T

(4)

where R is a random number between 0 and 1. The difference ΔEij must fulfill the detailed balance, and usually it is simply chosen as ΔEij = Ej
Ei (the energy of the final state minus the energy of the initial state). The Metropolis algorithm can be adapted to the simulations of the kinetic pathway, using realistic i ! j transition mechanisms and rates. A physical timescale may be defined from the attempt frequency v0 of the point defect jump frequencies (see Sect. 2.2) and the time of one Monte Carlo step, defined as one attempt per possible jump, is 1/v0. Contrary to the standard equilibrium Metropolis algorithm, the true migration barrier ΔEij = ESP
Ei must be used, where ESP is the energy of

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the transition state or saddle-point, not the difference between the final and the initial energies. The advantage is that only one barrier is computed at each step. The drawback is that with migration barriers of at least a few tenths of eV, which are typically encountered in solid-state diffusion, most of the exchanges are rejected. In residence time algorithms, one jump is chosen using a random number, according to relative probabilities given by the frequencies Γi of all the possible jumps. The associated time step is computed as 1/iΓi (the sum runs over all the possible jumps) as in the first implementation of the residence time algorithm [9] or, more rigorously, as
R/iΓi using a random number R between 0 and 1 [10]. Contrary to the Metropolis algorithm, one jump is necessarily accepted at each step, but for that one must compute the frequencies of all the possible jumps.

2.2

Cohesive and Diffusion Models

As mentioned in the introduction, precipitation results from thermally activated jumps of point defects. AKMC simulations of precipitation model thus the motion of those point defects, unlike the first simulations where direct exchanges between atoms were done (see many examples in Ref. [8]). This early approach is still valid in simple cases [8, 11, 12]; however, it cannot be applied in the case of radiation damage where point defect fluxes drive the microstructure evolution. Let us consider the relatively simple case of a vacancy in a substitutional A-B alloy. According to the harmonic transition state theory [13], the frequency of the exchange between a vacancy V and one neighboring atom A is given by: 3N Q

ΓAV ¼

vi

i¼1 3N
1 Q i¼1

Emig exp
AV kB T 0

! (5)

vi

where vi and v0i are respectively the normal frequencies of vibration of the system of N mig atoms in the initial stable configuration and at the stable-point. EAV ¼ ESP
Ei is the energy of migration, or migration barrier, i.e., the difference between the energies of the system at the saddle-point and in its initial configuration i. In an alloy, both Emig AV and the pre-exponential factor of the jump frequency depend on the local environment, i.e., on the distribution of A and B among the neighbors of the vacancy and the exchanging atom. Equation (5) may be rewritten as: mig

ΓAV

S ¼ v0 exp AV kB

!

mig

E exp
AV kB T

!

mig

F ¼ v0 exp
AV kB T

! (6)

mig mig mig where v0 is the “attempt” frequency, Fmig AV ¼ EAV
TSAV and SAV the free energy and entropy of migration. At constant pressure, one should indeed consider the enthalpies and free enthalpies of migration, but the difference is often negligible in

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709

solids. The jump frequencies obtained using this theory appear to be a very good approximation for the real jump frequencies up to at least half the melting point in most solid materials [14]. mig Usually, the vibrational, electronic, or magnetic entropic contributions in SAV are directly computed (or neglected), and not explicitly handled by the KMC method which is only used to explore the atomic configurations: in such as case, the KMC method presented in Sect. 2.1 is indeed used with the free energy of each configuration instead of the energy. The jump frequencies of direct interstitials (e.g., C or N in iron) or of the self-interstitials created by irradiation with their typical dumbbell configurations are given by similar expressions.

2.2.1 First Principles Calculations The most precise way to compute jump frequencies is to use first principles methods. With the generalization of standard codes based on the Density Functional Theory mig (DFT), it is nowadays possible to get a reliable estimation of EAV and even of the pre-exponential term (at a higher computational cost) for a given atomic distribution. The migration barriers can be computed using methods such as the nudged elastic band (NEB) [15], the dimer [16], autonomous basin climbing (ABC) [17] or eigenvector following methods such as ART-nouveau [18, 19]. However, these methods are computationally very demanding when applied to DFT calculations. Direct estimations of tracer diffusion coefficients, which require the calculations of only a few jump frequencies give results in good agreement with experimental data [20–23]. But in concentrated alloys, it is not possible to compute and tabulate the jump frequencies corresponding to all the possible environments (especially in multicomponent alloys and when the interactions exceed the first nearest neighbors). These methods are also too time consuming to be used “on-the-fly” in AKMC simulations that often require billions of point defects jumps. In practice, simpler models based on strong approximation must be made, but DFT calculations are nevertheless useful to fit the corresponding parameters. 2.2.2 Interatomic Potentials Empirical or semi-empirical interatomic potentials can be used to compute the migration barriers, using the same techniques as with first principles calculations (i.e., NEB, the dimer, the ABC methods). The computational cost decreases dramatically and on-the-fly calculations of the barriers become possible. Off-lattice AKMC simulations using interatomic potentials can take into account atomic relaxations, long-range elastic interactions and, in principle, can deal with non-coherent precipitates and point defect clusters. Another advantage for multiscale modeling is that a same potential can be used in molecular dynamics and AKMC simulations, thus decreasing all the uncertainties inherent to empirical models. In principle, the only limitation of this approach is the quality of the potential. Interatomic potentials can be significantly improved by fitting their parameters on DFT calculations; however, it remains difficult to develop potentials giving a good description of all the properties controlling the precipitation (the phase stability, the

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point defect formation, migration properties, etc.), especially in concentrated and multicomponent alloys. For both interatomic potentials and DFT approaches, the vibrational entropic contribution must be computed for each jump if one wants to get consistent thermodynamics properties. This is possible [24] but still time consuming and practically this is not done in simulations of precipitation kinetics. A typical consequence of neglecting the vibrational contribution may lead to significant underestimation of solubility limits (see e.g., a discussion for Fe-Cr alloys in [25]).

2.2.3 Effective Interactions and Rigid Lattice Approximations Due to the previous difficulties, the simplest approach is to compute the jump frequencies using simple “broken-bond models,” i.e., effective interactions on a rigid lattice. Cluster expansion methods can be used to get pair and many-body interactions [11] although the models used for precipitation under irradiation are often limited to pair interactions, between species i and j at nth neighboring positions (including atoms and defects) [5, 6, 26]. By comparison with interatomic potentials, the method has several advantages. On-the-fly calculations on jump frequencies are very rapid. The cohesive and diffusion models are often simple enough to derive analytical expressions of the alloy free energies, diffusion coefficients, or other properties controlling the precipitation kinetics. It is possible to take into account vibrational or magnetic contributions through temperature dependence of the effective interactions [25, 27] although these contributions have received less attention and are often simply included as a constant correction of the attempt frequency v0. Another advantage is that some parameters can be fitted independently on a given property, which makes the parameterization much simpler than with empirical potentials. Limitations are that the effective interaction models, especially when limited to pair interactions, are too simple to give a full and reliable description of the energetic landscape. Moreover, the rigid lattice approximation prevents a full description of atomic relaxations of elastic interactions, and it is impossible to deal with incoherent precipitation. 2.2.4 Saddle-Point Models Additional approximations are often used for the description of the saddle-point within the previous models [5, 6, 26]. According to the transition state theory, the mig ini migration barrier EAV ¼ ESP tot
Etot corresponds to the difference between the total energy of the system at the saddle-point and the initial configurations. This means that when one computes a migration barrier with an interatomic potential, one should compute the energy of the system at the SP (including relaxations, to find the true minimum barrier). For the sake of simplicity, this is often replaced by an extrapolation of the barriers from the properties of the initial and final configurations. A common approximation (model (1) on Fig. 1) is to write the migration barriers as [28, 29]:

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  mig ini Fig. 1 Saddle-point models. (1) Model with an average barrier Q: EAV ¼ Efin tot
Etot =2 þ Q P ðnÞ P ðnÞ ini and Efin tot
Etot ¼ ij eij ðfinÞ
ij eij ðiniÞ . (2) Model with pair interactions at SP. ContribuP ðnÞ tions 1: broken pairs in the initial configuration jk eik ðiniÞ, 2: non-broken bonds, 3: bonds created P ðpÞ at the SP: ieAi ðSPÞ

mig

EAV ¼

ini Efin tot
Etot þQ 2

(7)

where Q is a constant or  can depend  on the environment as proposed in [30]. The SP fin SP ini energy is then Etot ¼ Etot þ Etot =2 þ Q. The model fulfills the detailed balance (2) – and therefore the system must be driven towards its proper equilibrium state. But it imposes a somewhat arbitrary correlation between the barriers and the energy of stable positions. This may affect both the precipitation kinetics pathways and under irradiation, the possible steady-states. In broken-bond models, this approximation is often referred to as the “Kinetic Ising model” and Eq. (7) becomes: X mig

EAV ¼

ðnÞ

eij ðfinÞ


ij

X ij

2

ðnÞ

eij ðiniÞ þ Q,

(8)

where the two sums correspond, respectively, to the bonds created in the final configuration and the bonds broken in the initial one. Another approximation (model (2) in Fig. 1) is to introduce special effective ðpÞ interactions eAi (with ranges p) at the saddle-point: the migration barrier is then the difference between the bonds created at the saddle-point and the bonds destroyed in the initial position around the atom and the vacancy:

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EAV ¼

X i

eðAipÞ ðSPÞ


X

ðnÞ

eik ðiniÞ

(9)

jk

note that it does not depend on the bonds created in the final configuration. Equation (9) also obeys the detailed balance, takes explicitly in to account the configuration around the saddle-point, and gives more flexibility to fit the barriers computed by DFT. Equation (8) tends to reduce the variations of the barriers with the local environment since they only deviate strongly from the average value Q when the energies of the final and initial states are very different. This imposes an arbitrary condition, but may also prevent the apparition of strong and unphysical variations, which are possible with Eq. (9) when the SP pair interactions are fitted on a limited set of DFT barriers.

2.2.5 Point Defect Concentrations and Time Estimate The kinetics of precipitation does not depend only on the jump frequencies of vacancies and self-interstitials but also on their numbers, the diffusion coefficients of atoms being proportional to point defect concentrations (see Sect. 3.3). For thermal aging, AKMC simulations are usually performed with a given concentration of defects (very often, one vacancy). One must therefore rescale the Monte Carlo time tMC given by the residence time algorithm, according to t ¼ tMC  cMC v =cv , where cv is the real vacancy concentration that, in usual situations, remains equal to the equilibrium one, ceq v . It is worth noticing, may change during the precipitation because the compositions however, that ceq v and the volume fractions of the matrix and the precipitates change. During copper precipitation in iron, for example, simulations show that ceq v increases by orders of magnitudes because the vacancy formation energy is significantly lower in copper than in iron (Fig. 2). Therefore, the rescaling factor cMC v =cv is not a constant that can be fitted on the experimental kinetics. The difficulty can be easily overcome by measuring the evolution of vacancy concentration in a reference state (say, pure iron) in the AKMC simulation. Details can be found in Refs. [31, 32]. Under irradiation, such time-rescaling techniques should be used only when the only irradiation effect is to accelerate the diffusion: in all the other cases (e.g., when ballistic mixing or radiation-induced segregation phenomena are expected), it is better to explicitly introduce the formation and elimination of point defects to get realistic point defect concentrations.

3

Applications

3.1

AKMC Simulations of Precipitation Without Irradiation

The development of cohesive and diffusion models for KMC is a complex procedure and before to turn to the modeling of irradiation, it can be useful to validate it by checking that it gives reasonable behaviors in simpler situations. Typically, one can

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713

Fig. 2 Copper precipitation in Fe-1.34%Cu, during an isothermal annealing at 500  C. Comparison between AKMC simulations and experimental observations (symbols), with the evolution of the vacancy concentration, the density, and average radius of copper precipitates [40].

check that the model gives a realistic phase diagram and diffusion coefficients. For most systems of interest, the phase diagram, tracer diffusion, or interdiffusion coefficients are known, at least for some temperatures and compositions (although often not at the temperature of interest, where diffusion is too slow). The equilibrium phase diagram of the model can be determined using the AKMC itself, or mean-field models or equilibrium MC methods. In pure metals and dilute alloys, reliable multifrequency models can be used to compute the tracer diffusion coefficients [33], and the AKMC code can be used to simulate diffusion experiments. The modeling of precipitation without irradiation is probably the best validation: when the irradiation flux is set to zero, the simulation must drive the system to its proper equilibrium state, through its proper kinetic path. Of course, it is an

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important topic of its own: AKMC simulations have proven to be an efficient tool to study kinetics of precipitation during thermal aging in model systems, e.g., to test the theories of phase transformations [8, 34]. In what follows we will focus on simulations of kinetics of coherent precipitation in specific systems, usually binary and ternary alloys (dilute and concentrated) and to a few cases of dilute multicomponent alloys. One typical case is the modeling of copper-rich precipitates (CRPs) in bcc α-Fe. Copper has a very low solubility in α-Fe and CRPs has been identified as one of the major causes of embrittlement in reactor pressure vessel (RPV) steels, or as a way to improve the hardenability of some high-strength low-alloy (HSLA) steels [35]. In dilute Fe-Cu alloys (~1 at %Cu or below), the CRPs are fully coherent with the iron bcc lattice up to a size of approximately 4 nm [36], with little elastic deformations due to the similar size of Fe and Cu atoms: this makes rigid lattice approximations well suited to model precipitation kinetics. Several AKMC simulations have been proposed since the mid 1990s [29, 31, 37–40], with successive improvements of the Fe-Cu thermodynamic and diffusion properties (see ref. [5] for a review). The effect of additional solutes (Ni, Mn, Si, etc.) on the CRPs has been studied within similar approaches, motivated by the late blooming-phase problem (see Sect. 3.5) [39, 41, 42] and is of interest for other industrial applications (see ▶ Chap. 9, “Multiscale Simulation of Precipitation in Copper-Alloyed Pipeline Steels and in Cu-Ni-Si Alloys”). Other examples are the simulations of precipitation in concentrated binary iron or aluminum alloys [25, 27, 43–46] and of ordered phases in ternary alloys [34, 47, 48]. In such relatively simple systems, quantitative agreement with experimental kinetics has been achieved although often for limited times (see e.g., Fig. 2, [43]). Unexpected behaviors have been predicted, for example, the high mobility of small Cu clusters (due to the low vacancy formation energy in bcc CRPs) has been found to strongly accelerate the precipitation kinetics. In agreement with experimental observations, the addition of Ni has been found to promote the formation of a higher density of smaller CRPs in binary Fe-Cu alloys, and copper-shell microstructures have been observed in Fe-Cu, Mn, Si [39, 42], and Al-Zr-Sc alloys [34]. It has also been shown that complex magnetic properties of Fe-Cr alloys affect the precipitation kinetics, through their effect on the solubility limit and diffusion properties [25].

3.2

AKMC for Radiation Damage: Cascade Annealing

Under irradiation, thermally activated jumps of point defects are not the only mechanisms that contribute to mass transport: Frenkel pairs are created and atoms move from one lattice site to another within displacement cascades (or, under electron irradiation, replacement collision sequences). These events are sometimes referred to, respectively, as atomic displacements and replacements. The corresponding frequencies are proportional to the dose rate, in dpa.s
1. Self-interstitial atoms and vacancies recombine almost instantaneously below a given

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recombination distance, and point defects may eliminate, also almost instantaneously, on sinks such as dislocations, cavities, or grain boundaries. Note that by comparison with the jump frequencies of Eq. (1), the frequencies of these events do not obey the detailed balance: this is the cause of the alloy evolution towards nonequilibrium states. Radiation damage has thus a strong impact on precipitation as it controls the point defect concentrations as well as the ballistic/thermal jump ratio. On the short timescale, neutron or high energy particle interaction with matter leads to the formation of displacement cascades, thereby producing the so-called primary damage: small regions containing a core of vacancies, either isolated or grouped in small clusters, surrounded by SIAs, themselves either isolated or in clusters. This first stage (t < 10 ps) can be modeled by molecular dynamics. See ▶ Chap. 21, “Atomistic Modeling of Radiation Damage in Metallic Alloys”. The evolution of this primary damage at intermediate times, (times longer than the lifetime of displacement cascades, but shorter than the lifetime of the component), i.e., the annealing of the cascade, its interaction with other cascades, solute atoms, and elements of the microstructure will lead to changes in the microstructure and for instance the formation of solute clusters or precipitates. For instance, the first stage of precipitation may occur during cascade annealing as shown in a work coupling OKMC and AKMC methods to study the evolution of the primary damage in FeCu alloys [49]. OKMC on isolated cascade debris provided the SIA-vacancy recombination ratio and the description of the interstitial clusters escaping from the cascade core and thus participating in the defect flux in the materials. AKMC gives then the interaction of the vacancies with the solute atoms and shows the formation of vacancies and copper clusters as well as vacancy-copper complexes in the cascade area. A single 20 keV cascade typically leads to the formation of more than one large object, probable nuclei of the precipitates, or the dilute atmospheres observed to form under irradiation. The number and size of these objects were shown to strongly depend on temperature. At low temperatures, small clusters cannot emit vacancies and only local rearrangements by vacancy jumps and slow diffusion can take place. One observed thus the formation of small mixed objects (Cu-V complexes or clusters). At intermediate temperatures (temperatures close to the vessel in-service temperature), vacancies from the smallest clusters or complexes are emitted and contribute to the growth of larger mixed objects. At even higher temperatures, vacancies do not bind with Cu atoms long enough to transport them far and no large Cu clusters did form.

3.3

Acceleration of Precipitation Versus Ballistic Mixing

On the long run, the consequences of the previous radiation damage mechanisms are multiple. The diffusion coefficient of substitutional atoms is given by: Dirr th ¼ f v cv Dv þ f i ci Di where fv.i, cv.i, and Dv.i are, respectively, the correlation factors, concentrations, and diffusion coefficients of the point defects [50]. Except at high temperatures, the point defect concentrations are much higher than at equilibrium; therefore, eq eq eq the atomic diffusion is significantly accelerated: Dirr th >> Dth ¼ f v cv Dv . The first

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effect of irradiation is thus to accelerate the evolution towards the equilibrium and irradiation has indeed been used as a tool to determine the equilibrium phase diagram of alloys at temperatures where equilibrium diffusion is too slow. In FeNi alloys, for instance, two ordered compounds have been proposed in addition to the Ni3Fe ones, experimentally observed after thermal annealing. The acceleration is limited by mutual recombination and by annihilation at sinks (grain boundaries, dislocations, surfaces, point defect clusters). Since point defect clusters are formed in displacement cascades and grow during subsequent annealing, the sink microstructure and therefore the point defect concentrations change with the irradiation doses. Ballistic mixing in displacement cascades introduces another diffusion process, which is not thermally activated, does not fulfill the detailed balance condition and always tends to disorder the alloy, irrespective of its equilibrium properties. A 2 corresponding ballistic diffusion coefficient can be defined as Dirr bal ¼ a nrep G ,
1 where G is the dose (in dpa.s ) and nrep the number of replacement per displaceirr ment. Martin et al. [3] have defined the irradiation intensity as γ ¼ Dirr bal =Dth acceleration being dominant at low γ and ballistic mixing at high γ. It is still difficult to take directly into account in standard AKMC simulations all the diffusion mechanisms controlling the evolution of the sink density, especially that of the microstructure of small point defects clusters. One typical example is the rapid 1D motion of small SIA loops [51] which involves correlated motions of the atoms. Therefore, most AKMC simulations have been performed so far with a given sink density. Rate theory [50] and cluster dynamic models [52] are used to estimate the evolution of point defect clusters and concentrations under irradiation, and the resulting evolution of diffusion coefficients can be taken into account through a rescaling of the time. Simulations of precipitation using variable sink densities have been recently proposed to study the decomposition of Fe-Cr alloys (Fig. 3, [53]). The techniques described in Sect. 5.1, which do not presuppose any migration mechanisms, can be applied also, but they are very time consuming. AKMC simulations in binary model “A-B” alloys [54] have confirmed the trends predicted on irradiation intensities: full precipitation at low irradiation intensities, full disordering at high irradiation intensities. But they have also revealed a third kind of microstructure at intermediate γ values: a patterning of precipitates with a steady-state length scale, which could explain experimental observations of “inverse coarsening” or “fragmentation of precipitates.” They also show that an increase in the range of ballistic mixing favors the dissolution and affects the steady-state length scale of precipitates. In iron-based alloys, AKMC simulations suggest that Cu precipitation under electron irradiation [37] and Cr precipitation under neutron irradiation [52], at low or moderate dose rates, result in a simple acceleration of the precipitation kinetics. Recent simulations have highlighted the role of the sink densities on the balance between accelerated diffusion and ballistic disordering: in Fe-Cr alloys contrary to neutron or electron irradiation at moderate doses rates, ion irradiations at high fluxes produce a high density of vacancy and self-interstitial clusters, i.e., in a high sink densities, and a dissolution of Cr precipitates – as

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observed in some recent experiments (Fig. 3). Simulations also show precipitate-free zones in the vicinity of grain boundaries, which are due to the low point defect concentrations and therefore to the slower diffusion of solutes [26, 52, 55]

3.4

Flux Coupling: Radiation-Induced Segregation and Radiation-Induced Precipitation

Due to their formation and annihilation, permanent fluxes of point defects towards the sinks are sustained by irradiation. The flux of SIAs corresponds to a flux of atoms towards the sinks, the flux of vacancy corresponds to a flux of atoms away from the sinks. In an alloy, diffusion of the different elements within a given point defect mechanism does not occur, in general, at the same speed. As a consequence, the composition frequently changes near point defect sinks. Such radiation-induced segregation (RIS) are usually observed for moderate dose rates and intermediate temperatures. Thermodynamics of irreversible processes states that the flux of one component i (atom or point defect) depends on the gradient of chemical potentials of all the other j components, i.e., Ji =
jLij∇μj, where the Lij are the so-called Onsager (or phenomenological) coefficients and control the RIS. Lij are very sensitive to the point defect jump frequencies and are very difficult to measure experimentally (especially for SIA diffusion). DFT calculations coupled with multifrequency models or AKMC simulations provide a way to systematically compute the Lij coefficients [23] and thus predict the direction of solute fluxes. AKMC simulations have been also used to model the evolution of the RIS profiles near grain boundaries [26, 55, 56]. If RIS produces a sufficient enrichment of solute near a point defect sink, the local composition may reach the solubility limit even if the average composition is below it, and radiation-induced precipitation can be observed in a normally under-saturated alloy. A typical case is the precipitation of the Ni3Si ordered phase on dislocation loops, in Ni-4%Si alloys (at a temperature where the Si solubility limit is 10%): it is believed to be due to the positive coupling between the fluxes of SIA and Si atoms towards the loops. Few clear cases of similar radiation-induced precipitation have been modeled so far. AKMC simulations of RIS in Fe-Cr alloys predict a positive coupling between SIA and Cr fluxes, a negative coupling between fluxes of Vand Cr, the former effect being dominant below a critical temperature. AKMC simulations predict a radiation-induced precipitation in slightly under-saturated alloys [56] and some experimental observations [57, 58] confirm this (although it is difficult to clearly distinguish between a simple radiation-induced enrichment and an actual precipitation). Similarly, recent AKMC simulations in under-saturated W-Re alloys predict Re precipitation due to the positive coupling between fluxes of SIA and Re [59]. In a supersaturated alloy, RIS can reinforce or oppose the thermodynamically driven precipitation: the positive coupling between solutes and V in α-iron probably contributes to the formation of solute-V clusters [60] and plays a role in the formation of the

dp (cm-3)

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Fig. 3 (continued)

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late blooming phases (next section). Inversely, in case of negative couplings, RIS can lead to a solute depletion near point defect sinks and eventually to a radiation-induced dissolution of precipitates: the dissolution of Ni3Al near dislocation loops observed under ion irradiation of Ni-Al alloys has, for instance, been explained by this mechanism [61]. To our knowledge, this has not yet been modeled by AKMC simulations.

3.5

Examples of Complex Systems: Open Questions

In general, all these phenomena may occur at the same time and affect the evolution of the system. One typical example of situations where many mechanisms are probably involved is the formation of the so-called late blooming phases [62]. These phases are supposed to start to form above a threshold incubation dose in low-Cu, high-Ni (and/or high-Mn) RPV steels, thereby causing additional hardening. They are thus currently considered as a serious threat in connection with programs of extension of the lifetime of operating power plants (beyond doses ~0.1 dpa in the vessel, equivalent to 40 years of operation). Questions arise regarding their origin and more precisely whether they are stable phases, predicted by thermodynamics as proposed in [62] or radiation induced [63]. An atomistic kinetic Monte Carlo model parameterized on electronic structure calculations data was used to study the formation and evolution under irradiation of solute clusters in FeMnNi ternary and FeCuMnNi quaternary alloys. The results show that the thermodynamic stability of these clusters as phases is not a necessary condition for their appearance under irradiation and that a radiation-induced mechanism based on segregation of solute atoms at self-interstitial clusters via dumbbell transport was sufficient to explain their formation [63]. Later on, new DFT studies and Metropolis Monte Carlo simulations [64] also seem to indicate that irrespective of their thermodynamic stability, the formation of these clusters stem from the massive fluxes of point defects and by point defect cluster formation. They are thus the consequence of irradiation. Figure 4 (Fig. 3 of [60]) illustrates the intricate relationship between the point defects and the solute atoms as well as the importance of displacement cascades. When cascades are introduced, the clusters formed are usually smaller than when a homogeneous FP production is used because cascades favor the formation in a very localized area of small, stable, and immobile point defect clusters [5]. ä Fig. 3 AKMC simulations of Cr precipitation in iron, during an isothermal annealing at 300  C and under electron irradiation at 5.2  10
5 dpa.s
1 (left) and ion irradiation at 2.8  10
3 dpa.s
1 (right), compared with 3D atom probe observations [53]. Evolution of the density dp and radius R of precipitates, and of the sink strength k2tot (which is proportional to the sink density). Under electron irradiation, the precipitation kinetics ( full line) is accelerated by several orders of magnitude compared with isothermal annealing (dotted line). Under ion irradiation at high dose rates, simulations predict first a Cr precipitation, then a dissolution of precipitates when the sink density becomes high enough. At 120 dpa, the precipitates have completely dissolved, in agreement with the experiments

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Fig. 4 Point defect – solute complexes formed during AKMC simulations of neutron (introduction of cascade debris) and electron (introduction of FP) irradiations. The vacancies are yellow, Mn is black, Ni green, and Si are blue (From [60])

Another family of complex systems of interest in the nuclear field of materials is that of oxide dispersion strengthened (ODS) alloys which show great promises for increased radiation resistance [65]. They consist of a polycrystalline matrix containing different populations of tiny oxide particles. Despite a large body of experimental studies, the mechanisms responsible for the irradiation evolution of the nano-precipitates are far from being understood. Not one mechanism alone can explain all the experimental observations and a combination of these mechanisms must be thus active in order to explain experimental results [66]. KMC modeling in the continuation of the work in [67] is thus expected to be very challenging.

4

Beyond AKMC: Towards Larger Space and Timescales: OKMC/EKMC

The AKMC approach naturally takes into account the correlations between successive jumps, which play a critical role in diffusion properties [33]. The price to pay is that billions of jumps have often to be performed to model the very first stages of precipitation. AKMC becomes especially inefficient when point defects are trapped in specific configurations, or when the frequencies of the different diffusion mechanisms are significantly different (in such cases only the rapid events can take place on reasonable simulation times). To overcome these problems, mesoscale approaches of the KMC method, such as the OKMC or the EKMC have been used for some time now. They allow to take into account the point defect cluster evolution under irradiation. In fact, these techniques, as applied currently, aim at modeling the microstructure evolution under irradiation by looking at the point defect and point defect cluster distributions. Both KMC methods are defined as coarse-grained models because atoms are not explicitly treated. They are also based on the residence

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time algorithm. In the case of the long-term evolution of radiation damage, the objects are the intrinsic defects (vacancies and self-interstitials) or impurities, and their clusters which are located at known (and traced) positions in a simulation volume. The events are all the possible actions that these objects can perform and the reactions that they may undergo, such as: (a) migration, (b) dissociation (emission of a smaller defect from a bigger one), (c) aggregation of like defects or of defects and impurities, and (d) annihilation between opposite defects (self-interstitials and vacancies). As in the case of AKMC, the probability for a migration event is given by the corresponding jump frequency. For the dissociation events, the probability is also given in terms of a jump frequency, with an activation energy often taken to be equal to the sum of the migration energy plus the binding energy between the emitted defect and a cluster of the size that remains after emission, Eb:   Em þ Eb Γd ¼ Γ0 exp
kB T

(10)

Eb will in general depend on the type of emitted and emitting object, as well as on the emitting object’s size. Regarding the reaction between defects, it is generally assumed that they are diffusion limited. That is, the reaction occurs as soon as the two objects are within a predefined capture radius, but no energy barrier is associated with the reaction. These events occur therefore only on the basis of geometrical considerations (overlap of reaction volumes) and do not participate in defining the progressing of time. Events of this type also include the absorption of objects by sinks (impurities, dislocations, grain boundaries, . . .). On the other hand, events producing damage, such as the appearance of isolated Frenkel pairs (FP), as in electron irradiation, or of the debris of a displacement cascade, as in ion or neutron irradiation influence the rate at which time progresses. Since each of these external events occurs at a known rate, Pj, these rates are included in the computation of the total rate. Difficulties in applying the OKMC algorithm are not due to the method itself, which is fairly straightforward. They stem, from the fact that all the possible events that each object can undergo according to a specific physical mechanism, their probability, and their properties must be predefined. It is important to stress out that, similar to most cases of AKMC (the exceptions being on-the-fly AKMC techniques described in V.1), these methods do not have the capability to predict structures or events that are not explicitly included in the model (unlike ab initio or MD simulations). Furthermore, they are usually based on a “gray alloy” approach, where the effect of solutes is introduced assuming that it changes the properties of the objects. This approach has been used to study the microstructural evolution in reactor steels under neutron irradiation in FeCNiMn model alloys [68], considered as representatives for low-Cu reactor pressure vessel steels and FeCCr model alloys as model materials for high-Cr ferritic/martensitic steels [69]. Based on earlier works focusing on the role of C impurities in pure Fe, these models incorporate the influence of alloying chemical elements on the mobility and stability of point defect

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clusters, which was key for the development of the model. Direct comparison with experimental evidence was possible from two aspects: (a) the population of single vacancies and small vacancy clusters as deduced from PAS; (b) the direct observation of SIA loops in TEM. This approach is quite simple and has been refined in a model that contains a more explicit description of solute atoms [70]. While solute atoms are explicitly introduced in the matrix, in an AKMC fashion, only single point defects are allowed to interact with them in first approximation. Point defect clusters still assume the gray matrix. When either a single vacancy or a single SIA catches a solute atom (found within its radius of interaction), the latter is removed from the matrix and incorporated into the objects, thereby redefined into a new one. Finally, the solutes are reintroduced in the atomic matrix when the carrying point defect interacts with another object, i.e., either during a recombination event with an opposite kind of defect, or merging with another defect of the same kind. Globally, solute transport is thus implemented in the simplest way that can be regarded as a compromise between providing atomistic-level description while keeping the evolution of the predicted microstructure from the successful gray alloy model. If the model is consistent with the physics of the studied system, clusters of solute atoms should spontaneously be formed by the cumulated deposition in specific sites. One way to speed up the AKMC simulations is to couple both mesoscale and atomistic approaches as was done for instance in [71] to model the precipitation of Cu in Fe. In this study, Cu clusters above 15 atoms are considered as objects, for which migration and dissociation events are defined, based on specific, size-dependent, and thermally activated frequencies. Matching between the fully atomistic and the coarse-grained approach is achieved again by using a neural network, that provides all stability and mobility parameters for large Cu clusters, after a training on atomistically informed results. The same approach has been very recently developed to model the evolution of complex Fe alloys, representative of RPV steels, under irradiation [72]. Another possibility is to couple AKMC with the phase field approach as proposed in [73] (Fig. 5).

akmc

n

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akmc Em

iss

Single point defect jumps Fe Vacancy

ion

Object properties: Diffusion coefficient and Emission frequency

Fig. 5 Schematic description of the hybrid AKMC/OKMC method (From [72])

Hybrid

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5

Challenges

5.1

Towards More Physical Descriptions

723

To obtain a full description of the microstructure evolution during precipitation, one needs to be capable of calculating the energy of an assembly of atoms whatever their position (i.e., one needs to remove the rigid lattice constrain). This can only be done using DFT or empirical potentials. However, DFT methods are limited by the system size which can be modeled, whereas empirical potentials are limited by the chemistry. Indeed, it is very difficult to obtain, even for a pure element, an interatomic potential capable of describing correctly the point defect properties, the influence of temperature on these properties, the phase diagram, etc. The challenge for industrial alloys which contains many alloying elements and impurities is therefore tremendous. However, if one is capable of describing correctly all the interactions between all the elements, one can for instance take into account elastic strains using different techniques. The impact of elastic strain on precipitation has been nicely demonstrated in the case of Al-Cu-Mg ternary alloys [74]. Because of the compression around Mg atoms, a small concentration of these atoms rapidly precipitates out of solution in a rigid lattice, whereas in a flexible lattice, it does not, as if the solubility limited had been increased. Furthermore, it was shown that for a fixed number of vacancy–atom exchanges, larger clusters grow on a flexible lattice than on a rigid lattice owing to the reduction in elastic strain energy. New methods are thus nowadays developed to relieve the rigid lattice constrain.

5.1.1

Taking into Account Elastic Strains: Lattice Free and On-the-Fly Methods For most of the current AKMC models, a list of possible mechanisms is required a priori. This strongly limits the kind of mechanism that can be modeled as in some cases, atomic motion may not be intuitive and can be extremely difficult to guess or determine in advance. This is for instance the case in Si, where an on-the-fly KMC showed that two self-interstitial atoms (SIA) can aggregate into a structure that contains four SIAs and two vacancies, allowing the structure to diffuse with a barrier less than half that of a single SIA [75]. Furthermore, classic AKMC programs use an on-lattice approximation, which limits their ability to describe systems where defects are not exactly on-lattice such as interstitials or extended defects. In the case of extended defects, regions of the lattice undergo significant deformation (the dislocation core for instance) and the on-lattice approximation is no longer appropriate. A solution to both these issues is lattice free on-the-fly KMC methods, first proposed by Henkelman and Jónsson [76] to study the growth of Al. These methods are designed to handle very general cases without being bound to restricting hypotheses. At every step, they explore the local curvature of the potential energy surface, and find transitions to nearby basins looking for saddle points. A complete exploration of the potential energy surface is however very costly and in some cases, one can “help” the search by defining a so-called event definition generic procedure, from the

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knowledge of the diffusion mechanisms. Coupled with an artificial neural network (ANN), this method proved to be very efficient to investigate the migration of vacancies in the complex environment of a grain boundary in FeCr [77].

5.1.2 Vibrational Entropic Contributions The simulations presented in Sect. 3 still suffer important limitations: the contribution of finite temperature effects is still roughly taken into account. However, important progress has been recently achieved for the ab initio calculation of vibrational [20, 21, 78] and magnetic contributions [79] to transport coefficients in dilute alloys – and similar methods could be implemented in principle in AKMC simulations of precipitation. Although being often significant, vibration contributions are often ignored. This issue is related to the prefactor used in Eq. (6). For the sake of simplicity, one uses a constant prefactor of the order Debye’s frequency. This commonly used assumption is based on the fact that small variations of the activation energy are more likely to have greater impact than variations of the attempt frequency on the transition probability (because of its exponential dependency). However, it has been recently shown [80] using Vineyard approach [13] that the prefactor has an impact on the primary damage evolution and that the use of a constant prefactor may change the SIA migration mechanisms, the amount of vacancy SIA recombinations and enhances the difference between the diffusion rates of vacancies and interstitials. This work also concluded that the vibrational internal energy contributes little to the attempt frequencies of small point defect cluster migration mechanisms in Fe and that Vineyard approach was thus accurate enough to obtain them. 5.1.3 Incoherent Precipitation During the nucleation stage, the first precipitates are often coherent with the matrix (inducing possible elastic strains), they may become incoherent or semi-coherent when growing, sometimes going through a series of intermediate structures, such as GP zones-θ-θ’ in Al-Cu alloys or the 3R-9R-FCC sequence for Cu in Fe. These effects are known to affect the precipitation kinetics by changing the formation energies of the phases and the interface energies. Under irradiation, specific effects must be considered: the strain field can modify the point defect migration barriers, the Lij coefficients and therefore the radiation-induced segregation. Furthermore, an incoherent interface is a potential sink for point defects and can modify the balance between ballistic mixing and thermally activated diffusion, or even lead to a radiation-induced precipitation due to flux couplings [81]. AKMC simulations have only started to deal with these problems, and almost exclusively without irradiation. Strain effects may be taken into account in AKMC simulations with interatomic potentials and relaxation of the atomic positions, as it has been done to model the formation of coherent precipitates in Al-Cu [82, 83], Al-Zn [11], or other model alloys [12]. However with fully incoherent interfaces, AKMC must deal with the elimination of point defects and the apparition of new and possibly collective, transition mechanisms. The advanced methods described above seem especially promising to achieve these objectives.

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Reaching Realistic/Useful Times

Despite being faster than molecular dynamics, AKMC, because it models the jumps of individual point defects and atoms is still very limited in the amount of real time it can simulate. A few techniques have been proposed to speed up the calculations; we present them very briefly in the following sections.

5.2.1 Accelerating Methods Calculating all the possible migration barriers during the course of an AKMC simulation requires a lot of computational time, even with efficient methods such as the NEB for instance. One way of speeding the calculations is to store them using, for instance, a topological classification of the atomic environments, allowing configurations and events to be recognized and stored efficiently as proposed in [19]. Another possibility is the use of artificial neural networks trained on a large set of data, i.e., migration barriers, to estimate, in the course of the simulation, the migration barriers from the knowledge of the local chemical environment of the migrating species [84, 85]. One approach to speed up the AKMC simulations is to replace the individual atomic jumps by larger displacements as proposed in [86]. The method relies on time-dependent Green’s functions to propagate the diffusing species over long distances in a single KMC step. The space is partitioned into non-overlapping “protective domains” (PDs), each containing one diffusing species. Using appropriate first-passage Green’s functions the diffusing species are propagated to the boundaries of their respective PDs, one at a time. Thus, numerous diffusive hops are replaced by large “superhops” guaranteeing that the statistics of random walks remain unaltered. Unfortunately, the method becomes limited when the density of particles or diffusing species is high. In this condition, the protective zones become very small and the computational efficiency decreases in a radical way. Another issue frequently encountered in AKMC simulations is trapping. In some energy landscapes, basins can be found with small barriers, and the moving species may spend a lot of time jumping these barriers without escaping the basin. A typical case is the Fe-Cu system, in which diffusion of Cu atoms is driven by the motion of vacancies. Because the vacancy formation energy is smaller in bcc Cu than in Fe, the vacancy spends a lot of time in the Cu precipitates (which in the first stages of nucleation are bcc) and time runs very slowly. Ways around that are to, for instance, increment the Monte Carlo time only when the vacancy is in the matrix [31]. Methods to solve analytically the average residence time for the in-basin states such as the basin auto-constructing mean rate method (bac-MRM) can also be used [87]. Bac-MRM computes on-the-fly a statistically correct analytic solution of the connected states and their escape rate as the energy landscape is explored, at the cost of specific trajectories. Another method has been proposed recently by Athènes and Bulatov, based on path factorization of the evolution operator [88]. This method detects trapping and charts the trapping basin iteratively state by state. Applied to the Fe-Cu system, this method allowed these authors to reach later stages of precipitation and uncover new kinetics.

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5.2.2 Parallelization With the emergence of multicore systems, parallelization strategies for AKMC have been proposed. They are usually done by dividing the simulation box into domains [89–92]. Each of these domains runs independent KMC simulations. Communication between cores is imperative to ensure the passage of diffusing species from one domain to another. In addition, the simulated time flows differently from one domain to another. This forces the domain whose time flows the fastest to “wait” for those whose time runs slower. This is why a NULL event is commonly used in this type of simulations thereby causing a decrease in performance [93]. The main problem with this method is that the diffusing species have to be homogeneously distributed in the simulation box to really observe an increase of the KMC performances. In the case of thermal aging and precipitation, since the system evolves only when the vacancies move, scalability is not that good [92] as the number of vacancies is very small. It becomes worse in the case of irradiation, where the introduction of a new cascade leads to a very localized activity in a small portion of the box.

6

Conclusions

In the last decade, KMC simulations of precipitation under irradiation have made important progress, thanks to the introduction of realistic diffusion and radiation damage mechanisms. From this point of view, atomistic KMC methods have several key advantages. Based on an atomic description of the point defect energetic landscapes and taking naturally into account the correlation between successive point defect jumps, they can give a realistic description of diffusion properties. They also naturally take into account thermal fluctuations, which control the nucleation of precipitates. These potential advantages have become especially effective when coupled with first principle methods that give reliable estimations of the migration barriers. If DFT calculations remain too time consuming to be used on-the-fly in AKMC codes, they have been successfully used to improve simpler diffusion models, based on effective interactions and rigid lattice or on empirical potentials. In this chapter, we have presented several simulations of the first stages of precipitation that can be quantitatively compared with experimental studies, at least for relatively simple problems of coherent precipitation in alloys with a limited number of components. The energetic models can still be improved to give a better description, for example, of the vibrational or magnetic contributions, atomic relaxations, or collective transition mechanisms. Several advanced methods have already been proposed to deal with these issues and remained to be applied to the modeling of precipitation under irradiation. AKMC methods are nevertheless computationally expensive and in the foreseeable future will be probably limited to the modeling of the beginning of precipitation (the nucleation and growth of precipitates, or the first stages of spinodal decomposition). The coupling with other kinetic models is therefore of critical importance. Object and event KMC have proven to be very useful to model the annealing of

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cascades, the clustering of point defects, or precipitation kinetics in pure metals and dilute alloys. Here, the challenge is to establish the link between the atomic scale and the effective parameters of such mesoscale models, when dealing with concentrated and heterogeneous alloys. The same can be said of other kinetic models, e.g., cluster dynamics or phase field models that may be considered for the same objective. Our review points out two issues of practical interest that are especially challenging and have been barely addressed – even in the absence of irradiation: the developments of thermo-kinetic models for multicomponent alloys, and the kinetics of incoherent precipitation. In our opinion, this is the area where efforts should be pursued.

7

Cross-References

▶ Atomistic Modeling of Radiation Damage in Metallic Alloys ▶ Multiscale Modeling of Radiation Hardening ▶ Multiscale Simulation of Precipitation in Copper-Alloyed Pipeline Steels and in Cu-Ni-Si Alloys

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Mechanics of Auxetic Materials

23

Hyeonho Cho, Dongsik Seo, and Do-Nyun Kim

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Deformation Mechanisms of Auxetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Re-entrant Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Rotating Unit Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Chiral Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fibril/Nodule Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Miura-Folded Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Buckling-Induced Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Helical Auxetic Yarn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Crumpled Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Relationship Between the Deformation Mechanism and the Material Properties . . . . . . . . . 4 Expected Properties of Auxetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Synclastic Curvature in Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Variable Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 High Shear Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Enhanced Indentation Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 High Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Increased Energy Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Potential Applications of Auxetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Poisson’s ratio is a mechanical property that represents the lateral behavior of materials under an axial load. In contrast to typical natural materials with a positive Poisson’s ratio, auxetic materials have a negative Poisson’s ratio (NPR) H. Cho (*) · D. Seo · D.-N. Kim Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Republic of Korea e-mail: [email protected]; [email protected]; [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_25

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characterized by unilateral shrinkage or expansion against axial compressive or tensile loadings, respectively. Here, based on a comparison between conventional and auxetic materials, a review of auxetic materials that exhibit various types of deformation mechanisms and characteristics induced by a negative Poisson’s ratio is provided. First, the deformation mechanisms in auxetic materials resulting in lateral expansion under tensile loads are described. They are classified according to their deformation mechanism or the structural motif enabling the auxetic behavior including re-entrant structures, rotating unit structures, chiral structures, fibril/nodule structures, buckling-induced structures, helical yarn structures, Miura-folded structures, and crumpled structures. Then, the expected properties of auxetic materials in several aspects are discussed. The mechanical response of these materials can be drastically changed depending on the amount of applied loads, and auxetic materials are expected to have unusual, possibly enhanced geometrical and mechanical characteristics such as synclastic curvature in bending, deformation-dependent permeability, high shear stiffness, indentation resistance, and fracture toughness, and improved damping and sound absorption properties. Finally, some representative potential applications of auxetic materials are illustrated with a short discussion on the limitations and outlook of auxetic materials. Keywords

Auxetic materials · Negative Poisson’s ratio · Structural motif · Deformation mechanism · Mechanical properties

1

Introduction

When materials are stretched in a certain direction, they usually become thinner in directions orthogonal to the direction of the applied force (Fig. 1a). The degree of this transverse deformation with respect to the deformation along the loading direction is characterized by Poisson’s ratio, which is defined as the ratio of the compressive transverse strain to the tensile axial strain. According to the

Fig. 1 Deformation of a rod under tension when its Poisson’s ratio is (a) positive (as in ordinary materials) and (b) negative (as in auxetic materials)

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classical theory of elasticity, three-dimensional isotropic materials can have Poisson’s ratios ranging from
1.0 to 0.5, while they can vary between
1.0 and 1.0 for two-dimensional isotropic materials [1]. Most natural materials have a positive Poisson’s ratio. For example, metallic materials typically have a Poisson’s ratio of 0.3, while the Poisson’s ratio of rubbery materials is close to 0.5 due to their incompressibility. Materials with a negative Poisson’s ratio are usually referred to as auxetic materials, derived from the Greek word auxetikos meaning “that which tends to increase” [2]. These materials, by definition, expand in all directions when a tensile force is applied in one direction, as shown in Fig. 1b. Because auxetic behaviors originate from the deformation mechanism of particular geometries and internal structures in response to uniaxial loadings, a number of natural and synthetic materials have been found and developed across the length scales. Compared with conventional materials with a positive Poisson’s ratio, auxetic materials are expected to have several interesting characteristics in their geometrical and mechanical properties, including synclastic curvature in bending [3, 4], variable permeability [5], high shear stiffness [4, 6], enhanced indentation resistance [4, 7, 9], high fracture toughness [3, 8, 10, 11], and damping and sound absorption [12–15]. These extraordinary properties offer ample opportunities to use auxetic materials for various applications including, but not limited to, biomedical materials [16], safety equipment, shock absorbing materials [17], energy harvesting devices [18], sports equipment [9], tunable filters [19], soft robotics [20], fashion textiles [5, 21], and aerospace applications [6]. In nature, only a few auxetic materials have been found. The first experimental study suggesting the existence of auxetic materials in nature was reported in 1882 for iron pyrite monocrystals whose Poisson’s ratio was estimated to be
0.14 [22]. Other examples of natural auxetic materials are α-cristobalite, cow teat skin, pyrolytic graphite, polymorphic silicones, zeolites, silicates, and crystal cadmium [22]. Because only a limited number of auxetic materials are available in nature, despite their potential to provide exceptional material properties for a broad range of applications, a tremendous research effort has been devoted to the development of artificial materials and structures with auxetic properties. In particular, many researchers have investigated novel deformation mechanisms leading to auxetic behavior with high controllability and low manufacturing costs. Representative mechanisms are re-entrant structures [23–28], rotating rigid units [29–34], chiral structures [35, 36], fibril/nodule structures [37, 38], Miura-folded structures [39], buckling-induced structures [40–43], helical auxetic yarn [44], and crumpled structures [45, 46].

2

Deformation Mechanisms of Auxetic Materials

Since a pioneering study by Lakes in 1987 [3], various man-made auxetic structures have been studied with structural motif inspiration observed in the microscopic structures of natural and artificial materials with a negative Poisson’s ratio (NPR)

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[3, 23, 24, 37] or based purely on geometrical intuition from individual researchers [36–38, 45, 46]. The degree of auxetic behavior and the related material properties are highly dependent on the deformation mechanism. Therefore, depending on their deformation mechanism or structural motif, which induces expansion in all directions when pulled in a certain direction, these auxetic structures are classified. For example, some auxetic microstructures have a structural motif possessing initially re-entrant parts that are gradually straightened up when loaded [3, 23–27]. In this case, Poisson’s ratio may vary with the applied load as the re-entrance angle changes. When all the re-entrant parts are fully straightened the auxetic behavior no longer appears. Another case is the buckling-induced auxetic structure, where buckling occurs under compressive loads. Unlike the case mentioned above, this structure exhibits auxetic behavior only when subjected to a compressive force above a certain critical value, but not when subjected to a tensile force [40–43]. Therefore, understanding and investigating the deformation mechanisms or structural motifs, which differ according to each auxetic structure, can play a central role in the design and use of auxetic materials for diverse potential applications in engineering and science.

2.1

Re-entrant Mechanism

The most common type of auxetic materials is re-entrant auxetic structures. Simply, they are formed by truss structures that are composed of thin ribs and linking hinges. In a 2D re-entrant structure, a unit cell of a polygonal truss, which consists of ribs, is arranged repeatedly in the structure [25]. The unit cells of these structures are commonly folded inside the unit cells and the re-entrant sides and vertices form an important structural motif of the re-entrant structures. In addition, the re-entrant vertex is joined with one or more neutral ribs that are members of one neighboring cell, but do not belong to the original unit cell. For example, a re-entrant bowtie honeycomb structure, which is illustrated in Fig. 2a, is a truss structure in whose unit cell two facing vertices of a hexagonal structure are re-entrant inside the unit cell. Re-entrant vertices are linked to the horizontal ribs of neighboring cells to deliver a load in the lateral direction [23]. There is a close mechanical relationship between structural motifs and the auxeticity of re-entrant structures. When a tensile load is applied to the re-entrant structures, the re-entrant sides are unfolded with respect to the re-entrant vertex, and the vertex moves outward from the unit cell. This lateral deformation of the reentrant vertex is transmitted to neighboring cells through neutral ribs connected to the vertices and pushes nearby cells to induce vertical expansion. Typical examples of the transformation mechanism in auxetic re-entrant structures are the bowtie honeycomb structure, triangular re-entrant structure (Fig. 2b), and re-entrant star structure (Fig. 2c). In the bowtie honeycomb structure, when tensile loads are applied to horizontal ribs of a unit cell at the top and bottom, the tilted re-entrant sides are aligned and stretched in the tensile direction. This lengthening behavior causes the re-entrant vertices to move horizontally and push the neighboring cells through the horizontal neutral ribs causing them to expand laterally. The triangular

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Fig. 2 Two-dimensional re-entrant auxetic structures: (a) bowtie honeycomb, (b) triangular reentrant, and (c) re-entrant star structure. (d) Unit cell of a 3D re-entrant auxetic structure [28]. (e) 3D re-entrant auxetic structure [28] (Part (d and e) reprinted with permission from Ref. [28] # 2017 Elsevier)

re-entrant structure is a truss structure whose unit cell is an isosceles triangular structure with a re-entrant base side. When a tensile load is applied to a triangular reentrant structure, loads are transferred from the two neutral ribs connected to the reentrant vertices so that the re-entrant sides are unfolded. As a result, the triangular unit cell is widened in the lateral direction. Re-entrant star structures are truss structures consisting of unit cells of three-, four-, or six-tip star shapes and neutral ribs connecting their re-entrant vertices. The different star-shaped unit cells have isotropy in three, four, and six directions, respectively, and when a tensile load is applied through neutral ribs in a specific direction, not only the vertices jointed to the loading ribs but also the other re-entrant vertices are unfolded to the same extent [26, 27]. As a result, the unit cell expands in all directions to push neighboring cells outwards. The relationship between the NPR of the re-entrant auxetics and the structural motif, or re-entrance, is clear in that the auxeticity is extinguished when all re-entrant sides are unfolded to their maximum extent. Three-dimensional re-entrant structures are designed by transforming the structural motif of a 2D structure into a 3D structure [24]. Further, there are structures in which ribs and vertices are complexly interconnected, as well as simple applications

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at the level of simply extruding or three dimensionally stacking sheets of a 2D reentrant structure. In this complex example, as the unit cell of the 2D re-entrant structure is a polygonal truss with re-entrant vertices, the unit cell of the 3D reentrant structure (Fig. 2d) is a polyhedral truss structure with vertices which are dented inwards. Similar to the mechanism shown in the 2D structure, 3D re-entrant structures also cause lateral deformation due to re-entrant sides unfolding with respect to the tensile load. A typical 3D re-entrant structure is the 3D bowtie shape structure, which inherits the geometry and structural motif of the 2D re-entrant bowtie structure, as shown in Fig. 2e [28]. Another example is the 3D re-entrant pyramid structure that is a 3D transformation example of the triangular re-entrant structure. In addition to these structures, there are various other re-entrant structures. By using a re-entrant polygonal geometry that can be tessellated, microscopic structures with re-entrant motifs can be easily developed with unlimited design space. There are advantages and disadvantages of the structure resulting from such structural motifs. Because these structures have auxeticity not only for the tensile load but also for the compressive load, these structures have a wide applicability in comparison with a structure having a NPR only in one direction of the load. Furthermore, re-entrant structures have high porosity, or low density, which is useful from the standpoint of lightweight construction. However, re-entrant structures consist of complicated connections of thin ribs, making them difficult to fabricate with high accuracy and without defects. In the case of 3D re-entrant structures, additive manufacturing is essential because of the complicated structure including the internal cave. In addition, flexure and shear deformation of thin ribs are an obstacle to intended auxetic behavior. In particular, when a large compressive load is applied to a re-entrant structure, buckling, which is not considered in the kinematic model, may occur in the thin ribs. Further, the thin ribs are vulnerable to fatigue failure and may cause the overall structure to have low durability. Because of these pros and cons, re-entrant structures are mainly applied to the core structures of lightweight sandwich panels and analytical models for investigating the microscopic structure of auxetic foam.

2.2

Rotating Unit Mechanism

Another representative auxetic mechanism is the rotating unit mechanism. Structures with a rotating mechanism consist of rigid units connected by smooth hinges. Rigid units are arranged according to a consistent rule, and their initial positions are slightly tilted in a clockwise or counterclockwise direction, which is opposite to the tilting direction of the nearby units. A typical example with such a structural motif is a rotating square unit structure [29]. Square rigid units are placed repeatedly in longitudinal and lateral directions. Square units have slightly tilted initial positions and form rhombic voids. The initial tilt of units induces a rotation of the unit with respect to the tensile load and directly affects the lateral deformation. Because of the initial position of the units, applied tensile loads are not collinear between the upper and lower hinges. By applying a

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Fig. 3 Rotating unit auxetic structures: (a) square, (b) rectangular, (c) trans-rectangular, (d) bisquare, (e) triangular, (f) isosceles triangular, (g) bi-triangular, and (h) hexa-triangular rotating unit structures

tensile force, a torque is applied to the unit, and units rotate in the clockwise or counterclockwise direction opposite to their neighboring units. Because units are rigid and hardly deformed compared to hinges, local rotation of units causes lateral movement to the hinges on the flank side and lateral expansion. Local rotation creates the auxetic behavior in which the hinges connected to the left and right units move outward of the unit and expand in all directions. This rotation occurs until the tilted square is fully rotated and aligned with the tensile load, and a NPR can no longer be shown in the fully rotated state. The rotating square unit structure is the simplest 2D structure with a rotating motif, which can be used to construct various 2D unit structures. For example, scaling and tilting these basic square unit structures result in rectangle [31] and parallelogram unit structures [30], respectively. It is also possible to replace the rotating unit with two or more different types of unit. Representative examples of these heterogeneous rotating unit structures are the trans-rectangle (Fig. 3c) and the bi-square unit structures (Fig. 3d). In addition to quadrangular unit structures with four hinge points in one unit, 2D rotating structures using a triangular unit with three hinge points have been developed. Similar to the quadrangular type, the equilateral triangle unit is the simplest one, and rotating unit structures with arbitrary triangles or heterogeneous triangles have been developed from the basic geometry (Fig. 3e–h) [31]. A hexagon can be tessellated in two dimensions, like a quadrangle and triangle, but it cannot be used to make a rotating unit structure by itself alone. A hexagonal unit is used with other types of units to create a 2D heterogeneous rotating unit structure. Typical examples of this type are the hexagon-triangle rotating unit structure and the hexagon-triangle-rhombi rotating unit structure.

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Similar to re-entrant structures, 3D rotating unit structures can be designed using the structural motif of a 2D rotating unit structure. A typical example is a 3D cubic unit structure [34] that is derived from the geometry of a 2D rotating square unit structure. Like the 2D rotating square unit structure, tilted cubic units are arranged regularly and connected by joints. Therefore, when a tensile load is applied to the 3D rotating cubic unit structure, the tilted unit rotates in a certain direction and lateral deformation occurs. In addition to the structures developed so far, a structure having a rotating motif can be implemented in a wide variety of forms. In particular, heterogeneous rotating unit structures composed of more than two kinds of units have fundamentally unlimited design variations, like re-entrant structures. Moreover, there are many other advantages of rotating unit structures as well as a large design space. Unlike most other auxetic structures that contain thin ribbons or fibrils, rotating unit structures do not have slender parts in their geometry; they have very low porosity compared to structures with other structural motifs. This low porosity enables subtractive manufacturing and avoids the problems of additive manufacturing including the limits in selecting mother material, the manufacturing time, and the accuracy. In particular, they are easily fabricated by patterning slits or ellipses on a 2D sheet, which can give auxeticity to thin-walled structures such as shells [33]. These slit or ellipse patterned structures are used in applications that deal with compressive loads because they exhibit auxetic behavior stably under compressive loads. However, because units are rigid, they do not deform significantly, and most deformation occurs as the hinge bends. This deformation causes stress concentration in the hinge area and deteriorates the durability of the structure. In addition, because of the low porosity, there is a limit to how much the effective density can be lowered, which causes difficulty in engineering lighter structures.

2.3

Chiral Mechanism

A chiral motif with chiral arrangement of unit circles and ribs also causes auxetic behavior. The unit circles of chiral structures are uniformly arranged in tri-, tetra-, or hexa-tessellation, and the ribs chirally wrap around the circular units and link one circle unit to another. When a normal load is applied to an arranged chiral structure, the load is transferred to the circular units through chiral ribs, and the transferred load is deflected from the center of circle which generates a rotational torque. The circle unit then rotates in a certain direction and pulls or pushes the adjacent circular units through ribs connected in a different direction to the load direction [35]. Two-dimensional chiral structures are composed of circular units and ribs, and they are limited to five different types [36], unlike the other types of auxetic structures with unlimited variation. Chiral structures are largely classified as trichirals, tetrachirals, and hexachirals according to the unit circle arrangement rule, and chiral tessellations and anti-chiral tessellations according to the rotation direction of the auxetic circle unit. The arrangement rules of unit circles follow basic 2D polygon tessellation. That is, trichirals, tetrachirals, and hexachirals are determined by three types of tessellation which are triangles, squares, and hexagons,

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Fig. 4 Chiral auxetic structures: (a) tetrachiral structures, (b) anti-tetrachiral structures, (c) trichiral structures, (d) anti-trichiral structures, and (e) hexachiral structures

respectively. The orientation of ribs surrounding the unit circle depends on the rotation direction of a unit circle caused by a load. Thus, whole circular units of the chiral tessellated structures rotate in the same direction, yet the circular units of the anti-chiral tessellated structures rotate in the opposite direction to their neighboring units. A combination of these categories can create a total of five auxetic chiral structures: trichiral structures, anti-trichiral structures, tetrachiral structures, anti-tetrachiral structures, and hexachiral structures, as shown in Fig. 4a–e. Note that an anti-hexachiral structure does not exist because it cannot be arranged to rotate the three units of the triangle in different directions. The chiral mechanism consists of slender structures which have similar structural advantages and disadvantages of the re-entrant mechanism. The high porosity of the slender structures allows the structure to be lightweight, but has a detrimental effect on the durability and stability of the chiral structures. Although the structure also has NPR behavior under compressive loads, it is vulnerable to local buckling if compressive loads between circular units are transmitted through thin ribs. Further, owing to geometrical arrangement, chiral structures have limited structural variation and a narrow design space, unlike the re-entrant mechanism and the rotating mechanism.

2.4

Fibril/Nodule Mechanism

Structures with a fibril/nodule mechanism, as the name implies, consist of fibrils transferring tensile load and rigid nodule units placed between the fibrils. The fibrils are connected to the nodule units, and if there is no applied force, nodule units are

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Fig. 5 Typical shape of fibril-nodule structures: (a) single fibril-type structural model for liquid crystalline polymer (bundle type) [38], (b) Multi-fibril structures with rectangular nodules, and (c) circular nodules (network type) (Part (a) reprinted with permission from Ref. [38] # 1998 American Chemical Society)

intertwined with fibrils. However, when a tensile load is applied to a fibril-nodule structure, a tensile load straightens the fibrils. In this process, the fibrils push the solid nodule units in the perpendicular direction to the load, and increase the distance to the adjacent fibrils and nodule units. This tensile force on the fibril leads to expansion in that direction, resulting in a NPR. Various fibril/nodule structures are broadly categorized into two types depending on how the fibrils and nodules are connected: bundle type and network type. The bundle-type fibril/nodule structures are made up of single fibril/nodule chains in bundles, as shown in Fig. 5a. Each single chain has an anisotropic nodule attached to the fibril in a row. When the fibril is stretched, the fibril intertwined with the nodule unfolds to increase the effective radius of the fibril/nodule chain. This causes single chains to push each other, and the whole bundle indirectly expands in the perpendicular direction to the applied load. On the other hand, the network-type fibril/ nodule structures, in which one nodule unit is connected to several fibrils, is a nodule network with linking fibrils rather than a bundle of fibril/nodule chains, as shown in Fig. 5b, c. When a tensile load is applied to the network fibril/nodule structure, intertwined fibrils are unfolded between nodules, dropping the overlapping nodules. Because of the pushing nodules, the tensile load directly causes expansion in all directions. The fibril/nodule mechanism is one of the auxetic mechanisms developed in the early days of auxetic material research, and it is a mechanism suitable for describing the auxetic polymer materials due to its fiber-like nature. Liquid crystalline polymer is an auxetic polymer material whose auxetic behavior can be understood as a fibril/nodule mechanism. Liquid crystalline polymer has the bundle-type fibril/nodule structure that forms crystalline in stress-free states, but show auxeticity with decrystallized microscopic structure when a tensile load is applied [38]. Another example is polytetrafluoroethylene foam, which is a kind of auxetic foam material. The internal structure of polytetrafluoroethylene is composed of a network of nodules connected by fibrils [37]. Because the fibril/nodule mechanism includes fibrils that have no shear resistance, fibril/nodule structures cannot bear compression, under which microscopic fibrils warp. Therefore, the fibril-nodule mechanism only derives a NPR when stretched, and the auxeticity vanishes under compressive loads.

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2.5

Mechanics of Auxetic Materials

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Miura-Folded Mechanism

The Miura-folded mechanism easily assigns auxeticity by simply folding a 2D shell structure. In the Miura-folded structure, the concavities and convexities appear repeatedly along a zigzag fold line, as shown in Fig. 6. To create a Miura-folded structure, a symmetrical parallelogram grid is engraved repeatedly on a 2D shell structure and then the shell is folded regularly along the grid lines. The interior angle of the parallelogram grid is a key geometric parameter that determines the shape and the characteristics of a Miura-folded structure. In the maximally folded state, the Miura-folded structure has a very small thickness in a certain direction and facets overlap each other in the other direction. When a tensile load is applied in this state, the fold lines of the Miura-folded structure unfold and expand the overlapping facets along the loading direction [39]. By adjusting the angle of the parallelogram grid, which is a key parameter of this structure, geometry including the bottom shape and the height of the structures, and mechanical properties including NPR can be controlled. This can be used to create two different Miura-folded structures with the same bottom shape and top fold lines but different heights. These different structures can be alternately stacked to form a 3D cellular Miura-folded structure. The Miura-folded structure has the advantage of achieving a very wide range of NPRs. In addition, unlike other auxetic mechanisms that require additive or

Fig. 6 A unit cell of a Miura-folded structure. a, b, γ and dihedral fold angle θ are the geometric parameters that determine a parallelogram facet. The other parameters of the outer dimension H, S, V, L and angles φ, ξ, ψ are defined to describe the folded state usefully [39] (Reprinted with permission from Ref. [39] # 2013 PNAS)

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subtractive manufacturing processes, the Miura-folded mechanism simply has auxeticity by folding a 2D shell. Therefore, it is useful for applications, such as fluid transport, because the entire structure is connected continuously and the material cannot move through the structure. However, the Miura-folded structure is a thin shell with a fold line, so it is less rigid than other auxetic structures. In addition, its durability is low because deformation stress is concentrated on thin fold lines that behave like hinges [39].

2.6

Buckling-Induced Mechanism

The buckling-induced mechanism is a unique mechanism, quite unlike the other auxetic mechanisms described above. Because auxeticity is created by buckling, buckling-induced auxetic materials have a NPR only when the applied compressive force is greater than a critical value. In addition, unlike other auxetic mechanisms exhibiting initially auxetic behavior, auxetic material with the buckling-induced mechanism does not exhibit auxeticity in its initial shape or at small strain ranges. The buckling-induced mechanism is represented by simple structures in which circular patterns are arranged at regular intervals in vertical and horizontal directions on a 2D sheet (Fig. 7a) [40]. Despite their simple geometry, these structures exhibit complicated behavior with three phases depending on load magnitude. A buckling-induced auxetic structure behaves differently depending on the level of strain due to compressive load. At small strain ranges, owing to the symmetry of the placed circles, a buckling-induced auxetic structure undergoes linear elastic deformation, which is similar to that of conventional structures. In this linear deformation phase, or pre-buckling phase, no special lateral deformation appears. However, as the load increases and beyond the critical strain point, it shifts to the buckling phase where nonlinearity appears. The thin part between the circle patterns is buckled and bent symmetrically by the applied compressive load. As a result, the circle patterns become ellipse or dumbbell shape and alternate in longitudinal and lateral directions. In this process, the entire structure shrinks not only in the compressive load direction but also in the lateral direction, resulting in a NPR. In the post-buckling phase, the auxeticity remains until the load is further increased and the compression is progressed. In this inner contact phase, the structure no longer has auxetic behavior, like the linear deformation phase. The simple structural motif of the buckling-induced mechanism can be easily incorporated into 3D structures from 2D circle-patterned sheets. As a typical example, a circle-patterned cylinder is a structure in which circular patterns are arranged in the direction of the axis and tangent to the surface of a thick cylinder, as shown in Fig. 7c [41]. Similar to the 2D circle-patterned structures, the nonlinear buckling behavior is only shown with respect to a compressive load in the axial direction of the cylinder. An example of applying a 2D circle-patterned structure to a 3D structure in a more sophisticated manner is a 3D buckyball structure, as shown in Fig. 7b [42]. Like the 2D structure, it has a simple uniformly patterned structure and the nonlinear buckling behavior is shown above the critical point, after which the

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Fig. 7 Buckling-induced auxetic structures: (a) 2D circle-patterned sheet and its deformation [40], (b) 3D buckyball structure and its deformation [42], (c) circle-patterned cylinder [41] (Part (a) reprinted with permission from Ref. [40] # 2008 Elsevier. Part (b) reprinted with permission from Ref. [42] # 2012 PNAS. Part (c) reprinted with permission from Ref. [41] # 2016 Elsevier)

surface of the sphere is locally rotated. In this process, the structures show NPR behavior. In addition, a number of buckyball structures may be regularly stacked to form a 3D bulk structure [43]. When a compressive load is applied, the buckyball structures inside the 3D bulk structure shrink under load, resulting in a NPR bulk structure. A buckling-induced structure has the advantages of controlling the physical properties and performing motion actuating. However, as buckling is induced by compression, auxeticity does not appear with tensile loads. Furthermore, because it is very sensitive to loading and boundary conditions, when applying the bucklinginduced mechanism, devices that can set and control the load and boundary conditions accurately should be used.

2.7

Helical Auxetic Yarn

Helical auxetic yarn is a unique auxetic material that consists of two types of threads [44]. The core thread, one of the constituents, is a thick but soft thread that has straight shape in stress-free states. On the other hand, the wrap thread, the other constituent, is thin but has a high stiffness. This wrap thread helically coils around and attaches to the core thread in the initial state, and the two threads do not move relative to each other. In the initial state, the effective diameter of the entire helical yarn is defined as the diameter of the core thread plus twice the diameter of the wrap thread (Fig. 8a). When a tensile load is applied to the helical yarn, a dramatic shape

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a

Stiffer Wrap

c

Elastomeric Core

b Stiffer Wrap

Elastomeric Core

Fig. 8 Helical auxetic yarn structure: (a) Undeformed [44] and (b) deformed configurations of helical auxetic yarn [44]. (c) Auxetic fabric with helical yarn [44] (Part (a–c) reprinted with permission from Ref. [44] # 2009 Elsevier)

change occurs due to the difference in stiffness values between the two threads. Because the wrap thread is stiffer than the core thread, the helically coiled wrap thread straightens in the tensile load direction. In this process, the soft core thread is pushed by the wrap thread and twisted along the attached side of the wrap thread, as shown in Fig. 8b. As a result, the core thread is now helically coiled around the wrap thread as opposed to the condition in the stress-free state. In this state, the effective diameter of the deformed shape is defined as the diameter of the wrap thread plus twice the diameter of the core thread. Because the core thread is considerably thicker than the wrap thread, the effective diameter increases due to the tension load. As a result, an entire fabric composed of helical auxetic yarns has a NPR, and extends in a perpendicular direction to the applied load. Unlike other auxetic structures, helical auxetic yarns have the advantage that they can be easily fabricated to surround a free-form surface, such as a body. However, the structures are composed of core and wrap threads that have no resistance to shear forces, so they do not exhibit effective auxetic behavior when subjected to compressive loads. In addition, as the helical yarns have indirect auxetic behavior, the lateral expansion due to a tensile load is not large and does not produce a high NPR value compared to other mechanisms.

2.8

Crumpled Mechanism

A crumpled mechanism uses random and isotropic crumples as a structural motif. Crumpled structures have microscopic crumples due to omnidirectional compressions or chemical imperfections inside the structure. These crumples are randomly oriented, as shown in Fig. 9, so that when a tensile load is applied to the crumpled structure the crumples unfold and expand in all directions. In addition, when thin foil structures are compressed in all directions, natural random crumples are generated

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Fig. 9 A typical example of crumpled structures: (a) 3D [46] and (b) 2D crumpled aluminum foil [46] (Part (a) and (b) reprinted with permission from Ref. [46] # 2013 Elsevier)

by membrane buckling. Even after the compression load is removed, the crumples are retained by plastic deformation and the desired crumpled mechanism can be represented. In addition to the method of compressing the mother material, an imperfection in the mother material can be used to harness crumples. For example, a carbon imperfection in hexagonal tessellations of graphene is induced to create a 2D crumpled graphene. A carbon imperfection results in bending to minimize the internal energy, and harnessing crumples in load-free conditions [45]. Crumpled auxetics, like the Miura-folded structure, do not pattern on a 2D sheet, so fluid does not flow through the sheet. That is, unlike other auxetic mechanisms, crumpled structures can be easily applied to engineering problems dealing with fluids, such as fluid transportation. In addition, because of the use of random crumples without structural symmetry, unlike other auxetic materials that undergo cutting or additive manufacturing processes, they only undergo compression and imperfection creation. This makes it easy to obtain large quantities of high-quality auxetic materials compared to other structures that require high precision in the manufacturing process. Moreover, the structures can control the mechanical properties of the entire material through compression and imperfection control in the process without difficulty [46].

3

Relationship Between the Deformation Mechanism and the Material Properties

As described above, much research on the structural behavior of auxetic material and materials that are likely to become auxetic has been conducted. However, there was not as much information on the relationship between the mechanical properties of auxetic materials and their deformation mechanism. In 2012, Elipe et al. [47] performed a comparative study of two- and three-dimensional auxetic geometries

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using simulation tools. They found some relevant design properties of different auxetic geometries such as the Poisson’s ratio, the maximum volume, the area reduction, the equivalent Young’s modulus, and the density. In this study, they used computer-aided design tools to first design several auxetic structures with different geometries by replicating unit cells. These structures were then analyzed by using the finite element method (FEM) to derive the relevant properties. From these results, the relationship between some geometric parameters and material properties could be found including the Poisson’s ratio versus the maximum area (for 2D structures) or volume (3D structures) reduction, the equivalent Young’s modulus for different 2D/3D auxetic structures, and the Young’s modulus versus the density with respect to bulk properties. The characteristics of auxetic materials vary significantly depending on the geometry and shape of the structures. Although only few properties were considered, the results clearly demonstrate that auxetic materials offer a versatile way of achieving the specific target properties by employing an appropriate auxetic mechanism [47].

4

Expected Properties of Auxetic Materials

Because of the NPR effect of auxetic materials, there are various expected geometrical and mechanical properties hardly observed in natural, nonauxetic materials. The properties include, but are not limited to, synclastic curvature in bending in contrast to anticlastic curvature appeared in conventional materials, variable permeability, high shear stiffness, enhanced indentation resistance, high fracture toughness, and improved damping and sound absorption properties. Proper understanding of geometric, static, and dynamic properties of auxetic materials is therefore essential to fully utilize these extraordinary properties for specific target applications. In this section, we describe the fundamental reasons why auxetic materials are expected to have such properties by comparing them with the properties of conventional, nonauxetic materials.

4.1

Synclastic Curvature in Bending

When out-of-plane bending is applied to nonauxetic materials, the surface displays anticlastic curvature, whereby the material is curved from a vertical position to the opposite direction [3], as shown in Fig. 10a. This behavior leads to a requirement of additional forces for conventional materials when constructing convex shapes. However, the surface of auxetic materials has synclastic curvature if deformations occur in the same direction [4], as shown in Fig. 10b. Compared to the saddle shape with anticlastic curvature, the dome-like shape with synclastic curvature is expected to be useful because it can be used to construct complex structures through a relatively small number of machining steps and low manufacturing cost [4].

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Fig. 10 Curvature in bending: (a) anticlastic curvature of conventional, nonauxetic materials and (b) synclastic curvature of auxetic materials

Fig. 11 An illustration of variable permeability in a triangular re-entrant structure

4.2

Variable Permeability

In a porous material used for filtration, such as filters, controlling permeability is one of the key factors. Porous auxetic materials are excellent for varying permeability thanks to the NPR effect. These materials increase the size of their pores when a tensile load is applied in a certain direction. For example, a triangular re-entrant structure improves permeability by adjusting the size of its pores as each unit cell unfolds in all directions, as shown in Fig. 11. Because of auxetic behaviors, this variable permeability can be utilized from macroscale to nanoscale materials [5].

4.3

High Shear Stiffness

Shear modulus (G) is a property measured to determine the deformation that occurs when a force is applied parallel to one side of an object and the other side is fixed. If the value of shear modulus is large, the material is rigid and has high shear resistance. Thus, shear modulus is an important factor to select when designing and manufacturing any structure. In isotropic auxetic materials, shear resistance is expected to be more beneficial than in nonauxetic materials. This feature can be

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explained by the following equations consisting of the Young’s modulus (E), the shear modulus (G), the bulk modulus (K ), and the Poisson’s ratio (v) [4]. G¼

E 2ð 1 þ v Þ

(1)

K¼

E 3ð1
2vÞ

(2)

As can be seen in Eqs. (1) and (2), if the Poisson’s ratio (v) approaches 0.5, the value of the shear modulus is reduced but the bulk modulus is greatly increased. This means that the material easily undergoes shear deformation, but the shape of material does not change much; the material is incompressible like rubber. However, when the Poisson’s ratio is close to
0.5, the shear modulus (G) and the Young’s modulus (E) have almost the same value and a higher value than the bulk modulus (K ). Thus, in this case, it is easy to compress the material but hard to shear it. Further, as the Poisson’s ratio (v) approaches
1, the shear modulus (G) becomes infinite which is similar to the hardness becoming infinity [6] and the shear resistance gets significantly larger [48]. In other words, it is hard to shear but easy to undergo volumetric deformation. Therefore, by adjusting the Poisson’s ratio (v) it is possible to design a structure efficiently according to its purpose.

4.4

Enhanced Indentation Resistance

When an impact is applied to nonauxetic materials in a certain region, these materials tend to spread from the region to relieve the pressure, as shown in Fig. 12a. Thus, this behavior causes a decrease in density at the impact region. On the contrary, in the same case, isotropic auxetic materials move to regions where the impact is applied. This leads to an increase in indentation resistance by having regions of higher density [4], as shown in Fig. 12b. This phenomenon can also be confirmed by the relationship between the Poisson’s ratio (v), the Young’s modulus (E), and the Fig. 12 An illustration of the resistance to indentation for (a) nonauxetic and (b) auxetic materials

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material hardness (H ) associated with the indentation resistance. The relationship between material hardness, Young’s modulus, and Poisson’s ratio is given by


 E γ H/ , 1
v2

(3)

where γ is 1 or 2/3 for uniform pressure distribution or Hertzian indentation, respectively. As mentioned earlier, Poisson’s ratios for 3D isotropic materials range from
1.0 to 0.5. When the Poisson’s ratio approaches
1.0, the hardness becomes infinite and this leads to an increase in indentation resistance [4]. However, Poisson’s ratios for the 2D isotropic materials can change between
1.0 and 1.0, which means that the upper limit of the ratios is different from the 3D materials. Therefore, in 2D materials the hardness can be infinite even for positive Poisson’s ratio. This behavior is not presented in anisotropic materials because the above equation is only applicable for isotropic materials. Many studies have been carried out on various kinds of synthetic auxetic materials to investigate hardness properties. For instance, Alderson et al. found experimentally that auxetic ultra-high molecular weight polyethylene (UHMWPE) is twice as hard as conventional UHMWPE. This behavior specifically occurs at low loads (10–100 N) and in the radial direction. At low loads, because UHMWPE undergoes elastic deformation, the NPR effect is significant. Further, although indentation resistance increases only in the radial direction, there is no reduction to the hardness in the axial direction. Therefore, anisotropic as well as isotropic auxetic UHMWPE can be beneficial to use when considering hardness [7].

4.5

High Fracture Toughness

Fracture resistance is better for auxetic materials than for conventional materials [8]. Auxetic materials lead to low crack propagation and require more energy to expand materials. Thus, many studies have been conducted to find the difference in fracture toughness between auxetic and nonauxetic materials. Donoghue et al. demonstrated experimentally that auxetic materials need less energy for crack propagation than conventional materials [10]. Further, Maiti showed that the fracture toughness of a conventional material is determined by the ratio of its normalized density [11]. From the critical tensile stress equation, Lakes indicated that toughness can be changed as the Poisson’s ratio varies, and also if the Poisson’s ratio approaches
1 the material becomes very tough [3]. The critical tensile stress equation is as follows:
σ¼

πET 2r ð1
v2 Þ

1=2

,

(4)

where E is the Young’s modulus, T is surface tension, and r is the plane circular crack radius.

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Later, the fracture toughness of re-entrant foam and conventional foam was analytically derived and a comparison between the fracture toughness of the two foams was conducted by Choi and Lakes [8]. The fracture toughness ( K IC ) of conventional foam can be expressed by   K IC ρ pffiffiffiffi ¼ 0:19 , ρs σ f πl

(5)

where ρs is the density of the solid made from the foam, ρ is the density of the foam, σ f is the fracture strength of the cell rib, and l is the cell rib length. The fracture toughness (K TIC ) of the re-entrant foam is given by K TIC pffiffiffiffi ¼ 0:10 σ f πl

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 þ sin ð0:5π
φÞ ρ , 1 þ cos 2φ ρs

(6)

where φ is the rib angle which determines cell shapes. By combining the above two equations, Eqs. (5) and (6), the proportion of the fracture toughness between the re-entrant foam and the conventional foam can be derived. K TIC ¼ 0:53 K IC

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ sin ð0:5π
φÞ 1 þ cos 2φ

(7)

From Eq. (7), it can be seen that density is not needed when comparing re-entrant foam with conventional foam. Although these equations provide the basic insight into foam-type structures, the expressions are based on ideal models rather than actual ones, and so it is difficult to predict the behavior of real cell shapes from these analytical expressions.

4.6

Increased Energy Absorption

Compared to conventional materials, auxetic materials exhibit superior features in terms of damping and sound absorption. NPR foams are better at absorbing sound than conventional materials [12]. The ability to absorb sound depends on the size of the pore of an auxetic foam, which is excellent for NPR foams with smaller pores at frequencies above 630 Hz. In addition, NPR covered foams have better acoustic absorption capacity than NPR uncovered foams at the frequencies between 250 and 1000 Hz [13]. Alderson et al. demonstrated experimentally that the highest ultrasonic attenuation coefficient of auxetic UHMWPE was 1.5 times higher than the coefficient of the nonauxetic microporous UHMWPE and 3 times higher than foams that were processed conventionally [14]. Furthermore, auxetic foams have excellent resilience under dynamic impact loading, which is not noticeable in conventional foams [15].

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Potential Applications of Auxetic Materials

Auxetic materials, whose geometrical and mechanical properties have many novel characteristics due to the NPR effect, can be applied in various engineering and scientific fields, and many potential applications have been suggested. In this section, several applications with their basic mechanisms are described. In biomedical engineering, there are many possible applications for using auxetic materials, such as arterial prostheses, implants, stents, scaffolds, dilators, sutures, ligament/muscle anchors, bandages, and prosthetic linings [16]. One of the wellknown smart biomedical devices is the auxetic bandage. When the auxetic bandage is attached to an infected wound, the bandage expands and the pores extend. This mechanism, derived from shape memory ability and the NPR effect, increases porosity and breathability and allows the wound to heal faster. A filter using the variable permeability property is another example of an application. Because particulates trapped in the pores reduce the efficiency of filter systems, pore opening is important for effective filtration. If the filter is fabricated with the characteristics of an auxetic material, the size of its pores is extended in all directions as a tensile load is applied in one direction. Therefore, by controlling pore size, when compared to a conventional filter, the filtration efficiency and functional performance of the auxetic filter structure can be improved [19]. Another example is a press-fit fastener, which uses the auxetic material property to expand in the transverse direction when a tensile load is applied longitudinally. This characteristic makes it more difficult to remove the fastener [49, 50], and so the holding capacity of this fastener is increased with tensile load. Similar effects occur in fiber-reinforced composites, which enhance the pull-out resistance as the fiber expands [50]. When a fiber with a NPR is pulled along one axis, it will become wider perpendicular to the applied force, which can prevent loosening between the matrix and the fiber. The behavior of this reinforcing fiber is described in Fig. 13. In addition, by using auxetic materials, a soft metamaterial robot with only one actuator can be designed to move through a travel channel. This robot is made with Fig. 13 Pull-out resistance of a reinforcing fiber when a tensile load is applied

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two passive clutch components and a linear actuator. The two passive clutch components, connected to a linear actuator, possess opposite signs of Poisson’s ratio. Thus, the two components are composed of auxetic and normal materials, and they have different mechanical behaviors. When the actuator is expanded in an axial direction, the auxetic material is compressed but the normal material is expanded transversely. In this case, the limiting friction of the auxetic material is relatively reduced when compared to that of the normal material. Therefore, the normal material does not move but the auxetic material slides along the channel. On the contrary, if the actuator is compressed, the limiting friction is decreased in the normal material. This case has the opposite effect and so the normal material slips. With this simple mechanism controlling limiting friction, linear unidirectional motion is realized with only one actuator, and the difficulty of synchronization can also be solved [20]. Manufacturing clothes using auxetic fabrics could be a potential candidate for a solution to solve the problem of children growing quickly. Parents usually buy clothes that are much larger than their child’s size, taking into account the fact that they will grow. As a result, children wear loose clothes, which can cause serious injuries when playing. However, thanks to the behavior of auxetic fabrics, which expand in all directions when a tensile force is applied in one direction, auxetic fabric with a foldable structure could provide long-lasting clothing for children. This could relieve the discomfort and burden for both children and their parents [5]. Konakovic et al. introduced a computational method to approximate a nondevelopable surface with a double-curvature from flat rigid materials [21]. Although thin flat materials, which are inextensible but bend freely, are able to approximate developable surfaces, which are single curved surfaces, they cannot approximate nondevelopable surfaces. If rotating rigid units with triangular patterns are inserted into these flat materials, the material gets much more freedom and can be stretched. However, it is hard to intuitively predict flat domains that can construct target surfaces. Thus, by using a conformal map and numerical optimization, the flat domain that is the easiest to approximate a target surface is obtained. After deciding the cuts and orientations of a specific cutting pattern, a 2D region can be lifted onto target surfaces. These approximations provide new possibilities in material design and have the potential to be applied in many areas. Furthermore, auxetic material can be used for composite sandwich panels that are designed to resist high strain rate loads. A panel with multiple layers of the same auxetic unit cells and metal facets enhances its resistance to blast damage while remaining lightweight. By using this benefit, the panel can be applied to armored vehicles to increase their ballistic and impact resistance. Further, composite structures are used to prevent underwater explosions in ship hulls and improve crash absorption in automobile parts [18]. In addition, to protect soldiers on the battlefield thick protective clothing is required, which is normally stiff, heavy, and rigid. However, if protective clothing with auxetic tiles is produced, it can provide a similar level of protection but is lighter and thinner. In addition, body armor and combat jackets with auxetic material improves energy absorption and reduces wear

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resistance. These textiles also lead to changes in high volume and drapability due to the properties of auxetic materials [9]. Auxetic materials with synclastic curvature have been applied to the field of aerospace, especially in aircraft nose cones and wing panels. Helmets and kneepads are also manufactured with auxetic material, because it can better cover the body [6]. In addition to these applications, there are other potential applicable fields, including efficient piezoelectric sensors, sports equipment, and sound-absorbing materials [17].

6

Concluding Remarks

Because of the NPR effect, there are many unique, enhanced properties expected for auxetic materials. But, at the same time, certain physical properties can be deteriorated when incorporating the deformation mechanism of auxetic materials. For example, most auxetic structures have hinges and pores that often make these structures weaker and more flexible than conventional nonauxetic solids due to the nature of their geometry and deformation mechanism. Hence, the use of these materials for large load-bearing structures can be limited. Also, a significant amount of stresses can be concentrated near the hinge regions, rendering the auxetic structure vulnerable to cyclic fatigue loadings. Therefore, when designing and using auxetic materials, extreme care must be taken to consider the weakened properties due to the effect of negative Poisson’s ratio as well. Nevertheless, auxetic materials have been successfully applied to various fields of engineering applications and offer a great possibility to develop materials and structures with extraordinary mechanical properties. In order to further broaden the application of auxetic materials, more in-depth understanding and thorough investigation on the deformation mechanism and its link to the mechanical properties are essential. In particular, combining auxetic materials with conventional, nonauxetic materials would enlarge the design space of mechanical properties that have not been reached by using solely conventional materials or auxetic materials.

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8. Choi JB, Lakes RS. Fracture toughness of re-entrant foam materials with a negative Poisson’s ratio: experiment and analysis. Int J Fract. 1996;80:73–83. 9. Liu Q. Literature review: materials with negative Poisson’s ratios and potential applications to aerospace and defence. Defence Science and Technology Organisation Victoria (Australia) Air Vehicles Div; 2006. 10. Donoghue JP, Alderson KL, Evans KE. The fracture toughness of composite laminates with a negative Poisson’s ratio. Phys Status Solidi B. 2009;246:2011–7. 11. Maiti SK, Ashby MF, Gibson LJ. Fracture toughness of brittle cellular solids. Scr Metall. 1984;18:213–7. 12. Chekkal I, Bianchi M, Remillat C, Becot FX, Jaouen L, Scarpa F. Vibro-acoustic properties of auxetic open cell foam: model and experimental results. Acta Acustica United Acustica. 2010;96:266–74. 13. Howell B, Prendergast P, Hansen L. Examination of acoustic behavior of negative poisson’s ratio materials. Appl Acoust. 1994;43:141–8. 14. Alderson KL, Webber RS, Mohammed UF, Murphy E, Evans KE. An experimental study of ultrasonic attenuation in microporous polyethylene. Appl Acoust. 1997;50:23–33. 15. Scarpa F, Ciffo LG, Yates JR. Dynamic properties of high structural integrity auxetic open cell foam. Smart Mater Struct. 2003;13:49. 16. Bhullar SK, Lala NL, Ramkrishna S. Smart biomaterials - a review. Review on advanced. Mater Sci. 2015;40:303–14. 17. Novak N, Vesenjak M, Ren Z. Auxetic cellular materials - a review. Strojniški Vestnik – J Mech Eng. 2016;62:485–93. 18. Imbalzano G, Tran P, Ngo TD, Lee PV. Three-dimensional modelling of auxetic sandwich panels for localised impact resistance. J Sandw Struct Mater. 2017;19:291–316. 19. Alderson A, Rasburn J, Ameer-Beg S, Mullarkey PG, Perrie W, Evans KE. An auxetic filter: a tuneable filter displaying enhanced size selectivity or defouling properties. Ind Eng Chem Res. 2000;39:654–65. 20. Mark AG, Palagi S, Qiu T, Fischer P. Auxetic metamaterial simplifies soft robot design. In Robotics and Automation (ICRA), International Conference on Robotics and Automation. IEEE. 2016. 21. Konaković M, Crane K, Deng B, Bouaziz S, Piker D, Pauly M. Beyond developable: computational design and fabrication with auxetic materials. ACM Trans Graph (TOG). 2016;35:79–89. 22. Carneiro VH, Meireles J, Puga H. Auxetic materials - a review. Mater Sci – Poland. 2013;31:561–71. 23. Masters IG, Evans KE. Models for the elastic deformation of honeycombs. Compos Struct. 1996;35:403–22. 24. Yang L, Harrysson O, West H, Cormier D. Mechanical properties of 3D re-entrant honeycomb auxetic structures realized via additive manufacturing. Int J Solids Struct. 2015;69:475–90. 25. Larsen UD, Signund O, Bouwsta S. Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio. J Microelectromech Syst. 1997;6:99–106. 26. Theocaris PS, Stavroulakis GE, Panagiotopoulos PD. Negative Poisson's ratios in composites with star-shaped inclusions: a numerical homogenization approach. Arch Appl Mech. 1997;67:274–86. 27. Wang ZP, Poh LH, Dirrenberger J, Zhu Y, Forest S. Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Comput Methods Appl Mech Eng. 2017;323:250–71. 28. Wang XT, Wang B, Li XW, Ma L. Mechanical properties of 3D re-entrant auxetic cellular structures. Int J Mech Sci. 2017;131:396–407. 29. Grima JN, Evans KE. Auxetic behavior from rotating squares. J Mater Sci Lett. 2000;19:1563–5. 30. Grima JN, Farrugia PS, Gatt R, Attard D. On the auxetic properties of rotating rhombi and parallelograms: a preliminary investigation. Phys Status Solidi B. 2008;245:521–9.

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1. Int J Mech Sci. 1997;39:305–14. 36. Grima JN, Gatt R, Farrugia PS. On the properties of auxetic meta-tetrachiral structures. Phys Status Solidi B. 2008;245:511–20. 37. Evans KE, Caddock BD. Microporous materials with negative Poisson’s ratios II. Mechanisms and interpretation. J Phys D Appl Phys. 1989;22:1877–83. 38. He C, Liu P, Griffin AC. Toward negative Poisson ratio polymers through molecular design. Macromolecules. 1998;31:3145–7. 39. Schenk M, Guest SD. Geometry of Miura-folded metamaterials. Proc Natl Acad Sci. 2013;110:3276–81. 40. Bertoldi K, Boyce MC, Deschanel S, Prange SM, Mullin T. Mechanics of deformation-triggered pattern transformations and superelastic behavior in periodic elastomeric structures. J Mech Phys Solids. 2008;56:2642–68. 41. Javid F, Liu J, Shim J, Weaver JC, Shanian A, Bertoldi K. Mechanics of instability-induced pattern transformations in elastomeric porous cylinders. J Mech Phys Solids. 2016;96:1–17. 42. Shim J, Perdigou C, Chen ER, Bertoldi K, Reis PM. Buckling-induced encapsulation of structured elastic shells under pressure. Proc Natl Acad Sci. 2012;109:5978–83. 43. Babaee S, Shim J, Weaver JC, Chen ER, Patel N, Bertoldi K. 3D soft metamaterials with negative Poisson's ratio. Adv Mater. 2013;25:5044–9. 44. Miller W, Hook PB, Smith CW, Wang X, Evans KE. The manufacture and characterisation of a novel, low modulus, negative Poisson’s ratio composite. Compos Sci Technol. 2009;69:651–5. 45. Grima JN, Winczewski S, Mizzi L, Grech MC, Cauchi R, Gatt R, Rybicki J. Tailoring graphene to achieve negative Poisson's ratio properties. Adv Mater. 2015;27:1455–9. 46. Bouaziz O, Masse JP, Allain S, Orgéas L, Latil P. Compression of crumpled aluminum thin foils and comparison with other cellular materials. Mater Sci Eng A. 2013;570:1–7. 47. JCA E, Lantada AD. Comparative study of auxetic geometries by means of computer-aided design and engineering. Smart Mater Struct. 2012;21:104993–5004. 48. Saxena KK, Das R, Calius EP. Three decades of auxetics research - materials with negative Poisson’s ratio: a review. Adv Eng Mater. 2016;18:1847–70. 49. Choi JB, Lakes RS. Non-linear properties of polymer cellular materials with a negative Poisson's ratio. J Mater Sci. 1992;27:4678–84. 50. Evans KE. Auxetic polymers: a new range of materials. Endeavour. 1991;15:170–4.

Nanoindentation and Indentation Size Effects: Continuum Model and Atomistic Simulation

24

Chi-Hua Yu, Kuan-Po Lin, and Chuin-Shan Chen

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 MD for Quasi-Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Indenter Repulsive Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Dislocation Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Dislocation Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Simulation Details and Methodology Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Simulation System Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Simulation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dislocation Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Indentation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Indentation Size Effect (ISE)-Dislocation Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Indentation Size Effect (ISE)-Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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C.-H. Yu Division of Research and Development, CoreTech Corp. (Moldex3D), Hsinchu, Taiwan K.-P. Lin Structural Engineering, University of California, San Diego, CA, USA C.-S. Chen (*) Department of Civil Engineering, National Taiwan University, Taipei, Taiwan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_26

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Abstract

Nanoindentation is one of the most widely used methods to measure the mechanical properties of materials at the nanoscale. For spherical indenters, when radius decreases, the hardness increases. The phenomenon is known as the indentation size effect (ISE). Nix and Gao developed a continuum model to explain the ISE in microindentation. However, the model overestimates the hardness at the nanoscale. The objective of this study is to develop proper methods to probe key quantities such as hardness and geometric necessary dislocation (GND) density from the quasi-static version of molecular dynamics (MD) simulations and to develop a mechanism-based model to elucidate the ISE phenomenon at the nanoscale. A reliable method is presented to extract the GND directly from dislocation length and the volume of plastic zone in the MD simulations. We conclude that the hardness determined directly from MD simulations matches well with the hardness determined from the Oliver–Pharr method. The ISE can be observed directly from the MD simulations without any free parameters. The model by Swadener et al. rooted from the Nix and Gao model underestimates the GND density at the nanoscale. However, this model can accurately predict the hardness size effects in nanoindentation if it uses the GND density directly calculated from the MD simulations. Keywords

Nanoindentation size effect · Molecular dynamics simulations · Geometrically necessary dislocation density · Hardness

1

Introduction

Nanoindentation is one of the most widely used methods to measure the mechanical properties of materials at the nanoscale [1–5]. In the past three decades, significant advances in instruments make it possible to measure load and displacement throughout the process of indentation [5–8]. To resolve uncertainty in measuring the contact area at the nanoscale, Oliver and Pharr [7] proposed an ingenious way to probe nanohardness and Young’s modulus from the curve of load and penetration depth. In addition to hardness and Young’s modulus, the nanoindentation tests have been extended to characterize other mechanical properties such as visco-elastic properties and fracture toughness. Nanoindentation tests have been used in many applications, for example, mechanical integrity of interconnect circuits and microelectromechanical systems (MEMS), mechanical properties of cutting-edge materials, and biomaterials design etc. [9–11]. Among these applications, thin film or layered systems are often dominated. For these systems, nanoindentation tests provide an additional advantage as the film does not need to separate from the substrate for testing. Mechanical properties at the nanoscale often exhibit difference from bulk behavior [12–14]. Furthermore, some of these properties correlate well with the length scale and are termed as size effects. For nanoindentation testing, the indentation size

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effect (ISE) refers to the phenomena of increase in hardness when a shallower indentation depth is applied on crystalline metals. For spherical indenters, the ISE refers to a phenomenon in which hardness increases when the radius of a spherical indenter decreases. The phenomenon, mechanism, and theory for the ISE at the micron scale are now well-established. It is proposed to be induced by the strain incompatibility which is caused by geometrical condition. Nevertheless, it is not clear whether the same phenomenon occurs at the nanoscale. Ashby [15] and Nye [16] explained the deformation incompatibility with the concept of geometrically necessary dislocation (GND). The strain distribution becomes localized when it is subjected to an indentation. Given this circumstance, the GND should be generated to accommodate the incompatible deformation. Nix and Gao [17] introduced a theoretical strain gradient model of GND to interpret ISE in crystalline materials with a conical indenter at the micron scale. Following closely with the strain gradient plasticity model proposed by Nix and Gao [17], several studies were conducted to explain and interpret the ISE at the nanoscale [1, 18–33]. Swadener et al. [34] extended Nix and Gao’s model to spherical and Berkovich indenters at the nanoscale. Huang et al. [38] proposed that a maximum dislocation density may exist if the spherical indenter size was reduced to a submicron scale. Analogously, it was argued that if the indenter size was small, the dislocation density would reach a fixed value. Consequently, the nanohardness would stay in a steady state value and the ISE phenomenon may not exist at the nanoscale. Nix and Gao model has been extended to indenters of various shapes and has successfully modeled microindentation experiments [34–36]. However, recent studies showed deviation from the Nix and Gao model when the penetration depth is less than hundreds of nanometers. Feng and Nix [1] proposed a correction to modify the plastic zone size by a fudge factor ( f ). The overestimation of hardness has been improved by an expansion of plastic zone size. Durst [25, 37] also argued a correction of Nix and Gao model by expanding the plastic zone size. Huang et al. [38] introduced the maximum allowable geometrically necessary dislocation density (ρgmax) to explain that the density becomes unrealistically high when indenter is at the nanoscale. Abu Al-Rub [30] suggested a micromechanics model to improve the strain gradient plasticity model in micron and submicron length scale by altering the linear combination of geometrically necessarily dislocation density (GND) and statistically stored dislocation density (SSD). Molecular dynamics (MD) has been exploited in the study of the crystalline defects and mechanical properties [39–44]. Empowered by high performance computer and parallel computing technique, MD becomes a powerful computational tool to simulate the nanoindentation test with all atomistic information. Gannepalli et al. [45] have studied the nucleation and mobility of dislocation during nanoindentation process of Au (001). It has reported that the critical resolved shear stress of dislocation gliding was about 2 GPa. Lilleodden et al. [46] have proposed an atomistic study of gold subjected to a spherical indenter, and the results were compared with Hertzian contact theory. In their study, the Young’s modulus for Au (111) has been reported to be 110 GPa and Au (001) to be 89 GPa, which

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indicated a significant anisotropic effect of nanogold thin film. Liang et al. [42] employed the MD simulation to study the anisotropy of single crystalline copper. Nair et al. [47] has studied the nanoindentation of nickel thin films which contain low angle grain boundaries with different film thickness. The size effect for the thickness of thin film showed the decrease of thickness leads to an increase of the flow stress from their study. Urbassek et al. [48] provided a method to measure dislocation density in MD simulation. The method is to determine the largest distance of the dislocation still connected to the contact surface as the radius of plastic volume. The results of this method deviate significantly with different materials and indentation orientations. Retrieving the information of dislocation directly from experimental instruments is a nontrivial task. MD simulations are not only able to provide the information of crystalline structure during indentation process but also to reveal the evolution of dislocation nucleation and mobility. Several research efforts have been devoted to reveal the defect structure of crystalline materials. Honeycutt and Andersen [49] discussed the solid-like structure of HCP crystalline using Lennard-Jones particles. By defining several common types of atom structures and searching for near neighbors within a given cutoff, the defect structures can be identified. Kelchner et al. [50] proposed the centrosymmetric method to identify atoms undergone elastic deformation or plastic deformation for FCC metals. Adopted from Honeycutt and Andersen, Ackland and Jones [51] reported that the bond angular distribution method is able to recognize the local crystalline structure. Stukowski and Albe [52] presented the dislocation extraction algorithm which is able to extract the dislocation and associated Burgers vector directly from MD simulations. This algorithm enables one to visualize the defects and dislocation through the nanoindentation process and is adopted in the present study. In contrast to the strain-gradient theory that can explain the ISE phenomenon at the micron scale very well, the theory and mechanism for ISE at the nanoscale has not been well-established. For years, many studies tried to explain and modify Nix and Gao model for nanoindentation with free parameters. More theoretical development, experimental observations, and MD simulations are thus required. The objective of this study is to fulfill this gap and present a reliable method to determine key quantities such as hardness and GND density from MD simulations and to develop a mechanism-based model to elucidate the ISE phenomenon at the nanoscale.

2

Methodology

MD is an ideal tool to study the mechanism of indentation at the nanoscale. The methodology to perform nanoindentation simulations and to probe key quantities such as hardness and GND density using the quasi-static version of MD is described in details below.

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2.1

Nanoindentation and Indentation Size Effects: Continuum Model and Atomistic. . .

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MD for Quasi-Static Loading

In order to study the nanoindentation size effect, a quasi-static MD was employed in the present study. The simulations were performed with a constant number of atoms and the energy was minimized by the conjugate gradient method. Total potential energy in a MD system is the summation of the strain energy and the external work. Strain energy is the energy stored in the model. External work is from the indentation process. The total potential energy of the model can be derived as Π ¼ Em  W

(1)

where Π is the total potential energy of the system, W is the external work done by indentation and the negative sign indicates the potential loss due to external work, and Em is the strain energy of the model due to interatomic interactions. Differentiating Eq. 1 with the position of ith atom, we have: @Π @Em @W ¼  @xi @xi @xi

(2)

The total potential energy is at its minimum value in equilibrium and the derivative of the total potential energy is zero. Thus, we have: @Em @W ¼ @xi @xi

(3)

where the term on the left-hand side is the internal force between the ith atom with the rest of atoms within the interatomic potential cutoff, and the term on the right-hand side is the external force exerting on the ith atom. Eq. 3 also indicates that when the model is in equilibrium, the internal force is equal to the external force.

2.2

Indenter Repulsive Potential

In order to reduce the complexity of nanoindentation simulations, Lilleodden et al. [46] used a repulsive potential to simulate a spherical indenter. It enables one to ease the computational effort from constructing a real indenter with millions of atoms. The repulsive potential form can be described as: F ¼ Að R  r i Þ 2

(4)

where A is a constant based on the effective stiffness of indenter, R is the radius of the spherical indenter, and ri is the distance from indenter center to the ith atom inside the indenter. When ri is smaller than the indenter radius (R), the repulsive force will be applied to the ith atom by Eq. 4. The load of indentation can be recorded as the

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summation of force given by all the contacted atoms. Indentation simulations conducted by the indenter repulsive potential can save considerable computation resource, which is helpful to perform large-scale simulations.

2.3

Hardness

Hardness is one of the most measured mechanical properties using indentation experiments. Nanoindentation experiments usually rely on the Oliver–Pharr method [7] to estimate the hardness of materials. By definition, hardness should be computed by: H¼

P A

(5)

where P is the applied load on the specimen, and A is the projected contact area. In this study, three different methods were used to evaluate the hardness from MD simulations. One method is to determine the hardness directly from the MD simulations during the loading process of indentation. The second one is the Oliver–Pharr method and the hardness is evaluated after the unloading process. The third one is to derive the hardness from dislocation density by the Taylor’s dislocation model.

2.3.1 Hardness from Loading Process During indentation loading, the applied load can be evaluated directly from which atoms were contacted to the indenter. The repulsive force from each contacted atoms can be calculated from Eq. 4. Consequently, the total applied load P is able to compute from assembling the forces of all the atoms which were contacted by the indenter, i.e., F¼

n X i¼1

AðR  jxi  rjÞ2

xi  r j xi  r j

(6)

where F is the repulsive force of indentation, n is the number of the contacted atoms, x is the position of the contacted atoms, and r is the center position of a spherical indenter, as shown in Fig. 1. A novel technique has been employed to calculate the contact area in the present study. Contact area is determined by extracting the crystalline geometry directly Fig. 1 Indenter repulsive potential illustration

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Fig. 2 Triangle mesh of projected contact atoms

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10 8 6 4

Y

2 0 –2 –4 –6 –8 –10 –10 –8 –6 –4 –2

0 X

2

4

6

8

10 12

from MD simulations. Since the indenter has been modeled as a repulsive potential with a spherical shape, it is easier to identify all the atoms that are contacted to the indenter. Those atoms are projected into a horizontal plane that is perpendicular to the indent axis and connected by triangular mesh into a convex domain [53]. As shown in Fig. 2, the contact area is able to compute easily by evaluating the total area of these triangular elements.

2.3.2 Hardness from Oliver–Pharr Method The well-known Oliver–Pharr method [7] is the primary method to determine the hardness from indentation experiments. Figure 3 presents a schematic representation section of indentation experiments and shows the parameters used in the analysis. During the indentation process, the total indenter displacement h is h ¼ hc þ hs

(7)

where hc is the contact depth, and hs is the displacement of surface boundary of the contact. After the unloading process, the elastic displacements are recovered. The final depth of the residual indentation print is hf. The curve of load versus indenter displacement is shown schematically in Fig. 4. The curve can be analyzed according to the equation: S¼

dP dh

(8)

where S is the experimentally measured stiffness of unloading. The contact area at the peak load (Pmax) is determined by the contact depth (hc) by an area function f(hc).

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a

Underformed configuration

Indenter tip

Contact depth hc

Derformed configuration

b

Surface displacement hs

Contact diameter 2a Final depth hc

Fig. 3 A schematic representation of the material cross section when (a) loading and (b) unloading process

Fig. 4 A schematic representation of load versus penetration depth

A ¼ f ðhc Þ ¼ πh2c þ 2πRhc

(9)

where the area function can be fitted from experimental data. The contact depth can be written as: hc ¼ hmax  hs

(10)

in which hmax is the displacement at the peak load, and hs can be obtained from the unloading stiffness (S) and the geometry shape of indenter.

Nanoindentation and Indentation Size Effects: Continuum Model and Atomistic. . .

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hs ¼ e

Pmax S

767

(11)

where e is the geometric constant. For the spherical indenter, e = 0.75. Using Eqs. 9, 10, and 11, the contact area can be computed.

2.3.3 Hardness from Taylor Dislocation Theory According to the Taylor dislocation theory [54], the shear strength can be written as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi τ ¼ αμb ρT ¼ αμb ρg þ ρs

(12)

where μ is the shear modulus, b is Burger’s vector, ρT is the total dislocation density, ρs is the geometric necessary dislocation (GND) density, and ρs is the statistically stored dislocation density. Using Eq. 12 with von Mises flow rule and the Tabor’s factor of 3, the hardness can be converted by: pffiffiffi 3τ

(13)

H ¼ 3σ

(14)

σ¼ and

If there is only GND density existed, Eq. 13 can be written as: pffiffiffi pffiffiffiffiffi H ¼ 3 3αμb ρg

(15)

Using Eq. 15, the theoretical hardness can be determined from the GND density.

2.4

Dislocation Extraction

In this study, the dislocation extraction algorithm (DXA) is used to analyze the dislocations and crystalline defects. The algorithm contains three main steps. In the first step, DXA identify crystalline atoms based on the common neighbor analysis (CNA) method [49]. The method is based on the number of bonded atoms of a given central particle to assign a structure type. As shown in Fig. 5, the white atoms are the crystalline atoms, and the red atoms are the disordered atoms. In the second step, a closed four-node tetrahedral mesh (interface mesh) is constructed with all the disordered atoms, as can be seen in Fig. 6. In the third step, burger circuit is generated on every dislocation segment. Extend the closed circuit in both directions to construct one line to represent the dislocations as illustrated in Fig. 7. Then the dislocation path’s burger vector can be calculated by summing up the lattice vectors:

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Fig. 5 Common neighbor analysis

Fig. 6 Interface mesh of disordered atoms

manifold enclosing dislocation cores

Burgers circuit junction of three Burgers circuits

swept area

reverse Burgers circuit dislocation core atoms

Fig. 7 Illustration of constructing dislocation loop [52]

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Fig. 8 Interface mesh of dislocation loops

Fig. 9 Particle type of dislocation loops

b¼

n X

l ðei Þ

(16)

i¼1

where l(ei) is the relative displacement between the tetrahedral element and a reference structure. If the burger vector of the disordered structure is not zero, it is identified as dislocations, as shown in Fig. 8. The last procedure is to transform the interface mesh to particle type, as shown in Fig. 9.

2.5

Dislocation Density

2.5.1 Dislocation Density-Length/Volume Dislocation density is usually treated as an index of measuring the plasticity deformation of materials. During the indentation test, dislocations usually occur beneath the indenter tip when shear stress reaches a critical value for dislocation nucleation. When the shear stress keeps on rising, dislocations start to glide along each slip systems and accumulate as plastic deformation. Dislocation density is defined as:

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ρ¼

λ Vp

(17)

where λ is the total length of dislocation line, and Vp is the total volume of plastic zone. In general, dislocations can be categorized into geometrically necessary dislocation (GND) and statistically stored dislocation. It is rather difficult to distinguish one from the other by observation or measurement. We assume that all the dislocations generated in our simulations should be GND only. The density of GND has a strong relation to the nanoindentation size effect. During nanoindentation, materials are subjected to a localized, nonuniform deformation which causes localized strain incompatibility. Hence, extra dislocations occur to accommodate these geometrically inconsistence. The GND density ρg can be defined as: ρg ¼

λg Vp

(18)

The information of dislocation length can be extracted by the DXA algorithm. It is a nontrivial task to determine the plastic zone size during indentation process. For the crystalline-based dislocation mechanics, the mobility of dislocation is believed to be the main cause for plastic deformation. Hence, a realm contains all the dislocations is literally the plastic zone. However, it is rather difficult to monitor all the dislocations and identify their influence domain individually. Following the concept of Nix and Gao [17], the plastic zone size was determined from an equivalent radius of the contact area. Figure 10 demonstrates the idealized contact area and the plastic zone. Given the radius of the contact area, the volume of plastic zone can be expressed as: 2 V p ¼ πa3  V indenter 3

(19)

where a is the radius of the contact area, and Vindenter is the volume occupied by the indenter. It is worth mentioning here if the penetration depth is very shallow, Vindenter is negligible. Fig. 10 Ideal plastic zone esimation from Nix and Gao

Spherical Indenter a

Dislocation

a

Plastic Zone

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Fig. 11 Illustration of GND and the affective zone

To probe the hardness via MD simulations requires a deeper penetration depth at the nanoscale. When the penetration depth increases, the plastic zone reduces due to Vindenter. This may result in the overestimation of hardness and dislocation density. Thus, the idealized estimation of plastic region was no longer valid in the present study. Adopted from the works of Nix and Gao [17], and Swadener et al. [34], a reasonable range of radius of plastic zone is considered. Figure 11 illustrates a possible scenario of dislocation distribution during indentation test. Firstly, the GNDs should connect to the indented surface beneath the indenter tip. And secondly, the GND density should be weakly insensitive to the radius of plastic zone when a steady state is reached. Once a dislocation density is qualified for these two assumptions, the dislocation density is determined. The partial length of dislocation within the plastic zone is determined as λpartial. Using Eq. 18, the GND density can be derived as: ρg ¼

λpartial Vp

(20)

With Eq. 20, the density can be computed with numerous radius of plastic zone. The density is determined when their values reach a plateau with respect to the plastic zone radius. Figure 12 shows the GND density versus plastic radius of 20 Å indentation on Au in periodic boundary conditions. The ρg is 0.00495 Å2, 0.005141 Å2, 0.005224 Å2, 0.00558 Å2, 0.005214 Å2, and 0.005213 Å2 for plastic radius equal to 29, 30, 31, 32, 33, and 34, respectively. It is apparent that the dislocation density reaches a plateau.

2.5.2 Dislocation Density-Number/Surface Area Computing dislocation density through the ratio of the total dislocation length dividing by the volume of the plastic zone is rather straightforward in MD simulations. This is not the case for experimental measurements. Consider that dislocation length is very difficult to measure during the indentation experiments, dislocation

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Fig. 12 Illustration of density convergency with 20 Å indenter on Au (PBC)

density is usually measured purely from the specimen surface in the experiments with the etching method: ρ¼

n A

(21)

where n is the number of dislocation that cut through the indentation residual print, and A is the surface area of the indentation residual print [55].

3

Simulation Details and Methodology Verification

MD simulations were performed with the open-source program LAMMPS [56]. It is a full-function, large-scale classical MD code. LAMMPS has implemented many different potentials, and the potentials are constantly updated. Using numerous force fields and boundary conditions, LAMMPS can model atomic, biological, metallic, and coarse-grained systems. It is an open-source code and is designed to be modified or extended with new capabilities. Help and support from the community of the developers are very strong.

3.1

Simulation System Setup

Figures 13 and 14 show two simulation system setups of this study. The indenter pushed in the –Z direction on the +Z direction of the specimen. In the first setup, the periodic boundary conditions were applied in the X and Y directions. In the second

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Fig. 13 Simulation system setup of free boundary condition

Fig. 14 Simulation system setup of periodic boundary condition

setup, the free boundary conditions were applied in the X and Y directions. The top surface was imposed with a free boundary condition. In addition, three atom layers at bottom were fixed to provide sufficient constraints and to prevent rigid body motion. We studied indentation into three different FCC metals – Au, Cu, Ni with the surface orientation (001) along with the Z direction was studied. The EAM potential

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Table 1 Size of simulation model Indenter radius (Å) 20 40 60

Length in X(a) 80 80 95

Length in Y(a) 80 80 95

Length in Z(a) 60 60 70

Number of atoms 1,536,000 1,536,000 2,527,000

was used as the interatomic potential [57]. We used the potential developed by Foiles et al. for all the three materials [58]. The initial lattice constants were set to 4.08 Å for bulk gold, 3.615 Å for bulk copper, and 3.52 Å for bulk nickel. The radii of spherical indenters ranging from 20 Å to 60 Å was exploited. The size of simulation model is summarized in Table 1. The models contained about 106 atoms. The simulation was performed with a constant number of atoms and the energy minimized by the conjugate gradient method. Stopping tolerance for energy and force was set at 10–10.

3.2

Simulation Process

3.2.1 Indentation Process The center of the indenter was initially set at a height of an indenter radius (R) above the surface of specimen. After that, the indentation process was performed. Considering that minimization by the conjugate gradient method was to find the equilibrium state on one specific moment, the applied displacements of the indenter were divided into small incremental displacements. In this study, the indenter was set to move with a constant displacement 0.05 Å per step downward. Each step the indenter pushed 0.05 Å into the specimen, then some of the atoms entered the indenter repulsive force field. The minimization process made the atoms to move out of the indenter repulsive force field. After the minimization process was done, the positions of atoms were output to a file. This indentation process was repeated until the applied displacement reached the radius of the indenter. 3.2.2 Retraction Process The model setup was the same as the indentation process, only position of atoms was read from the last output file after the loading process. Then, the indenter was retracted from the specimen with 0.05 Å per step upward. The minimization process was performed on each step, and the specimen was relaxed from the indenter repulsive force field. This retraction process was repeated until the indenter moved to the initial height before the indentation process.

3.3

Hardness

To verify our method of computing hardness in the MD simulations, the results of different methods are compared in this section. Firstly, the simulation results of hardness determined from the MD simulations is reported with periodic and free

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boundary condition settings. Secondly, the MD results of hardness computed from the Oliver–Pharr method are reported. The results of two different methods to probe hardness are compared and discussed.

3.3.1 Hardness Determined from Atomistic Information One of the main advantages of MD simulations is that the hardness can be directly obtained. Figures 15, 16, and 17 show the calculating results of hardness with different boundary condition settings. There is no significant difference between the results with different boundary condition settings except for the case with a 60 Å indenter. One possible cause for this difference is that the model is not large enough to ignore the boundary effects. Periodic boundary conditions are the commonly used in MD simulations. However, the dislocations cannot pass through the periodic boundary and result in dislocation pileup in the model. Unlike the periodic boundary condition, dislocations can glide out of the free boundary. The difference between these two boundary conditions is significant when the model size is not large enough. 3.3.2 Hardness Comparison In this section, two methods of determining hardness are compared and discussed. With the unloading process, the hardness can be determined by the Oliver–Pharr method. The hardness determined directly from the MD simulations is compared with the hardness derived from the Oliver–Pharr method in Figs. 18, 19, and 20. As can be seen from the figures, we can conclude that the two different methods to

Fig. 15 Comparison of Au indent hardness with boundary setting

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Fig. 16 Comparison of Cu indent hardness with boundary setting

Fig. 17 Comparison of Ni indent hardness with boundary setting

compute the hardness agree with each other. Comparing both results, we can assume that this is due to the fact that there is very little elastic displacement after the full indentation. From these data, we conclude that hardness computation is reliable in both indentation and retraction processes.

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Fig. 18 Compare of hardness in different stages of indentation on Au

Fig. 19 Compare of hardness in different stages of indentation on Cu

3.4

Dislocation Density

Two different methods of calculating the dislocation density are used for comparison. Firstly, the results of dislocation density determined by length/volume from different boundary condition settings are reported. Secondly, the results of

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Fig. 20 Compare of hardness in different stages of indentation on Ni

dislocation density determined by the experimental method is compared with those obtained from length/volume.

3.4.1 Dislocation Density Determined by Length/Volume The results of dislocation density are compared with different boundary and loading settings. The results of the calculated GND density for spherical indenters are reported in Table 2 for Au, Table 3 for Cu, and Table 4 for Ni. In these tables, we note that there is no retraction case for the free boundary conditions since all the dislocations glide out through free boundary after retraction. It is worth noting that some results of dislocation density do not follow the desirable trend of ISE in periodic boundary conditions. A possible explanation is that the dislocation has piled up in the boundary. Because of the model is not large enough, it is saturated with dislocations that may result in a higher dislocation density. Consequently, we suggest that to study the ISE and to determine the dislocation density, the free boundary condition setting in the X and Y directions is a better choice. One interesting finding here is that there is no dislocations left near the indent surface after retraction in Au with a 20 Å indenter. All the dislocations retract back to the surface. A possible cause is that all the dislocations are not be blocked by other dislocations, thus all glide out through the surface in the retraction phase. To study the ISE at the nanoscale, these results suggest that the MD simulations should be performed with free boundary conditions along the X and Y directions (Figs. 21, 22, 23).

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Table 2 Geometrically necessary dislocation density (GND) of Au Radius (Å) 20

40

60

B.C. PBC PBC Free PBC PBC Free PBC PBC Free

Loading condition Indentation Retraction Indentation Indentation Retraction Indentation Indentation Retraction Indentation

GND(Å2) 0.00495 x 0.00336 0.00325 0.00346 0.00229 0.00361 0.00250 0.00209

Table 3 Geometrically necessary dislocation density (GND) of Cu Radius (Å) 20

40

60

B.C. PBC PBC Free PBC PBC Free PBC PBC Free

Loading condition Indentation Retraction Indentation Indentation Retraction Indentation Indentation Retraction Indentation

GND(Å2) 0.00470 0.00390 0.00630 0.00398 0.00301 0.00483 0.00334 0.00364 0.00260

Table 4 Geometrically necessary dislocation density (GND) of Ni Radius (Å) 20

40

60

B.C. PBC PBC Free PBC PBC Free PBC PBC Free

Loading condition Indentation Retraction Indentation Indentation Retraction Indentation Indentation Retraction Indentation

GND(Å2) 0.00591 0.00292 0.00512 0.00352 0.00207 0.00424 0.00319 0.00165 0.00241

3.4.2 Dislocation Density Determined by Number/Surface Area Two different methods of computing dislocation density are compared in this section. In indentation experiments, measuring the dislocation length inside the specimen during or after indentation is a very difficult task. That is the reason why the etching method is usually used to determine the dislocation density. Figure 24 provides the simulation results of dislocation density obtained from two different

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Fig. 21 Comparison of Au dislocation density from the method of length/volume

Fig. 22 Comparison of Cu dislocation density from the method of length/volume

computing methods on Ni with the free boundary condition setting. From these data, it can be seen that the difference between two methods is acceptable but with some discrepancies. The reason for these discrepancies might contribute to the surface area that cannot be easily quantified.

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Fig. 23 Comparison of Cu dislocation density from the method of length/volume

Fig. 24 Comparing of measuring dislocation density methods on Ni

3.5

Summary

The verification results using different methods to probe hardness and GND density enhance our understanding of the limitation of MD simulations. The hardness determined directly from MD simulations matches well with the hardness determined from the Oliver–Pharr method. Care must be taken when the model is not

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large enough. In this case, the difference might be considerable between two different boundary conditions. The method to determine dislocation density by length/volume might yield problems on the periodic boundary condition setting, if the model is not large enough. To remedy the boundary effects in MD simulations, the models have to be much larger in using the periodic boundary condition setting than those using the free boundary condition setting. Considering the limitation of computational source, the free boundary condition is used to study the ISE in this study.

4

Results and Discussion

4.1

Indentation Process

In this section, we present and discuss the effects of loading and unloading in the MD simulations. Elastic deformation may cause overestimation of the contact area. The main purpose of unloading is to release the elastic deformation. Using the effective stiffness (S) from the unloading load-displacement curve, the hardness can be determined by the Oliver–Pharr method. The indentation on Ni with a 40 Å indenter in periodic boundary conditions is used for discussion herein. During the indentation process, the indentation force and contact area increase as the indenter pushes into the specimen. It exhibits the Hertzian contact behavior when the penetration depth is less than 5 Å in Fig. 28. In this stage of indentation test, the local deformation resulted mainly from the bond stretch, as shown in Fig. 25. When the penetration depth is greater than 5 Å, there are very few dislocations nucleated beneath the indenter tip (Figs. 26 and 27). The hardness starts to decrease as shown in Fig. 29. Then the dislocations start to glide when the indenter penetrated much deeper and the

Fig. 25 The indentation process that atoms displaced without the occurrence of dislocations

Fig. 26 The indentation process that dislocations beneath the indenter started nucleating

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Fig. 27 The indentation process that dislocations reached a steady state Fig. 28 Evolution of applied load with a 40 Å indenter into Ni

hardness decreases as well. After a gradual decrease of hardness, the hardness fluctuates and converges to a mean value. In this stage of indentation, the nucleation and mobility of dislocation shows a self-similarity pattern and reaches a steady state. Consequently, the hardness reaches a steady state in this stage.

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Fig. 29 Evolution of hardness with a 40 Å indenter into Ni

Fig. 30 The retraction process after 5 Å

Upon retraction, the indentation force, contact area, and the total dislocation length start to decrease. Figure 30 shows the dislocation structure after retraction distance of 5 Å. In Fig. 28, the force drops significantly as soon as the indenter retracted and vanishes to zero when the retraction distance is less than 10 Å. The

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Fig. 31 The retraction process after the model reaches its equilibriunm state Fig. 32 Evolution of contact area with 40 Å indenter into Ni

contact area decreases smoothly in first 5 Å and declines to zero in Fig. 32. The indenter loses contact with the specimen when retraction distance is about 8 Å, and the specimen is relaxed to an equilibrium state. Figure 33 also shows that the total length of dislocation remains a constant at 5628.02 Å after the entire model is relaxed. Figure 31 shows the final dislocation structure after the retraction process.

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Fig. 33 Evolution of total dislocation length with 40 Å indenter into Ni

4.2

Indentation Size Effect (ISE)-Dislocation Density

In this section, the ISE of dislocation density is discussed. Figures 34, 35, and 36 show the dislocation density compared with the model presented by Swadener et al. [34]. It is apparent from all the simulation results that GND density at nanoscale is overestimated by the model by Swadener et al. [34] that is rooted from the Nix and Gao model [17]. The results are in a good agreement with the findings of Feng [1].

4.3

Indentation Size Effect (ISE)-Hardness

In this section, ISE of hardness is discussed. ISE is mainly caused by extremely locally strain distribution underneath the area subjected to the penetration of indenter. The free boundary condition results of hardness are plotted in Figs. 37, 38, and 39. The MD-direct line is the hardness directly obtained from the MD simulations. The MD-derived line is the hardness derived from the Taylor dislocation theory. As can be seen in these figures, although the hardness derived from dislocation density is much closer to the hardness directly obtained from the MD simulations than the model by Swadener et al., it still has a considerable difference between the two. The cause of the discrepancy may be associated with the burgerpvector in the ffiffi 2 Taylor dislocation model. The burger vector is assumed to be 2 a in FCC systems. However, simulation results implied that most of the dislocations are the partial dislocations in MD simulations. Considering this fact, the partial pffiffi burger vector 66 a is applied to the Taylor dislocation model. The results are plotted in Figs. 40, 41, and 42. The derived hardness now matches very well with the hardness directly obtained from the MD simulations.

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Fig. 34 Dislocation density of simulation indentation on Au

Fig. 35 Dislocation density of simulation indentation on Cu

The ISE of spherical indenters is only related to the radius of the indenter. Materials responded to the extremely confined penetration of smaller size of a indenter are stronger. This phenomenon reveals that the ISE is also valid in such a small scale. Moreover, the ISE exhibits a much more sensitive decaying rate at these extremely small sizes of indenters. The hardness reduces about 50% when the radius of indenter increases three times larger.

788 Fig. 36 Dislocation density of simulation indentation on Ni

Fig. 37 Hardness of Au (free boundary condition, whole)

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Fig. 38 Hardness of Cu (free boundary condition, whole)

Fig. 39 Hardness of Ni (free boundary condition, whole)

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790 Fig. 40 Hardness of Au (free boundary condition, partial)

Fig. 41 Hardness of Cu (free boundary condition, partial)

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Fig. 42 Hardness of Ni (free boundary condition, partial)

5

Conclusions

Using the MD simulations, three FCC single crystals were studied using spherical indenters. The effect of boundary condition setting, loading and unloading process, and calculating methods were discussed to determine their influence on computing the dislocation density and hardness. Four major findings are drawn from this study: 1. A reliable method is presented to extract the GND directly from the dislocation length and the volume of plastic zone in the MD simulations. This method works well in the indentation simulations if the model is large enough to ignore the boundary effects. The concept of determining dislocation density with continuous enlarging plastic zone is feasible for MD simulations. 2. The ISE is observed directly from the MD simulations without any free parameters. Using the Taylor dislocation theory, the hardness derived from GND agrees very well with the hardness directly obtained from the MD simulations. 3. The GND density is the main reason of ISE at the nanoscale. The relationship between the hardness and the GND density shows a great correlation. GND density is proved to occur underneath the indenter tip in the present study through the comparison with the analytical models. 4. The model by Swadener et al. [34] that is rooted from the Nix and Gao model [17] underestimates the GND density at the nanoscale. However, this model can

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accurately predict the hardness size effects in nanoindentation if it uses the GND density directly calculated from the MD simulations. In addition to these four major findings, two interesting observations are drawn: 1. The etching method used in experiments to compute the dislocation density correlates well with the proposed method through the ratio of dislocation length and the volume of plastic zone. 2. For spherical indenters, the hardness in the steady state is independent to the penetration depth. ISE correlates well with the radius of the indenter. These observations obtained from the MD simulations follow the current understanding of indentations with spherical indenters. Acknowledgments This research was supported by the Ministry of Science and Technology in Taiwan. We are also grateful to the National Center for High-Performance Computing for providing the computational resources required for this study.

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Continuum Theory for Deformable Interfaces/Surfaces with Multi-field Coupling

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linearized Theory of Piezoelectricity for Bodies with Electroelastic Coupling Under Biasing Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Three Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lagrangian Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Theory of Infinitesimal Biasing Fields in Lagrangian Description . . . . . . . . . . . . . . . . . . 3 State-Space Formalism for the Linearized Theory of Incremental Fields . . . . . . . . . . . . . . . . . 4 Interface/Surface Piezoelectricity Under Biasing Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Comparison and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Surface Effect on Wave Propagation in a Piezoelectric Nanocylinder . . . . . . . . . . . . . . . . . . . . . 7 Future Directions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Continuum mechanics is a well-demonstrated powerful tool for dealing with comprehensive physical problems of macroscopic bodies involving deformation and flow processes. It has also been successfully and extensively applied to deformable bodies at micro- and nano-scales without or with appropriate modifications. One particular and interesting phenomenon of small-sized subjects is that they usually exhibit sizedependent characteristics, which cannot be well explained within the conventional

B. Wu Department of Engineering Mechanics, Zhejiang University, Hangzhou, P. R. China e-mail: [email protected] W. Q. Chen (*) Department of Engineering Mechanics, Zhejiang University, Hangzhou, P. R. China State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, P. R. China e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_27

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framework of continuum mechanics. A modified version that accounts for the so-called surface elasticity has been developed. The rigorous derivation of surface elasticity is due to Gurtin and Murdoch as early in 1975 by creating a two-dimensional set of elasticity. This chapter aims to introduce a relatively traditional method to derive the interface/surface elasticity, which further incorporates the coupling among multiple physical fields (e.g., elastic, electric, and magnetic). The method is illustrated here by only considering the electroelastic coupling, but there is completely no difficulty to make a further step toward the situation where more fields are involved. The method first assumes a finite thickness of the interface/surface so that an interphase layer model is actually adopted, which is governed by the traditional three-dimensional theory of piezoelectricity. Then, the state-space formalism is derived, based on which a transfer relation between the state vectors at the upper and lower surfaces of the interphase layer can be easily obtained. By series expansion and truncation, various theories of interface/surface piezoelectricity of different orders can then be established. Comparison is made with those reported in the literature, which shows good agreement and hence validates the present approach. Keywords

Surface elasticity · Multi-field coupling · State-space formalism · Effective boundary conditions · Interface piezoelectricity

1

Introduction

Microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) nowadays have been widely used in multifarious advanced technologies such as cell phones, flexible electronics, soft robotics, fluidics, smart clothing, etc. These functional systems are small in size when compared to the traditional mechanical systems. There are two new prominent aspects associated with these tiny systems: one is that multiple physical fields are usually required to make them function in a different way from their precedents, and the other is that when the size becomes smaller and smaller, their mechanical behavior usually becomes sizedependent. Continuum theories involving interactions between elastic deformation and other physical fields (e.g., thermal, electric, and magnetic) can date back to the nineteenth century, but only in the 1960s was enormous interest aroused in the systematic development of generalized continuum mechanics that can accommodate size-dependent phenomena. The first effort to combine the two aspects was made by Mindlin, who included the polarization gradient in the stored energy function of elastic dielectrics and presented a modified version of piezoelectricity [1]. Mindlin’s theory is able to predict the size-dependent capacitance of thin dielectric films [2]. There are alternative ways or strategies to capture size-dependent phenomena that have been widely observed in experiments. Surface effect is one such candidate, which originates from the fact that the surface and bulk atoms experience different environmental conditions [3, 4]. It is obvious that surface effect will play a more significant role in a structure when its size becomes smaller while its aspect surface-to-volume ratio

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becomes larger. This has been confirmed by molecular dynamics simulations [5]. Although surface effect has long been a research topic since Gibbs’ pioneering work a century ago [3, 4], the rigorous theory of surface elasticity within the general framework of continuum mechanics appeared much later in 1975. It was established by Gurtin and Murdoch by developing a two-dimensional counterpart (i.e., curved surface) of the welladopted three-dimensional (3D) continuum [6]. Their arguments and derivations therefore seem much different from those in a standard textbook on continuum mechanics and are difficult even for an experienced researcher to grasp within a short time. This chapter aims to introduce a different procedure to derive the interface/surface elasticity on the basis of an interphase layer of finite thickness (see Fig. 1) This allows one to follow the standard language in traditional continuum mechanics, which is much easier to understand. Moreover, the effect of multi-field coupling can be readily incorporated into the analysis. This will be illustrated by assuming that the material is piezoelectric, within which the elastic and electric fields are coupled. Further incorporation of couplings among other physical fields can be done almost in the same way. It shall be noticed that the adopted interphase layer model was also initiated by Mindlin [7], who considered the effect of a thin coating on an elastic plate and derived the corresponding approximate or effective boundary conditions based on the arguments similar to those employed in the establishment of plate theories. Tiersten showed that Mindlin’s analysis is accurate enough when the thickness of the film is sufficiently thin compared to the characteristic length of the problem (e.g., the wave length in a wave propagation problem) [8]. Gurtin and Murdoch pointed out that for a plane boundary, the effective boundary conditions obtained using Mindlin’s analysis are exactly the same as the governing equations of surface elasticity when the residual surface tensions are absent and the surface parameters are defined appropriately [9]. Mindlin’s analysis has been extended to general anisotropic interphase layer [10] and surface layer [11] using different approaches: Benveniste based his analysis on Taylor series expansion, while Ting employed the Stroh formalism. They both assumed the absence of residual surface tension, which however is very important in many applications [3]. Later, Benveniste [12] extended his analysis to the

a

b 2#

v3

v3

v1 1#

v2

h v1

Fig. 1 The interphase layer model. When medium 1 or 2 becomes vacuum, it reduces to the model of a surface layer. (a) An interphase layer between two media. (b) 3D view of the interphase layer

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piezoelectric case. Based on the interphase layer model, this chapter will, in addition to the electroelastic coupling effect, incorporate the residual field (including residual surface tension and residual electric polarization or electric displacement) into the interface surface elasticity by following a different procedure [13–15]. To do so, it is necessary to start with the general nonlinear theory of piezoelectricity and the corresponding linearized theory for the incremental field superimposed on a finite biasing field. The state-space formalism for the equations of the linearized theory is then presented, which enables the derivation of a transfer relation between the state vectors at the upper and lower surfaces of the interphase layer. By expressing the transfer matrix in terms of power series, it is straightforward and very convenient to obtain the governing equations of different orders for a material interface/surface with piezoelectric effect and residual surface field. Comparison is then made to confirm the validity of the obtained results as well as the effectiveness of the proposed procedure.

2

Linearized Theory of Piezoelectricity for Bodies with Electroelastic Coupling Under Biasing Fields

2.1

Three Configurations

To include the effect of residual surface field, it is necessary to consider the general case of a piezoelectric body subjected to a biasing field. The general nonlinear theory of piezoelectricity and the corresponding linearized theory for the small field superimposed on a finite bias are best described using three different configurations [16–19], as shown in Fig. 2. The natural (or reference) configuration Br corresponds to an undisturbed state of the body, which undergoes no deformation and electric field. The unit outward normal on the boundary @Br is denoted as N. A material particle X in the natural

Fig. 2 Natural, initial, and current configurations

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configuration is identified by its position vector X, with its Cartesian components being XK. Here the capital subscript is used to denote the component in the natural configuration, in which the mass density is denoted as ρ0. The initial (or biased) configuration B~ corresponds to a disturbed state of the body, which undergoes a biasing field consisting of an electric field and a static deformation, both being of arbitrary magnitude. The unit outward normal on the boundary ~ and the differential surface area is dA. ~ The deformation of the @ B~ is denoted as N, ~ body from Br to B is described by the mapping x = x(X) or xγ = xγ (XK). Here and after, Greek subscripts are used to indicate the components of a physical quantity (vector or tensor) in the initial configuration. Furthermore, the tilde above a symbol indicates the quantity in the initial configuration. Thus, for the motion x = x(X), the ~ ¼ ðGradxÞT, and the volume ratio as J~ ¼ det F. ~ deformation gradient is denoted as F This chapter focuses on the case of static biasing field, i.e., all the field variables in the initial configuration are independent of time t. Therefore, the following equations of nonlinear piezoelectricity hold for the biasing field:

 ~ KL ¼ F~αK F~αL  δKL =2, CurlE ¼ 0, G K     @Ω @Ω   ~ K ¼ ρ0 e KL ¼ ρ0 Π , D , (1)  @GKL G~ KL , E~K @E K G~ KL , E~K
 ~ ¼ 0, ~ ¼ 0, DivD e KL F~αL , DivT T~Kα ¼ Π α

where Ω = Ω(GKL, E K) is the total energy density function per unit mass in the natural configuration; GKL is the Green-Lagrange strain tensor; E K and DK are the Lagrangian electric field vector and electric displacement vector, respectively; and TKi and ΠKL are the first and second Piola-Kirchhoff stress tensors, respectively. In this chapter, both tensor and component forms will be adopted in the mathematical formulations for the sake of easy reading. The current (or temporal) configuration B is also a disturbed state, in which a time-dependent incremental field (including elastic deformation and electric polarization) is further superimposed on the static biasing field in the initial configuration. In this chapter, it is assumed that the incremental field is infinitesimal. The unit outward normal on the boundary @B is denoted as n. The position vector of the material particle X in the current configuration becomes y = y(X, t) or yi = yi(XK, t). Here the lower subscript is used to denote the component in the current configuration. The deformation gradient in the current configuration with respect to the natural configuration is denoted as F(X, t) = (Grady)T, and we have J = det F(X, t). Since the incremental field is generally time-dependent, the following dynamic field equations for nonlinear piezoelectricity should be satisfied:
 GKL ¼ ðFiK FiL  δKL Þ=2, CurlE K ¼ 0,   @Ω  @Ω  ΠKL ¼ ρ0 , DK ¼ ρ0 , @GKL GKL , E K @E K GKL , E K T Ki ¼ ΠKL FiL , ðDivTÞj ¼ ρ0 y€j , DivD ¼ 0, where the dot over a quantity denotes the material time derivative.

(2)

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2.2

Lagrangian Description

By denoting the infinitesimal incremental displacement vector and electric field vector as u(X, t) and E ðX, tÞ, one has yðX, tÞ ¼ xðXÞ þ uðX, tÞ, E ðX, tÞ ¼ E ðX, tÞ þ E ðX, tÞ  
yi ðXK , tÞ ¼ δiα xα ðXK Þ þ uα XK , t , E K ðXL , tÞ ¼ E~K ðXL , tÞ þ E K ðXL , tÞ

(3)

where δiα is the translation operator, which translates a vector (or a vector-like quantity) from the initial configuration to the current configuration and vice versa. The gradient of Eq. 31 with respect to X, along with the transpose operation, gives
 FiK ¼ δiα F~αK þ FαK

~ þ F, F ¼ ðGradyÞT ¼ F

(4)

where F ¼ ðGraduÞT , FαK ¼ ðGraduÞKα

(5)

is the Lagrangian incremental deformation gradient tensor or the incremental displacement gradient tensor (with respect to the natural configuration). Note that the left gradient operator and the left divergence operator are adopted throughout this chapter. Since both u and E are infinitesimal, by omitting the high-order terms of F and E (keeping the terms up to the first order), one obtains from Eqs. 11, 21, and 4:
 ~ KL þ GKL , GKL ¼ F~αK FαL þ FαK F~αL =2 GKL ¼ G

(6)

where GKL is the incremental Green-Lagrange strain tensor. From Eqs. 12, 22, and 32, it is found that E satisfies




¼0 (7) K

 Since Π G, E and D G, E are functions of the finite Green-Lagrange strain tensor G and the Lagrangian electric
field vector
E , while the biasing field is infinitesimal, one may expand Π G, E and D G, E in Taylor series at the initial ~ and electric field vector E. Omitting the high-order terms of values of strain tensor G G and E, one obtains from Eqs. 13,4, 23,4, 32, and 61 CurlE

e KL þ ΠKL , ΠKL ¼ Π

~ K þ DK DK ¼ D

(8)

where the Lagrangian increment in the second Piola-Kirchhoff stress tensor Π and that in the electric displacement vector D are, respectively,

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  @ 2 Ω  @ 2 Ω  ΠKL ¼ ρ0 GMN þ ρ0 EM, @GKL @GMN G~ KL , E~K @GKL @E M G~ KL , E~K   @ 2 Ω  @ 2 Ω  DK ¼ ρ0 GMN  ρ0 EM @E K @GMN G~ KL , E~K @E K @E M G~ KL , E~K

801

(9)

where the symbol jG~ , E~ indicates that the second derivative of the total energy density ~ and E. Thus, from Eqs. 15, 25, 4, and 81, function (Ω) takes its value at the initial G one obtains
 T Ki ¼ δiα T~Kα þ T Kα ,

e KL FαL þ ΠKL F~αL T Kα ¼ Π

(10)

where T Kα is the increment in the Lagrangian first Piola-Kirchhoff stress tensor. In view of Eq. 62 and the symmetry property of G, it is found   @ 2 Ω  @ 2 Ω  F~αM FαN G ¼ MN @GKL @GMN G~ KL , E~K @GKL @GMN G~ KL , E~K

(11)

Substituting Eq. 91 into Eq. 102 and making use of Eq. 11 yields e KL FαL þ T Kα ¼ Π

  2  ρ0 @ 2 Ω  ~βN FβM F~αL þ ρ0 @ Ω  F E M F~αL (12)  @GLK @GMN G~ KL , E~K @GLK @E M G~ KL , E~K

e , it can be obtained from Then, noticing the symmetry properties of G and Π Eqs. 92 and 12 the following incremental constitutive relations in Lagrangian description: T Kα ¼ AKαLγ FγL  MMKα E M , DK ¼ MKLγ FγL þ RKL E L

(13)

where A, M, and R are defined in component form as  @ 2 Ω  e KL δαγ ¼ ALγKα , F~γN þ Π AKαLγ @GKM @GLN G~ KL , E~K  @ 2 Ω  F~γM , MKLγ ¼ ρ0 @E K @GML G~ KL , E~K  @ 2 Ω  RKL ¼ ρ0 ¼ RLK @E K @E L G~ KL , E~K ¼ ρ0 F~αM

(14)

A, M, and R are called the effective elastic constants, the effective piezoelectric constants, and the effective dielectric constants. It is seen from Eq. 13 that the Lagrangian increment in the first Piola-Kirchhoff stress tensor T and that in the

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electric displacement D linearly depends on the incremental deformation gradient tensor F and the incremental electric field vector E (with respect to the natural configuration). According to Eq. 14, the effective material constants are related to ~ and the initial electric field E. It should be pointed ~ (or G) the initial deformation F out that, even when only the fundamental material constants (please refer to the following section) are involved, the effective material constants will still vary with the biasing field. Generally speaking, the symmetry of the effective material constants in Eq. 13 is lower than that of the fundamental elastic, piezoelectric, and dielectric constants. This phenomenon is known as the bias-induced material anisotropy. In the most general case of anisotropy, A has 45 independent components, M has 27 independent components, while R has 6 independent components. Due to the dependence of the effective material constants on the biasing field, they can further vary with the position. That is to say, the incremental constitutive relations (13) are generally linear algebraic equations with variable coefficients. It is obvious that for an originally homogeneous piezoelectric material, when the biasing field is uniform, then the effective material constants will not change with the position, and the incremental constitutive relations (13) will be with constant coefficients. Since T and D in the current configuration satisfy Eq. 26,7, while the biasing field in the initial configuration satisfies Eq. 16,7, the subtraction of the two respective equations gives the following field equations that T and D should satisfy


 DivT α ¼ ρ0 u€α , DivD ¼ 0

(15)

where body forces and free electric charges are assumed to be absent during the incremental motion. e and the incremental electric By introducing the initial electric potential ϕ potential ϕ as follows:  
 e , E K ¼  Gradϕ (16) E~K ¼  Gradϕ K K

then the initial electric field E and its increment E automatically satisfy Eqs. 12 and 7, respectively. When the effect of exterior electric field is neglected, the incremental boundary conditions in Lagrangian description become uα ¼ uα , ϕ ¼ ϕ1 ; T Kα N K ¼ tα , N K DK ¼ qe

(17)

where uα and ϕ1 are the increments in displacements and electric potential on the boundary, respectively, and tα and qe are the incremental surface tractions and incremental surface electric charge density, respectively. It shall be pointed out that either uα or tα should be prescribed on a part of the boundary and similarly either ϕ1 or qe is known on a part of the boundary.

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803

Theory of Infinitesimal Biasing Fields in Lagrangian Description

Although the static biasing field (including the intrinsic state and the residual state [20]) is usually much larger than the superimposed dynamic incremental field, the biasing field can also be infinitesimal. In this case, only the linear effect of the biasing field needs to be considered. For an infinitesimal biasing field, denoting the initial infinitesimal displacement vector as u~ (see Fig. 2), one has ~ ðXÞ, xðXÞ ¼ X þ u

xα ¼ δαK ðXK þ u~K Þ

(18)

Here δαK is the translation operator enabling the switch of a vector between the initial and natural configurations. The gradient of Eq. 18 with respect to X, along with the transpose, gives ~ ¼ Ir þ d, ~ F


 F~αL ¼ δαK δKL þ d~KL

(19)

where ~ ¼ ðGrad~ d u ÞT ,

d~KL ¼ ðGrad~ u ÞLK

(20)

is the initial displacement gradient tensor with respect to the natural configuration. From Eq. 11 and in view of Eq. 19, it is obtained that   ~ ¼ d ~þd ~ T =2, G


 ~ KL ¼ d~KL þ d~LK =2 G

(21)

If only weak nonlinearity in the piezoelectric material is considered, then one can expand the total energy density function into a polynomial. In general, it is sufficient to have an expansion of Ω(G, E) up to the third-order terms of its arguments for an infinitesimal biasing field: 1 ρ0 ΩðGKL , E K Þ ¼ ρ0 Ωð0, 0Þ þ σ 0AB GAB  D0A E A þ cABCD GAB GCD 2 1 1 þ cABCDEF GAB GCD GEF  eABC E A GBC  eABCDE E A GBC GDE 6 2 1 1  κ ABC E A E B E C  lABCD E A E B GCD 6 2

(22)

where Ω(0, 0) is the intrinsic energy density per unit mass; σ 0AB and D0A are the intrinsic stress tensor and electric displacement vector, respectively; cABCD, eABC, and κ AB are the second-order elastic, piezoelectric, and dielectric constants; and cABCDEF, eABCDE, κ ABC, and lABCD are the third-order elastic, piezoelectric, dielectric, and electrostrictive constants, respectively. All these constants

804

B. Wu and W. Q. Chen

belong to the fundamental material constants, with the second-order ones characterizing the linear behavior of the piezoelectric material, while the third-order or high-order ones describing its nonlinear behavior. Substituting ~ and E, one obtains Eq. 22 into Eq. 13,4, and omitting the high-order terms of G ~ AB  eAKL E~A , e KL ¼ σ 0 þ cABKL G Π KL

~ K ¼ D0 þ eKBC G ~ BC þ κKA E~A D K

(23)

Substituting Eqs. 19 and 231 into Eq. 15 gives

 ~ AB  eAKL E~A δαM δML þ d~ML e KL F~αL ¼ σ 0 þ cABKL G T~Kα ¼ Π KL
 ~ AB  eAKL E~A þ σ 0 d~αL ¼ δαL Π e KL þ σ 0 d~αM ¼ δαL σ 0KL þ cABKL G KL KM   e KL þ σ 0 d~LM ¼ δαL Π KM

(24)

In addition, it is obtained from Eq. 22 that ρ0 @ 2 Ω ¼ cLNKM þ cABLNKM GAB  eALNKM E A , @GKM @GLN ρ0 @ 2 Ω ¼ eKML  eKBCML GBC  lAKML E A , @E K @GML ρ0 @ 2 Ω ¼ κ KL  κ AKL E A  lKLCD GCD @E K @E L

(25)

Then from Eq. 14, making use of Eqs. 19 and 25 and omitting the high-order ~ and E, one can obtain the effective material constants A, M, and R as terms of d follows: AKαLγ ¼ cKαLγ þ cKαLγ , MKLγ ¼ eKLγ þ eKLγ , RKL ¼ κKL þ κKL

(26)

~ AB  eAKαLγ E~A þ cKMLγ d~αM þ cKαLN d~γN , e KL δαγ þ cKαLγAB G cKαLγ ¼ Π ~ BC þ lAKLγ E~A þ eKLM d~γM , κKL ¼ κKLA E~A þ lKLCD G ~ CD eKLγ ¼ eKLγBC G

(27)

where

3

State-Space Formalism for the Linearized Theory of Incremental Fields

In view of Eqs. 5, 16, 20, and 21, and noticing the translation operator δαK between the initial and natural configurations as well as the symmetry properties of the fundamental material constants, one can transform the incremental field equations (15), the incremental constitutive relations (13, 26, and 27), as well as the incremental boundary conditions (17) to

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .

25




805

 DivT K ¼ ρ0 u€K , Div D ¼ 0


 T IJ ¼ AIJKL ðGraduÞKL þ MMIJ Gradϕ M ,
 DM ¼ MMIJ ðGraduÞIJ  RML Gradϕ L , ~ AB  eAIJKL E~A þ cIMKL d~JM þ cIJKN d~LN , e IK δJL þ cIJKLAB G AIJKL ¼ cIJKL þ Π ~ BC þ lAMIJ E~A þ eMIN d~JN MMIJ ¼ eMIJ þ eMIJBC G ~ CD , RML ¼ κML þ κMLA E~A þ lMLCD G     e , ~ G KL ¼ ðGradu~ÞLK þ ðGradu~ÞKL =2, E~K ¼  Gradϕ uK ¼ uK , ϕ ¼ ϕ1 , T KL N K ¼ tL , N K DK ¼ qe

(28)

K

In the case of the interphase layer shown in Fig. 1, note that the relation between the three-dimensional gradient and the surface gradient of a scalar and that of a vector are, respectively, given by


Gradϕ

 K


 @ϕ @uP ¼ GradS ϕ K þ δ3K , ðGraduÞKL ¼ ðGradS uÞKL þ θKLP (29) @v3 @v3

where the nonzero components of θKLP are θ3JJ = 1 (no summation over J ). Thus,
 @uP ¼ ℬPJ T 3J þ MPKL ðGradS uÞKL þ W PK GradS ϕ K þ F P D3 , @v3
 @ϕ ¼ GJ T 3J þ OKL ðGradS uÞKL þ LK GradS ϕ K þ ℋD3 , @v3

(30)

where B ¼ K1 , K JP ¼ A3J3P þ M33J M33P =R33 ¼ K PJ , F P ¼ ℬPJ M33J =R33 MPKL ¼ ℬPJ ðA3JKL þ M33J M3KL =R33 Þ,W PK ¼ ℬPJ ðMK3J  M33J R3K =R33 Þ, GJ ¼ ℬPJ M33P =R33 ,OKL ¼ ðM3KL þ MPKL M33P Þ=R33 , LK ¼ ðW PK M33P  R3K Þ=R33 , ℋ ¼ ðF P M33P  1Þ=R33 (31) In addition, one has




 @T 3J DivT J ¼ GradT KKJ ¼ GradS T KKJ þ ¼ ρ0 u€J , @v3

 @D3 DivD ¼ GradD KK ¼ GradS D KK þ ¼0 @v3

(32)



 @T 3J @D3 ¼ ρ0 u€J  GradS T KKJ , ¼  GradS D KK @v3 @v3

(33)

Thus,

806

B. Wu and W. Q. Chen

The incremental constitutive relations can be written as
 @uP @ϕ þ MMIJ GradS ϕ M þ MMIJ δ3M , @v3 @v3
 @uP @ϕ DM ¼ MMIJ ðGradS uÞIJ þ MMIJ θIJP  RML GradS ϕ L  RML δ3L @v3 @v3 (34) Thus, T IJ ¼ AIJKL ðGradS uÞKL þ AIJKL θKLP

0

00

0

T ¼ T þ T ,D ¼ D þ D

00


 0 00 ^ IJKL ðGradS uÞ þ M ^ MIJ GradS ϕ , T IJ ¼ p0IJK T 3K þ q0IJ D3 , T IJ ¼ A KL M
 0 00 0 0 ^ MIJ ðGradS uÞ  R ^ ML GradS ϕ DM ¼ gM D3 þ mMJ T 3J , D M ¼ M IJ L

(35)

where ^ IJ3P ℬPK þ M3IJ GK , q0 ¼ AIJ3P F P þ M3IJ ℋ, p0IJK ¼ A IJ ^ ^ KLIJ , A IJKL ¼ AIJKL þ AIJ3P MPKL þ M3IJ OKL ¼ A ^ KIJ ¼ AIJ3P W PK þ MKIJ þ M3IJ LK ¼ MKIJ þ MK3M MMIJ  RK3 OIJ , M g0I ¼ MI3P F P  RI3 ℋ, m0IJ ¼ MI3P ℬPJ  RI3 GJ , ^ IK ¼ RIK  MI3P W PK þ RI3 LK ¼ R ^ KI R (36) Thus,  i @T 3J @ 2 uJ h ^ : GradS u ¼  ½GradS ðp0 tÞKKJ þ ρ0 2  GradS A KKJ @v3 @t h  i 
0  ^  GradS MGradS ϕ  GradS q D3 , h

KKJ

KKJ

 i @D3 ^ ¼  ½GradS ðm0 tÞKK  GradS Mgrad Su KK @v3 
 
0  ^ þ GradS RGradS ϕ KK  GradS g D3 KK

(37)

where the colon indicates a double contraction according to the rule that can be easily identified from the component form in Eq. 35. In Dupin coordinates, one has tL ¼ T 3L , D3 ¼ qe

(38)

By utilizing the expressions of scalar, vector, and second-order tensor in Dupin coordinates, Eqs. 30 and 37 can be rewritten as @uP ¼ ℬPK T 3K þ I PK uK þ J P ϕ þ F P D3 @v3

(39)

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .

25

807

where ψ PJ ¼

2 X MPαJ @ MP12 @h1 MP13 @h1 MP22 @h2 , I P1 ¼ ψ P1   þ , @v h h1 h2 @v2 h1 @v3 h1 h2 @v1 α α α¼1

MP11 @h1 MP23 @h2 MP21 @h2   , h1 h2 @v2 h2 @v3 h1 h2 @v1 ! 2 X MP11 @h1 MP22 @h2 W Pα @ ¼ ψ P3 þ þ , JP ¼ h1 @v3 h2 @v3 hα @vα α¼1

I P2 ¼ ψ P2 þ I P3 and

@ϕ ¼ GJ T 3J þ KJ uJ þ ℒϕ þ ℋD3 @v3 where 2 X Oαj @ ηj ¼ , ℒ¼ h @vα α¼1 α

K2 ¼ η 2 þ

(40)

(41)

! 2 X Lα @ O12 @h1 O13 @h1 O22 @h2  þ , , K1 ¼ η1  h @vα h1 h2 @v2 h1 @v3 h1 h2 @v1 α¼1 α

O11 @h1 O23 @h2 O21 @h2 O11 @h1 O22 @h2   , K3 ¼ η 3 þ þ h1 h2 @v2 h2 @v3 h1 h2 @v1 h1 @v3 h2 @v3 (42)

and @T 3J @ 2 uJ ¼ N JI T 3I þ ρ0 2 þ OJI uI þ P J ϕ þ QJ D3 @v3 @t

(43)

where χ IJ ¼

2 ^ X A KLαJ @ @ ð1Þ , π KLJ ¼ , hα @vα hα @vα α¼1

2 X p0

αIJ

α¼1

p012J þ p021J @h1 p013J þ p031J @h1 p022J  p011J @h2 p031J @h2  þ  , h1 h2 @v2 h1 @v3 h1 h2 @v1 h2 @v3 p0  p022J @h1 p032J @h1 p012J þ p021J @h2 p023J þ p032J @h2    , N 2J ¼ χ 2J þ 11J h1 h2 @v2 h1 @v3 h1 h2 @v1 h2 @v3 p0 @h1 p011J  p033J @h1 p013J @h2 p022J  p033J @h2 þ  þ , N 3J ¼ χ 3J  23J h1 h2 @v2 h1 @v3 h1 h2 @v1 h2 @v3 ^ KL12 @h1 A ^ KL13 @h1 A ^ KL22 @h2 A ð2Þ  þ , π KL1 ¼  h1 h2 @v2 h1 @v3 h1 h2 @v1 ^ KL11 @h1 A ^ KL23 @h2 A ^ KL21 @h2 A ð2Þ π KL2 ¼   , h1 h2 @v2 h2 @v3 h1 h2 @v1   ^ KL11 @h1 A ^ KL22 @h2 A ð2Þ ð1Þ ð2Þ π KL3 ¼ þ , π^ KLI ¼ π KLI þ π KLI , h1 @v3 h2 @v3 (44)

N 1J ¼ χ 1J 

808

B. Wu and W. Q. Chen 2 ^ X @2 A KLαI @hα @ ð2Þ @  þ π KLI , 2 @v @v @vβ h @v @v h α α β β α α α¼1 α¼1

^ KL12 @ 2 h1 ^ KL22 @ 2 h2 ^ KL13 1 @h1 @h1 A A A @ 2 h1 ¼ þ þ  h1 @vα @v3 @vα @v3 h1 h2 @vα @v2 h1 h2 @v1 @vα h1 !

^ ^ @h1 @h2 A KL12 @h1 A KL22 @h2 þ , h2 þ h1  2 2 @vα @vα h21 h22 @v2 h1 h2 @v1

^ KL21 @ 2 h2 ^ KL23 1 @h2 @h2 ^ @ 2 h1 A A @ 2 h2 A ¼ KL11  þ  h2 @vα @v3 @vα @v3 h1 h2 @v1 @vα h2 h1 h2 @vα @v2 !

^ ^ @h1 @h2 A KL21 @h2 A KL11 @h1 h2 þ h1  , þ @vα @vα h21 h22 @v2 h21 h22 @v1

ξKLβI ¼ ζ KLα1

ζ KLα2

2 ^ X A KLαI

^ KL22 @ 2 h2 ^ KL11 @h1 @h1 A ^ KL22 @h2 @h2 ^ KL11 @ 2 h1 A A A þ  2  2 h2 @vα @v3 h1 @vα @v3 h2 @vα @v3 h1 @vα @v3  ξ111I þ ζ 111I ξ212I þ ζ 212I π^ 12I þ π^ 21I @h1 O1I ¼  þ þ h1 h2 h1 h2 @v2 π^ 13I þ π^ 31I @h1 π^ 11I  π^ 22I @h2 π^ 31I @h2 þ þ þ , h1 @v3 h1 h2 @v1 h2 @v3  ξ þ ζ 121I ξ222I þ ζ 222I π^ 22I  π^ 11I @h1 π^ 32I @h1 þ þ þ O2I ¼  121I h1 h2 h1 h2 @v2 h1 @v3 π^ 12I þ π^ 21I @h2 π^ 23I þ π^ 32I @h2 þ , þ h1 h2 @v1 h2 @v3  ξ þ ζ 131I ξ232I þ ζ 232I π^ 23I @h1 O3I ¼  131I þ þ h1 h2 h1 h2 @v2 π^ 33I  π^ 11I @h1 π^ 33I  π^ 22I @h2 π^ 13I @h2 þ þ þ , h1 @v3 h2 @v3 h1 h2 @v1 ! 2 2 2 ^ αKI @ ^ αKI @ 2 ^ αKI @hα @ X X X M M M ð1Þ ð2Þ ð3Þ , λKIβ ¼ , λKIβ ¼ , λKI ¼ 2 @v @v @v h h @v @v α α α β α β α α¼1 α¼1 α¼1 hα     2 ð2Þ ð3Þ ð2Þ ð3Þ ð1Þ ð1Þ λ111  λ111 λ212  λ212 λ þ λ21 @h1 P 1 ¼ 4 þ þ 12 h1 h2 h1 h2 @v2 # ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ λ13 þ λ31 @h1 λ11  λ22 @h2 λ31 @h2 , þ þ þ h1 @v3 h1 h2 @v1 h2 @v3    2 ð2Þ ð3Þ ð2Þ ð3Þ ð1Þ ð1Þ λ121  λ121 λ222  λ222 λ  λ11 @h1 P 2 ¼ 4 þ þ 22 h1 h2 @v2 h1 h2 # ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ λ32 @h1 λ12 þ λ21 @h2 λ23 þ λ32 @h2 , þ þ þ h1 @v3 h1 h2 @v1 h2 @v3    2 ð2Þ ð3Þ ð2Þ ð3Þ ð1Þ ð1Þ ð1Þ λ131  λ131 λ232  λ232 λ @h1 λ33  λ11 @h1 P 3 ¼ 4 þ þ 23 þ h1 h2 @v2 h1 @v3 h1 h2 # ð1Þ ð1Þ ð1Þ λ  λ22 @h2 λ13 @h2 þ 33 þ h2 @v3 h1 h2 @v1 ζ KLα3 ¼

(45)

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .

25

φI ¼

809

2 X q0 α¼1

αI @ , hα @vα

q012 þ q021 @h1 q013 þ q031 @h1 q022  q011 @h2 q031 @h2  þ  , h1 h2 @v2 h1 @v3 h1 h2 @v1 h2 @v3 q0  q022 @h1 q032 @h1 q012 þ q021 @h2 q023 þ q032 @h2    , Q2 ¼ φ2 þ 11 h1 h2 @v2 h1 @v3 h1 h2 @v1 h2 @v3 q0 @h1 q011  q033 @h1 q0 @h2 q022  q033 @h2 þ  13 þ Q3 ¼ φ3  23 h1 h2 @v2 h1 @v3 h1 h2 @v1 h2 @v3

Q1 ¼ φ1 

(46)

and @D3 ¼ X J T 3J þ W J uJ þ Uϕ þ ZD3 @v3

(47)

where 2 X m0

@ m0 @h1 m03J @h1 m01J @h2 m03J @h2 , X J ¼ τJ  2J    , h @vα h1 h2 @v2 h1 @v3 h1 h2 @v1 h2 @v3 α¼1 α ! 2 ^ 2 ^ 2 ^ X X X R Iα @ R Iα @ 2 R Iα @hα @ ð 1Þ ð2Þ ð3Þ , γ Iβ ¼ , γI ¼ , γ Iβ ¼ 2 @v @v @v h h @v @v α α α α β β α α¼1 α¼1 α¼1 hα     ð2Þ ð3Þ ð2Þ ð3Þ ð1Þ γ 11  γ 11 γ 22  γ 22 γ @h2 þ þ 1 U¼ h1 h2 h1 h2 @v1 ð1Þ ð1Þ ð1Þ 2 X γ @h1 γ 3 @h1 γ 3 @h2 g0α @ þ 2 þ þ , δ¼ , h1 h2 @v2 h1 @v3 h2 @v3 h @vα α¼1 α τJ ¼

αJ

g02 @h1 g03 @h1 g0 @h2 g03 @h2   1  , h1 h2 @v2 h1 @v3 h1 h2 @v1 h2 @v3 2 ^ PαJ @ X M ¼ , hα @vα α¼1

Z ¼ δ  ð1Þ

κPJ

^ P12 @h1 M ^ P13 @h1 M ^ P22 @h2 M  þ , h1 h2 @v2 h1 @v3 h1 h2 @v1 ^ P11 @h1 M ^ P23 @h2 M ^ P21 @h2 M ð2Þ κP2 ¼   , h1 h2 @v2 h2 @v3 h1 h2 @v1 ^ P11 @h1 M ^ P22 @h2 M ð2Þ ð1Þ ð2Þ κP3 ¼ þ , ^κ PK ¼ κPK þ κPK , h1 @v3 h2 @v3 2 2 ^ PαK @ 2 ^ PαK @hα @ X X M M ð1Þ ð2Þ @ υPβK ¼  þ κ PK , 2 @vβ hα @vα @vβ α¼1 hα @vβ @vα α¼1 ð2Þ

κP1 ¼ 

(48)

810

B. Wu and W. Q. Chen

^ P12 @ 2 h1 ^ P22 @ 2 h2 ^ P13 1 @h1 @h1 @ 2 h1 M M M þ þ  h1 @vα @v3 @vα @v3 h1 h2 @vα @v2 h1 h2 @v1 @vα h1 !

^ ^ P22 @h2 @h1 @h2 M P12 @h1 M h2 þ h1  2 2 , þ @vα @vα h21 h22 @v2 h1 h2 @v1

ð2 Þ

υPα1 ¼ 

ð2Þ

υPα2 ¼

^ P11 @ 2 h1 ^ P21 @ 2 h2 ^ P23 1 @h2 @h2 @ 2 h2 M M M  þ  h2 @vα @v3 @vα @v3 h1 h2 @vα @v2 h1 h2 @v1 @vα h2 !

^ ^ P11 @h1 @h1 @h2 M P21 @h2 M h2 þ h1  2 2 , þ @vα @vα h21 h22 @v1 h1 h2 @v2

^ P11 @ 2 h1 ^ P22 @ 2 h2 ^ P11 @h1 @h1 M ^ P22 @h2 @h2 M M M þ  2  2 , h1 @vα @v3 h2 @vα @v3 h1 @vα @v3 h2 @vα @v3 " # ð1Þ ð2Þ ð1Þ ð2Þ υ11J þ υ11J ^κ 2J @h1 ^κ 3J @h1 υ22J þ υ22J ^κ 1J @h2 ^κ 3J @h2 þ þ þ þ þ WJ ¼  h1 h1 h2 @v2 h1 @v3 h2 h1 h2 @v1 h2 @v3

(49)

ð2Þ

υPα3 ¼

Thus, Eqs. 39, 41, 43, and 47 can be written in matrix form as



 Y1 @ Y1 ¼ Mðρ0 , A, M, R; there is a blank. v1 , v2 ; @ 1 , @ 2 , @ t ; hi Þ @v3 Y2 Y2   M11 M12 Y1 ¼ M21 M22 Y2 (50)  T  T where Y1 ¼ u1 , u2 , u3 ,ϕ and Y2 ¼ T 31 , T 32 , T 33 , D3 are the generalized incremental displacement vector and the generalized incremental stress vector, respectively, and M is the system matrix of 8  8, with its four submatrices of 4  4 being given by 2

I 11

6I 6 21 M11 ¼ 6 4 I 31 2

M21

2

ℬ11

I 13

I 22 I 32

I 23 I 33

6ℬ J27 7 6 21 7, M12 ¼ 6 4 ℬ31 J35

K2

K3

ℒ

@ 6 ρ0 @t2 þ O11 6 6 6 6 O21 ¼6 6 6 6 O31 4 2

M22

K1

J1

3

I 12

ℬ13

ℬ22 ℬ32

ℬ23 ℬ33

F2 7 7 7, F3 5

G2

G3 3

ℋ

G1

2

O12 ρ0

@2 þ O22 @t2

O23 @ þ O33 @t2 W3 2

O32

W1 N 11 N 12

N 13

N 22

N 23

N 32 X2

N 33 X3

6N 6 21 ¼6 4 N 31 X1

O13

W2 3 Q1 Q2 7 7 7 Q3 5 Z

ρ0

P1 7 7 7 7 P 2 7, 7 7 7 P3 7 5 U

F1

3

ℬ12

(51)

25

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .

4

811

Interface/Surface Piezoelectricity Under Biasing Field

Now it is ready to start from the state equation (50) to establish the interface/surface piezoelectricity under biasing field based on the interphase layer model in Fig. 1. The interphase can be arbitrarily curved, but with a uniform thickness h. The uniformity of thickness is just for the sake of simplicity, not a must. Applying Eq. 50 to this interphase layer (or thin shell), one obtains @ @v3

Ys1 Ys2




 Ys1 ¼ Ms ρs0 , A s , Ms , Rs ; v1 , v2 ; @ 1 , @ 2 , @ t ; hi Ys2

(52)

where the superscript s indicates the quantity associated with the curved interphase. It is seen that the elements of the system matrix Ms are related to the material constants of the interphase ρs0 , A s , Ms , Rs, the two in-surface coordinates v1, v2, the partial differential operators @ 1, @ 2, @ t, and the Lamé constants hi. Since hi generally depends on the thickness coordinate v3, the state equation (52) is usually with variable coefficients. But in consideration of the fact that the curved interphase is very thin, as an approximate treatment, we can set v3   h/2 or v3  h/2 in the system matrix Ms so that @ @v3

Ys1 Ys2



¼ Ms0

 Ys1 , Ys2

@ @v3

Ys1 Ys2



¼ Ms1

Ys1 Ys2

 (53)

where  Ms0 ¼ Ms v3 ¼h=2 ,

 Ms1 ¼ Ms v3 ¼h=2

(54)

With respect to the thickness coordinate v3, Eq. 53 is the state equation with constant coefficients. Treating the partial differential operators @ 1, @ 2, @ t as the usual parameters like the interphase material constants ρs0 , A s , Ms , Rs and the in-surface coordinates v1, v2, one can write out the formal solution to the state equation (53) as

s  Y1 ðh=2Þ h v3 þ ¼ exp , 2 Ys2 ðv3 Þ Ys2 ðh=2Þ 

s

s   Y1 ðh=2Þ Y 1 ðv 3 Þ h ¼ exp Ms1 v3  s 2 Y 2 ðv 3 Þ Ys2 ðh=2Þ

Ys1 ðv3 Þ





Ms0

(55)

Setting v3 = 0 in Eq. 55 leads to  s  Y1 ðh=2Þ s h ¼ exp M , 0 2 Ys2 ð0Þ Ys2 ðh=2Þ  s

s   Y1 ðh=2Þ Y1 ð0Þ h s ¼ exp M1  2 Ys2 ð0Þ Ys2 ðh=2Þ

Ys1 ð0Þ



(56)

812

B. Wu and W. Q. Chen

Making use of the power series expansion of the matrix exponential, keeping the terms up to the first order, one obtains

s  Y1 ðh=2Þ s h ¼ I þ M0 , 2 Ys2 ð0Þ Ys2 ðh=2Þ

s

s   Y1 ðh=2Þ Y 1 ð 0Þ h ¼ I  Ms1 s 2 Y 2 ð 0Þ Ys2 ðh=2Þ



Ys1 ð0Þ

(57)

Then from Eq. 57, we obtain the relation between the state vectors on the upper and lower surfaces of the curved interphase:

Ys1 ðh=2Þ





Ys2 ðh=2Þ



Ys1 ðh=2Þ Ys2 ðh=2Þ

 s 

s  Y1 ðh=2Þ Y1 ðh=2Þ h s s ¼ M0 þ M1 2 Ys2 ðh=2Þ Ys2 ðh=2Þ (58)

By applying the Taylor’s theorem to the second term on the right-hand side of Eq. 58 with respect to the lower surface v3 =  h/2, Eq. 58 becomes, to the first-order accuracy,

Ys1 ðh=2Þ Ys2 ðh=2Þ





Ys1 ðh=2Þ Ys2 ðh=2Þ



¼

hMs0

Ys1 ðh=2Þ Ys2 ðh=2Þ

 (59)

which can also be directly obtained from Eq. 551 by setting v3 = h/2 and making use of Taylor series expansion. On the interfaces v3 =  h/2 and v3 = h/2 between the curved interphase and the bulk materials (medium 1 and medium 2), the continuity conditions hold, which require that the state vector in the interphase side should be equal to that in the bulk side, i.e., ð1Þ

Ys1 ðh=2Þ ¼ Y1 ðh=2Þ, ð2Þ

Ys1 ðh=2Þ ¼ Y1 ðh=2Þ,

ð1Þ

Ys2 ðh=2Þ ¼ Y2 ðh=2Þ,

(60)

ð2Þ

Ys2 ðh=2Þ ¼ Y2 ðh=2Þ

where the superscripts (1) and (2) indicate those physical quantities associated with bulk materials 1 and 2, respectively. In view of Eq. 60, Eqs. 58 and 59 can be written as (

ð2Þ

Y1 ðh=2Þ

)

ð2Þ

Y2 ðh=2Þ

( 

ð1Þ

Y1 ðh=2Þ ð1Þ

Y2 ðh=2Þ

)

" ( ð1Þ ) ( ð2Þ )# Y ð h=2 Þ Y ð h=2 Þ h 1 1 Ms0 ¼ þ Ms1 ð1Þ ð2Þ 2 Y ðh=2Þ Y ðh=2Þ 2

2

(61) and (

ð2Þ

Y1 ðh=2Þ ð2Þ

Y2 ðh=2Þ

)

( 

ð1Þ

Y1 ðh=2Þ ð1Þ

Y2 ðh=2Þ

)

( ¼

hMs0

ð1Þ

Y1 ðh=2Þ ð1Þ

Y2 ðh=2Þ

) (62)

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .

25

813

Equations 61 and 62 give the first-order approximations of 3D piezoelectricity (or the O(h) effective interface conditions) for a curved thin interphase. The two models in Eqs. 61 and 62 differ by an infinitesimal quantity, both accounting for the effects of biasing field, electroelastic coupling, and material anisotropy. They become the interface elasticity theories once the interface material parameters are defined properly. When the biasing field is absent, the effective material constants in Eq. 61 reduce to the second-order fundamental material constants, and Eq. 61 corresponds to the first-order piezoelectricity theory for a curved interface in the absence of biasing fields. Equation 61 is identical to that obtained by Benveniste [12], but his derivation looks more tedious. Furthermore, when either bulk material 1 or 2 becomes vacuum, the above interface piezoelectricity theories reduce to the surface piezoelectricity theories. Expanding Eq. 59 leads to
s s  s usP ðh=2Þ  usP ðh=2Þ ¼ h ℬsPJ T 3J þ I sPI usI þ J sP ϕ þ F sP D3 h=2 ,
s s s s s s  ϕ ðh=2Þ  ϕ ðh=2Þ ¼ h GsJ T 3J þ KsI usI þ ℒs ϕ þ ℋ D3 h=2 ,

 @ 2 us s s  s s s T 3J ðh=2Þ  T 3J ðh=2Þ ¼ h N sJI T 3I þ ρs0 2J þ OsJI usI þ P sJ ϕ þ QsJ D3  , @t h=2
s s s s  s D ðh=2Þ  D ðh=2Þ ¼ h X s T þ W s us þ U s ϕ þ Z s D  3

3

J 3J

J J

3

h=2

(63) In view of Eqs 33 and 35, the last two equations in Eq. 63 become h 2 s     i 0s 00s s s @ u  T 3J ðh=2Þ  T 3J ðh=2Þ ¼ h ρs0 @t2J  DivS T  DivS T  J J h=2   s s 0s 00s  D3 ðh=2Þ  D3 ðh=2Þ ¼ h DivS D þ DivS D 

(64)

h=2

When the residual field in the interphase layer is solely induced by the intrinsic stress and electric displacement, by applying Eqs. 32 and 33 to the initial residual field, one can rewrite Eq. 16,7 as   @ T~3I ~s , ¼  DivS T I @v3

~3 @D ~s ¼ DivS D @v3

(65)

~ 3 at the upper surface Making use of the Taylor series expansions of T~3I and D with respect to the lower surface, one obtains, to the first-order accuracy,   s ~ s  @ T~3J  @D s s s s 3 ~ ~ ~ ~ T 3J ðh=2Þ  T 3J ðh=2Þ ¼ h , D 3 ðh=2Þ  D 3 ðh=2Þ ¼ h   @v3  @v3  h=2

h=2

(66) Then from Eqs. 65 and 66, one arrives at the following first-order effective boundary conditions for the residual field:

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B. Wu and W. Q. Chen

  s s ~ s  T~3J ðh=2Þ  T~3J ðh=2Þ ¼ h DivS T ,  J h=2 ~ s ðh=2Þ  D ~ s ðh=2Þ ¼ hDivS D ~ s  D 3 3

(67)

h=2

The relation between the total stress (or total electric displacement) at the lower surface and that at the upper surface can be obtained from Eqs. 64 and 67 as 

   s   @ 2 usJ  0s 00s  ~  ¼h  DivS T  DivS T þ T J J h=2 @t2 h  s i 0s ~ þ D00s  Ds3 ðh=2Þ  Ds3 ðh=2Þ ¼ h DivS D þ DivS D T s3J ðh=2Þ

T s3J ðh=2Þ

ρs0

h=2

(68)

5

Comparison and Validation

If the residual field caused by the intrinsic stress and electric displacement is not considered, the initial configuration coincides with the natural configuration, ~ s ¼ Is , J~s ¼ 1 . In this case, there are only intrinsic stress and electric i.e., F displacement in the interphase layer. Thus, from Eq. 23, one knows that the first and second Piola-Kirchhoff stress tensors the total Cauchy stress tensor are all
0 and s s s es ¼ e σ ¼ σ , while the initial electric displacement identical, i.e., T~KL ¼ Π KL KL
 KL ~ s ¼ D0 s . The effective material constants are vector is D M M
s AsIJKL ¼ csIJKL þ σ 0IK δJL , MsMIJ ¼ esMIJ , RsML ¼ κsML

(69)

The constitutive relations of the interphase layer, when expressed in terms of the first Piola-Kirchhoff stress, are
s 
s  T sIJ ¼ σ 0IJ þ csIJKL þ σ 0IK δJL ðGradus ÞKL  esMIJ E sM ,
s DsM ¼ D0M þ esMIJ ðGradus ÞIJ þ κsML E sL

(70)

Notice the following formulae: F ¼ Ι þ H, Π ¼ TFT

H ¼ ðGraduÞT , J ¼ det F  1 þ trH,
  T I  HT ¼ TðI  GraduÞ

F1 ¼ I  H, (71)

1

τ ¼ J FT  ð1  trHÞðΙ þ HÞT Then, the constitutive relations can be expressed in terms of the second PiolaKirchhoff stress as follows:

25

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .


s ΠsIJ ¼ σ 0IJ þ csIJKL ðGradus ÞKL  esMIJ E sM ,
s DsM ¼ D0M þ esMIJ ðGradus ÞIJ þ κsML E sL

815

(72)

The constitutive relations in terms of the total Cauchy stress are
s 
s
s  σ sIJ ¼ σ 0IJ þ csIJKL þ σ 0IK δJL þ σ 0KJ δIL ðGradus ÞKL
s  σ 0IJ trðGradus Þ  esMIJ E sM
s DsM ¼ D0M þ esMIJ ðGradus ÞIJ þ κ sML E sL

(73)

In the absence of electroelastic coupling effect, the constitutive relations (70) in terms of the first Piola-Kirchhoff stress reduce to
s Ts ¼ σ0 þ A s : ðGradus Þ,


s AsIJKL ¼ csIJKL þ σ 0IK δJL

(74)

Here, once again, it is emphasized that the colon means double contraction as specified in Eq. 37 in component form. Now introduce the infinitesimal symmetric strain tensor Es and antisymmetric skew tensor Ws as h i Es ¼ Gradus þ ðGradus ÞT =2,

h i Ws ¼ ðGradus ÞT  Gradus =2

(75)

Thus, Eq. 74 becomes
s Ts ¼ σ0 þ A s : ðEs ÞT þ A s : ðWs ÞT

(76)

Noticing
s A s : ð Ws Þ T ¼ σ0 ð Ws Þ T ,

h i 

s  s skw A s : ðEs ÞT ¼ σ0 Es  Es σ0 =2 (77)

one can rewrite Eq. 76 as h i

s 
s
s  s Ts ¼ σ0 þ σ0 Es  Es σ0 =2 þ sym A s : ðEs ÞT þ σ0 ðWs ÞT

(78)

where sym and skw represent the symmetric and skew parts of a second-order tensor, respectively. Hoger [21] also obtained the constitutive relations for the residual stress in an elastic body, which when using the symbols adopted in this chapter read as
s
s 
s
s  Ss ¼ σ0 þ Ws σ0 þ Es σ0  σ0 Es =2 þ Ls ½Es 

(79)

where Ss is the first Piola-Kirchhoff stress tensor defined in the paper of Hoger. Note that Ss in Ref. [21] is the transpose of Ts in this paper, i.e., Ts = (Ss)T; Ls is the

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B. Wu and W. Q. Chen

incremental elasticity tensor, which is different from the fundamental elasticity tensor cs. Comparing Eq. 78 with Eq. 79 leads to h i 
s
s  Ls ½Es  ¼ sym A s : ðEs ÞT ¼ cs : Es þ σ0 Es þ Es σ0 =2

(80)

Therefore, the incremental elasticity tensor Ls depends not only on the fundamental elasticity tensor cs but also on the intrinsic stress (σ0)s. This result indicates that one should be very careful when experimental testing is carried out for the characterization of the material intrinsic stress or elastic constants. Now, in addition to the omission of the residual field induced by the intrinsic stress and electric displacement in the interphase layer as well as the electroelastic coupling effect, it is further assumed that the transverse components of the intrinsic es ¼ stress and the symmetric second Piola-Kirchhoff stress are negligible, i.e., Π 3L e s ¼ Πs ¼ Πs ¼ 0. Then, Eq. 102 indicates that the transverse components of Π L3 3L L3 the non-symmetric first Piola-Kirchhoff stress tensor vanish, i.e., s T 3K ¼ 0 ðK ¼ 1, 2, 3Þ. Thus, the incremental constitutive relations (35) which are expressed in terms of the first Piola-Kirchhoff stress become s ^ s ðGradS uÞ , T aJ ¼ A aJKL KL

ℬs ¼ ðKs Þ1 ,

K sJP

s s s s ^s A aJKL ¼ AaJKL  AaJ3P ℬPQ A3QKL ,
s ¼ As3J3P ¼ K sPJ , AsIJKL ¼ csIJKL þ σ 0IK δJL , ða ¼ 1, 2Þ

(81) When the elastic material is isotropic, one has csIJKL ¼ λs δIJ δKL þ μs ðδIK δJL þ δIL δJK Þ,


s AsIJKL ¼ λs δIJ δKL þ μs ðδIK δJL þ δIL δJK Þ þ σ 0IK δJL , 
s  K sIJ ¼ ðλs þ μs Þδ3I δ3J þ μs þ σ 033 δIJ , 1 λs þ μ s
s  δ3I δ3J ℬsIJ ¼ s
0 s δIJ   s
0 s  s λ þ 2μs þ σ 033 μ þ σ 33 μ þ σ 33

(82)

Moreover, in the case of in-surface isotropic intrinsic stress without transverse components, i.e.,
0 s
0 s σ ab ¼ σ δab ,




σ 03I

s


s ¼ σ 0I3 ¼ 0,

ða, b ¼ 1, 2Þ

(83)

one can obtain from Eqs. 81 and 83 the constitutive equations in terms of the first Piola-Kirchhoff stress as follows:
s ^ s ðGradS us Þ , T sab ¼ σ 0 δab þ A abKL KL s s s ^ ðGradS u Þ , ða, b ¼ 1, 2Þ T ¼A a3

a3KL

KL

Substituting Eq. 83 into Eq. 82 and then into Eq. 84 gives

(84)

25

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .


s
s 2μs λs T sab ¼ σ 0 δab þ s trð«s Þδab þ 2μs «s þ σ 0 ðGradS us Þab , λ þ 2μs
s T sa3 ¼ σ 0 ðGradS us Þa3

817

(85)

where the in-surface infinitesimal strain reads as h i «s ¼ P GradS u þ ðGradS uÞT =2

(86)

where P = I  N  N is the projection tensor, with N being the outward normal to the interface/surface. The projection tensor, when applied to any vector u, gives the projection of u onto the plane perpendicular to N. When omitting the residual field induced by the intrinsic stress and electric displacement in the interphase layer as well as the transverse stress components s s T~3K and T 3K , the effective boundary conditions (68)1 of the interphase layer become  T s3J ðh=2Þ



T s3J ðh=2Þ

¼h

ρs0

  @ 2 usJ s   ð Div T Þ S J  @t2 h=2

(87)

For an interphase layer with the upper surface tractions-free, one has T s3J ðh=2Þ ð1Þ ¼ 0. In view of T s3J ðh=2Þ ¼ T 3J ðh=2Þ, the effective boundary conditions (87) become


ð1Þ

DivS ðhTs ÞÞJ ¼ T 3J þ ρs0 h

@ 2 usJ @t2

(88)

It is noted that the constitutive relations and the equations of motion for a material surface derived by Gurtin and Murdoch [9] are

 Σsab ¼ τs0 δab þ 2 μs0  τs0 ð«s Þab þ λs0 þ τs0 trð«s Þδab þ τs0 ðGradS us Þab Σsa3 ¼ τs0 ðGradS us Þa3 ,


DivS ðΣÞÞJ ¼

ð 1Þ T 3J

ða, b ¼ 1, 2Þ

(89)

@ 2 us þ ρ0 2J @t

Comparing Eqs. 85 and 88 with Eq. 89, one finds that on letting
s τs0 ¼ σ 0 h,

μs0  τs0 ¼ μs h,

λs0 þ τs0 ¼

2μs λs h, λ þ 2μs s

ρ0 ¼ ρs0 h,

(90)

the two models are identical. Thus, the comprehensive interface piezoelectricity derived in the last section does recover the well-known results for an isotropic material surface established by Gurtin and Murdoch [6]. It is noted that, in the absence of intrinsic stresses (or residual stresses), the scaling relations similar to those in Eq. 90 have been noticed by Gurtin and Murdoch [9] when they compared

818

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their results with those of Mindlin [7]. Chen et al. [15] also found similar relations for an isotropic cylindrical surface. In the case of a spherically isotropic piezoelectric spherical surface layer, by following the same procedure as outlined above, Wu et al. [22] obtained the following first-order surface piezoelectricity theory in the spherical coordinate system (r, θ, φ):

s 

k1 þ ks2 h @ 2 2 ks1 þ ks2 h @ σ rr þ þ cot θ þ uθ ur þ @θ @t2 r 20 r 20
s  k þ ks h @uφ 2γ s h  þ 12 2 Dr ¼ 0, r 0 sin θ @φ αs r 0
s  

2 k1 þ ks2 h @ur h cs66 @ 2 2 s @ s s þ ρ 0 h 2  2 k 1 ∇4  k 2 þ σ rθ  uθ @θ @t r 20 r0 sin2 θ @φ2 
s  @2
 h @ γ s h @Dr  ks1 þ cs66 cot θ ¼ 0,  2 uφ þ s k2 þ cs66 @φ α r 0 @θ @φ@θ r 0 sin θ
s  
s  @2
 k þ ks h @ur h @  2 þ ks1 þ cs66 cot θ σ rφ  12 2 uθ k2 þ cs66 @φ @θ @φ r 0 sin θ @φ r 0 sin θ

 s  2
2  h k1 @ 2 γ s h @Dr s @ s ¼ 0, þ ρ0 h 2  2 þ c ∇ þ 1 u þ φ 66 4 αs r 0 sin θ @φ @t r 0 sin2 θ @φ2 ks h 2h Dr þ 32 ∇21 ϕ  Dr ¼ 0 r0 r0 (91) 

ρs0 h

where the effects of the residual field and three transverse stress components in the surface layer have been all neglected, but the radial electric displacement is kept. In Eq. 91, r0 is the radius of the lower or inner surface of the spherical surface layer, and α ¼ c33 κ33 þ e233 ,

β ¼ c13 κ33 þ e33 e31 ,

γ ¼ c33 e31  e33 c13 ,

c13 β e31 γ c13 β e31 γ e2 þ , k2 ¼ c12  þ , k3 ¼ 15 þ κ11 , α α α α c44 2 2 2 @ @ 1 @ @ @ ∇21 ¼ 2 þ cot θ þ , ∇24 ¼ 2 þ cot θ  cot2 θ @θ sin2 θ @φ2 @θ @θ @θ k1 ¼ c11 

(92)

When the following relations hold Y

Y

ρY0 ¼ ρs h, k1 ¼ ks1 h, k2 ¼ ks2 h, cY66 ¼ cs66 h, κY11 ¼ ks3 h, eY31 γ s ¼ , DYr ¼ hDr jr¼r0 , κ Y33 αs

(93)

where the quantities with superscript Y are the surface material constants defined in Ref. [23], it is found that Eq. 91 gives the same results as the surface piezoelectricity

25

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .

819

suggested by Huang and Yu [23]. Thus, the comprehensive interface piezoelectricity theory derived in this chapter is again validated for a spherical surface. It is noted that the surface radial electric displacement DYr in the surface piezoelectricity of Huang and Yu [23] appears to be the bulk counterpart multiplied by the thickness of the surface layer, as seen from the last one of Eq. 93. The physical reality of this surface electric displacement for a zero-thickness mathematical surface still needs to be explored since, unlike other surface parameters in Eq. 93 that are material-related, it is an external stimulus.

6

Surface Effect on Wave Propagation in a Piezoelectric Nanocylinder

For numerical illustration, wave propagation in a piezoelectric nanocylinder is considered in the absence of any residual field. The governing equations of surface piezoelectricity in cylindrical coordinates can be easily derived and are not shown here. The following three dimensionless quantities are introduced for the characterization of the piezoelectric surface: r h ¼ h=r 0 , r ρ ¼ ρs =ρ, r c ¼ csij =cij ¼ esij =eij ¼ esij =eij

(94)

where the material parameters without superscript are those for the bulk material and r0 is the radius of the piezoelectric cylinder, which is axially polarized. The material constants of the bulk material may be found in Ref. [24]. Obviously, rh = 0 corresponds to the case of no surface effect. Figure 3 displays the first three branches of the dispersion spectrum of a PZT-4 cylinder with surface effect (rc = rρ = 1) for the axisymmetric wave n = 0. Here n is the circumferential wave number. The results for the non-axisymmetric flexural wave n = 2 are shown in Fig. 4. In the two figures, the dimensionless wave numberpξffiffiffiffiffiffiffiffiffiffiffi and frequency Ω have been adopted, which are defined as ξ = kr0 and Ω ¼ r 0 ω ρ=c44 , with k and ω being the axial wave number and circular frequency, respectively. The results clearly indicate that surface piezoelectricity can have a significant effect on the wave propagation behavior in nanosized structures.

7

Future Directions and Open Problems

This chapter presents a different approach to the derivation of the governing equations of interface/surface piezoelectricity including the effects of biasing field. The approach is based on the thin interphase layer model and directly employs the conventional 3D theory of piezoelectricity which can be readily found from the related textbooks. The corresponding equations are rearranged in a mixed form in the state space, which enables the derivation of a formal solution and hence a transfer relation between the state vectors at the upper and lower surfaces of the interphase layer. The series expression for the transfer matrix with truncation leads to the

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Fig. 3 Surface effect on waves propagating in a PZT-4 cylinder (n = 0, rc = rρ = 1)

Fig. 4 Surface effect on waves propagating in a PZT-4 cylinder (n = 2, rc = rρ = 1)

governing differential equations for an arbitrarily curved piezoelectric interface at any desired order. These governing equations correspond to the 2D interface/surface piezoelectricity with biasing field. Comparison with existing results confirms the derived formulae and validates the approach. The most critical research in the future should be directed to the accurate characterization of surface parameters by well-designed experiments, in addition to the molecular dynamics or first-principle simulations. Moreover, the following two theoretical problems are also of significant interest: (1) How will the surface effect play the role in soft materials? For soft materials, large deformation is usually involved. Although the general framework of interface/surface elasticity has been

25

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field. . .

821

established in this chapter, the analysis of a concrete problem will be particularly complex; (2) Will the surface effect couple with other effects (like gradient, nonlocal, etc.) and how to develop a theory accounting for all the acting effects?

References 1. Mindlin RD. Polarization gradient in elastic dielectrics. Int J Solids Struct. 1968;4:637–42. 2. Mindlin RD. Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Int J Solids Struct. 1969;5:1197–208. 3. Cammarata RC. Surface and interface stress effects in thin films. Prog Surf Sci. 1994;46:1–38. 4. Wang JX, Huang ZP, Duan HL, Yu SW, Feng XQ, Wang GF, Zhang WX, Wang TJ. Surface stress effect in mechanics of nanostructured materials. Acta Mech Solida Sin. 2011;24:52–82. 5. Miller RE, Shenoy VB. Size-dependent elastic properties of nanosized structural elements. Nanotechnology. 2000;11:139–47. 6. Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal. 1975;57:291–323. 7. Mindlin RD. High frequency vibrations of plated, crystal plates. In: Progress in applied mechanics. New York: MacMillan; 1963. 8. Tiersten HF. Elastic surface waves guided by thin films. J Appl Phys. 1969;40:770–89. 9. Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct. 1978;14:431–40. 10. Benveniste Y. A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J Mech Phys Solids. 2006;54:708–34. 11. Ting TCT. Mechanics of a thin anisotropic elastic layer and a layer that is bonded to an anisotropic elastic body or bodies. Proc R Soc A. 2007;463:2223–39. 12. Benveniste Y. An interface model for a three-dimensional curved thin piezoelectric interphase between two piezoelectric media. Math Mech Solids. 2009;14:102–22. 13. Chen WQ. Surface effect on Bleustein–Gulyaev wave in a piezoelectric half-space. Theor Appl Mech Lett. 2011;1:041001. 14. Chen WQ. Wave propagation in a piezoelectric plate with surface effect. In: Fang DN, Wang J, Chen WQ, editors. Analysis of piezoelectric structures and devices. Berlin/Boston: De Gruyter; 2103. 15. Chen WQ, Wu B, Zhang CL, Zhang C. On wave propagation in anisotropic elastic cylinders at nanoscale: surface elasticity and its effect. Acta Mech. 2014;225:2743–60. 16. Tiersten HF. Electroelastic interactions and the piezoelectric equations. J Acoust Soc Am. 1981;70:1567–76. 17. Yang JS, Hu YT. Mechanics of electroelastic bodies under biasing fields. Appl Mech Rev. 2004;57:173–89. 18. Yang JS. An introduction to the theory of piezoelectricity. New York: Springer; 2005. 19. Wu B, Zhang CL, Zhang C, Chen WQ. Theory of electroelasticity accounting for biasing fields: retrospect, comparison and perspective. Adv Mech. 2016;46:201604. (in Chinese) 20. Tiersten HF, Sinha BK, Meeker TR. Intrinsic stress in thin films deposited on anisotropic substrates and its influence on the natural frequencies of piezoelectric resonators. J Appl Phys. 1981;52:5614–24. 21. Hoger A. On the determination of residual stress in an elastic body. J Elast. 1986;16:303–24. 22. Wu B, Chen WQ, Zhang C. On free vibration of piezoelectric nanospheres with surface effect. Mech Adv Mater Struct. 2017. https://doi.org/10.1080/15376494.2017.1365986. 23. Huang GY, Yu SW. Effect of surface piezoelectricity on the electromechanical behaviour of a piezoelectric ring. Phys Status Solidi B. 2006;243:22–4. 24. Ding HJ, Chen WQ. Three dimensional problems of piezoelasticity. New York: Nova Science Publishers; 2001.

Part II Micromechanics

Interaction Between Stress and Diffusion in Lithium-Ion Batteries: Analysis of Diffusion-Induced Buckling of Nanowires

26

F. Q. Yang, Yan Li, B. L. Zheng, and K. Zhang

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Governing Equations in Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Governing Equation in a Large Deformed Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Governing Equation in a Small Deformed Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lithiation-Induced Buckling of a Nanowire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mechanical Equations for the Deformation of an Elastic Nanowire . . . . . . . . . . . . . . . . . 3.2 Concentration Distribution of Lithium in a Core-Shell Nanowire . . . . . . . . . . . . . . . . . . . 3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Lithiation-Induced Buckling of an Elastic-Perfectly Plastic Nanowire . . . . . . . . . . . . . . . . . . . . 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Postbuckling Analysis of an Elastic Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

826 827 827 831 832 832 835 836 838 841 842 843

F. Q. Yang (*) Department of Chemical and Materials Engineering, University of Kentucky, Lexington, KY, USA e-mail: [email protected] Y. Li School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, People’s Republic of China e-mail: [email protected] B. L. Zheng School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China e-mail: [email protected] K. Zhang Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_59

825

826

F. Q. Yang et al.

Abstract

This chapter summarizes the frameworks of the diffusion-induced stress both in the theory of linear elasticity and the theory of nonlinear elasticity, which provide the theoretical principles to investigate the coupling between stress and diffusion in analyzing the elastic deformation of materials induced by the migration/ diffusion of solute atoms. The buckling behavior of an elastic core-shell nanowire induced by the migration/diffusion of lithium during lithiation is analyzed by using the theory of linear elasticity and diffusion equation. Closed-form solution is obtained for the calculation of the critical time for the onset of buckling of the nanowire. The effects of current density, length, and the radius ratio on the critical concentration and critical time for the occurrence of the buckling are studied. For an elastic-perfectly plastic core-shell nanowire, numerical analysis of the lithiation-induced buckling of the core-shell nanowire is performed, using the commercial finite element software ABAQUS. The effects of the radius ratio and yield stress of the outer shell on the critical average concentration and critical time for the occurrence of the buckling are investigated.

1

Introduction

Understanding the stress evolution in active materials in electrochemical environment is one of the important steps in developing lithium-ion batteries of high energy density and capacity. During lithiation and de-lithiation, local deformation in the active materials of electrodes in lithium-ion batteries will occur, which leads to the evolution of local stress in the active materials. The stress, which is controlled by the volume change associated with migration/diffusion of lithium, has been referred to as chemical stress or diffusion-induced stress. The conventional method to investigate the diffusion-induced stress in solid is based on thermal analogy, as proposed by Prussin [1] in the analysis of the diffusion-induced stress in a thin plate. The stress generated due to the lithiation or de-lithiation of the active materials will lead to surface cracks and cavitation [2], and the concept of the diffusioninduced stress has been extensively used to explain mechanical degradation in lithium-ion batteries [2–6]. There are extensive studies analyzing the stress evolution in the electrodes of different shapes in lithium-ion batteries. Cui et al. [4] used the stress-dependent chemical potential in diffusion equation and the theory of large deformation in analyzing the stress in a spherical silicon electrode. Hao and Fang [5] incorporated the surface stress and stress-assisted diffusion in analyzing the chemical stress in a core-shell spherical electrode. Zhang et al. [7] studied the stress evolution in an ellipsoidal electrode and considered the contribution of hydrostatic stress in chemical potential. Song et al. [6] obtained the analytical solutions of diffusioninduced stress in a cylindrical multilayer electrode during lithiation. Yang [8] studied the contribution of solid reaction to the transport of solute atoms in the analysis of the coupling between diffusion and stress. Haftbaradaran et al. [9] considered the effect of strain energy on diffusivity in analyzing the diffusion of lithium during lithiation.

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Interaction Between Stress and Diffusion in Lithium-Ion Batteries: Analysis. . .

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Yang [10] incorporated the contribution of solute atoms to the Helmholtz free energy of solid solution, and derived a linear relation between Young’s modulus and the concentration of solute atoms. Yang [11] recently analyzed the effects of stress and electric field on the migration of solute atoms from the framework of thermal activation process, and derived a new relationship between the diffusion flux of solute atoms, hydrostatic stress, and electric field intensity. Zhao et al. [2] investigated the large plastic deformation in a spherical silicon electrode, using nonequilibrium thermodynamics and diffusion equation. Sethuraman et al. [12] reported real-time measurement of average stress in a composite silicon electrode during electrochemical lithiation and de-lithiation, and suggested that the stress evolution depends strongly on the mechanical properties of binder. Drozdov [13] investigated the viscoplastic response of a host material induced by the diffusion of solute atoms and extended it to the analysis of the self-limiting lithiation in a spherical silicon electrode. Bhandakkar and Johnson [14] simulated and analyzed the diffusion-induced stresses in honeycomb electrodes undergoing elastic-plastic deformation during electrochemical cycling. It is known that slender structure will buckle under the action of compressive load before reaching failure. Such behavior has been observed during the lithiation of Si nanowires [15] and SnO2 nanowires [16]. The lithiation-induced buckling plays a vital role in determining the mechanical durability of low-dimensional structures, such as nanowires and nanotubes, used in lithium-ion batteries, which necessitates an investigation to examine the effect of lithiation on the buckling behavior of nanowires and nanotubes. In this chapter, the state of the interaction between stress and diffusion is summarized, and the buckling behavior of a core-shell nanowire induced by the migration/diffusion of lithium during lithiation is analyzed. Such a structure allows one to introduce an interface between a Li-free phase and Li-rich phase, associated with self-limiting behavior [15]. Two types of materials are considered; one is linearly elastic, and the other is elastic-perfectly plastic. The effects of the nanowire length, radius of the inner Li-free core, and yield stress on the critical concentration and critical time for the occurrence of buckling are analyzed.

2

Governing Equations in Solid

2.1

Governing Equation in a Large Deformed Solid

The migration/diffusion of solute atoms into a host material will introduce local deformation due to the mismatch of atomic sizes between the solute atom and the host atom or the residing of the solute atoms in the interstitial sites in the host material of crystal phase. There are two approaches of Lagrangian description and Euler description, which can be used to describe the deformation of the host material. The Lagrangian description is based on initial state, and the Eulerian description is based on the current state. The corresponding coordinates are expressed as X and x, respectively. The Lagrangian description and Eulerian description are related to each other by a uniquely invertible vector field of χ as

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x ¼ χ ðX, tÞ,

(1)

X ¼ χ 1 ðx, tÞ,

(2)

where χ is referred to as the motion of the material. The displacement vector of U in the Lagrangian description is UðX, tÞ ¼ xðX, tÞ  X:

(3)

and the deformation gradient tensor of F in terms of the displacement vector is calculated as F ¼ Gradx ¼ I þ GradUðX, tÞ,

(4)

where Grad is the gradient operator in the Lagrangian description and I is a unit tensor of second order. The total deformation gradient tensor is calculated from the elastic deformation tensor of Fe and diffusion-induced deformation tensor of Fi as F ¼ Fe Fi :

(5)

Assume that the diffusion-induced deformation due to the insertion or de-insertion of solute atoms is isotropic. The deformation gradient tensor due to the insertion or de-insertion of solute atoms is [4] Fi ¼ ð1 þ Ω1 CÞ1=3 I,

(6)

where C is the concentration of solute atoms in the Lagrangian description, and Ω1 is the volumetric strain per unit mole fraction of the solute atoms. The Green–Lagrange strain tensor of E in the Lagrangian description is adopted, and can be expressed as E¼

 1
T F FI : 2

(7)

Following the form of the Green–Lagrange strain tensor in Eq. 7, the elastic counterpart of Ee and diffusion-induced counterpart of Ei are expressed, respectively, as  1 e T e Ee ¼ ðF Þ F  I , (8) 2  1 
i T i F F I , (9) Ei ¼ 2 In this chapter, the elastic strain energy density of W in the Lagrangian description is assumed as a quadratic function of the Green–Lagrange strain tensor,
 W ¼ det Fi

  Eh ν e 2 e e ½tr ðE Þ þ trðE E Þ : 2ð1 þ νÞ ð1  2νÞ

(10)

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Interaction Between Stress and Diffusion in Lithium-Ion Batteries: Analysis. . .

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Here, Eh and ν are Young’s modulus and Poisson’s ratio of the material, respectively, and det(Fi) is the determinant of the deformation gradient tensor for the diffusion-induced deformation. From the strain energy density, the constitutive relation can be found as [17] P¼

@W @W @Ee @Fe ¼ , @F @Ee @Fe @F

(11)

where P is the first Piola–Kirchhoff stress. Substituting Eq. 10 into 11 yields
 P ¼ det Fi

  e Eh ν e e F tr ðE Þ þ 2E : 2ð1 þ νÞ ð1  2νÞ Fi

(12)

The equilibrium equation at a static state without the action of body force is DivP ¼ 0,

(13)

where Div is the divergence operator in the Lagrangian description, and the counterpart is div in the Eulerian description. Let J be the resultant nominal flux of solute atoms, and C be the concentration of solute atoms, which are functions of coordinates in the Lagrangian description of X and time of t. The conservation of solute atoms gives @C þ DivJ ¼ 0: @t

(14)

The concentration, c, and resultant flux, j, in the Eulerian description are related to their Lagrangian counterparts by C=det(F)c and J=det(F)jF-T, respectively. Note that no solid reaction is considered in Eq. 14. When there exists a solid reaction, as discussed by Yang [8], Eq. 14 becomes @C þ DivJ ¼ generation=loss amount: @t

(15)

For the irreversible solid reaction of aA+bB!fF, Eq. 15 gives @C þ DivJ ¼ kCa ð1  CÞb , @t

(16)

with k being the rate constant for forward reaction. The analysis of the kinematics of solute atoms must be based on the current state in the Eulerian description. In general, the true diffusion flux, j, is proportional to the gradient of chemical potential of μ as [8], j ¼ Mc  gradμ,

(17)

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where grad is the gradient operator in the Eulerian description, and M is the mobility of the solute atoms. For a dilute solution, the chemical potential per mole can be expressed as [18, 19] μ ¼ μ0 þ Rg Tlnc  σ m Ω1 þ wΩ2 ,

(18)

with μ0 being a constant, Rg being the gas constant, T being absolute temperature, σ m being the Cauchy hydrostatic stress in the Eulerian description, Ω2 being the partial molar volume of the solute atoms, and w being the strain energy density in the Eulerian description. The Cauchy hydrostatic stress is  1 1
σm ¼ trðσÞ ¼ tr det1 ðFÞPFT 3 3 
i   e det F 1 Eh ν e e F tr ðE Þ þ 2E FT , ¼ tr 3 detðFÞ 2ð1 þ νÞ ð1  2νÞ Fi

(19)

and the strain energy density in the Eulerian description is w¼

  W Eh ν
i ¼ ½tr ðEe Þ2 þ trðEe Ee Þ : 2ð1 þ νÞ ð1  2νÞ det F

(20)

Substituting Eqs. 18, 19, and 20 into Eq. 17 yields 

  Ω 2 Eh ν e 2 e e ½trðE Þ þ trðE E Þ j ¼ Mc  grad μ0 þ Rg Tln c þ 2ð1 þ νÞ ð1  2νÞ (21)    e Ω1 detðFi Þ Eh ν F trðEe Þ þ 2Ee i FT  tr detðFÞ 2ð1 þ νÞ ð1  2νÞ 3 F According to the relationships between the Lagrangian description and the Euler description for the concentration of solute atoms and flux, C = det(F)c and J = det (F)jFT, the diffusion flux in the Lagrangian description is 

C Þ MC  Grad μ0 þ Rg Tln detðFÞ   Ω2 E h ν e 2 e e ½trðE Þ þ trðE E Þ þ 2ð1 þ νÞ ð1  2νÞ    e Ω1 detðFi Þ Eh ν F  trðEe Þ þ 2Ee i FT : tr detðFÞ 2ð1 þ νÞ ð1  2νÞ 3 F

J ¼ ðF

T 2

g

(22)

Substituting Eq. 22 into Eq. 14, one obtains the diffusion equation in the Lagrangian description as

26

Interaction Between Stress and Diffusion in Lithium-Ion Batteries: Analysis. . .

 
T 2 @C C þ Div  F MC Grad μ0 þ Rg Tln @t detðFÞ 
i   e det F 1 Eh ν F tr ðEe Þ þ 2Ee i FT  tr 3 detðFÞ 2ð1 þ νÞ ð1  2νÞ F   Ω 2 Eh ν ½tr ðEe Þ2 þ trðEe Ee Þ ¼ 0: Ω1 þ 2ð1 þ νÞ ð1  2νÞ

2.2

831

(23)

Governing Equation in a Small Deformed Solid

For small deformation, the theory of linear elasticity [20] can be used in analyzing the evolution of stresses and strains during lithiation. In the theory of linear elasticity, there is no difference between the Lagrangian description and Eulerian description for all the quantities, that is, the stress tensor P, strain tensor E, and coordinate X in the Lagrangian description are the same as the stress tensor σ, strain tensor e, and coordinate x in the Eulerian description, respectively. There are J=j and C=c, and Eqs. 14 and 16 remain the same. The constitutive relation proposed by Prussin [1] in the 1D case can be extended to 3D case as ε¼

1 ΩC I, ½ð1 þ νÞσ  ν  trσI þ Eh 3

(24)

where Ω= Ω1. The values of Ω and Young’s modulus are assumed to be independent of C in the following analysis. In comparison with the diffusion process, the elastic deformation can be approximated as quasi-static [21]. The equilibrium equations are @σ þ Fv ¼ 0 @x

(25)

where Fv represents body force. Equation 25 can be rewritten as G∇2 u þ

G ∇ð∇  uÞ ¼ KΩ∇C  Fv 1  2ν

(26)

where u is the displacement vector, and G and K are the shear and bulk moduli of the solid, respectively. In the absence of body force, there is ∇2 ðσ þ αCÞ ¼ 0 with σ being the hydrostatic stress. The parameter of α is calculated as

(27)

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α¼

2EΩ 9ð 1  ν Þ

(28)

The diffusion equation considering the effect of stress-induced diffusion can be written as   Ω ΩC 2 @C 2 D ∇ C ∇C  ∇σ  ∇ σ ¼ Rg T Rg T @t

(29)

where D is the diffusion coefficient of the solute atoms in the active material.

3

Lithiation-Induced Buckling of a Nanowire

3.1

Mechanical Equations for the Deformation of an Elastic Nanowire

There exists an interface between the lithiated phase and Li-free phase during the migration/diffusion of lithium into an active material. As shown schematically in Fig. 1, α phase represents the lithiated phase and β phase represents the Li-free phase. Assume that the β phase remains Li-free during lithiation. The continuous migration/diffusion of Li will lead to the increase of the concentration of lithium in the α phase. The constitutive equation of Eq. 24 can be written as eR ¼

1 ΩC , ½σ R  νðσ Θ þ σ Z Þ þ E 3

Fig. 1 Schematic of a cylindrical core-shell nanowire in a cylindrical coordinate system

R1

Lithiated phase (α Phase) Li-free phase (β Phase)

(30) R2

Z Θ

R

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Interaction Between Stress and Diffusion in Lithium-Ion Batteries: Analysis. . .

833

eΘ ¼

1 ΩC , ½ σ Θ  ν ðσ R þ σ Z Þ þ E 3

(31)

eZ ¼

1 ΩC , ½σ Z  νðσ R þ σ Θ Þ þ E 3

(32)

where ei and σ i (i = R, Θ, Z) are the strain components and stress components, respectively. If the end effect is negligible, the strain components of ei can be calculated from the radial displacement of u and axial displacement of w as eR ¼

@u u @w , eΘ ¼ , eZ ¼ : @R R @Z

(33)

The mechanical equilibrium equation in absence of body force is dσ R σ R  σ Θ þ ¼ 0: dR R

(34)

It should be mentioned that the effect of the interface between the inner core and outer shell on the evolution of shear stress is assumed to be negligible in Eq. 34. From Eqs. 30, 31, 32, 33, and 34, the radial displacement of u and axial displacement of w can be found as 1þv Ω 1 u¼ 1v 3 R

ðR

CRdR þ AR þ

Ri

B @w , ¼ q, R @Z

(35)

where Ri is the radius of the inner core. A, B, and q are constants to be determined from the boundary conditions. From the radial displacement and axial displacement of Eq. 35, one can calculate the stress components as ΩE 1 σR ¼  3ð 1  v Þ R2 ΩE 1 σΘ ¼ 3ð 1  v Þ R2 σZ ¼

ðR Ri

ðR Ri

CRdR þ

CRdR þ

EðA þ vqÞ EB  , ð1 þ vÞð1  2vÞ ð1 þ vÞR2

EðA þ vqÞ EB ΩEC þ ,  ð1 þ vÞð1  2vÞ ð1 þ vÞR2 3ð1  vÞ

2Ev ΩE Eð1  vÞ A Cþ q: ð1 þ vÞð1  2vÞ 3ð 1  v Þ ð1 þ vÞð1  2vÞ

(36)

(37)

(38)

For a nanowire with length much larger than radius, the deformation state of the nanowire can be assumed as plane strain, which gives q=0. For the β phase, the condition of |u| 0.5 and is 841.5 for ξ = 0.9. Figure 6 shows the effect of yield stress on the critical average concentration of lithium and the outer radius of the nanowire for the buckling of a core-shell silicon nanowire for ξ = 0.7, in which ΔRcr is the change of the outer radius of the core-shell nanowire at the onset of the buckling of the nanowire. The critical average concentration of lithium decreases with the increase of the yield stress. It is known that the critical load for the initiation of the buckling of a nanowire is dependent on the compressive stress applied to the nanowire in the longitudinal direction and the corresponding cross-sectional area. The ratio of the lithiationinduced change of the outer radius of the nanowire to the outer radius (ΔRcr/R2) at the onset of the buckling of the nanowire is also depicted in Fig. 6. It is interesting to note that the value of ΔRcr/R2 also decreases with the increase of the yield stress, suggesting that less compressive load is likely needed for the onset of the buckling of the nanowire. This result indirectly is in accord with the decrease of the critical average concentration of lithium with the increase of the yield stress, as shown in Fig. 6.

Interaction Between Stress and Diffusion in Lithium-Ion Batteries: Analysis. . .

841

350

L/R2=30

300

1.5

Ea/Eb =0.5 ξ=0.7

250

1.0

200

ΔRcr /R2

Fig. 6 Effect of the yield stress of α phase (shell) on the critical average concentration of lithium and the outer radius of the nanowire for the buckling of a core-shell silicon nanowire (ξ = 0.7)

CcrD/i0 R2

26

0.5 150 100

100

200

300

400

500

0.0

σ Y (MPa)

5

Summary

In summary, this chapter presents the nonlinear framework of the diffusion-induced stress in the theory of nonlinear elasticity, which provides the basis to investigate the coupling between stress and diffusion for analyzing the large deformation of materials induced by the migration/diffusion of solute atoms. Both the nonlinear mechanical and diffusion equations reduce to the corresponding equations, which are based on the theory of linear elasticity. For a core-shell nanowire whose deformation is described by the theory of linear elasticity, closed-form solution is obtained for the calculation of the critical time for the occurrence of the buckling of the nanowire. The effects of current density, length, and the ratio of R1/R2 (ξ) on the critical concentration and critical time for the occurrence of the buckling were studied. The results show that the compressive load applied to the nanowire increases with the increase of the current density for the same lithiation time and the same nanowire length. Less time is needed for the compressive load to reach the critical load for the occurrence of the buckling of longer nanowires. The dimensionless critical buckling time decreases with the increase of the radius ratio of R1/R2, which suggests that the critical buckling time decreases with the decrease of the shell thickness. For the same lithiation time, a core-shell nanowire with a thinner shell reaches the larger critical load at the smaller lithiation time due to a shorter diffusion path. For an elastic-perfectly plastic core-shell nanowire, numerical analysis of the lithiation-induced buckling of the core-shell silicon nanowire was performed by using the commercial finite element software ABAQUS. The effects of the radius ratio of R1/R2 and yield stress of the outer shell on the critical average concentration and critical time for the occurrence of the buckling were investigated. The critical average concentration of lithium for the occurrence of buckling increases with the increase of the ratio of R1/R2, as expected, while the critical average concentration of lithium decreases with the increase of the yield stress due to the decrease of ΔRcr/R2 with the increase of the ratio of R1/R2.

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Understanding the electrochemical-mechanical behavior of electrode materials of LIBs in the service environments plays an important role in designing better LIBs with prolonged and reliable service life. This chapter explores the possible failure mode of electrode materials of slender shapes induced by the diffusion of lithium from the viewpoint of diffusion-induced stress. It is known that there exist electrochemical reactions on the surface of electrode materials and inside electrode materials, which will alter the diffusion of lithium and the mechanical properties of the electrode materials. Both the electrochemical reactions and the variation of the mechanical properties of electrode materials with local electrochemical reactions will contribute to the structural degradation, including fracture and buckling. A unified theory, which consists of electrochemical reactions and the variation of the mechanical properties with local electrochemical reactions, needs to be developed for quantitatively predicting the electrochemical-mechanical behavior of advanced material architectures and designing better electrode materials of lithium-ion batteries.

Appendix: Postbuckling Analysis of an Elastic Rod A pin-end column with a radius of 10 nm and a length of 500 nm was created to validate the imperfection method used in the finite element analysis. The elastic modulus and Poisson’s ratio of the column are 200 GPa and 0.3, respectively. The first-order buckling mode was used to analyze the post buckling of the column, which was obtained from the buckle step in the finite element analysis. Figure 7 shows the relationship between the reaction force at the boundary and the arc length, in which the maximum reaction force is found to be 608  1010N. The critical buckling load calculated from Eq. 48 is 620.126  1010N. There is only ~1.9% 700

Reaction Force (10-16 N)

Fig. 7 Reaction force versus arc length for the post buckling of a pin-end column

600 500 400 300 200 100 0 0

1

2

3

4

5

Arc length (1nm)

6

7

8

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difference. Such a result validates the finite element model used in the analysis of the lithiation-induced post buckling of the nanowire.

References 1. Prussin S. Generation and distribution of dislocations by solute diffusion. J Appl Phys. 1961;32:1876–81. 2. Zhao K, Pharr M, Cai S, Vlassak JJ, Suo Z. Large plastic deformation in high-capacity lithiumion batteries caused by charge and discharge. J Am Ceram Soc. 2011;94:s226–35. 3. Brassart L, Suo Z. Reactive Flow in Large-Deformation Electrodes of Lithium-Ion Batteries. Int J Appl Mech. 2012;04:1250023. 4. Cui Z, Gao F, Qu J. A finite deformation stress-dependent chemical potential and its applications to lithium ion batteries. J Mech Phys Solids. 2012;60:1280–95. 5. Hao F, Fang D. Diffusion-induced stresses of spherical core-shell electrodes in lithium-ion batteries: the effects of the shell and surface/interface stress. J Electrochem Soc. 2013;160: A595–600. 6. Song Y, Lu B, Ji X, Zhang J. Diffusion induced stresses in cylindrical lithium-ion batteries: analytical solutions and design insights. J Electrochem Soc. 2012;159:A2060–8. 7. Zhang X, Shyy W, Marie Sastry A. Numerical simulation of intercalation-induced stress in Liion battery electrode particles. J Electrochem Soc. 2007;154:A910. 8. Yang FQ. Effect of local solid reaction on diffusion-induced stress. J Appl Phys. 2010;107:103516. 9. Haftbaradaran H, Gao H, Curtin WA. A surface locking instability for atomic intercalation into a solid electrode. Appl Phys Lett. 2010;96:091909. 10. Yang FQ. Diffusion-induced stress in inhomogeneous materials: concentration-dependent elastic modulus. Sci China Phys Mech Astron. 2012;55:955–62. 11. Yang FQ. Field-limited migration of Li-ions in Li-ion battery. ECS Electrochem Lett. 2015;4: A7–9. 12. Sethuraman VA, Srinivasan V, Bower AF, Guduru PR. situ measurements of stress-potential coupling in lithiated silicon. J Electrochem Soc. 2010;157:A1253–61. 13. Drozdov A. Constitutive equations for self-limiting lithiation of electrode nanoparticles in Liion batteries. Mech Res Commun. 2014;57:67–73. 14. Bhandakkar TK, Johnson HT. Diffusion induced stresses in buckling battery electrodes. J Mech Phys Solids. 2012;60:1103–21. 15. Liu XH, Fan F, Yang H, Zhang SL, Huang JY, Zhu T. Self-limiting lithiation in silicon nanowires. ACS Nano. 2013;7:1495–503. 16. Huang JY, Zhong L, Wang CM, Sullivan JP, Xu W, Zhang LQ, Mao SX, Hudak NS, Liu XH, Subramanian A, Fan H, Qi L, Kushima A, Li J. In situ observation of the electrochemical lithiation of a single SnO2 nanowire electrode. Science. 2010;330:1515–20. 17. Cui Z, Gao F, Qu J. Interface-reaction controlled diffusion in binary solids with applications to lithiation of silicon in lithium-ion batteries. J Mech Phys Solids. 2013;61:293–310. 18. Larché F, Cahn JW. Overview no. 41 the interactions of composition and stress in crystalline solids. Acta Metall. 1985;33:331–57. 19. Larché FC. What can the concept of a perfect chemoelastic solid tell us about the mechanical and thermodynamic behaviour of a solid? Le J de Phys IV. 1996;06:C1-03–9. 20. Timoshenko SP, Goodier JN. Theory of elasticity. 2nd ed. New York: McGraw-Hill; 1951. 21. Yang FQ. Interaction between diffusion and chemical stresses. Mater Sci Eng A. 2005;409:153–9. 22. Gere J, Goodno B. Mechanics of materials. Toronto: Cengage Learning; 2012.

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23. Hahn HT, Tsai SW. Introduction to composite materials. Lancaster: Technomic Publishing Company Inc; 1980. 24. Crank J. The mathematics of diffusion. New York: Oxford University Press; 1979. 25. Pal S, Damle SS, Patel SH, Datta MK, Kumta PN, Maiti S. Modeling the delamination of amorphous-silicon thin film anode for lithium-ion battery. J Power Sources. 2014;246:149–59. 26. Bucci G, Nadimpalli SP, Sethuraman VA, Bower AF, Guduru PR. Measurement and modeling of the mechanical and electrochemical response of amorphous Si thin film electrodes during cyclic lithiation. J Mech Phys Solids. 2014;62:276–94. 27. Mises RV. Mechanics of solid bodies in the plastically-deformable state. Göttin Nachr Math Phys. 1913;1:582–92.

Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials: Magnesium Single Crystal, Polycrystalline Magnesium, and Magnesium Alloy

27

Qizhen Li

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Typical SHPB System and its Data Processing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Constitutive Models for Dynamic/High Strain Rate Behaviors of Materials . . . . . . . . . . . . . . 3.1 Zerilli–Armstrong Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Johnson-Cook Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Steinberg–Cochran–Guinan–Lund (SCGL) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Mechanical Threshold Stress (MTS) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Preston–Tonks–Wallace (PTW) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Experimental Results, Theoretical Modeling, and Computational Simulations of Compressive High Strain Rate Mechanical Behavior of Magnesium-Based Materials (Magnesium Single Crystal, Polycrystalline Magnesium, and AZ31 Magnesium Alloy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Dynamic Stress-Strain Curves of Magnesium Single Crystal, Polycrystalline Magnesium, and AZ31 Magnesium Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Yield Strength, Maximum Stress and Fracture Strain of Magnesium Single Crystal, Polycrystalline Magnesium, and AZ31 Magnesium Alloy . . . . . . . . . . . . . . . . . 4.3 Theoretical Stress-Strain Relations of Magnesium Single Crystal, Polycrystalline Magnesium, and AZ31 Magnesium Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Finite Element Modeling (FEM) of Stress-Strain Relations of Magnesium Single Crystal, Polycrystalline Magnesium and AZ31 Magnesium Alloy . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

846 848 850 851 852 852 853 854

855 856 857 861 864 869 870

Abstract

This book chapter focuses on compressive mechanical behavior of magnesiumbased materials under dynamic loadings, especially high strain rate loadings. The content is organized into five sections: Section 1 provides an introduction of Q. Li (*) School of Mechanical and Materials Engineering, Washington State University, Pullman, WA, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_61

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related literature results, Section 2 presents an overview of the split Hopkinson pressure bar technique for high strain rate tests and its fundamental data processing method, Section 3 delineates some widely used constitutive models for predicting mechanical responses of materials under high strain rate loadings, Section 4 reports the experimental results, theoretical modeling, and computational simulations of compressive dynamic mechanical behavior of magnesiumbased materials under different high strain rate loadings, and Section 5 presents the conclusions. In Section 4, three types of magnesium-based materials (i.e., magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy) were tested at both quasistatic and dynamic loading strain rates to investigate their compressive mechanical behavior. The employed strain rates are in the range of 0.001~3600 s
1. Theoretical stress-strain relations based on the empirical Johnson-Cook model were also derived for each type of the studied materials. The theoretically predicted stress-strain curves agree well with the experimental curves for these three types of materials. Finite element modeling was also performed to investigate the dynamic compressive behavior of the three types of the studied materials. The computational stress-strain curves match the experimental data for magnesium single crystal and polycrystalline magnesium, while a simulation was conducted to predict the compressive properties of AZ31 magnesium alloy at a randomly chosen strain rate. Keywords

Dynamic Mechanical Behavior · High Strain Rate · Magnesium · Theoretical Models · Finite Element Modeling

1

Introduction

Many industries such as automotive, aerospace and airplane, and defense industries are in great need of light-weight structural materials to reduce the weight of transportation vehicles and improve fuel efficiency. The specific benefits with the application of lightweight structural materials are as follows: (a) the weight reduction of civilian transportation vehicles will allow more cargo weight per transporting cycle; (b) the weight reduction of racing cars will increase their agility and speed; (c) the weight reduction of military vehicles and armors can lower the burden carried by soldiers and improve their moving agility; (d) the weight reduction can lead to the improvement of fuel efficiency and reduce the amount of greenhouse gas emission; and (e) the emission reduction will contribute to environmental protection. Materials often experience dynamic and high strain rate loadings during some of the transportation and defense service situations such as crashing of automobiles during accidents, bird striking onto aircrafts, collision of racing cars, and bullet striking onto armors. It is important to know the dynamic/high strain rate mechanical properties of the materials in the services. Magnesium and its alloys are lightweight structural metals. In addition, they possess good damping properties and high shock energy absorption capability. Thus, they are highly attractive to be utilized in the aforementioned industries.

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Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

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Prior to their practical applications in various transportation tools, we should have a good knowledge of their mechanical behaviors during various crashing/striking events. A crash/strike often corresponds to dynamic/high strain rate loading and leads to high strain rate deformation of materials. Generally, many materials respond differently to the loadings at different strain rates, and it is important to study these different material responses. Among various high strain rate testing techniques, the split Hopkinson pressure bar system (SHPB) is popularly used to explore the dynamic mechanical response of materials at the strain rate of 102 ~ 104 s
1 since its creation and usage by Hopkinson [1], Kolsky [2], and Davies [3]. Many researchers devoted their efforts to investigate mechanical responses of magnesium and its alloys under high strain rate loadings through a SHPB system (e.g., [4–25]). The primarily studied materials are AZ31, AZ61, AZ80, AZ91, AM20, AM50, AM60, and ZK60 magnesium alloys. Polycrystalline magnesium, magnesium single crystal, Mg–Gd–Y, and Mg–8Li–1Al–1Ce were also investigated. Among the various types of magnesium alloys, those in the AZ series (especially AZ31, AZ61, and AZ91) are extensively studied. The following are just some examples of the reported research results. Song et al. [7] performed compression testing on magnesium-aluminum die casting alloys at various strain rates in the range from 0.001 s
1 to 1700 s
1 to uncover the material’s strain rate sensitivity. Their results showed that at strain rates above 1500 s
1, there was substantially higher work hardening and strain rate sensitivity compared with that observed for the quasistatic tests at strain rates from 0.001 s
1 to 1 s
1 [7]. For extruded AZ31 magnesium alloys, a number of research efforts found that their high strain rate behavior was anisotropic [6, 8, 24]. Wan et al. [6] prepared three types of cylindrical samples and impacted the samples along their axial direction using the SHPB system. The axial directions of their three types of samples are along the normal direction (ND), the transverse direction (TD), and the extrusion direction (ED), respectively, and their work indicated that the beginning flow stress for the samples impacted along ND was higher than those impacted along the other two directions; strain hardening was positive for all the three directions; the hardening rates in ND are high at the beginning but decrease continuously; and in TD and ED, the hardening rates are low at beginning but then increase rapidly [6]. Yang et al. [8] investigated the dynamic compressive behavior of AZ31 samples oriented at 0 , 45 , and 90 away from the normal direction at the strain rate of 1200 s
1. The sample with the axial direction along ND had the highest strength compared with the samples oriented along the other two directions [8]. Matthew et al. [24] also showed that the stress–strain responses of AZ31 magnesium alloy at high strain rate in ND, TD, and the rolling/ extrusion direction (RD/ED) were evidently different from each other. The effect of alloying element aluminum on high strain rate mechanical behavior was explored through compressive testing of the samples made of AZ31, AZ61, and AZ91, respectively, at strain rates ranging from 1000 to 1400 s
1 [11]. El-Magd and Abouridouane [10] performed dynamic testing on AZ80 magnesium alloy at room temperature and elevated temperatures such as 150, 200, and 450  C. When the testing strain rate is larger than 1000 s
1, the ductility of AZ80 magnesium alloy at room temperature significantly increases with the increase of strain rate and the

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dynamic ductility is also largely affected by the testing temperature [10]. The stressstrain responses of AZ80 magnesium alloy are controlled by both strain rate sensitivity and adiabatic heating [10]. Some attempts were also conducted to setup the constitutive models and describe the stress-strain curves using various equations theoretically [10, 26–37]. It is critical to construct a good constitutive model since the applications of magnesium and its alloys require the capability to predict their high strain rate behavior. Some examples of the reported constitutive models are Zerilli-Armstrong model [27–29], JohnsonCook model [30, 31], Steinberg–Cochran–Guinan–Lund (SCGL) model [32, 33], mechanical threshold stress (MTS) model [34–36], and Preston–Tonks–Wallace (PTW) model [37]. In this book chapter, Section 2 will introduce the SHPB testing technique and its fundamental data processing method, Section 3 will present a variety of widely used constitutive models for dynamic/high strain rate behaviors of materials, Section 4 will report the experimental results, theoretical modeling, and computational simulations of compressive dynamic mechanical behavior of magnesium-based materials (magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy) at a number of loading strain rates, and Section 5 will be the conclusions.

2

Typical SHPB System and its Data Processing Method

The primary composing elements of a typical compression SHPB system are schematically shown in Fig. 1. These elements are a striker bar, an incident bar, a transmission bar, a strain gage #1 on the incident bar, and a strain gage #2 on the transmission bar. The testing data are measured by the strain gages and collected by a recording system that is not shown in Fig. 1. During a test, the compressive loading surfaces of a sample are often covered by a small amount of lubricate grease to lower the friction. A pulse shaper in Fig. 1 is often placed at the interface between the striker bar and the incident bar. This shaper is a deformable shim of 0.1~2 mm thick, and it deforms plastically upon loading to transfer an incident pulse with a relatively long rise time [38, 39]. When performing an SHPB test, the general working principle is as follows. First, the compressed gas from a gas tank will push the striker bar out of a barrel. The velocity of the striker bar is often measured by a sensor at the end of the barrel. Second, the striker bar with a velocity V hits the incident bar to generate an incident stress wave σ in the incident bar. Third, this wave travels in the incident bar and results in the incident strain eI(t). Fourth, the wave interacts with the specimen at the interface between the incident bar and the specimen, and a part of the wave is reflected back to the incident bar and forms the reflected strain eR(t) and the other part is transmitted through the specimen and forms the transmission strain eT(t). Both eI(t) and eR(t) are collected by the strain gage #1 on the incident bar, and eT(t) is collected by the strain gage #2 on the transmission bar. These three strains (i.e., eI(t), eR(t), and eT(t)) are the functions of time t. In general, the same material is used for the striker bar, the incident bar, and the transmission bar; the cross section diameter and area are

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Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

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Fig. 1 Schematic drawing of a typical split Hopkinson pressure bar system for dynamic compression testing. V is the velocity of the striker bar, V1 is the velocity of the interface i between the incident bar and the specimen, and V2 is the velocity of the interface ii between the specimen and the transmission bar

also the same for these three bars; and the length of the incident bar is the same as that of the transmission bar. When the striker bar hits the incident bar, a compressive pulse will be generated in the two bars and the pulse moves with a velocity of C in the two bars along the direction that is away from their impacting interface. C is determined by the following relation [40, 41]: C¼

pffiffiffiffiffiffiffiffiffiffi Eo =ρ

(1)

where Eo is the Young’s modulus of the incident bar and ρ is the density of the incident bar. According to the one-dimensional wave theory, when the pulse reaches the nonimpacting and free end of the striker bar, a tensile pulse will be reflected and move back towards the impacting interface; once the tensile pulse arrives at the impacting interface, the striker bar will separate from the incident bar. The generated compressive pulse in the incident bar will produce an elastic strain that is represented as eI(t) and the corresponding stress wave σ in the incident bar. This σ is related to the velocity V of the striker bar, the density ρ of the incident bar, and the speed C of the wave in the incident bar according to the following form [40, 41]: σ ¼ ρCV=2

(2)

With the recorded eI(t), eR(t), and eT(t), the strain rate e_ðtÞ of the testing can be computed through the following expression [40, 41]. e_ðtÞ ¼ de=dt ¼ C½ðeI ðtÞ
eR ðtÞÞ
eT ðtÞ=L

(3)

where L is the length of the specimen. With the e_ in Eq. 3, the strain e(t) of the sample at an instantaneous time t can be computed through the integration of e_ðtÞ from 0 to t [40, 41]: ðt eðtÞ ¼ 0

C ½eI ðtÞ
eR ðtÞ
eT ðtÞdt L

(4)

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Based on the Hooke’s law and the equilibrium, the stress σ(t) in the specimen is σ ðt Þ ¼

Eo Ao ½eI ðtÞ þ eR ðtÞ þ eT ðtÞ 2A

(5)

where Ao is the cross-section area of the incident bar and the transmission bar, and A is the cross-section area of the specimen. Therefore, the Eqs. 3, 4, and 5 are employed to calculate strain rate e_ðtÞ, strain e(t), and stress σ(t) of the specimen for a SHPB testing. In addition, the equilibrium implies that eI(t) þ eR(t) = eT(t). These relations can simplify the Eqs. 3, 4, and 5 for calculating e_ðtÞ, e(t), and σ(t) of the specimen to the following expressions [40, 41]: e_ðtÞ ¼
2CeR ðtÞ=L
2C eðtÞ ¼ L σ ðtÞ ¼

(6)

ðt eR ðtÞdt

(7)

0

Eo Ao eT ðtÞ A

(8)

Although the raw data include the incident strain, the reflected strain, and the transmission strain, only two of these three strains are needed in the computation of strain rate, strain, and stress of the specimen for an SHPB test. Aluminum and steel are often utilized as the materials for the incident bar and the transmission bar. When aluminum bars are used, Eo and ρ are 75 GPa and 2700 kg/m3, respectively, and the wave speed C in the incident bar is calculated to be 5270 m/s when inputting the corresponding values of Eo and ρ into Eq. 1. When steel bars are used, C in the incident bar is 5160 m/s since the values of Eo and ρ for steel are 210 GPa and 7900 kg/m3, respectively.

3

Constitutive Models for Dynamic/High Strain Rate Behaviors of Materials

It is critical to explore constitutive models for dynamic/high strain rate behavior of materials because of the following reasons. First, the testing is often costly and it is destructive to the specimen. Second, it is almost impossible to perform the testing under all different strain rates. However, the actual strain rate applied to the materials can be any value and there are an infinite number of strain rates in the reality. If a reliable constitutive model is established for a type of material, the mechanical behavior can be predicted for a random strain rate. To date, there are extensive research efforts devoted to this area and many constitutive models are developed and studied. These models can be classified into two categories. One category is physically based, and the other is phenomenological based. Some examples of the

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Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

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physically based models are Zerilli-Armstrong model [27–29], mechanical threshold stress (MTS) model [34–36], and Preston–Tonks–Wallace (PTW) model [37], while some examples of the phenomenological based models are Johnson-Cook model [30, 31] and Steinberg–Cochran–Guinan–Lund (SCGL) model [32, 33]. This book chapter will introduce these aforementioned models in this section.

3.1

Zerilli–Armstrong Model

The Zerilli–Armstrong model proposed by Zerilli and Armstrong [27–29] is a microstructure-based constitutive model on high strain rate mechanical response of materials. The framework of this model is based on thermal activation analysis of dislocation motion, and the plastic shear strain rate γ_ is [27–29] γ_ ¼ m0 bNv

(9)

where m0 is a tensor orientation factor, b is the Burgers vector, N is the dislocation density, and v is the average dislocation velocity [27–29]. v is related to the shear stressdependent Gibbs free energy of activation G and the absolute temperature T [27–29]. 00 v ¼ v0 expð
G=kT Þ ¼ v0 exp@@G0


τðth

1

1

A bdτ0th A=ðkT ÞA

(10)

0

where v0 is the reference dislocation velocity, G0 is the reference Gibbs free energy at T = 0 k, A* is the area of activation, and τth is the thermal component of the shear stress [27–29]. Since the shear stress τth and shear strain γ are related to the normal stress σ th and normal strain e through σ th = mτth and e = γ/m, σ th is as follows [27–29] σ th ¼

mG0 expð
βT Þ A0 b

_ ðm0 bNv0 =mÞÞ β ¼ β0
β1 lnðe=

(11) (12)

where A0 is the dislocation activation at T = 0 k [27–29]. Since A* is independent of plastic strain for the body-centered cubic (BCC) metals and it is proportional to the parameter e
0.5 for the face-centered cubic (FCC) metals, the Zerilli–Armstrong model for both BCC and FCC metals is as follows [27–29] σ th ¼ c1 expð
c3 T þ c4 Tlne_Þ, for BCC metals

(13)

σ th ¼ c2 e1=2 expð
c3 T þ c4 Tlne_Þ, for FCC metals

(14)

where c1, c2, c3, and c4 are material constants that can be obtained through data fitting.

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3.2

Johnson-Cook Model

The Johnson-Cook model was created by Johnson and Cook in 1983 to investigate metals experienced large strains, high strain rates, and high temperatures [30, 31]. This constitutive model is phenomenological and empirical. With the goal of making the model easy for the computation purpose, this model separates the strain hardening effect, the strain rate hardening effect, and the thermal softening effect, and therefore has the following form [30, 31]:   ε_ σ ¼ ðA þ Ben Þ 1 þ C0 ln ð1
T m Þ ε_ 0

(15)

The first term (A + Ben) in the Eq. 15 is a power law strain hardening relationship between stress and strain. This term by itself is usually used to describe mechanical behavior of metals loading conditions and at room temperature.  under quasistatic  ε_ The second term 1 þ C0 lnε_ 0 in the Eq. 15 is on the effect of strain rate, while the

third term (1
Tm) is to capture the thermal softening effect. In addition to the free variables e, e,_ T, there are five material constants (i.e., A, B, n, C0, and m). A, B, and n are the three coefficients describing the quasistatic behavior of the material according to the first term, while C0 and m account for the strain rate hardening and thermal softening effects. e_0 is a user-defined reference strain rate, and is taken to be 0.001 s
1 for the quasistatic loading in this work. T* is the homologous temperature defined by T* = (T – Troom)/(Tmelt – Troom) in which T is the instantaneous temperature during a test, and Troom and Tmelt represent the room temperature and the melting temperature of the tested material, respectively [30, 31].

3.3

Steinberg–Cochran–Guinan–Lund (SCGL) Model

This model was initially proposed by Steinberg, Cochran, and Guinan (Steinberg–Guinan model) to define the dependence of the shear stress G and the yield strength Y on pressure P, temperature T, and the effective plastic strain e [32]. The authors assumed that the strain rate e_ has a negligible effect on G and Y when e_ is larger than some critical value, and this critical value is 105 s
1 [32]. The dependences of G and Y on P, T, and e at high strain rate are as follows [32], " G ¼ G0 1 þ " n

Y ¼ Y 0 ½ 1 þ β ðe þ ei Þ 1 þ  Y 0 ½ 1 þ β ðe þ ei Þ n

0 ! Gp P þ G0 η1=3

! # 0 GT ðT
300Þ G0

(16)

# ! 0  0 Yp P YT þ ðT
300Þ , while Y max Y 0 η1=3 Y0 (17)

27

Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

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where G0 and Y0 are the values for the reference state with P = 0, T = 300 K, and e = 0 0 0 0 0; Gp and Y p are the derivatives of G and Y with respect to pressure; GT and Y T are the derivatives of G and Y with respect to temperature; η is the ratio between the initial specific volume and the specific volume; n and β are the work-hardening parameters; ei is the initial effective plastic strain, and Ymax is the work hardening maximum of the model [32]. To extend the validity of this model to the low strain rates such as 10
4 s
1, Stenberg and Lund modified the Steinberg–Guinan model to the following form [33],     GðP, tÞ Y ¼ Y T e_p , T þ Y A f ep G0

(18)

  where Y T e_p , T and e_p are the thermally   activated portion of the yield strength and plastic strain rate, respectively; Y T e_p , T is a function of e_p and temperature; YT  Yp and Yp is the Peierls stress [33]. The second term in Eq. 18 is athermal and similar to the Steinberg–Guinan model [33]. This Steinberg–Cochran–Guinan–Lund (SCGL) model is valid for the strain rates in the range of 10
4 to 106 s
1 [33].

3.4

Mechanical Threshold Stress (MTS) Model

This model is based on thermally activated dislocation motion and the energy barrier a moving dislocation needs to overcome at various obstacles such as long range obstacles and short range obstacles [34–36]. Taking into account these obstacles, the MTS model has the following form [34–36],   μ σ y ep ,e,_ T ¼ σ a þ ðSi σ i þ Se σ e Þ μ0

(19)

  where σ y ep ,e,_ T is the yield strength and it is a function of the equivalent plastic _ and temperature T; σ a is the athermal component due to the strain ep, strain rate e, athermal obstacles; σ i is the stress component for thermally activated dislocation motion and the interactions among dislocations; σ e is the stress component for the microstructural evolution during the deforming process; Si and Se are the scaling factors depending on temperature and strain rate, respectively; μ is the temperaturedependent shear modulus; and μ0 is the shear modulus at 0 K [34–36]. Additionally, the two scaling factors are in the Arrhenius form as below [34–36], "



e_0 kb T Si ¼ 1
ln 3 g0i b μ e_ "



kb T e_0 Se ¼ 1
ln 3 g0e b μ e_

1=qi #1=pi

1=qe #1=pe

(20)

(21)

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where kb, b, and e_0 are the Boltzmann constant, the magnitude of the Burgers’ vector, and the reference strain rate; g0i and g0e are the normalized activation energies; and qi, pi, qe, and pe are material constants [34–36]. For the materials with body-centered cubic crystal structure, the temperature-dependent shear modulus is [34–36], μ ¼ ðC11
C12
C44 Þ=3

(22)

For the materials with face-centered cubic crystal structure, the temperaturedependent shear modulus is [34–36], μ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C44 ðC11
C12 Þ=2

(23)

This MTS model can provide high accuracy when modeling the dynamic responses of materials at high strains [34–36].

3.5

Preston–Tonks–Wallace (PTW) Model

This model attempted to establish a theoretical model with a linear Voce hardening law for the stress of materials experienced extremely high strain rates (i.e., up to 1011 s
1) in the following form [37], 8     θep < 2 τs þ αln 1
ϕexp
β
μ, thermal regime (24) σ y ep , e_p , T ¼ αϕ : 2τs μ, shock regime 



s0
τ y d τs
τy β≔ α

α≔

ϕ≔expðβÞ
1

(25) (26) (27)

where τs and τy are the normalized work hardening saturation stress and the normalized yield stress, respectively; s0 = τs(T = 0 K); θ is the hardening constant of the Voce hardening law; and d is a dimensionless material parameter for the modification of the Voce hardening law [37]. The following equations are the normalized saturation stress and the normalized yield stress [37],

  _  s1  ^ ln γ ξ , s0 e_p τs ¼ max s0
ðs0
s1 Þerf κ T e_p γ ξ_

(28)

  _

  y2  s1  ^ ln γ ξ , min y1 e_p , s0 e_p τy ¼ max y0
ðy0
y1 Þerf κT e_p γ ξ_ γ ξ_ (29)

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Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

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where s1 is the value of the normalized saturation stress at the temperature that is close to the melting temperature; y0 is the value of the normalized yield stress at the temperature of 0 K; y1 is the value of the normalized yield stress at the temperature    1=2 ^ ¼ T=T melt ; ξ_ ¼ 1 4πρ 1=3 μ ; Tmelt is that is close to the melting temperature; T 2 3M

ρ

the melting temperature; ρ is the density; M is the atomic mass; and κ, γ, s1, y1, and y2 are material constants [37]. The benefits of this model include the following: (a) it is applicable for strain rates in the range of 10
3 to 1012 s
1 through the merge of the thermal activation regime and the nonlinear dislocation drag regime; and (b) the variables are dimensionless since the stress is normalized by shear modulus and the temperature is normalized by the melting temperature [37].

4

Experimental Results, Theoretical Modeling, and Computational Simulations of Compressive High Strain Rate Mechanical Behavior of Magnesium-Based Materials (Magnesium Single Crystal, Polycrystalline Magnesium, and AZ31 Magnesium Alloy)

Three different magnesium-based materials were prepared, tested, and analyzed to explore their mechanical behaviors under compressive high strain rate loadings. The materials are magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy. Magnesium single crystal with a purity of 5 N was in a cylindrical bar shape with its [0001] crystal direction at about 10 away from the sample’s cylindrical bar axis. Polycrystalline magnesium with a 99.99% purity was extruded and in a cylindrical bar shape. AZ31 magnesium alloy has the nominal composition of ~3% Al, ~1% Zn, and ~96% Mg. The mechanical testing samples of AZ31 magnesium alloy were in a cubic shape. The dimensions of the mechanical testing samples for the three types of studied materials were as follows: the magnesium single crystal samples had the diameter of 9.5 mm and the length of 10 mm; the polycrystalline magnesium samples had the diameter of 6.35 mm and the length of 6.35 mm; and the AZ31 magnesium alloy samples had the length and width of 6.35 mm and the height of 10 mm. For these materials, both quasistatic and dynamic compression tests were performed at room temperature and in the air. The quasistatic compression tests were conducted using a universal mechanical testing machine under a strain rate of about 10
3 s
1, and the dynamic compression tests were done using an SHPB system at University of California, San Diego under various high strain rates. The strain rates employed for magnesium single crystal were 430, 1000, and 1200 s
1; those for polycrystalline magnesium were 800, 1000, 2000, and 3600 s
1; and those for AZ31 magnesium alloy were 1300, 2600, and 3500 s
1. For each quasistatic compression test, both load and displacement data were recorded and then computed to obtain a stress versus strain curve. For each SHPB dynamic test, the incident strain, the reflected strain, and the transmission strain were collected, and these data were input into the Eqs. 6, 7, and 8 to obtain the strain rate,

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the strain, and the stress of the sample. Based on the quasistatic and dynamic stressstrain curves, yield strength, maximum stress, and fracture strain can be extracted and mechanical responses of materials at a variety of strain rates can be investigated for each type of the studied materials. In the following, Subsection 4.1 will report the experimental stress-strain curves from both quasistatic and dynamic tests for magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy; Subsection 4.2 will study yield strength, maximum stress, and fracture strain of these materials and the effect of strain rate on these material properties; Subsection 4.3 will derive empirically analytic stress-strain expressions based on the JohnsonCook model for each type of materials and the experimental and theoretical stressstrain relations will be compared with each other; and Subsection 4.4 will present the computational stress-strain curves for magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy obtained through finite element modeling of the dynamic testing at high strain rates.

4.1

Dynamic Stress-Strain Curves of Magnesium Single Crystal, Polycrystalline Magnesium, and AZ31 Magnesium Alloy

Figure 2 reports the experimental true stress-true strain curves from both quasistatic and dynamic testing for magnesium single crystal [22], polycrystalline magnesium [21], and AZ31 magnesium alloy. The data at quasistatic strain rate are included for the comparison purpose. The curves are symbol-coded to represent the data for magnesium single crystal with the solid square symbols, those for polycrystalline magnesium with the open star symbols, and those for AZ31 magnesium alloy with the open triangle symbols. As reported in reference [22], for magnesium single crystal, the samples fractured at relatively small strains for both quasistatic and dynamic testing strain rates; the quasistatic stress-strain curve shows a clear Fig. 2 Experimental true stress-true strain curves from both quasistatic and dynamic testing for magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy (Note: the curves with solid square symbols are for magnesium single crystal, the curves with open star symbols are for polycrystalline magnesium, and the curves with open triangle symbols are for AZ31 magnesium alloy)

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Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

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transition from elastic deformation to plastic deformation with the increase of strain, while the dynamic stress-strain curves do not have a clear transition; the strain hardening rate for quasistatic testing was much lower than that for dynamic testing; the values of strain hardening rate were 3782 and 4578 MPa for the quasistatic testing and the dynamic testing, respectively; the strain hardening rate increased by about 20% when the testing strain rate changed from quasistatic to dynamic; and the strain hardening rate did not vary with the variation of strain rate for the studied dynamic strain rates (i.e., 430, 1000, and 1200 s
1). As reported in reference [21], for polycrystalline magnesium, the stress-strain curves for all testing strain rates have an inflection point on the curves; the samples experienced pretty large strain before fracture for each testing strain rate; the fracture strain for quasistatic testing was lower than that for dynamic testing; the strain hardening rate for quasistatic testing was about 1600 MPa, while that for dynamic testing (i.e., e_ = 800, 1000, 2000, and 3600 s
1) was about 3800 MPa; and the strain hardening rate for dynamic testing was about 2.4 times of that for quasistatic testing; and the strain hardening rate did not vary with the variation of strain rate in the studied dynamic strain rate range. For AZ31 magnesium alloy, the stress-strain curves for all testing strain rates also have an inflection point on the curve; the samples experienced pretty large strain before fracture for dynamic testing strain rates; the fracture strain for quasistatic testing was much lower than that for dynamic testing; the strain hardening rate did not change with the variation of strain rate in the studied range and the value was about 4500 MPa for all tested strain rates. To make a comparison among the three types of materials, magnesium single crystal with the studied orientation was the most brittle material; polycrystalline magnesium and AZ31 magnesium alloy had comparable ductility; magnesium single crystal had the highest strain hardening rate for dynamic testing; polycrystalline magnesium and AZ31 magnesium alloy had an inflection point on their stressstrain curves, which implied possible twinning dominated deformation mechanism; magnesium single crystal had no inflection point on the stress-strain curves, which indicated that the deformation mechanism was probably dislocation dominated; the strain hardening rate was low at low strain regime and increased to a high value at high strain regime for polycrystalline magnesium; and the strain hardening rate started from a high value at low strain regime and stayed approximately constant with the increase of strain for AZ31 magnesium alloy.

4.2

Yield Strength, Maximum Stress and Fracture Strain of Magnesium Single Crystal, Polycrystalline Magnesium, and AZ31 Magnesium Alloy

Both quasistatic and dynamic mechanical properties (i.e., yield strain, maximum stress, and fracture strain) are obtained for magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy. These data are listed in Table 1 and

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Table 1 Quasi-static and dynamic mechanical properties (i.e., yield strain, maximum stress, and fracture strain) of magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy (Note: the following samples did not fracture at the end of testing so that there were no fracture strain data for these cases: magnesium single crystal sample at 430 s
1, polycrystalline magnesium sample at 800 s
1, and polycrystalline magnesium sample at 1000 s
1) Material Magnesium single crystal

Polycrystalline magnesium

AZ31 magnesium alloy

Strain rate (s
1) 0.001 430 1000 1200 0.001 800 1000 2000 3600 0.001 1300 2600 3500

Yield strength (MPa) 230 – – – 140 145 145 145 140 100 80 91 110

Maximum stress (MPa) 380 – 481 475 265 – – 560 564 310 561 567 590

Fracture strain 0.06 – 0.064 0.068 0.15 – – 0.18 0.20 0.07 0.17 0.163 0.175

presented in Figs. 3, 4, and 5 for yield strength, maximum stress, and fracture strain, respectively. As reported in reference [22], for magnesium single crystal, the yield strength, the maximum stress, and the fracture strain were 230 MPa, 380 MPa, and 0.06, respectively, for quasistatic testing; the (maximum stress, fracture strain) pairs were (481 MPa, 0.064) for 1000 s
1, and (475 MPa, 0.068) for 1200 s
1 as listed in Table 1. The change of testing strain rate from quasistatic to dynamic largely increased the maximum stress (i.e., by a percentage of ~25%). However, the maximum stress for 1200 s
1 was slightly lower than that for 1000 s
1 and the difference of 6 MPa was only about 1% of their maximum stresses. The fracture strain for dynamic testing was slightly higher than that for quasistatic testing. Overall, the material was pretty brittle. As reported in reference [21], for polycrystalline magnesium, yield strength was roughly constant and had a value of about 140~145 MPa for all the testing strain rates (i.e., e_ = 0.001, 800, 1000, 2000, and 3600 s
1). On the contrary, maximum stress increased significantly from 265 MPa for e_ = 0.001 s
1 to 560 MPa for e_ = 2000 s
1 and 564 MPa for e_ = 3600 s
1. Thus, maximum stress of dynamic testing was more than twice of that for quasistatic testing, while the value of maximum stress stayed almost constant when the testing strain rate increased from 2000 s
1 to 3600 s
1. Fracture strain for both quasistatic and dynamic testing shows that the material was pretty ductile. Fracture strain also increased with the increase of strain rate, although this increase was not as significant as that for maximum stress. The increase was about 20% when the testing strain rate increased from 0.001 s
1 to 2000 s
1, and it was about 33% when the testing strain rate increased from 0.001 s
1

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Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

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Fig. 3 Yield strength of magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy at various testing strain rates ranging from 0.001 to 3600 s
1

Fig. 4 Maximum stress of magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy at various testing strain rates ranging from 0.001 to 3600 s
1

to 3600 s
1. This increase of fracture strain followed a linear relation as shown in Fig. 5, which was also different from the behavior of maximum stress. For AZ31 magnesium alloy, the yield strength, the maximum stress, and the fracture strain were 100 MPa, 310 MPa, and 0.07, respectively, for quasistatic testing. The (yield strength, maximum stress, fracture strain) triplets were (80 MPa, 561 MPa, 0.170) for the strain rate of 1300 s
1, (91 MPa, 567 MPa, 0.163) for 2600 s
1, and (110 MPa, 590 MPa, 0.175) for 3500 s
1 as listed in Table 1. Yield strength decreased from 100 MPa to 80 MPa when the loading changed from quasistatic to dynamic, and then increased when the dynamic strain rate changed from 1300 s
1 to 3500 s
1, as shown in Fig. 3. Maximum stress increased significantly when the loading changed from quasistatic to dynamic. In the

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Fig. 5 Fracture strain of magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy at various testing strain rates ranging from 0.001 to 3600 s
1

dynamic strain rate range, the maximum stress value continued to increase only slightly as shown in Fig. 4. The fracture strains for high strain rates were much larger than that for the strain rate of 0.001 s
1. The material was pretty ductile during dynamic testing. In the studied dynamic strain rate range, fracture strain slightly fluctuated around 0.17. Among the three types of materials, magnesium single crystal had the highest yield strength, the highest maximum stress, and the lowest fracture strain for the quasistatic testing strain rate. This indicates that the tested magnesium single crystal had a hard crystal orientation. Quasistatic yield strength and fracture strain of polycrystalline magnesium was higher than that of AZ31 magnesium alloy. However, quasistatic maximum stress of polycrystalline magnesium was lower than that of AZ31 magnesium alloy, which was due to the higher strain hardening rate in the low strain regime for AZ31 magnesium alloy. The strain hardening rate increased significantly for magnesium single crystal and polycrystalline magnesium when the testing strain rate changed from quasistatic to dynamic, while it did not vary for AZ31 magnesium alloy. Among all the tests, magnesium single crystal at dynamic testing strain rates had the highest strain hardening rate, and polycrystalline magnesium at quasistatic testing strain rate had the lowest strain hardening rate. For each of the three studied materials, both maximum stress and fracture strain were much higher for the dynamic testing cases than those for the quasistatic testing cases, and yield strength only slightly varied with the change of the testing strain rate. For the dynamic testing cases, magnesium single crystal had the lowest ductility, while polycrystalline magnesium and AZ31 magnesium alloy had comparable high ductility. In the studied dynamic high strain rate range, maximum stress and fracture strain varied slightly with the strain rate for all studied materials. Yield strength was the least sensitive property to the variation of strain rate, and maximum stress was the most sensitive property to the variation of strain rate.

27

4.3

Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

861

Theoretical Stress-Strain Relations of Magnesium Single Crystal, Polycrystalline Magnesium, and AZ31 Magnesium Alloy

It is important to establish a good constitutive model since the applications of magnesium alloys require the capability to predict their dynamic mechanical behavior at various loading strain rates. The theoretical analysis of magnesium single crystal and polycrystalline magnesium were reported in references [21, 22], based on the Johnson-Cook model. Additional theoretical analysis of magnesium single crystal was performed on the basis of the ZerilliArmstrong model, and the results showed that the analyses from both models were matching [23]. In this section, the Johnson-Cook model is selected for the data analysis. The constitutive or empirical relations are mainly derived through fitting the experimental data into a single mathematic equation. The derivation of empirically analytical equations for the prediction of stress-strain responses under a range of strain rates generally requires the consideration of strain hardening, strain rate sensitivity, thermal softening, and damping [21]. Strain hardening is obtained from fitting to the quasistatic stress-strain curve. Strain rate sensitivity and damping are represented by the parameters in the equation terms containing strain rate. Thermal softening is described by a parameter in the temperature term. A general assumption is that most of the strain energy (about 90%) is transformed into heat and lead to the change of the sample temperature. The temperature increase ΔT can be calculated as follows [40] ð ΔT ¼

0:9σde ρc

(30)

where ρ and c are the density and the specific heat capacity of the tested materials, respectively. The temperature increase (ΔT ) for each dynamic strain rate can be computed according to Eq. 30 and listed in Table 2. For magnesium single crystal, ΔT was 4.2, 10.7, and 11.5 K for the strain rates of 430, 1000, and 1200 s
1, respectively. For polycrystalline magnesium, ΔT was 15.0, 19.0, 32.4, and 38.2 K for the strain rates of 800, 1000, 2000, and 3600 s
1, respectively. For AZ31 magnesium alloy, ΔT was 30.6, 33.0, and 43.0 K for the strain rates of 1300, 2600, and 3500 s
1, respectively. The temperature change for magnesium single crystal was small and can be ignored, and the empirical expression without thermal effect can provide a good prediction of stress-strain curves for the different strain rates [22]. As reported in reference [22], for magnesium single crystal, the temperature effect was ignored for deriving the empirical theoretical expression, and the obtained values for A, B, n, and C0 were
78.4, 1509, 0.413, and 0.009778, respectively, as listed in Table 3

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Table 2 Temperature increases for the dynamic tests for magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy Material Magnesium single crystal

Polycrystalline magnesium

AZ31 magnesium alloy

Strain rate (s
1) 430 1000 1200 800 1000 2000 3600 1300 2600 3500

ΔT (K) 4.2 10.7 11.5 15.0 19.0 32.4 38.2 30.6 33.0 43.0

Table 3 Fitting parameters for magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy based on the Johnson-Cook model Magnesium single crystal A =
78.4 B = 1509 n = 0.413 C0 = 0.009778

Polycrystalline magnesium C0 = 0.033675 k0 = 10 k1 = 9521 m = 1.14 k2 =
213,010 m0 = 0.061564 k3 = 2,076,000 m1 = 91.267 k4 =
6,390,700 m2 = 1241.2 β = 0.0132

AZ31 magnesium alloy k0 = 4 C0 = 0.0591443 k1 = 10,834 m = 0.23855 k2 =
34,893 m0 = 0.24 k3 = 5,276,600 m1 = 25.18 k4 =
26,433,000 m2 = 1675 β = 0.304615

    e_ σ ¼
78:4 þ 1509e0:413 1 þ 0:009778ln e_0

(31)

The static strain rate (0.001 s
1) was used as the reference, i.e. e_0 = 0.001. When using this equation to fit the stress-strain curves for different strain rates, we obtained a good agreement between the experimental curves and the predicted curves for magnesium single crystal, as shown in Fig. 6a [22]. These curves were positioned crowdedly at the left portion of Fig. 6a. For polycrystalline magnesium [21] and AZ31 magnesium alloy, a new empirical formula is required for both quasistatic and dynamic stress-strain curves since these curves have a reflection point in the plastic deformation region. A fourth-order polynomial was used to fit the quasistatic curve, and both strain rate sensitivity and thermal softening were included in the expression for the dynamic curves. For strain rate sensitivity, a linear term was employed to account for the damping mechanism [42]. The equation has the following form [21]

27

Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

863

Fig. 6 Experimental and theoretical stress-strain relations of dynamic testing for (a) Magnesium single crystal and AZ31 magnesium alloy (Note: the curves with the solid square symbols, the open star symbols, and the open triangle symbols are for AZ31 magnesium alloy. The curves for magnesium single crystal do not have symbols and they are positioned in a crowded manner to the left portion of the figure.); (b) Polycrystalline magnesium (Note: the curves for ε_ = 800, 1000, and 2000 s
1 are those with the solid square symbols, the open star symbols, and the open triangle symbols, respectively, and the curves for ε_ = 3600 s
1 do not have symbols.). (Note: Exp. means experimental results, Theoretical in (a) means the prediction using Eq. 31, Theoreticala in (a) and (b) means the prediction using Eq. 32a, and Theoreticalb in (a) and (b) means the prediction using Eq. 32b)

    e_ σ ¼ k0 þ k1 e þ k2 e2 þ k3 e3 þ k4 e4 1 þ C0 ln ð1
T m Þ, for e_ e_0  2000 s
1 σ ¼ k0 þ k1 e þ k2 e2 þ k3 e3 þ k4 e4 þ βðe_
e_0 Þ  2000 s
1

(32a)   1
T ðmÞ , for e_ (32b)

The values of the parameters k0, k1, k2, k3, k4, β, C0, and m are listed in Table 3 for both polycrystalline magnesium and AZ31 magnesium alloy. This table also has the fitting parameters (m0, m1, m2) for calculating the strain energy through using a second-order polynomial [21]. The static strain rate (0.001 s
1) was used as the

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reference, i.e., e_0 = 0.001. When using this equation to fit the stress-strain curves for different strain rates, we obtained a good agreement between the experimental curves and the predicted curves for polycrystalline magnesium shown in Fig. 6b and AZ31 magnesium alloy shown in Fig. 6a [21].

4.4

Finite Element Modeling (FEM) of Stress-Strain Relations of Magnesium Single Crystal, Polycrystalline Magnesium and AZ31 Magnesium Alloy

There are some commercial software such as ABAQUS, ADINA, and ANSYS, and free software such as Afros2D and FEATool for conducting finite element modeling. ABAQUS is one of the most broadly used for the investigation of mechanical responses of materials. In this work, ABAQUS was chosen to simulate dynamic mechanical behavior of magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy. On one hand, the dynamic tests with the same strain rates as those of the experimental tests were simulated through ABAQUS\Explicit for magnesium singe crystal samples and polycrystalline magnesium samples with the goal of matching their experimental testing data. On the other hand, a randomly chosen strain rate was utilized for the simulation of AZ31 magnesium alloy to execute the prediction of the material’s dynamic mechanical properties without performing an actual experimental test. In the simulations, the simulated samples for the materials had the same geometry as the experimental samples. Aluminum was chosen to be the material utilized for the striker bar, the incident bar, and the transmission bar to match the actual experimental SHPB testing. The samples and bars in the simulations were discretized to eight-node solid elements. It was assumed that the bars were elastic during a simulated dynamic test. For magnesium single crystal, three simulations were performed and the nominal strain rates for these simulations were 430, 1000, and 1200 s
1. For polycrystalline magnesium, four simulations were performed and the nominal strain rates were 800, 1000, 2000, and 3600 s
1. The simulations were targeted to match the corresponding experimental strain rate cases. For AZ31 magnesium alloy, the simulated strain rate was 1700 s
1. For each simulation case, the incident strain εI(t) in the incident bar and the transmission strain εT(t) in the transmission bar were collected and reported in Fig. 7. Figure 7a presents εI(t) and εTt) for magnesium single crystal samples tested at the strain rates of 430, 1000, and 1200 s
1 respectively, while Fig. 7b presents εI(t) and εT(t) for polycrystalline magnesium samples tested at the strain rates of 800, 1000, 2000, and 3600 s
1, respectively. For the simulations of both materials, the impacting of the striker bar on the incident bar generated an incident strain with approximately constant amplitude in the incident bar. Here, the amplitude is defined to be the absolute value of the strain. For magnesium single crystal, the amplitudes of these incident strains were ~0.0007, ~0.0012, and ~0.0015 for the strain rates of 430, 1000, and 1200 s
1, respectively. Similarly for polycrystalline magnesium, the incident strain amplitudes were ~0.0011, ~0.0013, ~0.0025, and ~0.004 for the strain rates of 800, 1000, 2000, and 3600 s
1, respectively. Therefore, the increase

Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

Fig. 7 (a) The strain versus time histories of the incident strain and the transmission strain for the simulations with the high strain rates of 430, 1000, and 1200 s
1 performed on magnesium single crystal samples; (b) the strain versus time histories of the incident strain and the transmission strain for the simulations with the high strain rates of 800, 1000, 2000, and 3600 s
1 performed on polycrystalline magnesium samples (Note: the incident strain is the solid line and the transmission strain is the dashed line for each strain rate case)

a

865

0.0000 –0.0002 –0.0004 –0.0006

Strain

27

–0.0008 –0.0010 430/s, Incident 430/s, Transmission 1000/s, Incident 1000/s, Transmission 1200/s, Incident 1200/s, Transmission

–0.0012 –0.0014 –0.0016 –0.0018 0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

Time (sec)

b 0.000

Strain

–0.001

–0.002

–0.003

–0.004 0.0000

800/s, Transmission 800/s, Incident 1000/s, Transmission 1000/s, Incident 2000/s, Transmission 2000/s, Incident 3600/s, Transmission 3600/s, Incident 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

Time (sec)

of the incident strain amplitude resulted in the increase of the strain rate for both tested materials. The strain rates from both experimental testing and finite element modeling are reported in Fig. 8. The curves with the solid square symbols, the open star symbols, and the open triangle symbols in Fig. 8 show the strain rate versus time curves of different nominal strain rates (430, 1000, and 1200 s
1) for magnesium single crystal, while the curves without symbols in Fig. 8 show these curves of different nominal strain rates (800, 1000, 2000, and 3600 s
1) for polycrystalline magnesium. For the 430 s
1 case of magnesium single crystal, the strain rate versus time curve from the simulation matched well with that from the experimental testing. For the other two cases of magnesium single crystal and all cases of polycrystalline magnesium, the strain rate versus time curves from the simulations show that the strain rate reached the nominal value faster than the corresponding curves from the experimental testing. This may be due to the material inertia during the actual experiments.

866

Q. Li

Fig. 8 Strain rate versus time for high strain rate compression testing of magnesium single crystal (at strain rates of 430, 1000, and 1200 s
1) and polycrystalline magnesium (at strain rates of 800, 1000, 2000, and 3600 s
1) from experimental testing and finite element modeling (Note: Exp. means experimental results and FEM means simulation results. The curves with the solid square symbols are for the 430 s
1 case of magnesium single crystal, the curves with the open star symbols are for the 1000 s
1 case of magnesium single crystal, the curves with the open triangle symbols are for the 1200 s
1 case of magnesium single crystal, and the curves without symbols are for polycrystalline magnesium)

When the testing strain rate was low, the material can respond quickly enough so that the effect of inertia was small. When the testing strain rate was high, the material response was delayed and the effect of inertial became obvious. For all the simulated and experimental strain rates, the strain rate decreased after reaching a maximum value, which relates to the incident strain wave and the transmission strain wave. As shown in Fig. 7, the impacting of the striker bar on the incident bar generated a constant-amplitude incident strain pulse and this pulse was partially reflected to the incident bar and partially transmitted to the transmission bar. The incident pulse equals to the summation of the absolute values of the reflected and transmission pulses. Figure 7 shows that the transmission strain wave pulse was not constant; thus, the strain rate was not constant for both experimental and simulated data. The incident strain and the transmission strain in Fig. 7 can be input into Eqs. 7 and 8 to compute the stress and strain for each simulation case. The stress versus strain curves from both experimental testing and finite element modeling are reported in Fig. 9. The curves with the solid square symbols, the open star symbols, and the open triangle symbols in Fig. 9 show the curves of different nominal strain rates (430, 1000, and 1200 s
1) for magnesium single crystal, while the curves without symbols in Fig. 9 show these curves of different nominal strain rates (800, 1000, 2000, and 3600 s
1) for polycrystalline magnesium. The computed stressstrain curves for both materials had a good agreement with the corresponding curves from the experimental testing. The curves from the simulations also showed the similar shapes as the corresponding experimental curves.

27

Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

867

Fig. 9 Stress versus strain curves for high strain rate compression testing of magnesium single crystal (at strain rates of 430, 1000, and 1200 s
1) and polycrystalline magnesium (at strain rates of 800, 1000, 2000, and 3600 s
1) from experimental testing and finite element modeling (Note: Exp. means experimental results and FEM means simulation results. The curves with the solid square symbols are for the 430 s
1 case of magnesium single crystal, the curves with the open star symbols are for the 1000 s
1 case of magnesium single crystal, and the curves with the open triangle symbols are for the 1200 s
1 case of magnesium single crystal)

Since most of the work required for plastic deformation during high strain rate loading was converted to heat, it is expected that the sample temperature would increase for the tests. It was assumed that 90% of the energy was converted to heat in the simulations. The temperature increase (ΔT ) for each dynamic strain rate can be computed. For magnesium single crystal, the ΔT values were 3.9, 10.2, and 11.3 K from the simulations at the strain rates of 430, 1000, and 1200 s
1, respectively. The corresponding values from the experimental testing were 4.2, 10.7, and 11.5 K for the strain rates of 430, 1000, and 1200 s
1, respectively. The differences between the experimental data and the simulation data were less than 10%. Overall, ΔT was small and can be ignored in the tested strain rate range. For polycrystalline magnesium, the temperature increases based on the experimental stress-strain curves were 15.0, 19.0, 32.4, and 38.2 K for the strain rates of 800, 1000, 2000, and 3600 s
1, respectively. The temperature increases from the computational modeling were 15.0, 19.0, 32.4, and 38.2 K for the strain rate of 800, 1000, 2000, and 3600 s
1, respectively. The ΔT predicted by finite element modeling matched with those obtained from experimental data. For the case of 3600 s
1, T/Tmelt = (300 þ 38.2)/933 = 0.36. This is less than 0.5 and dynamic recrystallization generally does not operate. The FEM simulations for magnesium single crystal and polycrystalline magnesium produced high strain rate mechanical responses that agreed with the corresponding experimental data for each tested strain rate. A strain rate of 1700 s
1 was chosen to be the loading strain rate for the FEM simulation of AZ31 magnesium alloy. This simulation provided the incident strain history, the

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Fig. 10 (a) Strain versus time histories of the incident strain and the transmission strain, (b) strain rate versus time history, and (c) stress versus strain history for the FEM simulation with the high strain rate of ~1700 s
1 performed on AZ31 magnesium alloy

transmission strain history, the strain rate history, and the stress versus strain relation as shown in Fig. 10. The incident strain had approximately constant amplitude of ~ 0.0018. The strain rate increased steeply to ~ 1700 s
1 at the initial stage of testing and then gradually decreased with the progression of testing. The stress-strain curve in Fig. 10c shows that the sample yielded at about 85 MPa had a maximum stress of about 700 MPa, and fractured at a strain of about 0.175. The strain rate history and the stress-strain relation from the FEM simulation were plotted together with the data for the experimental tests with the strain rates of 1300, 2600, and 3500 s
1 in Fig. 11. This simulated strain rate history had a similar decreasing portion as those for the experimental tests after reaching the maximum strain rate. However, the initial portion of the strain rate history increased at a higher rate for the simulated case than those experimental cases as shown in Fig. 11a. This may be due to the inertia of material response in actual experimental tests. Figure 11b shows that the stress-strain curve of the simulated case almost overlapped with the curves from the experimental tests. This curve became different from the experimental curves when the strain reached about 0.125 and it predicted a higher maximum stress than the experimental tests. The fracture strain was almost the same for all cases (i.e., both the simulated

Dynamic Compressive Mechanical Behavior of Magnesium-Based Materials:. . .

Fig. 11 (a) Strain rate versus time history and (b) stress versus strain history for the simulation with the high strain rate of ~1700 s
1 and the experimental tests with the strain rates of ~1300, ~2600, and ~3500 s
1 performed on AZ31 magnesium alloy

869

a 4000 Strain Rate (s–1) 1300, Exp. 2600, Exp. 3500, Exp. 1700, FEM

3500 3000

Strain rate (s–1)

27

2500 2000

Finite element modeling

1500 1000 500 0 0.00

0.05

0.10

Time

0.15

(×103

0.20

sec)

b Compression stress (MPa)

Finite element modeling 600

400

Strain Rate (s–1) 1300, Exp. 2600, Exp. 3500, Exp. 1700, FEM

200

0 0.00

0.05

0.10

0.15

0.20

0.25

Strain

case with a strain rate of ~1700 s
1 and the experimental cases with the strain rates of ~1300, ~2600 and ~3500 s
1).

5

Conclusion

This book chapter started with introducing some related literature results on dynamic mechanical behavior of magnesium-based materials and the split Hopkinson pressure bar testing technique, described several widely used theoretical models for predicting mechanical responses of materials at high strain rates, and then analyzed the experimental results, theoretical modeling, and computational simulations of compressive dynamic mechanical behavior of three types of magnesium-based materials under different high strain rates. Specifically, quasistatic and dynamic

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Q. Li

compression tests were performed on samples made of magnesium single crystal, polycrystalline magnesium, and AZ31 magnesium alloy. The strain rate for dynamic testing is in the range of 430 ~ 3600 s
1. Comparatively speaking, the results show that yield strength was the least sensitive to strain rate and maximum stress and fracture strain were highly sensitive to strain rate. When the loading strain rate changed from quasistatic to dynamic, maximum stress increased significantly for the three types of materials, the strain hardening rate increased significantly for magnesium single crystal and polycrystalline magnesium, and the strain hardening rate stayed approximately constant for AZ31 magnesium alloy. Magnesium single crystal was more brittle than the other two materials. This is possibly because of the crystal orientations in the materials. For magnesium single crystal, the [0001] direction was almost parallel to compression loading direction and this was unfavorable for the operation of twinning. The plastic deformation would be mainly through hard non-basal dislocation slip. Thus, the yield strength was the highest for magnesium single crystal compared with the other two materials. For both polycrystalline magnesium and AZ31 magnesium alloy, they were extruded and should have [0001] direction perpendicular to the compression loading direction, which was favorable for twinning operation. The Johnson-Cook model was employed for the theoretical modeling of the three types of materials. The stress-strain curves for magnesium single crystal can be fit well theoretically with considering strain hardening (power law) and strain rate sensitivity, while the analytical expressions for polycrystalline magnesium and AZ31 magnesium alloy needed to consider strain hardening, strain rate sensitivity, thermal softening, and damping. FEM also provided well-matched predictions of the material responses such as the strain rate-time history, the stress-strain curve, and the temperature increase compared with experimental data for magnesium single crystal and polycrystalline magnesium. A simulation of AZ31 magnesium alloy at a randomly chosen strain rate validated the prediction capability of finite element modeling. The usage of finite element modeling to obtain material properties under various testing strain rates can reduce the cost incurred for performing actual experimental tests. Acknowledgments The author greatly appreciates the support for the related research from the US Department of Energy, Office of Basic Energy Sciences under Grant No. DE-SC0002144.

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29. Zerilli FJ, Armstrong RW. Dislocation mechanics based analysis of material dynamics behavior: enhanced ductility, deformation twinning, shock deformation, shear instability, dynamic recovery. J Phys IV France. 1997;01(C3):637. 30. Johnson GR, Cook WH. A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures. 7th international symposium on ballistics, Hague, Netherlands; 1983. p. 541. 31. Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech. 1985;21:31. 32. Steinberg DJ, Cochran SG, Guinan MW. A constitutive model for metals applicable at highstrain rate. J Appl Phys. 1980;51:1498. 33. Steinberg DJ, Lund CM. A constitutive model for strain rates from 10
4 to 106 s
1. J de Phys Colloq. 1988;49:3. 34. Follansbee PS, Kocks UF. A constitutive description of the deformation of copper based on the use of the mechanical threshold. Acta Metall. 1988;36:81. 35. Goto DM, Bingert JF, Reed WR, Garrett RK Jr. Anisotropy-corrected MTS constitutive strength modeling in HY-100 steel. Scr Mater. 2000;42:1125. 36. Kocks UF. Realistic constitutive relations for metal plasticity. Mater Sci Eng A. 2001;317:181. 37. Preston DL, Tonks DL, Wallace DC. Model of plastic deformation for extreme loading conditions. J Appl Phys. 2003;93:211. 38. Frew DJ, Forrestal MJ, Chen W. Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar. Exp Mech. 2002;42:93. 39. Gray III GT. Kuhn H, Medlin D, editors. Classic Split-Hopkinson Pressure Bar Testing. ASM handbook: mechanical testing and evaluation, vol. 8. Metals Park (OH): ASM International; 2000. p. 462. 40. Boyer HE, Gall TL, editors. Metals handbook. 9th ed. Materials Park (OH): American Society for Metals; 1985. p. 190. 41. Meyers MA. Dynamic behavior of materials. 1st ed. New York: Wiley; 1994. 42. Sakino K, Shiori J. Dynamic flow stress response of aluminium to sudden reduction in strain rate at very high strain rates. J Phys IV France. 1991;01(C3):35–42.

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 More-Than-Moore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 TSV Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Wafer Thinning Process and Redistribution Layer Deposition . . . . . . . . . . . . . . . . . . . . . . 1.4 Microbump Formation and Joining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Nanoindentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Micropillar Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Micropillar Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Synthesis of Sn-Based Intermetallic Compound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Micropillar Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Micropillar Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Micropillar Compression of Single-Crystalline Cu6Sn5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Micromechanical Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Effect of Anisotropy in Cu6Sn5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Micropillar Compression of Single-Crystalline Ni3Sn4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Micromechanical Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Slip System Existing in Ni3Sn4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Choice of Potential Structure Material in Microbumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Issues in Micropillar Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Damaged Layer Caused by Ion-Bombardment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Misalignment Between Flat Punch and Top Pillar Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Geometry of Pillar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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J. J. Yu Department of Materials Science and Engineering, National Taiwan University, Taipei City, Taiwan Department of Materials Science and Engineering, University of California at Los Angeles, Los Angeles, CA, USA J. Y. Wu · L. J. Yu · C. R. Kao (*) Department of Materials Science and Engineering, National Taiwan University, Taipei City, Taiwan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_62

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8 Future Directions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897

Abstract

The semiconductor industry is exploring a new scheme called “More-thanMoore” to overcome the high cost and difficulty of smaller node fabrications when facing the end of Moore’s law. For this purpose, the three-dimensional integrated circuit (3D IC) architecture is favored by most major semiconductor companies. The stacking of chips in 3D IC architecture with design of microbumps and through silicon vias (TSVs) allows integration of heterogeneous function within a single chip for enhancing performance with small form factor by increasing I/O density, shortening interconnect distance. Due to this reduction in solder volume, it is anticipated that Sn-based solder in microbumps will be totally converted into intermetallics (IMCs) during the assembly or the operation of further process. Sn-based IMCs therefore dominate the properties of microbump and become potential candidates of structural materials. The mechanical behaviors of IMCs and the preferred types of IMC become questions of importance. This chapter focuses on the micromechanical behaviors of single-crystalline Cu6Sn5 and Ni3Sn4 by micropillar compression. The failure mode of single-crystalline Cu6Sn5 and Ni3Sn4 are cleavage, but they both performed strain bursts as a result of dislocation gliding with certain slip system before failure. They are not as brittle as people thought and have adequate mechanical properties. For Cu6Sn5, grains with the c-axis aligned with the load direction have better mechanical properties. The compound Ni3Sn4 can withstand more than 4% strain along the slip system, (100)[010]. Compared to Cu6Sn5, Ni3Sn4 has better mechanical performance as well as toughness and should be more favorable to be adopted as structure materials of 3D IC microbumps.

1

Introduction

1.1

More-Than-Moore

Demands of consumer electronic products highly increased with the progress of time and the development of technology. It is regarded that a mobile product should be both versatile and convenient to carry. As a consequence, semiconductor industries are all committed to developing a small and multifunctional merchandise to attract consumers and to increase its competitiveness. To achieve these results, semiconductor industries focuses on increasing the density of transistors inside the chip by minimizing size of node to improve the performance of IC chip. Intel founder, Gordon E. Moore, boldly predicted at 1965: the number of transistors within an integrated circuit doubles every 18 months, and it also implies effectiveness of a chip

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twice increases while unit costs are halved. After that, Moore’s Law has become the guideline of later growth in the semiconductor industry. Nowadays, it is close to the end of Moore’s law because the node fabrication eventually meets the physical limit of node miniaturization at atomic levels by lithography. To achieve required pitch reduction, multiple patterning was used; however, the fabrication cost highly increased. The demand on lower power consumption and higher speed device will never stop by the end of Moore’s law. Semiconductor industries are devoted to the development of advanced schemes called “More-than-Moore” to surmount the physical limit for the next generation by merging frontend technology and backend/packaging technology together to solve problems [1]. The concept of 3D IC arises as a kind of More-than-Moore with several upcoming designs [2–5]. The stacking of chips in 3D IC architecture allows for a higher performance with a smaller form factor by increasing I/O density and shortening interconnect distance. Multiple chips with different function can be integrated in vertical stacking to improve its performance by functional diversification. Most importantly, it is able to integrate heterogeneous functions, e.g., logic, memory, analog, RF, and microelectrical mechanical systems (MEMS), into a single chip. Since different chips with different functions are made by different process nodes, implementation of system-on-a-chip (SoC) is unrealistic. The development of 3D IC greatly enhances the value of chip, makes it more competitive and achieves SoC. Fig. 1 shows the architecture of 3D IC. The pivotal design for 3D IC include (1) through silicon vias (TSVs) which provides the connection from upper to lower side of chip; (2) wafer thinning process to enable drilling through silicon die; (3) redistribution layers (RDL) to reroute circuit from TSV to microbumps; (4) microbumps to provide the connection between two chips; and (5) flip-chip joint to connect 3D IC stack to package laminate.

1.2

TSV Formation

A TSV is a vertical electrical connection passing entirely through a silicon chip by DRIE (deep reactive ion etch) or laser drilling with metal filling in via to connect circuit on both sides of chip and provide short vertical interconnection. Generally, the filling materials of via in 3D IC is Cu [6–10] or W þ Cu [10–12]. Dielectric, such as SiO2 or SiN, is used to provide good electrical isolation from the Si or other TSVs. The TSV can be fabricated in three different sequences: (1) via-first process (before transistor formation); (2) via-middle process (after transistor formation but before RDL formation); and (3) via-last process (after all of device fabrication process). The via-first can only be filled with poly-Si because of the temperature requirements in front-end-of-line (FEOL) processing. The via-middle and via-last processes are compatible with the use of Cu or W fill. Via-last that drills through whole chip and lands on the upper layers of RDL is the easiest way to fabricate, but it typically consumes the most space and limits the number of transistors.

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Fig. 1 Schemetic figure showing 3D IC structure and SEM image showing microstrucutre of microbump and flip-chip joint. SEM image are taken courtesy of ITRI and J. H. Ke

The TSV fabrication is significantly limited by the aspect ratio of the via which is 1:10 by current technology. It is important to reduce the diameter of via to increase unaffected Si area to accommodate more transistors. However, handling very thin wafer is difficult. Generally, the thickness of wafer is limited to around 50 μm and diameter of TSV is about 5 μm [13].

1.3

Wafer Thinning Process and Redistribution Layer Deposition

The typical thickness of 12-inch (300 mm) Si wafer is about 760 μm. Therefore, thinning of Si wafers to reveal the bottom of the TSV is an intrinsic requirement for the TSV depth and is only 50 μm in 3D IC. In order to handle delicate thin wafers for further processing, the Si wafer is bonded to a rigid carrier substrate before the backthinning process. The originally thick Si wafer is mounted with its active surface onto a temporary carrier using an adhesive bonding layer. Grinding is a conventional method which is fast and produces low variation and good surface finishes. Grinders are controlled to rotate the wafer on a vacuum head at a precise rate while grinding with diamond abrasives embedded in specially engineered binders on the wheel. Chemical mechanical polishing (CMP) is then employed to reveal TSVs, to remove grinding damage and to improve the final surface finish. RDL requires not only Cu wiring patterned and metallized over the wafer surface to provide new micropads as well as connection to bottom of TSVs but also insulator deposited on wafer surface provide electrical isolation. The principal purpose for RDL is to create distributed power and ground contacts and provide unique address

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lines so that each die can be placed on a specific location in order to move the micropad to a more convenient or accessible position for bonding process. This makes the layout of an old die replaced by a newer design. The use of RDL allows more degrees of freedom in design though it increase the overall process cost and complexity. After back-end-of-line (BEOL) process, the thin device wafer has to be demounted from the carrier for further processing, such as dicing, stacking, and packaging assembly. Since the wafer undergoes processing on the carrier, the adhesive bonding layer has to endure the severe BEOL processing condition. Temperature tolerance, chemical resistance, and plasma and mechanical stress from grinding are some of the parameters that need to be considered for choosing appropriate adhesive bonding materials which should provide strong bond of wafers to carriers and easy way to debond. After the wafer is mounted on the rigid carrier substrate, thermal curing or ultraviolet (UV) exposure is utilized to active temporary bonding between the wafer and the carrier. To demount the wafer, several approaches can be employed: (1) UV release adhesives debond after exposure to UV light at room temperature and adhesive layer can be removed either by chemical stripping or mechanical peeling, (2) thermal release adhesives debond after heating above a softening temperature, and then the wafer can be separated from the carrier, and (3) for those devices which prefer to demount at low temperature, solvent release adhesives can be used by dissolving adhesive layer in a chemical solvent for debonding of the device wafer from the carrier.

1.4

Microbump Formation and Joining

To form microbumps, solder caps are, first, deposited on the bond pads after RDL formation, and then thermal compression bonding is used to connect two chips. Çonventionally, the bonding pad in 3D IC is Cu [6–10]. In some products, Ni is preferred to be deposited on Cu as a diffusion barrier [14–17]. Several major reaction systems, Cu/Sn/Cu, Ni/Sn/Ni, and Ni/Sn/Cu, can be found in commercial products. The size of microbumps is only a few microns and thus the solder volume in 3D IC microbumps is much smaller than that in flip-chip joint. As shown in micrographs in Fig. 1, solder volume in 3D IC micro joint of which opening is 5 μm is about 100 μm3, only about 1/8000 compared to that in flip-chip joint of which opening is 100 μm. After reflow or bonding, about 1–10 μm thick IMCs, products of interfacial reaction and interdiffusion between pad and solder, form on the interface. In same bonding process, these IMCs occupied less than 10% of flip-chip joint while they occupied over 50% of 3D IC micro joint. Solder will be totally converted into IMCs during the assembly or the operation of further process in most of 3D IC microbumps. Major reaction products, Sn-based IMCs, dominate the properties of microbump and become the mainly potential candidates of structural materials. The trend of shrinkage of micro joint also satisfyingly meets the present demand of 3D IC. Due to the fact that IMCs have higher melting temperature than solder, it is desirable to expect IMCs to serve as structural materials in 3D IC micro joint. It can

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be bonded at low temperature and operate at high temperature in order to stack more chips on stacking chips without re-melt of micro joint in the formerly stacked chip. This concept is also called SLID (solid-liquid interdiffusion) or TLP (transient liquid phase) bonding. IMCs become main materials in micro joints to support the strength of structure and thus dominate the properties. Vertically stacking process leads to high process temperature and pressure simultaneously on the microbump because the microbumps height variation and die warpage required external pressure applied to flatten the thin die at high temperature. Due to the fact that IMCs occupy the large portion of microbumps in 3D IC, there is limited amount of Sn to absorb deformation as in conventional joint. Previous studies also pointed out that too thick IMCs likely cause a decline of mechanical properties of solder joint. The main reason is supposed that IMCs are relatively hard and brittle to lead to failure by crack propagation in IMCs [18], along the interface between IMCs and bond pad [18] or between IMCs and solder [18–21]. Reliability of IMC-based microbump will suffer more severe challenge. It becomes a really important issue to evaluate the practicality of this new structure. Whether the strength of IMCs is sufficient enough to support microbump and what kind of IMCs is appropriate to be utilized as structure material in 3D IC has drawn the attention of people. Since size of 3D IC microbumps has downsized to only a few microns, the size effects play an important role on mechanical properties [22]. The movement of dislocation and the plastic behavior at micron-to-submicron length scales are not well understood. The scientific community has started to pay attention and to explore size effects of micro-scaled to submicron-scaled system. Traditional mechanical measurement methods are no longer applicable.

1.5

Nanoindentation

Nanoindenter is developed to carry out ex situ mechanical test by equipping the diamond indenter tip and nanoindenter drives the tip vertically into and withdraws it from sample surface. The coarse positioner driven by a stepper motor is designed to execute coarse adjustment of distance between tip and specimen. The precision driven unit consisting of the MEMS-based transducer and the flexure hinge can monitor precise loading and unloading of the indenter tip. During the indentation test, the loads and displacements of the tip are recorded by the load sensor and the position sensor, respectively. The load transducer has force measurement ability to μN range and the displacement sensor has resolution down to angstrom scale. Environmental isolation is critical to nanoindentation operation to avoid significant errors. Though the measured load is the real load on specimen, the measured displacement requires correction of instrument compliance, fluctuation in atmospheric temperature and pressure, vibration, and thermal drift of the components. Nanoindentation tests are generally made with equipment of either spherical or pyramidal indenter tips. There are five common tips: (1) spherical tip, (2) Vickers tip,

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(3) Berkovich tip, (4) cube corner tip, and (5) Knoop tip. Berkovich tip is a threesided pyramid which has a very flat profile, with a half-angle of 65.27 . It is the most popular tip to be employed to study Young’s modulus and hardness owing to following advantages. Sharp and well-defined tip geometry is constructed to more easily meet at a single point ensuring well-defined plastic deformation into the surface and a more precise control over the indentation process. The Oliver–Pharr method [23] is the most frequently adopted method in nanoindentation testing to study Young’s modulus and hardness of single-phase materials. Fig. 2 showed a typical load-displacement curve carried out by a Berkovich tip. The parameter P represents the load and h represents the displacement relative to the initially contacted specimen surface. From load-displacement curves, the maximum indentation load (Pm), the displacement at maximum indentation load (hm), the initial unloading stiffness S (=dP/dh), and the final depth of the contact impression after unloading (hf) can be measured. The load-displacement relationships of unloading curve which is in elastic contact condition can conveniently be written as:
m P ¼ α h  hf

(1)

where α and m represents constants for experimental fitting. Value of exponent m is related to the geometry of indenter tip. For the Berkovich tip, m is 2. hf represents the final depth of the contact impression after unloading. (h - hf) is the elastic displacement. The initial unloading stiffness S is analyzed by the upper portion of the unloading data according to the equation:

Fig. 2 A schematic figure showing a typical loaddisplacement curve for an indentation experiment equipped with a Berkovich tip

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S¼

pffiffiffiffi dP 2 ¼ β pffiffiffi Er A dh π

(2)

where dimensionless parameter β is a geometry factor of indenter tip [24] and Er is defined as a reduced elastic modulus owing to effects of nonrigid indenters on the load-displacement behavior through the equation [25]:
 1  ν2i 1 ð1  ν 2 Þ ¼ þ Er E Ei

(3)

where νi represents the Poison’s ratio of indenter tip and ν represents the Poison’s ratio of target material. Ei represents the Young’s modulus of indenter tip and E represents the Young’s modulus of target material. A is the projected area of elastic contact at the contact depth h, and generally approximated by a fitting polynomial expressed as for a Berkovich tip [23]: AðhÞ ¼ 24:5h2 þ C1 h þ C2 h1=2 þ C3 h1=4 þ    þ C8 h1=128

(4)

After the contact area at peak load is measured, Young’s modulus then can be derived by equation mentioned above. In addition to the modulus, the data can be used to determine the hardness, H, computed from definition as followed [23]: H¼

1.6

Pm A

(5)

Micropillar Compression

Micropillar compression is first proposed by Uchic et al. to execute uniaxial compression test for studying size effect and microscale mechanical behavior [26]. FIB (focused ion beam) is an outstanding machine to fabricate micron-to-submicronscaled complex 3D structure with less restriction on materials. With the aid of secondary electron, the chance of failure significantly reduces without any special pretreatment. As shown in Fig. 3, any diameter and height of micropillars of interest can be manufactured by FIB. In Fig. 3b, the dimension of micropillar is very similar to the dimensions of 3D IC. The only sample preparation is a piece of arbitrary bulk material of interest greater than the diameter and length of desired micropillar. Use of bulk materials reduces the difficulty of the test sample manufacturing. Consequently, it starts to be widely utilized to study diverse materials like nanomaterials, singlecrystalline metals, superalloys, metallic glass, multilayer structure, bio-materials. Because of these advantages, hundreds of studies have adopted this technique to dig in the material behaviors and to explain the observed phenomenon. Generally, in 2000s, micropillar compression was carried out by nanoindenter equipped with flat-end punch to measure considerable varieties of mechanical

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Fig. 3 (a) Single-crystalline Ni3(Al,Hf) micropillar whose diameter is 43 μm and height is 90 μm. (b) Single-crystalline Ni super alloy (UM-F19) micropillar whose diameter is 2.3 μm and height is 4.6 μm. Ni3Al precipitation can be observed on surface of pillar. Adapted from [26], with permission from Elsevier

properties such as Young’s modulus, yield strength, fracture strength, strain to failure, flow stress. However, it is unable to monitor what is happening inside the material during testing, such as specific phenomena associated with discontinuities in the loading curves. Ex situ observation of failure structure only provides indirect evidence so that the proposed deformation mechanisms are merely seen as speculative hypothesis without in-situ observation. Recently, Picoindenter, a cutting-edge mechanical testing equipment placed either in scanning electron microscopy (SEM) or transmission electron microscopy (TEM) to visualize the mechanical response of testing structure, was developed to go beyond nanoindenter for superior result. The deformation process can be monitored directly during the micropillar compression test. In situ SEM or TEM images and load-displacement curve can be integrated to correlate the real-time mechanical behavior with numerical interpretation. For micro-scale testing specimen, SEM provides high quality of in situ images to identify onset and location of where shear bands, cracks, or buckling start to appear and to interpret the discontinuity events concerning deformation and failure modes. SEM equipped with electron back-scattered diffraction (EBSD) spectrum can further provide orientation gradients due to plasticity [27]. For submicron-scale testing specimen, TEM provides not only high in situ image quality but also crystal structure information, dislocation plasticity, phase transformations, and twinning [28]. In situ technique provides an excellent solution of research on deformation mechanism. Nowadays, micropillar compression is already a quite well-developed technology. Therefore, this technique is also well suited to explore mechanical properties of 3D IC microbumps and its size effects.

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2

Micropillar Fabrication

2.1

Synthesis of Sn-Based Intermetallic Compound

There are two major way in several kind of method to synthesize Sn-based intermetallic compound. One is by interfacial reaction. Solders are placed on metal substrates, and then samples are reflowed with a typical profile. After reflow, the samples were subjected to further reaction at either solid-solid state or solid-liquid state for a long time to form thick IMCs layer. The other one is by casting. Calculated weight ratios of Sn and other elements are evacuated in quartz ampoules under vacuum. Then, those sealed ampoules are put into oven with temperature higher than liquidus line, and agitated several times for homogenization. After quenching, IMCs are annealed at temperature lower than melting point about 30  C. After annealing for long time, ampoules were cooled in the air. Then, ampoules were smashed to reveal ingots. When we compare these two methods, interfacial reaction is a much easier way to prepare samples while casting provides specimen with larger grain size up to 1 mm rather than 100 μm. Fig. 4 showing the typical casting flowchart of large grain sample preparation and grain orientation identification. Cu6Sn5 and Ni3Sn4 are prepared by melting elements in their respective weight ratios (Cu:Sn = 39.1:60.9 for Cu6Sn5 and Ni: Sn = 27.0:73.0) in sealed, evacuated quartz ampoules under vacuum, followed by agitation (for homogenization) and subsequently solidified by water quenching. Cu6Sn5 was melted at 800  C for 5 days and then annealed at 380  C for 4 weeks while Ni3Sn4 was melted at 1000  C for 7 days and then annealed at 760  C for 4 weeks. Alloy ingots were subsequently cut into small pieces and metallographically polished to expose the cross-sectional microstructure, which was examined using a scanning electron microscope (SEM) with an electron back-scattered diffraction (EBSD) detector to acquire information of grain orientation. For convenience, the anisotropy of IMCs, Φc, is defined as the angle between c-axis of ηCu6Sn5 and the normal of sample surface which is also the direction of load. ΦN is defined as the angle between (100) pole of Ni3Sn4 and the normal of sample surface.

2.2

Micropillar Fabrication

Micropillars can be fabricated by FIB out from the top surface of specimen. The diameter of micropillar depends on requirement of study; however, the aspect ratio should be controlled in 2–5 (2–3 recommended) to prevent possible occurrence of plastic buckling if it is larger than 5 and strain hardening of substrate if it is smaller than 2 [29]. Taper angle should be limited as small as possible to avoid uneven stress distribution. There are generally two method to fabricate micropillars by FIB. One is by annular milling, a method by executing high current for high milling rate to create a circular trench from the top surface of specimen at first, and then applying low current to tailor pillars down to desired diameter, eliminate taper angle, and minimize

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Fig. 4 Flowchart showing the process of large grain IMCs preparation and orientation determination

the Ga+ ion implement. The other one is lathe milling developed from annular milling. After a high current annular milling, pillars are tilted to 28 and the side of pillar is milled after each rotation of 5–10 instead of simply utilizing low current. Though lathe milling may create 0 taper and better view of fillet, the extensive explosion of Ga+ ion beam cause a seriously damaged layer outside of pillar. For some materials, Ga-containing phases may form [30], and other materials tend to encounter Ga embrittlement [31]. Lathe milling also wastes more time to produce a pillar. However, it is easier to control aspect ratio of pillar and utilized to fabricate

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polycrystalline or polyphase pillar whereas the annular milling may encounter problems with differential milling rates [32].

3

Micropillar Compression

Specimen with multiple micropillars is first mounted on sample holder by silver paste and installed on testing equipment. Micropillar compression is performed with the flat punch in the direction perpendicular to top of specimen surface. The measured displacement by testing equipment (xm) literally combined three components: displacement of pillar (x), displacement of flat punch (xt), and displacement of base material (xb), given by xm ¼ x þ xt þ xb

(6)

To obtain the real displacement of pillar, deformation of flat punch and base material should be subtracted. Sneddon’s equation [25, 29, 33] is applied to extract the displacement of elastic half-space under compression of the cylinder flat punch:


1  ν2i x ¼ xm  Ei

 
  1  ν2b Fm Fm  Eb 2r t 2r b

(7)

where νi represents the Poisson’s ratio of diamond punch and νb represents the Poisson’s ratio of base material. Ei represents the Young’s modulus of diamond punch which is and Eb represents that of base material. Fm represents the measured load by picoindenter. rt and rb represent the contact radius of pillar on the top and in the bottom, respectively. Considering that material of flat punch is much harder than Sn-based IMCs and Sn-based IMCs tend to be brittle, plastic deformation of both flat punch and base material can be neglected. The measured displacement and load from Picoindenter should be converted into engineering stress and strain to carry out mechanical analysis. The engineering stress can be calculated as σ¼

Fm πr t 2

(8)

where σ represents the engineering stress if pillar is straight. Considering taper existing in micropillar, there are several equations to correct the engineering stress to approach the true stress. In first equation, engineering stress could be obtained by load divided by the average area of top and bottom of pillar, 2Fm  σ¼
2 π r t þ r 2b

(9)

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In the second equation, engineering stress could be obtained by load divided by the area calculated from average radius of top and bottom of pillar, Fm σ¼
 rt þrb 2 π 2

(10)

In the third equation, engineering stress could be obtained by load divides average area calculated from pillar volume divided by height, σ¼

Fm h V

(11)

where V represents the volume of pillar that can be calculated by integrating the cross-sectional area, given by, V¼

ðh 0

πr 2t ð1 þ xtan θÞ2 dx

(12)

where θ represents the taper angle and h represents the height of pillar. In the fourth equation, engineering stress could be obtained by averaging stress at the top and bottom surface,   Fm 1 1 σ¼ þ 2 πr t 2 πr b 2

(13)

Previous study [33] by finite element analysis demonstrated that for elastic material, Eqs. 8 and 9 show the better correction, and for plastic material, Eqs. 8–13 show the better correction. Engineering strain could be obtained by displacement of pillar dividing the height of pillar, ϵ¼

xp h

(14)

where ε represents the engineering strain.

4

Micropillar Compression of Single-Crystalline Cu6Sn5

4.1

Micromechanical Behavior

Recent studies by Prof. Chawla’s group [34, 35] and Prof. Kao’s group [36–38] examined the mechanical properties of single-crystalline Cu6Sn5, by micropillar compression. The typical stress-strain curve of single-crystalline Cu6Sn5 is performed by micropillar compression, as shown in Fig. 5. Several mechanical

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Fig. 5 (a) Typical micropillar compression strain-stress curve of single-crystalline Cu6Sn5. 15 tilted-view SEM images of micropillar (b) before compression and (c) after failure. Adapted from [38]

properties can be defined in Fig. 5a. First slip represents the point of first strain burst occurring in stress-strain curve. Final cleavage represents the point of failure of pillar and the maximum stress pillar can withstand. The stress and strain at first slip should be yield strength and strain to yield, and the stress and strain at final cleavage should be failure strength and strain to fail. Nonlinear part may occur due to the misalignment between the flat punch surface and the top pillar surface. This nonlinear part introduces the strain error which is the intercept of tangent line of linear part with x axis. To acquire the accurate result, the strain should be corrected by subtracting the strain error. Generally, for Cu6Sn5 micropillar, strain bursts only occur 1 to 5 times before final cleavage. In Fig. 5a, there is only one slight strain burst occurring. In situ image

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Fig. 6 (a) SEM image shows the slip trace occurring in pillar. (b) SEM cross-section view revealed by FIB milling shows no crack forming on slip plane. Adapted from [38]

indicated that one trace forming on the surface of pillar column corresponding to one strain burst [38] as predicted in [34]. However, FIB results in Fig. 6 showed that there is no any crack inside the pillar and the trace is the intercept of slip plane and pillar column. As a result, strain bursts are not the behavior of cleavage as described in [34], but the behavior of plastic deformation induced by slip. This is the first time it has been known that single crystalline Cu6Sn5 is capable of plastic strain accommodation. Though Cu6Sn5 can plastically deform, in general, it still is a strongly elastic material that the slope of compression curve keeps the same whether before or after strain bursts and keeps integrity until the stress up to failure strength. As a result, Cu6Sn5 micropillars do not perform significant continuous gliding and area change before its failure. The smooth cleavage plane revealed that the final failure mode of single crystalline Cu6Sn5 is cleavage as shown in Fig. 5c, and is still a brittle failure mode. Parallel multiple cleavage planes also existed in Cu6Sn5. Because of nature of IMCs crystal structure, the slip in ordered IMCs is by means of the movement of superlattice dislocations. Two partial superlattice dislocations must glide as a group and form antiphase boundary between them. The structure of η-Cu6Sn5 is the NiAs type; however, ratio of Cu and Sn is 1:1 in unit cell. An extra Cu atom occupying the trigonal bipyramidal interstices randomly can cause the disordered structure; therefore, ordinary dislocation can also form. Ordinary dislocation which has lower gliding activation energy can initiate plastic deformation while superlattice dislocation retards the further gliding of ordinary dislocations and it end up with dislocation piling followed by brittle fracture. Compared to FCC metal which is capable of plastical deformation by slip before failure, IMCs are, in terms of slip-induced plasticity, inferior to FCC metal due to the lack of slip systems, a higher critical shear stress and deformation constrained by superpartial dislocation. As a result, plastic deformation in IMCs is quite different from that in metal in the way that strain bursts occur instead of large slip deformation accompanied by strain hardening. Though plastic deformation strain is quite small compared to metal, it still arouses the interest that Cu6Sn5 which is originally regarded asa completely brittle material [34, 39–41] is capable of plastic deformation.

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Effect of Anisotropy in Cu6Sn5

The previous study [37] showed the relation between mechanical properties and the orientations of η-Cu6Sn5 which are 21 , 47 , 66 , and 76 . Fig. 7 showed the plot of various mechanical properties of single-crystalline Cu6Sn5 versus the angle between direction of load and c-axis of target grain. Though there is no obvious trend in yield strength and failure strength due to large deviation, strain to yield and strain to fail still show strong trend that the strain increases as the angle increases as shown in Fig. 7a, b. The anisotropy affects strain by around 0.3%. On the other hand, Young’s modulus decreases as the angle increases by value of 10 GPa as shown in Fig. 7c [37]. Similar trends are also demonstrated in [35] which only studied the load mainly parallel to c-axis and perpendicular to c-axis and in [36] which studied the angle from 0 to 90  . It is worth noting that Young’s modulus measured by nanoindentation shows consistent values of around 110–130 GPa which is believed to be more close

Fig. 7 Plot of (a) yield strength (blue), failure strength (red), (b) modified strain to yield (blue), modified strain to failure (red), and (c) Young’s modulus measured by nanoindentation (orange) and micropillar compression (green) versus angle ϕ [37]

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Table 1 Anisotropy mechanical properties of Cu6Sn5

The higher value, Stresskc-axis, or Stress⊥c-axis

Yield strength Similar

Failure strength Similar

Strain to yield Stress⊥caxis

Strain to fail Stress⊥caxis

Young’s modulus Stresskc-axis

to theoretical value. Considering the anisotropy, results from picoindentation also provided a similar trend in Young’s modulus as results from nanoindentation [35–37, 42–44]. All results still show the consistent trend as shown in Table 1.

5

Micropillar Compression of Single-Crystalline Ni3Sn4

5.1

Micromechanical Behavior

The typical stress-strain curve of single-crystalline Ni3Sn4 is performed by micropillar compression, as shown in Fig. 8. Generally, for Ni3Sn4 micropillar, strain bursts only occur 1 to 3 times before final cleavage. In Fig. 8a, there is only one slight strain burst occurring. Like Cu6Sn5, Ni3Sn4 pillar can also keep its integrity after first strain burst until failure strength and the slope of curve does not change significantly before and after strain bursts. This indicated that Ni3Sn4 is also capable of plastic strain accommodation. The smooth cleavage plane again revealed that the final failure mode of single crystalline Ni3Sn4 is cleavage as shown in Fig. 8c. There are also multiple cleavage planes parallel to each other in Ni3Sn4. Besides the higher fracture strength and strain to failure, the micromechanical behavior of Ni3Sn4 is almost identical with that of Cu6Sn5 [38].

5.2

Slip System Existing in Ni3Sn4

It is worth noting that Ni3Sn4 micropillars of specific grain orientations can deform up to 4% strain after stress is larger than yield strength as shown in Fig. 9a. It is about two times higher than predicted strain to failure. After compression, plastic deformation principally occurred at the upper part of the pillar that causes somewhat larger diameter after compression. It can be found that the extrusion and deformation traces form on the edge of pillar as shown in Fig. 9c. TEM analysis was carried out and result is shown in Fig. 10. Intensive shearing strain regions can be found as the result of plastic deformation. Shear bands which are parallel to each other with an angle around 45 with respect to the direction of load are found in the area marked in yellow. There is no evidence showing that deformation twins form in Ni3Sn4 during the compression test. As a result, it can be inferred that the plastic deformation is attributed to dislocation gliding. Selected area

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Fig. 8 (a) Typical micropillar compression strain-stress curve of single-crystalline Ni3Sn4. 15 tilted-view SEM images of micropillar (b) before compression and (c) after failure. Adapted from [38]

diffraction was carried out to identify the slip direction and slip plane. Diffraction pattern shows that the slip plane is (100) and slip direction is [010]. This result implies that there is at least one preferred slip system, (100)[010], in Ni3Sn4. Multiple cleavage planes in Fig. 8c should be related with the preferred slip system. Such slip system is responsible for the fact that Ni3Sn4 is able to accommodate high mechanical strain.

6

The Choice of Potential Structure Material in Microbumps

Table 2 shows the various mechanical properties of Cu6Sn5 and Ni3Sn4 in [38]. Ni3Sn4 has better yield and failure strength, about 34% and 43%, respectively, higher than those of Cu6Sn5, and also shows better yield strain and failure strain of about

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Fig. 9 (a) Micropillar compresion strain-stress curve of single-crystalline Ni3Sn4 in certain slip system. 15 tilted-view SEM images of micropillar (b) before compression and (c) after compression stop. Adapted from [38]

7% and 14%, respectively, higher than those of Cu6Sn5. Young’s modulus of Ni3Sn4 also shows 22% higher than that of Cu6Sn5. Considering the higher failure strength, strain to failure and Young’s modulus in Ni3Sn4, the absorbed deformation energy of Ni3Sn4 calculated from area below the strain-stress curve is also higher than that of Cu6Sn5, and therefore Ni3Sn4 has better toughness. In addition, if we carefully control the grain orientation of Ni3Sn4 to initiate preferred slip system in Ni3Sn4, it can provide higher absorbed energy than usual one with better strain to fail. Its plastic performance is competitive with ductility in pure metal. It is demonstrated that the mechanical performance of Ni3Sn4 is better than Cu6Sn5. Ni3Sn4 should be more promising to be utilized as structure materials of 3D IC microbumps.

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Fig. 10 TEM image of single-crystalline Ni3Sn4 micropillar in (100)//[010] slip system. Adapted from [38]

Table 2 Value of various mechanical properties in Cu6Sn5 and Ni3Sn4

Cu6Sn5 Ni3Sn4

Yield strength (MPa) 964  155 1289  295

Failure strength (MPa) 1306  143 1865  419

Strain to yield (%) 1.52  0.26 1.62  0.38

Strain to fail (%) 2.06  0.29 2.35  0.39

7

Issues in Micropillar Compression

7.1

Damaged Layer Caused by Ion-Bombardment

Young’s modulus (GPa) 66  5.6 81  8.3

It should be noticed that not all of Cu6Sn5 micropillars perform strain burst [37]. Though over 60% of specimen can deform plastically, the rest of pillars still fail directly without any plastic deformation. The amorphous layer introduced by over exposure to Ga+ ion during fabrication with lathe milling might be the reason for the suppression of slip propagation through the pillar surface [31]. For pure Mg, the yield strength increases up to 4 times. Studies by Motz et al. [45] and Bei et al. [46] also showed that the hardness of ion-damaged surfaces is about 2 times higher than the hardness of nondamaged surfaces. Since the amorphous layer is unavoidable in FIB process, the annular milling is favored due to lower exposure dose of Ga+ and thinner amorphous layer, i.e., most of micropillars in [37] has less ion-damaged. Considering the difficulty of dislocation movement in intermetallic compounds, it is more reasonable that extremely thin amorphous layer can cause the inhabitation of

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slip. The occurrence of amorphous layer can make failure strength of Cu6Sn5 increase slightly and strain to failure decrease slightly.

7.2

Misalignment Between Flat Punch and Top Pillar Surface

Though misalignment between flat punch and top pillar surface is carefully controlled in experiment, very small misalignment can lead to seriously underestimatation of Young’s modulus. H. Zhang et al. indicated that 1 misalignment can deteriorate Young’s modulus to 45% of theoretical value by finite element method [29] as shown in Fig. 11. In the experiment data of [37] as mentioned earlier in Fig. 7c, Young’s modulus measured by micropillar compression is approximately 55% lower than that measured by nanoindentation with the angle of misalignment which can be calculated approximately 1 by considering the strain error caused by misalignment. It should be noticed that Young’s modulus in [29] is without consideration of Sneddon’s correction and measured Young’s modulus is 10% smaller than corrected Young’s modulus in [37]. Consequently, results in [37] showed good consistence with the prediction in [29] in 5% deviation. Since misalignment is inevitable, finite element method on micropillar compression should be further developed on the combination of Sneddon’s correction and misalignment correction.

7.3

Geometry of Pillar

Besides damage layer and misalignment, geometry of pillar is required to be evaluated on data correction in micropillar compression. Micropillars fabricated by Fig. 11 The effect of misalignment on Young’s modulus. Adopted from [29], with permission from Elsevier

Normalized Modulus, E/E Ni

1

0.8

0.6

0.4

0.2

0

0

1

2 3 Misalignment, q (°)

4

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annular milling have taper angle which is at least 1.5 and also have fillet in the bottom of pillar due to intrinsic limit of FIB. Tapering can cause the inhomogeneous distribution of stress and deformation. Young’s modulus and yield strength can be overestimated. Slips congregating at the top of pillar result from tapering which gives rise to increasing stress gradient from bottom to top of micropillars as predicted in previous numerical studies [29, 33] and analytical study [47]. H. Fei et al. [33] suggested that higher taper angle can result in larger strain in plastic material. The straightforward calculation of stress does not fully capture the complexity of the situation. Even modified calculation of stress still does not compensate the effect of tapering owing to material-based problem. The presence of fillet might increase the average area of pillar. H. Zhang et al. [29] demonstrated that the overestimation of the stress of the material in the plastic range depends on the ratio of fillet radius/pillar radius which causes the greater error when it is larger. However, the sharper fillet can cause strong stress concentration at fillet and result in localized specimen failure before yielding. Simulation from [29] suggested that ratio of fillet radius/pillar radius should be controlled at 0.2–0.5. Practically, lathe milling method provides a better control of fillet shape compared to annular milling method. In addition, the view of the pillar height blocked by the protrusion on the milling area owing to the essence of annular milling causes the difficulty in measuring the exact pillar height, thus possibly leading to overestimation of pillar height and underestimation of strain. Those are reasons inducing overestimation and deviation of Young’s modulus in micropillar compression. The aspect ratio of pillars also plays an important role in measured stress-strain curve. If aspect ratio is less than 2, the strain hardening of pillar deviates from theoretical prediction due to the excessive influence of substrate. On the other hand, aspect ratio larger than 5 would cause the severe plastic buckling. If friction between punch and pillar is neglected, aspect ratio larger than 3 can cause severe plastic buckling. The occurrence of buckling leads to increase of strain and decrease of stress. Consequently, H. Zhang et al. [29] suggested that the aspect ratio of pillar should be controlled in 2–3.

7.4

Strain Rate

Micropillar compression can faithfully capture the strain rate sensitivity of the material. Fig. 12 showed that strain rates have a strong effect on micromechanical behavior of micropillar and slower strain rate brings about larger unrecovered inelastic deformation of micropillar than faster one. It is believed that there might be two probable reasons. One is that creep deformation might play an important role in deformation of micropillar, while the other is the geometry of micropillar is similar to the indentation tip to perform plastic deformation. However, it is required further effort to investigate relationship of creep deformation in IMC and testing geometry. Though there is almost no inelastic deformation under strain rate of 1  102, it is so fast for Picoindenter response that deformation cannot reach desired target strain.

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Fig. 12 Stress-strain curve of single-crystalline Cu6Sn5 micropillar at strain rate of (a) 1  102, (b) 1  103, and (c) 1  104

It is suggested that strain rate should be faster than 1  103 to avoid error in strain induced by inelastic deformation. Strain rate of 1  103 is favorable to study the mechanical properties of micropillars with strain error within 0.002.

8

Future Directions and Open Problems

There are still some unsolved problems in mechanical properties of Sn-based intermetallic compounds. Since both Cu6Sn5 and Ni3Sn4 fail by brittle cleavage, whether preferred cleavage plane exists and the corresponding slip system are worth further investigation. Previous studies [48–50] showed that Ni-doped Cu6Sn5 have better Young’s modulus and hardness. Whether there is any other strengthening element and the optimal amount of addition to enhance plasticity are worth exploring.

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The grain orientation plays an important role in the mechanical behavior and activation of certain deformation mechanisms. For Cu6Sn5 grains oriented with the c-axis almost parallel to the loading direction, the Schmid factor is very low to activate basal plane sliding. Other than basal slip, both prismatic and pyramidal sliding systems should also be considered. In Ni3Sn4, preferred slip system provides high strain accommodation though there are a lot of efforts needed to establish the slip system in Cu6Sn5. If the grain orientation can be controlled, the mechanical properties of microbumps can be further improved. Due to the small size of the real 3D IC microbumps, slight variation of temperature, bonding time, or joint size may lead to significant changes in the microstructure during bonding. A full-IMC joint can be easily achieved by bonding or annealing at high temperature for a long time; however, it is not an economic way without evaluating bonding or annealing parameter. Based on different processes, it is likely that solder joint consists of more than single crystal of Sn-based IMC. Grain boundaries and phase boundaries may exist. Interfaces between two different phases should be also considered, e.g., Cu/Cu3Sn, Cu3Sn/Cu6Sn5, Ni/Ni3Sn4. As a result, the mechanical properties of microbumps should not be as favorable as singlecrystalline Sn-based IMC, and reliability of IMC-based micro joint will face severe challenges. To evaluate the practicality of this new structure becomes a really important issue. Micropillars consisting of two IMC grains can be designed to observe the mechanical behaviors of grain boundary, and those of multiple layers can be designed to observe the mechanical behaviors of phase boundary. Up to now, micropillar compression has left something to be desired to provide consistent data of strength and strain points of both yield and failure. This problem is complicated; however, some hypotheses are given as follows. Defects in singlecrystalline intermetallic compounds which are invisible under SEM and FIB can either facilitate or deteriorate the failure of pillar. Also, the formation of amorphous layer can enhance the strength and diminish the strain while the density of dislocation in intermetallic compounds affects the deformation behavior of pillar. Unpredictable distribution and undetectable number of defects causes the wild deviations of mechanical properties.

9

Summary

In this chapter, micropillar mechanics of Sn-based intermetallic compounds is well summarized. The failure mode of single-crystalline Cu6Sn5 and Ni3Sn4 are cleavage; however, the cross-sectional view of pillar showed that the strain burst is not the mechanical behavior of cleavage but behavior of slip. Dislocation gliding is the deformation mechanism responding to plastic deformation. Both Cu6Sn5 and Ni3Sn4 preform plastic deformation through slip on certain crystallographic planes along the certain crystallographic direction before failure. They have superior mechanical properties and are not as brittle as thought before. For Cu6Sn5, grains with larger angle between c-axis and load direction tend to exhibit better strain to yield, strain to fail and Young’s modulus, and thus has better

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mechanical compatibility. Young’s modulus from micropillar compression shows good consistence and similar trend with that measured by nanoindentation, but systematically smaller about half of value from nanoindentation due to the misalignment between the flat punch and pillar top surface. The correction of Young’s modulus on misalignment is able to approach the theoretical value. Ni3Sn4 has at least one preferred slip system, (100)[010]; therefore, it can withstand more than 4% strain in this system. The anisotropy becomes so important that the specific grain orientation shows the better plastic behavior and mechanical properties. Compared to Cu6Sn5, Ni3Sn has better mechanical performance including yield strength, failure strength, strain to yield, strain to failure, and Young’s modulus. Considering competitive amount of plastic deformation in Ni3Sn4 preferred slip system, Ni3Sn4 possesses higher toughness and should be more advantageous to be adopted as structure materials of 3D IC microbumps. Several issues in micropillar compression should be carefully dealt with including damaged layer raised by FIB, misalignment, geometry of pillar, and strain rate. There are also several prospective directions worth further investigatation. All in all, micropillar compression with in situ observation technology provides an enabling technique to explore mechanical behavior of material in microscale and is therefore suited to conduct advanced studies of novel structure in 3D IC microbumps.

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31. Hütsch J, Lilleodden ET. The influence of focused-ion beam preparation technique on microcompression investigations: lathe vs. annular milling. Scr Mater. 2014;77:49–51. 32. Volkert CA, Minor AM. Focused ion beam microscopy and micromachining. MRS Bull. 2007;32(05):389–99. 33. Fei H, Abraham A, Chawla N, Jiang H. Evaluation of micro-pillar compression tests for accurate determination of elastic-plastic constitutive relations. J Appl Mech. 2012;79(6):061011. 34. Jiang L, Chawla N. Mechanical properties of Cu6Sn5 intermetallic by micropillar compression testing. Scr Mater. 2010;63(5):480–3. 35. Jiang L, Jiang H, Chawla N. The effect of crystallographic orientation on the mechanical behavior of Cu6Sn5 by micropillar compression testing. J Electron Mater. 2012;41(8):2083–8. 36. Yu JJ, Wu JY, Yu LJ, Kao CR. In situ observations of micromechanical behaviors of intermetallic compounds for structural applications in 3D IC micro joints. Electronics Packaging and iMAPS All Asia Conference (ICEP-IACC), 2015 International Conference on; 14–17 Apr. 2015. 37. Yu JJ, Wu JY, Yu LJ, Yang HW, Kao CR. Micromechanical behavior of single-crystalline Cu6Sn5 by picoindentation. J Mater Sci. 2017;52(12):7166–7174. 38. Yu JJ, Wu JY, Yu LJ, Kao CR. Choice of Intermetallics for Structural Applications in Micro Joints of Three-Dimensional Integrated Circuits (3D ICs). Electronic Components and Technology Conference, 2016 ECTC 2016 65th; 31st May–3rd June 2016. 39. Deng X, Sidhu RS, Johnson P, Chawla N. Influence of reflow and thermal aging on the shear strength and fracture behavior of Sn-3.5Ag solder/Cu joints. Metall Mat Trans A. 2005;36(1):55–64. 40. Chong DYR, Che FX, Pang JHL, Ng K, Tan JYN, Low PTH. Drop impact reliability testing for lead-free and lead-based soldered IC packages. Microelectron Reliab. 2006;46(7):1160–71. 41. Zhang N, Shi Y, Xia Z, Lei Y, Guo F, Li X. Comparison of impact toughness and fracture morphologies between Pb-containing and Pb-free solder joints subject to the Charpy impact test. J Electron Mater. 2008;37(10):1631–9. 42. Yang P-F, Lai Y-S, Jian S-R, Chen J, Chen R-S. Nanoindentation identifications of mechanical properties of Cu6Sn5, Cu3Sn, and Ni3Sn4 intermetallic compounds derived by diffusion couples. Mater Sci Eng A. 2008;485(1–2):305–10. 43. Song J-M, Huang B-R, Liu C-Y, Lai Y-S, Chiu Y-T, Huang T-W. Nanomechanical responses of intermetallic phase at the solder joint interface – crystal orientation and metallurgical effects. Mater Sci Eng A. 2012;534:53–9. 44. Mu D, Huang H, Nogita K. Anisotropic mechanical properties of Cu6Sn5 and (Cu,Ni)6Sn5. Mater Lett. 2012;86:46–9. 45. Motz C, Schöberl T, Pippan R. Mechanical properties of micro-sized copper bending beams machined by the focused ion beam technique. Acta Mater. 2005;53(15):4269–79. 46. Bei H, Shim S, Miller MK, Pharr GM, George EP. Effects of focused ion beam milling on the nanomechanical behavior of a molybdenum-alloy single crystal. Appl Phys Lett. 2007;91(11):111915. 47. Shan ZW, Mishra RK, Syed Asif SA, Warren OL, Minor AM. Mechanical annealing and source-limited deformation in submicrometre-diameter Ni crystals. Nat Mater. 2008;7(2):115–9. 48. Mu D, Yasuda H, Huang H, Nogita K. Growth orientations and mechanical properties of Cu6Sn5 and (Cu,Ni)6Sn5 on poly-crystalline Cu. J Alloys Compd. 2012;536:38–46. 49. Tsukamoto H, Dong Z, Huang H, Nishimura T, Nogita K. Nanoindentation characterization of intermetallic compounds formed between Sn–Cu (–Ni) ball grid arrays and Cu substrates. Mater Sci Eng B. 2009;164(1):44–50. 50. Xu L, Pang JHL. Nano-indentation characterization of Ni–Cu–Sn IMC layer subject to isothermal aging. Thin Solid Films. 2006;504(1–2):362–6.

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Thermodynamic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Elastic Lattice Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 State Functions for Transport of Neutral Species Within an Elastic Material Having a Network Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 State Functions for Transport of Multiple Ionic Species Subject to Mass and Charge Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Cohesive Zone Constitutive Theory for Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Example Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Ambipolar Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Kinetic Demixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Mechanically Driven Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Mechanical Degradation of All-Solid-State Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Incremental Boundary Value Problem in Discrete Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Static Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Diffusion Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Diffusion Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

902 906 910 910 913 916 918 919 919 925 932 936 939 942 942 945 948 950

G. Bucci (*) · W. C. Carter Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_63

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Abstract

A framework is presented that treats the combined effects of nonlinear elastic deformation, lattice constraints, and electrochemical potentials. The electrochemo-mechanical diffusion potential is derived, and the particular case where ionic species are subject to a crystal lattice constraint is also derived. By combining energy balance and local entropy production, the framework provides a consistent method to treat the evolution of charged species which also carry anelastic deformations in a crystal lattice. The framework is used to derive a finite element formulation that applies to general cases of interest for diffusion of active species in battery electrodes and in fuel cells. We demonstrate the application of the finite element formulation first with two cases: ambipolar diffusion and kinetic demixing. The predicted system response demonstrates how mechanical effects cannot be disregarded, even in the presence of dominant electrostatic forces acting on ion transport. A cation-rich surface layer is predicted when elastic forces, due to Vegard’s stress, participate in the migration of defects in multicomponent oxides. Simulations show how stress plays a central role in the cation segregation to interfaces, a phenomenon regarded as critical to power and durability of solid oxide fuel cells. Then, we analyze the interplay between electro-chemo-mechanics and fracture in battery electrodes. The presence of fracture in the electrode particles perturbs the stress field and results in stress concentration around the crack tips, which locally affects lithium concentration. This may prevent full lithiation and promote further fracture propagation. Finally, we simulate mechanical degradation of all-solid-state batteries via fracture within the solid electrolyte material. Such cracks would block Li diffusion and reduce the composite electrode’s effective ionic conductivity.

1

Introduction

Materials for energy conversion and storage have complex microstructures and their physical behavior depends on coupling of multiple mechanisms and driving forces. Macroscopic models that account for such complex phenomena are a materials modeling challenge. For instance, modeling of electrode and electrolyte materials for Li-ion batteries or solid oxide fuel cells (SOFCs) involves diffusion of multiple charged species in a solid medium in the presence of stress, electrostatic, and chemical potential gradients. Inclusion of these multiple fields and transport mechanisms into microstructural models can be performed by finite element methods. However, as a first step in the prediction of time-dependent species distribution, induced electric fields, deformation of the solid, and the effect of stress on diffusion, it is necessary to construct a rigorous formulation of the coupled physics of mass transport, induced electric fields, and nonlinear continuum mechanics. The purpose is to predict the overall electrochemical performance – including limiting factors, failure mechanisms, sensitivity to external conditions, and material properties – for electro-chemo-mechanical systems with complex microstructures.

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The present article proposes a quantitative description of such multifunctional materials drawn from the study of a general nonhomogeneous irreversible thermodynamic system under the action of multiple physical phenomena. From this analysis, we develop a finite element formulation and present simulations that demonstrate its applicability to ambipolar diffusion and kinetic demixing [1]. The predictive capability of the model depends on a consistent derivation and on the choice of appropriate numerical procedures to solve it. The formulation provides a foundation for subsequent models which treat the disparate length and time scales in complex battery and SOFC microstructures. We present a derivation of the driving forces for the time-dependent evolution of the state variables. At each instant in time, the system is considered to be in the state of local equilibrium. Mechanical equilibrium will be treated as instantaneous (i.e., static), and the transport of species is transient. To that end, we follow Larché and Cahn (see §7 in [3]): the most common partial equilibrium occurs when all processes except diffusion have relaxed to equilibrium. The local equilibrium is still guaranteed and “The only suppressed condition is that [the diffusion potential]1 MIK, need be constant, but MIK, remains continuous across all interfaces that have reached equilibrium [3].” Because the system under study is nonconservative, we use the entropyproduction formulation stated in the form of the Clausius–Duhem [4] inequality. In particular, we will consider natural isothermal processes, according to the definition proposed by Planck [5] and adopted by several authors, including Guggenheim [6]. For such cases, the Clausius–Duhem inequality reduces to the Clausius–Planck inequality [4]. The dissipation-rate density is the difference between the external power expenditure and the rate of change of free energy of the system per unit volume. It includes contributions from the transport of chemical species and that of charges. It may incorporate mechanical dissipation typical of viscous and plastic materials; however, the treatment of such constitutive behaviors is not included in the finite formulation below. The thermodynamic framework is constructed along the lines of theories formulated by Larché and Cahn [3], Truesdell [7], and Ortiz [8]. It also borrows from more recent application to the analysis of electrochemical systems, in particular Li-ion batteries, proposed by Bower [9, 10], Anand [11], Suo [12], and Wu [13]. As observed by Larché and Cahn [3], in many cases, the mutual interaction between stress and diffusion cannot be ignored; therefore, it will solve the equations of mechanical equilibrium and diffusion simultaneously. Furthermore, our model extends previous work by accounting for electrostatic forces and multiple charged mobile species. This includes the internal electric field induced by the transport and distribution of ions and vacancies in order for the system to maintain local electro-

In this paper, we use ϖi as notation instead of the Larché–Cahn notation MiK. The second dependent species K is treated explicitly.

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neutrality. The nonlinear theory of electro-elasticity cast into the Coleman–Noll formalism was originally proposed by Toupin [14], followed by several other contributions, including papers by Ogden and coauthors (see [15] for a literature review). We demonstrate the electro-chemo-mechanical applications of our model by presenting simulations of ambipolar diffusion and kinetic demixing (as defined in [16]) – both being relevant phenomena in SOFCs. Kinetic demixing (i.e., the production of nonhomogeneous compositions and defect populations due to mobility differences between species) can lead to interfacial segregation. For example, some chemistries and extreme operating conditions will be shown to produce demixing at the surfaces of SOFCs’ cathodes. Under operation, SOFC electrodes experience electrical biases that can redistribute their charged defects (e.g., oxygen and cation vacancies [17]). While the oxygen vacancy rapidly moves in response to this bias, cation defects, due in large part to their much lower mobility in these systems, migrate very slowly to establish a new equilibrium. Over time, a cationrich surface layer is often observed to develop in many of the advanced cathodes, blocking the critical oxygen reduction reaction and thus reducing SOFC power [18–21]. With changes in the oxidation state of the electrode (i.e., vacancy content), the relative amount of cation segregation has been shown to be modified [17]. In extreme conditions, this may even result in decomposition of the electrode material [22]. In another recent example, the oxidation state of the electrode has been controlled to intentionally segregate catalytically active cations from the bulk oxide to the surface, with corresponding improvements in SOFC power [23]. Elastic and electrostatic interactions of dopants with the surrounding lattice have been described as the driving forces for segregation on perovskite compounds of the type Ln1–xDxMnO3 (host cation Ln = La, Sm, dopant D = Sr, Ca, Ba) [17] which are among the most commonly used SOFC cathode materials. Elastic energy differences are created by differences in the atomic volume of dopants compared to the host cations (La, Sm). Gradients in elastic energy density can affect the stability of cathode surfaces. Atomistic studies also provide insight for some experimentally observed demixing phenomena [17, 24]. Kinetic demixing has also been investigated at the continuum level for oxide solid solutions in which oxygen ions can be regarded as an immobile species [25, 26]. However, these continuum approaches do not account for stress effects on diffusion and segregation of species. Stress-driven kinetic demixing and decomposition has been treated by Dimos and coauthors [27]. However, a combined electro-chemo-mechanics is necessary to model complex dynamics of these multifunctional materials under a variety of external conditions and microstructures. The rigorous continuum model proposed herein is amenable for numerical treatment and for the investigation of a wide range of technologically relevant systems in a rigorous quantitative manner. Central to the formulation is the derivation of a diffusion potential for each independent species – the gradients of which drive mass transport. Below, we simplify the formulation by assuming that the diffusion potential can be written as a sum of the chemical, electrostatic, and stress potentials and adopt a mixed finite element method [28]. Two example applications have been selected for demonstration of the formulation’s capability in the context of SOFCs:

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• In Sect. 4.2, simulations of kinetic demixing, in the presence of Vegard’s stresses and in the absence of stress, are presented. The simulation shows that stress plays a central role in the segregation to interfaces: the time-dependent evolution of stress should not be ignored in the formulation, even when there are large electric fields. The results have direct applicability to electrode surface properties in SOFCs. • Simulations that illustrate stress-effect ambipolar diffusion are presented in Sect. 4.1. These simulations show the predominance of electrostatic compared to chemical and mechanical driving forces: stresses on the order of 1 GPa are insufficient to decouple the transport of charges of opposite sign. All-solid-state rechargeable lithium-ion batteries have attracted much interest because they have features particularly favorable for large-scale application. The replacement of an organic liquid electrolyte with a nonflammable and more reliable inorganic solid electrolyte (SE) simplifies the battery design and improves safety and durability of the system [29–31]. However, the mechanical behavior of such electrodes will be considerably different than their liquid electrolyte counterparts. Direct stacking of solid-state cells enables the achievement of high operating voltages in a reduced volume. Furthermore, all-solid-state batteries may allow the use of combination of electrode materials, difficult to employ in conventional liquid electrolyte batteries. A key development to the success of all-solid-state batteries is a SE with high Li-ion conductivity at room temperature. In recent years, several SEs having level of conductivity comparable to organic liquid electrolytes have been discovered and tested with many active materials [32–34]. In all-solid cells, rate capability and the overall performance throughout the expected life cycles depend in larger measure on the mechanical integrity of the composite system. The durability of a cohering solid–solid interface is likely to be important practical consideration. Two examples are proposed here to investigate phenomena related to the mechanical reliability of all-solid-state Li-ion batteries: • The example of mechanically driven diffusion in Sect. 4.3 illustrates the role of stress as a driving force in cases where intercalation is coupled with anelastic deformation of the host lattice. Furthermore, it treats a highly nonhomogeneous stress field in addition to the Vegard’s stresses.2 • Finite element analyses of mechanical degradation of all-solid-state battery electrodes. The model predicts the evolution of fracture across the solid electrolyte material within a composite microstructure. At various states of charge, the active

2

Larché and Cahn’s work mostly focused on the effect of self-stress, i.e., stress induced by nonhomogeneous concentration. In [35] Larché and Cahn took into account problems, which require “the simultaneous solution of the equations of elasticity and those of chemical equilibrium.” However, they were able to decouple the chemo-mechanical problem under the assumption of constant diffusion potential. The use of numerical methods permits solutions to a fully coupled electrochemical–mechanical problem with external applied loads.

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material changes in volume to accommodate Li intercalation, generating stress concentration and promoting the formation of cracks [2]. The range of applications that can be approached goes well beyond the one proposed here and can be further extended in a straightforward manner. The development of this model has been motivated by the need for a rigorous and robust tool to solve a variety of problems in the field of energy storage systems. This article is organized as follows: Sect. 2 introduces the general theory and its finite element formulation; Sect. 3 presents the constitutive laws for the transport of ionic species in an elastic material and the modeling of fracture via an extended cohesive zone model; Sect. 4 illustrates applications to the solution of ambipolar and kinetic demixing problems, to a case of mechanically driven diffusion and fracture is battery microstructures; the numerical implementation is detailed in Sect. 6. A glossary of symbols can be found in Nomenclature section.

2

Thermodynamic Framework

We will develop a system of differential equations by considering the variations undergone by a continuous body, having reference configuration B  ℝ3 . The motion of the body is described by a time-dependent deformation mapping χ : B  ½a,b ! ℝ3 , where [a, b] is the time interval elapsed during the motion. The variations will include changes in the deformation, concentration of the various chemical species, and the charge of the chemical species. We call this an “electro-chemo-mechanical” variation. Let ci denote the total number of moles of the chemical species i per unit reference volume V0. Changes in ci in a part P are brought about by the diffusion of the species across its boundary @P. The diffusion is characterized by a flux Ji(X, t), i.e., the number of moles of the species entering P across @P, measured per unit area per unit time.3 To simplify the form of the equations that follow, all system variables will be measured with respect to an initial reference state with volume V0 and not in the electro-chemo-mechanical varied state (the Eulerian state). The results are independent of this choice of frame. Let P be an arbitrary part of the reference body B with N the outward unit normal on the boundary @P of P. In the finite element formulation that follows, the part P will correspond to the finite element. Let ci denote the total number of moles of the chemical species i per unit reference volume V0. Changes in ci in a part P are brought about by diffusive processes. Each diffusive process is characterized by the integral

Notation: ∇ and Div denote the gradient and divergence with respect to the material point in the reference configuration; a superposed dot denotes the material time-derivative. Vectors, second-, and fourth-order tensors are marked with bold and blackboard bold symbols, to distinguish them from scalar values. Throughout, we assume Fe  1 = (Fe)1 and Fe  T = ((Fe)1)T.

3

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of fluxes Ji(X, t)  N (i.e., Ji(X, t)  N is the number of moles of each species i entering P normal to @P, measured per unit area per unit time). We shall write the local form of the mass balance for the species i 4 c_i ¼ Div Ji

(1)

and analogously, the balance of charge q_ i ¼ Div Jqi

(2)

where qi the charge of species i per unit volume and Jqi the flux of charges across @P per unit area per unit time. We will also assume that the body is in local equilibrium so that any gradients of intensive variables are negligible within P. Z Z Z Z Z d _ þ _ U dV ¼  JQ  NdA þ QdV þ B  χdV PN  χdA dt P @P P P @P XZ XZ þ i ϖi Ji  NdA  ϕJqi  NdA (3) i @P

@P

where U is the internal energy per unit of P ’s undeformed volume; Q is the distributed heat source per unit of volume; JQ is the outward heat flux; B is the body force per unit volume5; P is the first Piola–Kirchhoff stress

The rate of change of species i in P is given by

4

d dt

Z

Z P

ci dV ¼ 

@P

Ji  NdA

Bringing the time derivative inside the integral and applying the divergence theorem, Z P

ðc_i þ Div Ji Þ dV ¼ 0

which leads to the local form of the mass balance for the species i. In the proposed application B will be taken as zero. No gravitational effects are considered, and electrostatic forces are not treated as body forces. A different approach to electromechanical coupling takes into account electric forces into the linear momentum balance [15]. Those forces are included in the form of body forces or by defining a total stress, as the sum of the classical mechanical stress and the Maxwell stress. Several contributions using this approach can be found in recent literature in applications to elastomeric and polymeric materials wherein electric field produces very large deformations. The key point is that the mechanical properties of these materials can be changed rapidly and reversibly by externally applied electric fields. Our treatment of electrostatic forces is limited to their impact on the transport of ionic species. We neglect their effect on the mechanical properties of the material and, therefore, assume negligible the contribution of the Maxwell stress in the equilibrium equation.

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tensor6; ϖi is the diffusion potential of the ith chemical species; and ϕ is the electropotential. The last two terms in Eq. (3) are the chemical and the electrostatic work performed on P. The fourth term is the mechanical work carried out by the traction T = PN on the surface @P, and the third term is the work performed by any external field via body forces which are conjugate to the deformation.7 All of the above are the rate of total work transferred to P. The total heat is composed of the first two terms in Eq. (3). The second law of thermodynamics for irreversible systems is stated in the form known as Clausius–Duhem inequality d dt

Z

Z P

sdV 

Z Q_ dV þ PT

@P

JQ  N dA  0 T

(5)

with s being the entropy per unit of undeformed volume and T the absolute temperature. In general, the appropriate free energy density for P depends on which of each conjugate state variable pair are allowed to vary in the electro-chemomechanical variation. The free energy density is constructed by performing Legendre transform of the internal energy density U with those conjugate pairs for which the fixed parameter is the intensive state variable. Cahn and Larché used the fixed stress generalization of the Gibbs free energy [3, 36], which, in our notation, reads Γ ðP, ci ,T Þ ¼ U ðF, ci ,T Þ  sT  P : F

(6)

with the deformation gradient F being defined as F  ∇ χ. Other appropriate work conjugate variables can be used in place of stress P and the deformation F. For a displacement-based finite element model of isothermal systems, the free energy density must allow adjacent elements to exchange mechanical work at a constant temperature. Furthermore, to study the transport of ionic species, we

6 The Piola–Kirchhoff stress tensor P is related to the Cauchy stress tensor σ via the Piola transformation

P ¼ J σFT

(4)

with J = det (F) being the determinant of the deformation gradient. The Cauchy stress is defined in the current (or deformed) configuration, while the Piola–Kirchhoff stress is defined in the reference one. We make use of the Piola–Kirchhoff stress in compatibility with the other quantities expressed in the reference configuration. In classical linear mechanics the distinction between reference and Eulerian configuration is not taken into account, and the Cauchy stress is the most commonly used stress quantity. 7 If there were fields that could perform work through variations in any other quantity (such as dipoles, magnetic moments, total charge) within P, then these should be included in Eq. (3). We ignore such fields in this formulation. If the fields were included, they would appear in the coupled differential equations that are derived below in a manner analogous to B.

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construct a free energy density which takes the electropotential as independent variable. It is derived by a Legendre transform of the internal energy as follows: " Υ ðχ, ci ,ϕ,T Þ ¼ inf U ðχ, ci , qi ,sÞ  sT 

X

qi

# ϕqi

(7)

i

The free energy density Υ can be regarded as analogous to the “grand potential density” (Eq. (41) [37]), in the way Leo and Sekerka treat the chemical potential, rather than concentration, as independent variable. From the charge balance Eq. (2) and applying the divergence theorem, we can write @ @t

Z

X P

ϕ qi dV ¼

i

XZ i

P

ϕ_ qi dV 

XZ @P

i

ϕJqi  N dA

(8)

Multiplying the entropy imbalance Eq. (5) by T and subtracting the result from the energy balance Eq. (3), and taking into account Eq. (8) yields the free energy imbalance of Υ d dt R

Z P

Υ dV

P B  χ_ dV þ

R

@P PN  χ_ dA 

XZ @P

i

ϖi Ji  NdA 

XZ P

i

ϕ_ qi dV

(9)

The right hand side of Eq. (9) represents the power expended on P by the material surrounding P. Bringing the time derivative inside the integral and applying the divergence theorem for P R P

Υ_  Β  χ_  Div P  χ_  P : ▽_ χþ

X

ϖi Div Ji þ

i

X i

XZ P

i

Ji  ▽ϖi þ !

(10)

ϕ_ qi dV 0

Assuming force balance Div P þ B ¼ 0

(11)

and inserting mass balance Eq. (1) into Eq. (10) Υ_  P : F_ 

X X X ϖi c_i þ Ji  ▽ϖi  ϕ_ qi 0 i

i

(12)

i

The rate of dissipation density is the difference between the external power expenditure and the rate of change of free energy density of the element P. From Eq. (12):

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D ¼ P : F_ þ

X

ϖi c_i 

X

i

Ji  ▽ϖi þ

i

3

Constitutive Equations

3.1

Elastic Lattice Material

X

ϕ_ qi  Υ_  0

(13)

i

We consider a multiplicative decomposition of the deformation gradient. An intercalating species will be treated as a substitutional atom according to Larché and Cahn model [38], and it will be assumed that the intercalating species has a fixed valence. Treating electrochemical intercalation in this way, the fraction of intercalation sites that are occupied by vacancies will vary from 0 to cmax. The deformation that arises from Vegard’s law (i.e., the strain associated with the concentration of intercalating atoms) is the anelastic deformations Fa, while the recoverable lattice distortion is the elastic deformation Fe. We assume a multiplicative decomposition of the total deformation gradient F ¼ Fe Fa

(14)

according to elastic–anelastic kinematics. The anelastic deformation gradient Fa ¼ Fa ðci Þ

(15)

is defined as function of the local concentration of the intercalating species.8 In the case of isotropic Vegard’s law volumetric expansion (similarly with isotropic thermal expansion) Eq. (15) is an extension of Eq. (4.1) in [3] to nonlinear kinematics. It is also related to the partial molar volume of the intercalating species with respect to the host lattice. We restrict the examples herein to the case of an isotropic volumetric expansion of the form9

It follows that the elastic deformation is also a function of the species concentration Fe(χ,ci) = F(χ) Fa  1(ci). 9 The coefficients βi correspond to the ratio between the lattice parameter of the intercalating species i and the one of the hosting compound. The relative volume change upon intercalation can be computed as the determinant of the anelastic deformation tensor. In linear kinematics, it can be approximated as follows 8

V þ dV ¼ detðFa Þ ¼ V

1 X 1þ β ci ρh i i

!3

1 X !1þ3

β ci ρh i i

! (16)

such approximation is valid in the limit βi ! 0. The factor 3βi/ρh corresponds to the partial molar volume of the intercalating species with respect to the hosting compound.

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F ð ci Þ ¼ a

911

! 1 X 1þ β ci I ρh i i

(17)

where the molar volume of the hosting compound is ρh (in number of moles per unit of undeformed volume). Let us consider the following generic forms for the free energy density, the first Piola–Kirchhoff stress tensor, the diffusion potential and the electropotential Υ ¼ Υ ðFe , ci ,ϕÞ

(18)

P ¼ PðFe , ci ,ϕÞ

(19)

ϖi ¼ ϖi ðFe , ci ,ϕÞ

(20)

ϕ ¼ ϕðFe , ci ,ϕÞ

(21)

Substituting the constitutive equations above into the free energy imbalance, Eq. (12) leads to the following 0

  @Υ ðFe , ci ,ϕÞ _ e X @Υ ðFe , ci ,ϕÞ @Υ ðF, ci ,ϕÞ _ _ : F þ c ϕþ þ i @Fe @c @ϕ i i X X X  e a  P : F_ Fa þ Fe F_  ϖi c_i þ J i  ▽ϖi  ϕ_ qi i

i

(22)

i

In order for inequality Eq. (22) to hold for arbitrary combinations of the independent variables (Fe, ci, ϕ), the coefficients of Eq. (22) must vanish. We are therefore led to the thermodynamic restriction that relates the free energy density Υ to the stress P the diffusion potential ϖi10 and the electropotential ϕ P¼

@Υ ðFe , ci ,ϕÞ aT F @Fe

(24)

10

Given the following identities

@ @ @ci @ 1 ¼ ¼ (23) @ni @ci @ni @ci V 0 the first term in the state equation Eq. (25) of the diffusion potential i (in the case of transport of species uncoupled from mechanical and electrostatic effects) is compatible with the classical definition of the chemical potential as derivative of the internal energy (integrated over the reference volume) with respect to the number of moles ni of species i; where entropy, volume and number of moles of species k 6¼ i are held constant. The second term in Eq. (25) corresponds to the stressdependent component of the diffusion potential introduced by Larché and Cahn in [38]. If we used the partial molar volume, instead of the relative lattice parameter βi, in the definition of the anelastic deformation, a 13 factor would appear in the expression.

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G. Bucci and W. C. Carter

ϖi ¼

@Υ ðFe , ci ,ϕÞ βi  eT   Tr F P @ci ρh

(25)

@Υ ðFe , ci ,ϕÞ @ϕ

(26)

qi ¼

In the literature, this procedure is sometimes referred to as the Coleman–Noll procedure [39]. For the derivatives with respect to ci, only those ci which are independently variable are included. For example, a network constraint (conservation of a particular lattice position) produces an additional equation of the form: ci þ cj ¼ constantij

(27)

In this work, the constraints will be inserted into Υ explicitly; for example: Υ (P, c1, c2, ..., cj–1, constantij – ci, cj + 1, ..., qi). The dissipation in Eq. (13) reduces to D¼

X

J i  ▽ϖi  0

(28)

i

Equation 28 consists of the energy density dissipated in transport of chemical species. In Eq. (28), extra terms will arise in presence of mechanical dissipation, for example, in plastic or viscous materials. The inequality in Eq. (28) demands that the projection of the fluxes onto the gradients of their conjugate fields is negative. We now seek to find a description of the rate problem, i.e., finding the rate of electro-chemo-mechanical variation of P given its boundary conditions. To develop the finite element formulation for an electro-chemo-mechanical system, we will choose a less general system, with a free energy density that can be decomposed as follows:   Υ Fe , ci, ϕ ¼ Υ M ðFe , ci Þ þ Υ EC ðci ,ϕÞ

(29)

The function Υ M determines the elastic response of the material, Υ EC represents the electrochemical free energy density. In the presence of only anelastic deformation (Fe = I; F = Fa), we assume that the mechanical free energy vanishes Υ M ðFe ¼ I, ci Þ ¼ 0

(30)

Assuming that there is no residual stress, any choice of mechanical free energy density requires that a global minimum appears when there is no elastic deformation (Fe = I and F = Fa), in other words, the mechanical energy minimizing state is characterized by stress-free kinematics.

29

Micro-mechanics in Electrochemical Systems

913

Any concentration gradient that develops will produce incompatible inelastic deformations which generate stress. Objectivity (frame indifference) of P ’s energy imposes a restriction on the dependence of the elastic energy density on the elastic strain tensor Υ M ðFe , ci Þ  Υ M ðC e , ci Þ

(31)

with Ce = FeTFe being the elastic right Cauchy–Green strain tensor [40]. In other words, rigid body motion does not change the energy density of the element P. Given the form of the free energy density Eq. (29), the ith diffusion potential of Eq. (25) is decomposed into the sum of mechanical and electrochemical terms ϖ i ð C e , ci Þ

¼

@Υ M ðC e , ci Þ @Υ EC ðci ,ϕÞ βi  eT  þ  Tr F P @ci @ci ρh e EC ¼ μM i ðC , ci Þ þ μi ðci ,ϕÞ

(32)

with the following definitions of mechanical and electrochemical components of the diffusion potential e μM i ðC , ci Þ ¼

@Υ M ðC e , ci Þ βi  eT   Tr F P @ci ρh

(33)

@Υ EC ðci ,ϕÞ @ci

(34)

μEC i ðci ,ϕÞ ¼

3.2

State Functions for Transport of Neutral Species Within an Elastic Material Having a Network Constraint

Let us consider first the treatment of transport of an uncharged chemical species within an imposed network constraint; this is, for instance, the case of the diffusion by vacancy exchange mechanism. In the case of nonideal solution conditions and constant temperature, the electrochemical free energy per unit of reference volume for the system under study takes the form [41] Υ EC ðci , cV i Þ ¼ ci RT lnci þ cV i RT lncV i þ Υ EC II ðci , cV i Þ

(35)

where ci represents the number of moles per unit of reference volume of the diffusing species, and cV i is the molar density of the corresponding vacancies; Υ EC II is the excess of free energy caused by the interactions among intercalated species. The network constraint results in the following condition c_i þ c_V i ¼ 0

(36)

914

G. Bucci and W. C. Carter

It follows that ci þ cV i ¼ cimax ¼ constant

(37)

i.e., the total number of sites cimax is conserved. Furthermore, we can express the molar density of vacancies as cV i  cV i ðci Þ. The electrochemical potential μEC, derived from Eq. (35) according to Eq. (25), takes the form (as in Eq. (5.16) in [3]) μEC i ð ci Þ ¼

@Υ EC @Υ EC @cV i þ @ci @cV i @ci

Υ EC Υ EC ¼ RT ln ci  RT ln cV i þ II  II @ci @cV i ci γ ¼ RT ln þ RT ln i cimax  ci γV i

(38)

In Eq. (38), we assume a definition of the excess of free energy such that Υ EC II ¼ RT ln γ i @ci

(39)

Υ EC II ¼ RT ln γ V i @cV i

(40)

The activity coefficient γ i ðci Þ ¼ γγ i represents the deviation from the infinite dilute Vi

solution condition. In nonisothermal systems, γ i will also be dependent on the temperature. We shall consider a Neo–Hookian energy functional as a suitable form for the elastic free energy density [40] 1 1 Υ M ðI 1 ðC e Þ, J ðC e Þ, ci Þ ¼ λ ðlog J Þ2  G log J þ GðI 1  3Þ (41) 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In Eq. (41), we make use of the definitions of the Jacobian11 J ¼ detðC e Þ, and of the first invariant of the elastic right Cauchy–Green strain tensor I1 = Tr(Ce). The first Lamé constant λ and the shear modulus G in Eq. (41) can be defined as concentration-dependent elastic properties of the material according to the following constitutive law λ ð ci Þ ¼ λ 0 þ λ 1 ð ci  ci 0 Þ Gðci Þ ¼ G0 þ G1 ðci  ci0 Þ

(42)

Not to be confused with J, a vector indicating the flux. A prescribed flux on the boundary will be  represented with the scalar valued quantity J.

11

29

Micro-mechanics in Electrochemical Systems

915

where ci0 is the reference concentration at which λ0 and G0 are defined. The first Piola–Kirchhoff stress can be derived from Eq. (41) according to Eq. (24) P¼

 @Υ M aT  F ¼ ðλ log J  GÞFeT þ GFe FaT @Fe

(43)

Analogously, the elasticity tensor is computed as follows12 ℂ¼

@ 2 Υ M a1 @ 2 Υ M aT F F ¼ Fa1 ℂe FaT @F@F @Fe @Fe

ℂe ¼ λFeT  λFeT  ðλ log J  GÞFe1 λFe1 þ G

(44) (45)

By making use of Eq. (38), in absence of electrostatic effects, the diffusion potential in Eq. (32) takes the form   @Υ M ðC e , ci Þ @Υ EC ðci Þ þ  βi Tr FeT P @ci @ci 1 1 ¼ λ1 ðlog J Þ2  G1 log J þ G1 ðI 1  3Þþ 2 2  ci β  þ RT ln þ RT ln γ i  i Tr FeT P cimax  ci ρh

ϖðC e , ci Þ ¼

(46)

This is a generalization of Eq. (5.16) of [3] to nonlinear kinematics. If the elastic properties vary with composition according to Eq. (42), the elasticity tensor ℂe in Eq. (45) will be composition dependent, analogously to the open system compliance defined in Reference [3]. Because the elasticity tensor ℂ in Eq. (44) contains the anelastic deformation Fa, it remains function of ci, even in the case of constant elastic properties. The derivative of the mechanical free energy density with respect to composition @Υ M ðC e , ci Þ 1 1 ¼ λ1 ðlog J Þ2  G1 log J þ G1 ðI 1  3Þ @c 2 2

12

Operators used in Eq. (45) have the following definition

ðA

J

AÞijkl

N ðA AÞijkl ¼ Aij Akl  1 ¼ Aik Ajl þ Ail Ajk 2

and the fourth-order identity tensor  is ðÞijkl ¼ δik δjl

A ¼ A :

916

G. Bucci and W. C. Carter

is null in the case elastic properties independent from concentration, namely if λ1 = G1 = 0.

3.3

State Functions for Transport of Multiple Ionic Species Subject to Mass and Charge Balance

In ionic systems, it is necessary to account for the effects of gradients in the electrostatic potential on the flux of charged species. Consequently, the total driving force for mass transport includes the electrochemical potential and the stressdependent contribution associated with changes in local molar volume. The electrochemical free energy density for this case will include an additional term respect to Eq. (35) to account for migration effect Υ EC ðci , cV i Þ ¼ ci RT ln ci þ cV i RT ln cV i þ Υ EC II ðci , cV i Þ þ zi Fci ϕ

(47)

F = eNa is the Faraday constant.13 The ion valence is zi. The diffusion potential for the charged species i, in presence of a network constraint on the number of i-sites available, is derived from Eq. (25) and can be expressed as   @Υ M ðC e , ci Þ @Υ EC ðci ,ϕÞ þ  βi Tr FeT P ¼ @ci  @ci  @Υ M ðC e , ci Þ ci β  þ RT ln ¼ þ RT ln γ i þ zi Fϕ  i Tr FeT P @ci cimax  ci ρh ϖ i ð C e , ci Þ ¼

(48)

with the dependence of the elastic properties on concentration given by the term @Υ M ðC e , ci Þ 1 1 ¼ λ1 ðlog J Þ2  G1 log J þ G1 ðI 1  3Þ @ci 2 2 In a multispecies diffusion process, the ionic fluxes are computed in the deformed configuration according to FJi ¼ 

Mi ci ▽ ðϖi Þ RT

(49)

or equivalently in reference configuration14 13

Not to be confused with F, a tensor values variable, representing the deformation gradient. Notation: the symbol ∇ denotes the gradient with respect to the deformed configuration. The following transformation between reference and spatial configuration (coordinate system marked respectively with X and x) applies

14

• ∇f = FT ∇ f, f being a generic scalar function; • u = FU, u being a vector defined in the spatial configuration and U the corresponding vector in the material configuration.

29

Micro-mechanics in Electrochemical Systems

Ji ¼ 

Mi ci C 1 ▽ðϖi Þ RT

917

(50)

assuming that the fluxes are independent; that is, there are no cross-correlation terms in the Onsager coefficients (i.e., the flux on one sub-lattice does not affect fluxes on other sub-lattices). Substituting Eq. (48) into Eq. (50), we obtain an extended form of the Nernst–Planck equation, which includes mechanical effects  J i ¼ M i

 cimax F 1 C ▽ϕ C 1 ▽ci þ zi ci RT cimax  ci

  β ci  i C 1 ▽ FeT P : I RT

(51)

Here, the treatment of electrostatic forces is limited to their impact on the transport of ionic species. We disregard, for instance, their effect on the mechanical properties of the material. In the literature [42, 43], the condition15 X

zi J i ¼ 0

(52)

i

has been used to impose local balance of charges and derive the ambipolar diffusivity, an effective diffusivity that characterizes the transport of ionic species. Our numerical approach is to permit a transient electrostatic field to arise such that Eq. (52) is satisfied at numerical convergence (i.e., it is a weak formulation of the electroneutrality condition). A (numerical or fictive) transient electric field is produced from charge separation and forces short-range fluxes to reestablish electroneutrality. However, local polarization is not treated here, and treatment of piezoelectric effects (local charge polarization related to the occurrence of electric dipole moments) and electrical bias is left to future publications. The internal electric field E with respect to the reference configuration can be computed by inserting the expression for the ionic fluxes Eq. (50) into the charge balance equation Eq. (52)16,17

15 For a discussion on the validity of the electroneutrality condition Eq. (52), we refer the reader to Sect. 11.8 of [43]. 16 The chemo-mechanical component of the diffusion potential appearing in Eq. (53) corresponds to the following definition

e μCM i ðC , c i Þ ¼

    @Υ M ðC e , ci Þ ci β þ RT ln γ i  i Tr FeT P þ RT ln @ci cimax  ci ρh

In Eq. (53), we made use of the relation E = FTe between the electrostatic field e(x,t) = ∇ϕ defined in the deformed configuration, the corresponding field E(X,t) = ∇ϕ in the Lagrangian configuration (see [15] for the complete derivation).

17

918

G. Bucci and W. C. Carter

P RT EðX ,t Þ ¼ ▽ϕ ¼ C F

i zi

Mi ci C 1 ▽μCM i RT P 2 i z i M i ci

(53)

As part of the numerical procedure, the local electric field is computed according to Eq. (53). The gradient of the electrostatic potential is then inserted into the expression of the flux Eq. (51) for each of the charged species. This numerical procedure is repeated at each assembly of the stiffness matrix Kcc (see Eq. (116)) at each iteration in the solution of the nonlinear coupled problem. At convergence, because local charge balance is reached, the local electric field converges to zero.

3.4

Cohesive Zone Constitutive Theory for Fracture

The cohesive zone model of fracture is a method, alternative to the continuum damage approach, that represents a crack as a sharp discontinuity of codimension one. The theory, due to Barenblatt [44, 45], postulates a continuum description of the process of separation of the material adjacent to the tip of an advancing crack (see Fig. 1a). Gradual separation of this material is resisted by tractions according to a traction–separation law (TSL). The TSL is based on the hypothesis that cohesive traction forces depend only on the crack profile in the process zone: the restoring forces depend only on the separation of nodes that are colocated at zero stress. The tractions are also independent of the properties of the surrounding bulk material. Generally, the tractions decay with increasing separation, until they vanish at a critical opening displacement, resulting in the creation of new free surface (see Fig. 2). The TSL is a phenomenological representation of the underlying fracture mechanism and requires empirical fitting of parameters through experiments. The energy per unit area expended to advance the crack is a well-defined constitutive property represented the integral of the TSL. The critical opening displacement naturally introduces an intrinsic length scale and embeds a size effect into the fracture problem [46]. Cohesive fracture simulation within an initially continuous body was pioneered by Ortiz and Suresh [47] using interface elements. In this technique, the interelement boundaries represent the possible crack paths, and fracture is represented by displacement jumps across these interfaces (see Fig. 1b). The cohesive tractions are applied to the adjacent element faces to resist the jumps. The topology of the mesh is affected by the presence of a cohesive zone, because the nodes of each element sharing the interface are cloned at their zero-stress-state positions in order to allow displacement jumps and tractions at finite stress. Nucleating fracture on element boundaries without resort to cohesive zone element would require changes in the mesh topology and recomputation of the shape functions. To avoid this complication and computational expense, cohesive zone models of the so-called intrinsic [48] type have been proposed (i.e., the interface elements are inserted prior to the start of the

29

Micro-mechanics in Electrochemical Systems

919

Traction

b

4

a

physical crack

cohesive zone

3-

1

dis

pla

ce

2-

me

nt

3+ 6

2+ 5

Traction Fig. 1 (a) Cohesive zone model treats fracture as a sharp discontinuity in the continuum. Cohesive forces resist the gradual separation of the material ahead of the crack tip with a force law indicated by Fig. 2. (b) In its finite element implementation, the cohesive zone model simulate crack nucleation at the interelement boundaries

simulation). The TSL of intrinsic cohesive zones contains a linear elastic regime prior to the softening regime that describes the separation process (see Fig. 2). The elastic regime is included to approximately maintain compatibility prior to fracture criterion. This sort of penalty approach is subject to a few drawbacks; in particular, it leads to wave propagation issues in dynamic problems. With the cohesive element approach, the fracture path is constrained by being restricted to element interfaces. The path constraint increases the required work of fracture. If the fracture pattern is known a priori, as is the case of interfaces weaker with respect to the bulk material, the mesh can be designed so to include those patterns and remove bias effects. Furthermore, a physical interpretation may be attributed to the elastic regime of the TSL, as describing a property of the interface. For the examples presented here, we choose an intrinsic history-dependent constitutive equation. Following [10, 49], we assume the bilinear TSL represented in Fig. 2. The flux across the interface is irreversibly set to zero at the onset of damage (i.e., once the opening displacement reaches the value corresponding to the elastic limit; see Fig. 2). One difference between our work and that of Bower et al. [10] is we do not allow Li flux to be reestablished when crack flanks are brought back into contact during unloading: that is, an infinite diffusive barrier remains at crack closure. Real systems are likely to have finite diffusive barrier.

4

Example Applications

4.1

Ambipolar Diffusion

Ambipolar diffusion is an example of coupled transport of charged species. The rate of transport of both species is characterized by an apparent diffusion coefficient (i.e., the ambipolar diffusion coefficient), whose value is bounded by the diffusivities of the individual species involved. Charge neutrality is maintained through an internal

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G. Bucci and W. C. Carter

Traction (GPa)

limit traction

0.4 0.3

normal displacement

δN

critical opening displacement

0.2 0.1 1

2

3

4

5

Displacement (nm)

Damage 1.0

tangential displacement

δT

0.8 0.6 0.4 0.2 1

2

3

4

5

Displacement (nm)

δ=

damage onset

2 + (bδ )2 δN T

Fig. 2 Bilinear TSL adopted in the finite element implementation of an intrinsic cohesive zone model. A linear elastic regime is included to maintain compatibility prior to the softening regime that describes the separation process. Irreversibility is produced by implementing lower elastic stiffness upon reloading. Dissipation of fracture energy starts at the onset of damage at the limit traction displacement. Irreversibility, fracture energy dissipation, and damage are correlated. The values for displacements and traction forces indicated here are chosen as an example for the case of fracture energy of 1 J/m2. The opening displacement is a scalar valued composition of the normal and tangential component of the displacement at the interface. The contribution from the tangential component is scaled by a factor b < 1

electric field, which fixes the separation of cations and anions even if their mobilities and driving forces differ. This phenomenon is observed in the case of oxidation or reduction of an oxide or a metal, formation of an oxide film in metal corrosion, mass transport under stress gradients during sintering, diffusional creep, etc. The simultaneous preservation of electrical neutrality and stoichiometry comes from the coupled fluxes of anions and cations from the same source to the same sink. Anion- and cation-rich regions may form, instead, in the case of solid demixing as presented in the following section. When the stress effect can be considered negligible and in the limit of a dilute solid solution, the internal electric field of Eq. (53) (written in the spatial configuration) takes the form e¼

RT F

P i zi Di ▽ ci P 2 i z i D i ci

typically used in literature (see, for instance, [42]).

(54)

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Micro-mechanics in Electrochemical Systems

921

By substituting Eq. (54) into the Nernst–Planck Eq. (51), it is possible to compute an ambipolar diffusivity that characterizes combined anion and cation transport. For instance, in the presence of two diffusing species, a cation A and anion B of valence +1 and 1, respectively, the ambipolar diffusivity is [42]: DAþ DB D~ ¼ DAþ þ DB

(55)

The extension of the result in Eq. (55) to a general case (electro-chemo-mechanical diffusion potential and arbitrary number of species, defects, and lattice constraints) is presented in Sect. 4.1.1. The numerical method permits material parameters to be varied and explore the relative weight of the different components of the diffusion potential. To illustrate the numerical method and simplify the discussion, one-dimensional models are presented for a series of physical cases. Three-dimensional models with microstructures will be presented elsewhere.

4.1.1

Examples of Ambipolar Diffusion: Analysis of the Electrostatic Effect on Transport Consider an elastic thin film deposited on a substrate with two species A and B with differing mobility. The film is assumed to be rigidly bounded to the substrate, so that only out-of-plane deformations are allowed, and a plane stress condition is established. Electrostatic effects can be quantified by comparison of the transport of two neutral species to the transport of two charged species. Suppose the mobility of B is two orders of magnitude smaller than that of A. Diffusion is assumed to occur by a vacancy mechanism; therefore, there is a network constraint. We will suppose that A and B occupy different lattice positions; so their network constraints are independent. This means that A only moves on the α-lattice, B only moves on the β-lattice. Flux of A-vacancies on the α-lattice is always opposite to the flux of A. Because A and its vacancy have opposite effective charges, each sub-lattice maintains electroneutrality. Time-independent boundary fluxes JA and JB of equal magnitude are imposed at the interface away from the substrate (top surface in Fig. 3), and JA = JB = 0 at the substrate interface. Boundary and initial conditions for this set of problems are illustrated in Fig. 3, and a list of input parameters is collected in Table 1. All results below are produced by a one-dimensional finite element model. Spatial, temporal, and concentration variables are nondimensionalized (Time: τ  tDA/L2, L being the film thickness and Depth: ξ  X/L, and Concentration: ζ i  ci =cimax where cimax is the limiting value of the ith species solubility). Details of the finite element formulation and its numerical solutions are presented in Sect. 6 and in the online supporting material. The problem is discretized in space (with 100 finite elements within the film thickness) and in time. To find the solution of the nonlinear coupled problem at

922

G. Bucci and W. C. Carter

Boundary conditions

Initial conditions

JA+ = JB − = 0

Depth cA+ = cB − = 0.1

JA+ = JB − = 0 0.1

Concentration

Fig. 3 Schematic representation of the set up for the numerical tests presented in Sect. 4.1.1

Table 1 Material parameters for problems in Sect. 4.1.1 Input value F = 96485.3365 C mol1 R = 8.314 J K1 mol1 T = 298 K MA = 1015 m2 s1 MB = 1017 m2 s1 γA ¼ 1

Description Faraday’s constant Gas constant Temperature Mobility species A Mobility species B Activity coefficient species A

γB ¼ 1

Activity coefficient species B

cAmax ¼ 1

Total number of α-lattice sites

cBmax ¼ 1

Total number of β-lattice sites

v = 0.22 βA = βB = 0.0 βA = 0.1; βB = 0.0 E = 100 GPa

Poisson’s ratio Relative lattice constants for the baseline case Relative lattice constants rate for the Ab case Young’s modulus

each time step, a Newton–Raphson procedure is employed. The electrochemical–mechanical problem to be solved at each iteration of the Newton–Raphson algorithm is given by Eq. (127) of Sect. 6. Constitutive laws for this problem can be found in Sect. 3 and listed again in Sect. 6. The panels in Fig. 4 illustrate the difference between the concentration evolution for charged and neutral species and serve as examples to illustrate the effect of the electrostatic potential in addition to the chemical potentials in coupling the transport of two charged species. Figure 4a shows that high-mobility neutral species A has a very small concentration polarization: ζA(ξ, τ) (the normalized concentration field for species A) is approximately uniform through the film thickness; ξ evolves linearly with time τ as dictated by the intensity of the external flux. Because the diffusion length is large compared to the film thickness, the steady-state condition appears rapidly. In Fig. 4a (left image), the small nonuniform concentration gradient is not resolved.

29

Micro-mechanics in Electrochemical Systems

923

a

Diffusion of neutral species

b

Diffusion of charged species Fig. 4 Comparison of ambipolar diffusion of neutral and charged particles (panels A and B, respectively). The mobility of species A is 100 times that of B; this emphasizes the effect of the electrostatic coupling. The enforcement of electroneutrality is expressed as a local constraint on the fluxes of cations and anions. The rightmost picture in (b) (orange line) shows how the presence of stress slightly changes the concentration profile of A+ and B. However, the electrostatic driving force remains dominant and it couples the transport of the two ions

Conversely, the contour plot (middle figure in Fig. 4a) for the neutral species B reveals concentration polarization, i.e., the presence of a diffusion front that slowly moves across the sample, as soon as the region above it has reached saturation. The low-mobility species B concentration polarization is illustrated by plotting the concentration evolution at the top and bottom interfaces (right portion of Fig. 4a). An identical numerical experiment (with two substitutional charged species A+B) is shown in Fig. 4b. We will assume that vacancies are created as a Schottky defect; in the Kröger–Vink notation [50] is written as null ! V0A þ VB•

924

G. Bucci and W. C. Carter

where V0A and VB• are the respective charged vacancies on the cation (A) and anion (B) sites. The network constraint on the total number of lattice sites available for ions and vacancies is expressed. cAA þ cV0A ¼ cAmax ¼ constant

(56)

cBB þ cVB• ¼ cBmax ¼ constant

(57)

and the corresponding electrochemical potentials take the form μEC A ðcA Þ ¼ RT ln

cA þ RT ln γ A cAmax  cA

(58)

μEC B ðcB Þ ¼ RT ln

cB þ RT ln γ B cBmax  cB

(59)

In this example, we will consider the solution to be ideal (i.e., γ A ¼ γ B ¼ 1). The gradient of the electrostatic potential is computed according to Eq. (53), then inserted into the expression of the flux Eq. (51) for each of the charged species. This procedure is repeated during the assembly of the stiffness matrix Kcc (see Eq. (116)) at each iteration in the solution of the nonlinear coupled problem of Eq. (127) of Sect. 6. The coupling in the transport of the two ionic species is illustrated in the contour plot of Fig. 4b. The effective diffusivity is bounded by DAþ and DB , but is approximately equal to the lowest diffusivity DB in this case. Furthermore, if Vegard strains are introduced along with the charged species, the effect of stress on ambipolar diffusion can be simulated. The orange lines that are superimposed in the rightmost graph of Fig. 4b result from a simulation that includes the stress-dependent term of the diffusion potential in Eq. (48) by associating a 10% isotropic expansion to the intercalation of species A+ (Ab case). The Vegard strain is described by the anelastic deformation of Eq. (15), with βA being a constant value set to 0.1, while βB is zero (Table 1). Young’s modulus and the Poisson’s ratio are taken to have the constant values of E = 50 GPa, v = 0.22. The film develops a compressive stress as high as of 1 GPa, but produces a negligible deviation of the concentration profile of both A+ and B. The electroneutrality condition imposes a local constraint on the fluxes and therefore on the relation among the rates of change of species concentration X i

zi J i ¼ 0 )

X i

zi ▽  J i ¼ 0 )

X @ci ¼0 zi @t i

(60)

Note that izi@ci/@t = 0 implies that izici = constant. This constant is typically zero because the initial stoichiometry is charge neutral; therefore, izici = 0. This is different from the steady state condition, which would read Ji = 0 for all i.

29

4.2

Micro-mechanics in Electrochemical Systems

925

Kinetic Demixing

In compounds containing multiple cations and anions, differences in the transport rates can lead to compositional separation during ambipolar diffusion. Kinetic demixing is an example of this phenomenon (distinguished from thermodynamic demixing such as spinodal decomposition) in which deviations from homogeneous compositions develop in ionic compounds [16]. We will treat the case where demixing is caused by electro-chemo-mechanical potential gradients of oxygen such as in SOFCs. Kinetic demixing can couple to thermodynamic demixing (e.g., spinodal decomposition) through gradients of the diffusion potential. In literature, a classification of kinetic demixing (or decomposition) phenomena based on the type of driving force has been proposed [25]. However, in most systems, multiple effects and physics interact, so that this sort of distinction may become arbitrary and not very representative. We prefer to classify multifunctional oxide materials, and the eventual demixing (or decomposition) that they develop, according to the following categories: • oxygen-preferred carriers, M O2 >> M cations • cation-preferred carriers, M O2 0). Consequently, Li is driven toward areas of higher tensile stress ahead of the crack. The sign of the stress-dependent driving force for diffusion depends on the sign of βLi, while its value is a function of the absolute value of βLi as Table 3 Material parameters for problems in Sect. 4.3 Input value F = 96485.3365 C mol1 R = 8.314 J K1 mol1 T = 298 K MLi = 0.601012 m2 s1 γ Li = 1 ζLi = 0.76 v=0 βLi = 0.085 βLi = 0.0164 βLi = 0.0477 E = 75 GPa L = 0.625103 m

Description Faraday’s constant Gas constant Temperature Mobility Activity coefficient Relative initial concentration Poisson’s ratio Relative lattice constant for the baseline case (28% swelling) Relative lattice constant for case with 5% swelling Relative lattice constant for case with 15% swelling Young’s modulus for the baseline case Half of the sample edge length

Fig. 11 (continued)

b

no external Li flux

initial uniform Li concentration

traction 10 MPa

stress

traction 10 MPa

a

0.494

0.592

1e-7 0 -1e-7

0.52 0.5

c

2e-7

3e-7

Displacements Z

0.54

-1.3e-07

3.46e-07

0.56

0.58

Li concentration

934 G. Bucci and W. C. Carter

29

Micro-mechanics in Electrochemical Systems

935

well as on the elastic properties of the material (refer to Eq. (46) for the diffusion potential of neutral species). If open-circuit potential versus concentration data are available, the term RT lnγ would be included in the diffusion potential by following the procedure described in [52]. The simulation conditions are illustrated in Fig. 11a. A constant load of 10 MPa was applied, and the resulting fluxes were computed until the composition reached a steady-state distribution. The configuration in Fig. 11a was discretized with linear quadrilateral elements, and with a nonuniform refinement in the crack tip region. The use of nonconforming elements is handled via the deal.II finite element library [53]. Figure 11a shows a contour plot of Li concentration and out-of-plane displacement field for the baseline case. In the image, the sample’s deformation is scaled by a factor of 50 in order to make it clearly visible. An initial uniform concentration ðζ Li  cLi =cLimax ¼ 0:5Þ is assumed throughout the sample, and the material properties are E = 75 GPa and β = 0.085. The contours represent the steady-state condition. Ahead of the crack tip, the relative concentration increases to ζLi = 0.592, an 18.4% relative enhancement. The time dependence of this enhancement is shown in Fig. 11b. Multiple sets of results are presented in Fig. 11b, c. Each test is characterized by a different value of Young’s modulus or lattice misfit parameter with respect to the baseline example. An additional test was performed by adding a plain strain constraint to the reference case, with the aim of investigating the effect of a simplified two-dimensional treatment of the problem. For the case of less stiff (E = 15 GPa) material, the maximum concentration reached at the crack tip shows a reduced, but significant, 16.8% enhancement. Lower values of volumetric expansion of 5% and 15% produce, respectively, a 3% and 10% Li redistribution. These results show a stronger dependence on the Vegard’s lattice parameter, rather than on the Young’s modulus. ä Fig. 11 Simulation of a traction load applied to a notched sample. From the initial uniform concentration ðζ Li  cLi =cLimax ¼ 0:5Þ, the 3D contour plot in (a) shows the steady state distribution of Li for a material with Young’s modulus E = 75 GPa. Having assumed Poisson’s ratio equal to zero, the out-of-plane displacement field shown in (a) arises from Li nonuniform distribution. Zero external Li flux is applied, and the change in concentration arises only due to the nonuniform stress field, and in particular to the localization of stress ahead of the crack tip. The development of an area of high tensile stress and of high stress gradient produces a driving force for the diffusion of lithium. Graphs in (b) show how Li concentration evolves over time at the crack tip. For the case of E = 15 GPa, the maximum concentration reached at the crack tip is equal to ζLi = 0.584, 16.8% higher than the initial value. In the analysis with E = 75 GPa, the relative concentration grows up to ζLi = 0.592, a 18.4% increment respect to the value away from the crack. If a plain strain boundary condition is imposed, out-of-plane expansion is prevented and Li concentration only raises up to 9%. (b) Compares results from analysis characterized by various volumetric expansion rates, thus showing how the lattice mismatch parameter largely affects the chemo-mechanical coupling. (c) Shows the time evolution of the hydrostatic stress at the crack tip. The development of compressive out-of-plane stress is responsible for the initial hydrostatic stress relaxation. If out-of-plane deformation is allowed, the hydrostatic stress tends to grow back to its initial value

936

G. Bucci and W. C. Carter

The out-of-plane displacement field in Fig. 11a is caused by Li nonuniform distribution, since the Poisson’s ratio is fixed to zero. By constraining the displacement in the Z direction, we recreate plain strain condition, one of the assumptions in [51]. As a consequence of the plain strain condition, a larger out-of-plane compressive stress develops upon Li redistribution around the crack tip. This, in turn, inhibits further diffusion, and a lower value of Li concentration is reached at the crack tip. The plot in Fig. 11b shows how the concentration for this case attains values close to the example with 15% volumetric expansion. The analyses were performed in finite strain, and the approximation of the stress in the area surrounding the crack tip is expected to improve upon mesh refinement. However, it is beyond the scope of the present work to accurately predict the stress at the crack tip. We propose this example mainly to illustrate how mechanical stress can produce a nonnegligible driving force for diffusion. The model is capable to capture Li redistribution under the sole effect of a stress gradient field. In Fig. 11b, we intend to compare the time evolution of the hydrostatic stress at the crack tip for problems with different input parameters or boundary conditions. In particular, we are interested in observing the effect of the Young modulus, Vegard’s lattice parameter, and plane strain boundary conditions. The initial relaxation of the hydrostatic stress observed in Fig. 11c is due to the development of a compressive out-of-plane stress, while the Cauchy stresses in the XY plane stay tensile and only moderately change over time. As expected, the largest out-of-plane stress arises in the constrained plane strain condition. The results for the test with lower Young’s modulus (15 GPa vs. 75 GPa) or lower anelastic deformation (5% and 15% of swelling vs. 28%) are rather similar in value and they follow the trend of the baseline case, showing and initial drop followed by a gradual increase. At steady state, the hydrostatic stress recovers approximately the value computed at the beginning of the simulation, before diffusion comes into play. The system requires a longer time to reach steady state when the material is stiffer. In [51], the authors predict a monotonically decreasing hydrostatic stress, this is due to the imposed plain strain condition. The examples presented in this section are meant to emphasize the role of stress as driving force in cases where intercalation is coupled with anelastic deformation of the hosting lattice. Even in presence of a moderate value of volumetric expansion rate, mechanical stress can be responsible for a driving force large enough to induce a considerable Li redistribution.

4.4

Mechanical Degradation of All-Solid-State Batteries

The last application is a 2D model of an all-solid-state battery electrode. All-solid-state batteries differ from traditional batteries in that the traditional battery’s liquid electrolyte is substituted with a solid electrolyte (SE). All-solid-state rechargeable lithium-ion batteries have attracted much interest [29–31] because the SE is nonflammable and thus safer than the flammable organic liquid electrolytes in traditional batteries. Composite electrodes are made of particles of active material (e.g., LixPO4, graphite, Si),

29

Micro-mechanics in Electrochemical Systems

937

embedded in a matrix consisting of a mixture of a solid electrolyte and electron conducting phase. When a cell is assembled, anode particles are typically in their delithiated state – they are fully charged. During cell discharge, the anode materials tend to increase their stress-free strain as they absorb lithium. The discharging anode is analogous to putting a dry sponge in contact with water. The particles’ chemical expansion is constrained by the solid electrolyte matrix and produces compressive stress. Stiffer matrix material results produce larger compressive stress. Reliability of all-solid batteries may be influenced by mechanical degradation of the SE and its interfaces with the active particles. While the active particles in liquid electrolyte batteries can undergo fracture [54] but interparticle mechanical coupling is absent. We simulate mechanical degradation via fracture within the solid electrolyte material. Such cracks would block Li diffusion and reduce the composite electrode’s effective ionic conductivity. The geometry of the particles distribution and its finite element discretization with linear quadrilateral elements are represented in Fig. 12. The finite element mesh to be representative of a portion of the composite electrode (square highlighted in Fig. 12). The microstructure includes 36 randomly oriented square particles in a region of dimensions 11 μm  11 μm. The average particle size is 1 μm. The volume ratio of active material is about 45% which is typical for commercial Li-ion batteries. We consider shapes with sharp corners (like the squares chosen here) to be a more realistic representation of particles than circles (see, for instance, Fig. 6 in Ref. [55]). Flaws and stresses are more likely to accumulate around sharp corners and therefore corners are likely be associated with damage. The grid is managed with deal.II finite element library, an open-source library widely used in scientific computing [53, 56]. We employ an irreversible intrinsic cohesive zone model to simulate fracture across the solid electrolyte. The traction Li flux

current collector

Lithium solid electrolyte electrode + SE + carbon current collector

Fig. 12 Geometry, discretization, and boundary conditions of a finite element model of a composite electrode. Electrode particles are embedded in a mixed conductor, consisting of a solid electrolyte and an electronically conductive additive

938

G. Bucci and W. C. Carter

Table 4 Material parameters for problems in Sect. 4.4 Input value F = 96485.3365 C mol1 R = 8.314 J K1 mol1 T = 298 K MEl = 1015 m2 s1 MSE = 1013 m2 s1 cmaxEl ¼ 1

Description Faraday’s constant

cmaxSE ¼ 0:25

Gas constant Temperature Mobility of Li in the electrode material Mobility of Li in the solid electrolyte material Maximum relative number of mole of Li per mole of electrode compound Maximum relative number of mole of Li per mole of solid electrolyte

i = 10 A m2 γ Li = 1 v=0 βEl = 0.1 βSE = 0 EEl = 100 GPa ESE = 15 GPa G = 0.25 J m2 δ0 = 5 nm δcr = 20δ0

Maximum relative number of mole of Li per mole of solid electrolyte Activity coefficient Poisson’s ratio for both materials Relative lattice constant for Li in electrode material Relative lattice constant for Li in the solid electrolyte material Young’s modulus of the electrode material Young’s modulus of the solid electrolyte material Fracture energy of the bulk solid electrolyte material Opening displacement at the onset of damage Critical opening displacement (complete interface separation)

separation law is a bilinear form, and Li flux across the interface is irreversibly turned off at the onset of damage. The constitutive equations for the cohesive model are detailed in Sect. 3.4. Galvanostatic tests are performed by applying a constant and uniform lithium flux (corresponding to a constant current density) at the interface in contact with the solid electrolyte (top edge in Fig. 12). A zero flux is assumed on the remaining edges, as Neumann boundary conditions for the diffusion problem. Model the matrix as a homogenized mixture of ionic and electronic conductor: Li ions transport is charged compensated by electrons diffusing through the electronically conductive material embedded in the matrix itself. For the mechanical problem, zero-displacement Dirichlet boundary conditions are applied on the left and right boundaries. The displacement is considered fully constrained of top edge by the presence of the solid electrolyte, and at the bottom edge by packaging or neighbor cells. Particles and solid electrolyte material are modeled as elastically and diffusively isotropic. This investigation is limited to the linear elastic regime. The input parameters for this problem are summarized in Table 4. As illustrative examples, we choose an intercalating compound with stress-free-strain volumetric expansion of 30% at full lithation (βLi = 0.1). However, our simulation shows that fracture initiates before particles have reached 3% swelling and 10% of the total lithiation. The Young’s modulus of Li2S–P2S5 sulfide solid electrolytes has been estimated (by ultrasonic sound velocity measurements) in the range of 18–25, for

29

Micro-mechanics in Electrochemical Systems

939

hot-pressed pellets, and 14–17 GPa for pellets pressed at room temperature [57]. We set the Young’s modulus Ematrix = 15 GPa. Figure 13 illustrates the result for the case of matrix with fracture energy G = 0.25 J/m2. Analogous tests have been performed with electrolyte material having fracture energy of G = 1.0 J/m2 and higher. The extension of the damage predicted at various states of charge strongly depends on the assumed SE toughness. Fracture nucleation is delayed to higher lithiation values or prevented as the toughness of the solid electrolyte material increases. However, to our knowledge, no experimental data are available for the fracture toughness of sulfide solid electrolyte materials. Highly conductive sulfide electrolytes react in contact with the atmosphere, making mechanical tests very challenging. A sequence of snapshots in Fig. 13 illustrate the state of charge (left column), the hydrostatic Cauchy stress (right column), and the progressive propagation of cracks within the solid electrolyte material. Cracks are represented as thin black lines at the onset of fracture and block Li diffusion. Accumulating damage is visualized with thickening black lines. The simulation shows regions of tensile stress developing in the area around the particles’ corners. The misalignment of these corners creates matrix shear- and tensile-stresses. Under the cohesive law’s assumptions (refer to Table 4 for the parameters and to Sect. 3.4 for their interpretation), the tension nucleates fracture at the diffusion front at short times. Fracture is allowed to propagate along a subset of finite element interfaces. Those interfaces lie along crack pattern that run between particles’ corners, where fracture is most likely to develop. The preinsertion of cohesive elements along potential crack paths follows from the “intrinsic” approach chosen here and discussed in Sect. 3.4. As the state of charge progresses, the pressure in the particles increases. In the last time frame in Fig. 13c, the compressive stress in the active material is higher than 1 GPa, for the particles that have stored 50% of their total Li capacity.

5

Conclusions

We have developed a continuum-level theory which couples the transport of multiple charged species with the deformation of the solid medium. It includes a fully nonlinear kinematic description of the anelastic volumetric expansion of the hosting lattice, caused by the intercalating particles with lattice network constraints. Electrostatic forces couple the diffusion of ions, as an internal electric field arises to counteract long-range charge separation. The system kinetics are driven by the concurrent action of electro-chemo-mechanical forces and are able to simulate the behavior of multifunctional materials. The purpose of this development is the study of the technologically relevant cases of diffusion of ions in electrode and solid electrolyte materials for lithium-ion batteries and solid oxide fuel cells. The theory, its reduction to a finite element formulation, and the numerical methods for its solution are targeted to perform solutions on the complicated microstructures that occur in batteries and SOFCs. For problem concerning SOFCs, we demonstrate that the method reproduces sensible physics in two simple cases. However, these simple cases do shed light on relevant phenomena.

940

G. Bucci and W. C. Carter

0.1

Concentration Li 0.2 0.3

Hydrostatic Stress (GPa) -1.25 -1.00 -0.75 -0.50 -0.25 0

0.4

4

4

2

2

0

0

-2

-2

-4

-4

-4

(a) t = 50 s 0.1

-2

0

2

Concentration Li 0.2 0.3

4

-4

4

2

2

0

0

-2

-2

-4

-4 -4

-2

0.1

0

2

Concentration Li 0.2 0.3

4

-4

4

2

2

0

0

-2

-2

-4

-4

-4

-2

Fig. 13 (continued)

0

2

4

2

4

-2

0

2

4

Hydrostatic Stress (GPa) -1.25 -1.00 -0.75 -0.50 -0.25 0

0.4

4

(c) t = 250 s

0

Hydrostatic Stress (GPa) -1.25 -1.00 -0.75 -0.50 -0.25 0

0.4

4

(b) t = 150 s

-2

-4

-2

0

2

4

29

Micro-mechanics in Electrochemical Systems

941

In the case of two-species ambipolar transport, where diffusion takes place by vacancies produced by Schottky defects, no charge imbalance or kinetic demixing can occur except for extraordinarily high stresses or chemical driving forces. This results from the relative magnitude of the electrostatic energy, with respect to the mechanical and chemical energy stored. Conversely, kinetic demixing is observed in the diffusion of three ionic species within an oxide material. The combined effect of Vegard’s stresses and different cations diffusivity results in a nonhomogeneous electrode composition. In the absence of mechanical coupling, the transient demixing vanishes as the system approaches its steady state. The results presented in Sect. 4.2 emphasize the relevance of mechanical coupling in the treatment of defect-migration problems. The proposed examples show how the fully coupled model is an appropriate tool to explain the formation of a cation-rich surface layer in SOFC cathodes. This phenomenon is detrimental to the cathode oxygen-reduction catalytic efficiency, and it is considered to be the main barrier for implementation of high-performance SOFCs at intermediate temperatures 500–700 C. To study diffusion under the effect of a highly nonhomogeneous stress field, in Sect. 4.3, we proposed a 3D simulation of a notched sample under tensile loading. A significant Li redistribution around the crack tip is observed in consequence of stress localization. Simulations of mechanically loaded cracks in electrodes indicate that significant intercalated Li concentration gradients can develop in the vicinity of the crack tip. Such inhomogeneous concentrations will affect fluxes from the electrolyte into the electrode. Furthermore, cyclic loading would incur energetic losses due to the redistribution of Li at each cycle. We investigated crack propagation across the solid electrolyte phase in an allsolid-state battery microstructure. During cell discharge, the anode materials tend to increase their stress-free strain as they absorb lithium. The particles’ chemical expansion is constrained by the solid electrolyte matrix and produces on average a state of compressive stress. The simulation shows, however, regions of tensile stress developing in the area around the particles’ corners. The misalignment of these corners creates matrix shear- and tensile-stresses. Under the assumption of a sulfide SE with fracture energy G = 0.25 J/m2, the cohesive zone model employed here predicts widespread fracture propagation as the state of charge progresses. Such cracks would block Li diffusion and reduce the composite electrode’s effective ionic conductivity.

ä Fig. 13 Evolution of damage in the solid electrolyte material at subsequent states of charge. With increasing Li content, the active material undergoes chemical expansion. As the particles’ stressfree strain increases, compressive stress develops in most of the microstructure. However, a few regions in proximity of the particles’ corner are under tensile stress (brown regions in the right contour plots above). This tension grows large enough to initiate fracture in the solid electrolyte matrix and captured in the simulation at cohesive law elements. Cracks propagate from corner to corner, cutting off diffusion paths for Li within the electrolyte

942

G. Bucci and W. C. Carter

The theory and its numerical implementation are quite general, and their application is not limited to the examples illustrated herein. A variety of electrochemical–mechanical systems and materials can be modeled at the mesoscopic scale; rigorous investigations can be carried out to identify the underlying circumstances which compromise their life and performance.

6

Incremental Boundary Value Problem in Discrete Form

The purpose of this supporting section is to offer a complete derivation of the finite element model in a notation consistent with the rest of the chapter. The system presented herein is governed by two equations: the linear momentum balance and the mass balance, having displacements and concentration as unknowns. In the standard formulation, the diffusion potential ϖ in Eq. (46) can be computed from the concentration and the stress field, which in turn is recovered from the displacements. Their computation requires the derivatives of the displacement field u. Furthermore, the increment of concentration ċ is function of the gradient of the diffusion potential. From a numerical point of view, differentiating implies a loss of precision. It is therefore appealing to look for a formulation in which constraints are readily accessible. Given the physical “importance” of ϖ, whose gradient appears as the driving force for diffusion, we choose to adopt a mixed finite element method [28] by introducing the diffusion potential (including its stress-dependent component) as a third independent variable. A numerical approximation of the system governing equations (here written, for simplicity, by considering only one diffusing species) is constructed according to the finite element theory. The static and transport problem, with their relative state functions, boundary, and initial conditions, will be introduced and analyzed separately for the sake of clarity.

6.1

Static Problem

The linear momentum balance in strong form can be written as 8 < Div P ¼ 0 χ¼χ : PN ¼ T

on @ D B on @ N B

(69  71)

with Neumann and Dirichlet boundary conditions imposed on subsets of the domain boundary. The corresponding weak form reads Z B

½Div P υ dV ¼ 0

(72)

29

Micro-mechanics in Electrochemical Systems

943

with υ being a test function. Integration by part leads to Z

Z B

½P  ▽υ dV 

@N B

PN υ dA ¼ 0

(73)

Let us consider the spatial discretization of the body B in its reference configuration into E disjoint finite elements such that E

B ¼ [ Ωe

(74)

e¼1

The deformation mapping and the test function can be written as a linear combination of the nodal values. We consider the same shape functions N to interpolate the nodal unknowns and the test functions υ χ’

N X

χbN b ¼

N X

χbN b

(75)

υa N a

(76)

e¼1 b¼1

b¼1

υ’

E X n X

υa N a ¼

a¼1

E X n X e¼1 a¼1

In Eqs. (75) and (76), the summation over total number of nodes is split into a sum over the number of nodes per element and a sum over all the finite elements composing the mesh. The gradient of the test functions becomes ▽υ ’ ▽

N X

! υ N a

¼

a

a¼1

E X E X

υa ▽N a

(77)

e¼1 a¼1

Substituting relations of Eqs. (75), (76), and (77) into the weak form leads to the equation "Z E X e¼1

B

P

n X

! υ ▽N a

a

Z

dV 

a¼1

@Ωe \@ N B

PN

n X

# υ N dA ¼ 0 a

a

(78)

a¼1

which must hold 8va, leading to E X n Z X e¼1 a¼1

Z B

ðP  ▽N a Þ dV 

@Ωe \@ N B

P N N a dA ¼ 0

(79)

The linear momentum balance in discrete form Eq. (79) can be condensed into ext F int χ ðχ ðt ÞÞ ¼ F χ ðt Þ

(80)

944

G. Bucci and W. C. Carter

with the following definitions of internal and external forces18 Z F int χ ðχ ðt ÞÞ ¼

B

Z F ext χ ðt Þ

¼

P  ▽N a dV

(81)

TN a dA

(82)

@Ωe \@ N B

T being the traction applied over the Newman boundary, and the index A = 1, ... N.

Let t0, t1, ..., tn + 1 be discrete sampling points on time. We aim to compute χ 0, χ 1, ... , χ n + 1 by the Newton–Raphson solution procedure. Assuming χ n known, we employ an iterative algorithm to find χ n + 1 by imposing   int ext F int χ χ nþ1 ¼ F χ ðχ n þ uÞ ¼ F χ ðt nþ1 Þ

(83)

u = χ n + 1  χ n being the incremental displacement. At each time step, the nonlinear system of Eq. (83) is, in turn, solved by the Newton method; i.e., the values ðkþ1Þ ðk Þ unþ1 relative to iteration k + 1 are computed by linearizing Eq. (83) about unþ1 and ðk Þ cnþ1 . To this end, we substitute into Eq. (83) the relations ðkþ1Þ

ðk Þ

ðkþ1Þ

ðk Þ

unþ1 ¼ unþ1 þ Δu

(84)

cnþ1 ¼ cnþ1 þ Δc

(85)

ðk Þ

ðk Þ

and the vector of internal forces linearized about unþ1 and cnþ1 becomes19 

ðkþ1Þ  F int  χ

nþ1

¼ þ



ðk Þ  F int  χ



þ nþ1     O juj2 þ O jΔcj2



ðk Þ  DF int  χ

nþ1

,u þ





ðk Þ  DF int  χ

nþ1

, Δc þ

(87)

We write |n and |n + 1 to indicate evaluated at time step n and n + 1, respectively. At each iteration, the following problem is solved Kuu u þ Kuc Δc ¼ Ru

(88)

where the residual Ru is computed as 18

In order to avoid confusion with other symbols, calligraphic notation is used within this section to indicate the vector of nodal forces and the stiffness and mass matrix.

19

ðk Þ

The directional or G^ateaux derivative of a generic function f about xnþ1 is defined as   d  ðk Þ  < D f ðk Þ nþ1 ,Δx >¼ f xnþ1 þ e Δx  de e¼0

(86)

29

Micro-mechanics in Electrochemical Systems

945

 int ðk Þ  Ru ¼ F ext ð t Þ  F  nþ1 χ χ

(89)

nþ1

and the blocks of stiffness matrix are (with indices A = 1, ... N, B = 1, ... N) Z K

uu

¼

Ωe



Z K ¼ uc

6.2

▽N b : ℂ : ▽N a dV

Ωe

@P b N @c

(90)

 : ▽N a dV

(91)

Diffusion Problem

The strong form of the diffusion problem consists of solving Eq. (92) with Dirichlet Eq. (93), Neumann boundary condition Eq. (94), and the initial condition Eq. (95). 8 > > < > > :

on @ D B : on @ N B at t ¼ 0

c_ þ Div J ¼ 0c ¼ cJ  N ¼ Jc ¼ c0

(92  95)

The weak form of the stated problem reads Z B

½c_ þ Div J  υ dV ¼ 0

(96)

Integration by part leads to Z Z c_ υ  J  ▽υ dV þ B

@N B

J  N υ dA ¼ 0

(97)

Following the procedure presented above, we introduce the finite element approximation c’

N X

cb N b ¼

N X

c_b N b ¼

N X a¼1

(98)

E X n X

c_b N b

(99)

υa N a

(100)

e¼1 b¼1

b¼1

υ’

cb N b

e¼1 b¼1

b¼1

c_ ’

E X n X

υa N a ¼

E X n X e¼1 a¼1

946

G. Bucci and W. C. Carter

where cb ,c_b represents the nodal concentration and concentration rate unknowns and Nb are the shape functions chosen to discretize the problem in space. Substituting Eqs. (98), (99), and (100) into Eq. (97), we write "Z E X e¼1

¼

Ωe

E X n X e¼1 a¼1

! Z n n n X X X b b a a a a υ N J  υ ▽N dV þ c_ N a¼1

b¼1

"Z

υa

n X Ωe

a¼1

!

Z

c_b N a N b  J  ▽N a dV þ

b¼1

@Ωe \@ N B

@Ωe \@ N B

n X J υa N a dA

#

a¼1

#

JN a dA ¼ 0 (101)

Equation 101 must hold for any choice of υa leading to the following approximated form of the diffusion problem ext M c_ þ F int c ¼ Fc

(102)

with c_ being the vector of nodal concentration rates and with the following definitions of mass matrix and vectors of internal and external forces Z M¼

Z

F int c ¼ Z F ext c

¼

Ωe

N a N b dV

(103)

J  ▽N a dV

(104)

Ωe

@Ωe \@ N B

JN a dV

(105)

6.2.1 Time Discretization of the Diffusion Problem For the time-dependent problem stated in Eq. (102), let us assume that the solution is known at the time step n and needs to be computed at the time step n + 1. To this end, we shall consider the trapezoidal rule as a time stepping algorithm [58] and solve the following system of equations at each time step. n

 ext  M c_nþ1 þ F int c jnþ1 ¼ F c nþ1 cnþ1 ¼ cn þ Δt c_nþα c_nþα ¼ ð1  αÞ c_n þ α c_nþ1 : (106  108)

By introducing the definition of the concentration predictor c~nþ1 ¼ cn þ ð1  αÞΔt c_n Eq. (107) can be rewritten as

(109)

29

Micro-mechanics in Electrochemical Systems

947

cnþ1 ¼ c~nþ1 þ α Δt c_nþ1

(110)

In particular, among the members of the generalized trapezoidal family, we choose the implicit Backward Euler iteration scheme, which corresponds to assuming α = 1. Let us rewrite Eq. (106) in order to solve for cn + 1   1 ext   M cnþ1 þ F int c nþ1 ¼ F c nþ1 Δt

(111)

At each time step, we adopt the iterative Newton method to solve the nonlinear ðkþ1Þ system of Eq. (111); i.e., we seek to compute the values cnþ1 relative to iteration ðk Þ ðk Þ k + 1 by linearizing Eq. (111) about cnþ1 and ϖnþ1 . To this end, we substitute into Eq. (111) the relations ðkþ1Þ

ðk Þ

ðkþ1Þ

ðk Þ

cnþ1 ¼ cnþ1 þ Δc

(112)

ϖnþ1 ¼ ϖnþ1 þ Δϖ ðk Þ

(113) ðk Þ

and the vector of internal forces linearized about cnþ1 and ϖnþ1 becomes  ðkþ1Þ  F int c nþ1

¼ ¼

D E D E    ðk Þ  ðk Þ  ðk Þ  F int þ D F int , Δc þ D F int , Δϖ c c c nþ1 nþ1 nþ1  F intðk Þ  þ Kcc Δc þ Kcϖ Δϖ c

nþ1

(114) The diffusion problem discretized in space and time has become equivalent to the following equation in the unknowns Δc, Δϖ 

 M cc þ K Δc þ Kcϖ Δϖ ¼ Rc Δt

(115)

the stiffness matrix blocks Kcc and Kcϖ in Eq. (115) can be computed as

Z Kcc ¼

Ωe

Z Kcϖ ¼

M b 1 N C ▽ϖ  ▽N a dV RT

M c C 1 ▽N b  ▽N a dV RT Ωe

(116)

(117)

and the residual Rc has the following expression    M  ðk Þ int ðk Þ   cnþ1  cn  Rc ¼ F ext c nþ1  F c nþ1 Δt

(118)

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G. Bucci and W. C. Carter

6.3

Diffusion Potential

From the diffusion potential of Eq. (46), we define the third variable in our system of equations as   β  eT  c @ΥM ðC e ,cÞ ¼0 ϖ þ Tr F P  RT ln  RT ln γ  ρh cmax  c @c

(119)

The choice of ϖ, together with displacement and concentration, as unknown variables will lead us to a mixed finite elements formulation. To simplify the derivation, we assume here constant elastic properties, and @Υ M(Ce, c)/@c = 0. The corresponding weak form of Eq. (119) (Z E X e¼1

" Ωe

n X

ϖb N b þ

b¼1

# )   n X β  eT  c  RT ln γ Tr F P  RT ln υa N a dV ¼ 0 ρh cmax  c a¼1

(120) must hold 8υa, leading to the discrete equation M ϖ þ F int ϖ ¼0

(121)

The internal forces vector has the following definition

Z F int ϖ ¼

Ωe

  β  eT  c Tr F P  RT ln  RT ln γ N a dV ρh cmax  c

(122)

We linearize the problem in order to make it amenable to be solved by Newton’s method.     ðk Þ ðk Þ ðk Þ M ϖðkþ1Þ þ F int þ D F int ,u u þ D F int , Δc Δc ¼ 0 ϖ ϖ ϖ (123) ðk Þ M Δϖ þ Kϖu u þ Kϖc Δc ¼ F int  M ϖðk Þ ¼ R ϖ ϖ In order to make the notation clearer, we left out the indication of the time step at which the iteration is performed. The blocks of the stiffness matrix are computed according to Z    β  ϖu K ¼ ℂ : ▽N b : Fe þ P : Fa1 ▽N b N a dV (124) Ωe ρ h  

 Z   β @P e cmax 1 @ ln γ ϖc  : F  RT  þ K ¼ N a N b dV (125; 126) @c cðcmax  cÞ c @ ln c Ω e ρh Complete System of Equations in Discrete Form In summary, the electrochemical–mechanical problem to be solved at each iteration of the Newton–Raphson algorithm consists of the following nonsymmetric system of three coupled equations in the unknowns u, Δc, Δϖ

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Micro-mechanics in Electrochemical Systems

8 uu K u þ Kuc Δc ¼ Ru > < M þ Kcc Δc þ Kcϖ Δ ϖ ¼ Rc Δt > : M Δϖ þ Kϖu u þ Kϖc Δc ¼ Rϖ

949

(127)

The model has been implemented in an in-house object oriented C++ numerical code. Grid handling and discretization make use of the deal.II finite element library, an open-source library widely used in scientific computing [53–56]. Furthermore, it employs parallel processing, i.e., the simultaneous use of more than one CPU to execute the most time consuming calculations, i.e., assembly of stiffness matrix and residual, and the solution of the linear system at each Newton iteration. Nomenclature Glossary of Symbols

B Ce ℂ Di D E F F ext F int F Fe Fa G I1 J J Ji JQ J qi K L Mi N M P Pe Q R R T T

body force per unit of reference volume elastic right Cauchy–Green strain tensor elasticity tensor diffusivity of species i dissipation density electric field in the reference configuration Faraday’s constant vector of external forces in the finite element discretization vector of internal forces in the finite element discretization deformation gradient elastic deformation gradient anelastic deformation gradient shear modulus or second Lamé constant first invariant of the elastic right Cauchy–Green strain tensor I1 = tr(Ce) Jacobian, J = det(Fe) imposed normal flux on the Neumann boundary outward flux of species i, measured per unit of undeformed area per unit time outward heat flux, measured per unit of undeformed area per unit time outward flux of charge i per unit of undeformed area per unit time stiffness matrix in the finite element discretization film thickness mobility of species i finite element shape function mass matrix in the finite element discretization first Piola–Kirchhoff stress tensor equilibrium first Piola–Kirchhoff stress tensor distributed heat source per unit of reference volume gas constant vector of residuals in the finite element discretization absolute temperature applied traction

950

U ci cimax c0 c c0 e ni qi t u v Wext zi ΥM Υ EC Υ EC II α βi γ s λ μΘ μEC i μM i v ωi ρh ϕ χ χ

G. Bucci and W. C. Carter

internal energy per unit of undeformed volume V0 number of moles of the chemical species i per unit reference volume V0 total number of sites per unit reference volume V0 available to the species i initial concentration imposed concentration on the Dirichlet boundary initial concentration electric field in the spatial configuration number of moles of the chemical species i charge of the chemical species i per unit volume time displacement finite element test function external power valence of species i elastic free energy density electrochemical free energy density excess of electrochemical free energy density for nonideal solution parameter in the trapezoidal time stepping algorithm relative lattice constant of species i intercalating into hosting material activity coefficient entropy per unit of reference volume first Lamé constant chemical potential at the system reference state electrochemical potential of the i  th chemical species stress-dependent component of the diffusion potential of species i Poisson’s ratio diffusion potential of the i – th chemical species molar density of the hosting compound electropotential deformation mapping imposed value of the deformation mapping on the Dirichlet boundary

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53. Bangerth W, Heister T, Heltai L, Kanschat G, Kronbichler M, Maier M, Young TD. The deal.II library, version 8.2, Archive of Numerical Software. 2015;3. https://doi.org/10.11588/ ans.2015.100.18031 54. Woodford WH, Chiang Y-M, Carter WC. Electrochemical shock of intercalation electrodes: a fracture mechanics analysis. J Electrochem Soc. 2010;157(10):A1052–9. 55. Sakuda A, Hayashi A, Tatsumisago M. Sulfide solid electrolyte with favorable mechanical property for all-solid-state lithium battery. Sci Rep. 2013;3:2261. https://doi.org/10.1038/ srep02261. 56. Bangerth W, Hartmann R, Kanschat G. Deal.II – a general purpose object oriented finite element library. ACM Trans Math Softw. 2007;33(4):24/1–24/27. 57. Sakuda A, Hayashi A, Takigawa Y. Evaluation of elastic modulus of Li2S – P2S2 glassy solid electrolyte by ultrasonic sound velocity measurement and compression test. J Ceram Soc Jpn. 2013;121(11):946–9. 58. Hughes TJR. The finite element method. New York: Dover Publications; 2000.

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics-Based Approach

30

Jacques Lamon

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Probabilistic Statistical Approaches to Brittle Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Flaw Strength Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Determination of Statistical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Damage Modes by Successive Cracking in Continuous Fiber-Reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Stochastic Models of Matrix Fragmentation in 1D Composite and Minicomposite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Single Filament and Multifilament Tows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Single Filament and Multifilament Tows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Bundle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Ultimate Failure of Single Filaments in the Presence of Fragmented Matrix . . . . . . . 5 Application to the Prediction of Tensile Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Unidirectionally Reinforced Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 2D Woven Composite: Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

956 957 960 962 966 967 974 974 975 976 979 979 979 982 982 983

Abstract

Being damage tolerant, the CMCs exhibit nonlinear deformations as a result of cracks that form in the matrix, in the interfaces, and in the fibers. The sequence of cracking modes displays several features that depend on the arrangement of fibers, the microstructure, and the respective properties of constituents. Being ceramic materials, the constituents are highly sensitive to inherent microstructural flaws generated during processing. The flaw populations govern matrix cracking J. Lamon (*) LMT (CNRS-ENS Cachan-University Paris Saclay), Cachan, France e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_64

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and fiber failures, so that strengths of constituents exhibit statistical distributions. A bottom-up multiscale approach based on micromechanics must account for the contribution of inherent fracture-inducing flaws, variability of constituent strengths, and associated size effects. The chapter deals with modeling of the stochastic processes of multiple fracture of the matrix and the fibers that govern damage and failure on fiberreinforced ceramic matrix composites. The models are based on probabilistic approaches to brittle fracture, including the Weibull phenomenological model and the physics-based elemental strength model that considers the flaws as physical entities. The probabilistic models that are discussed permit determination of stresses at crack initiation from microstructural flaws and resulting crack pattern. Applications to the prediction of tensile behavior of unidirectional or woven composites are then discussed.

1

Introduction

Micromechanics of failure (MMF) theory perform a hierarchy of micromechanical analyses, starting from mechanical behavior of composite constituents (the fiber, the matrix, and the interface) and then going on to the mechanical behavior of a ply, of a laminate, and eventually of an entire structure. The proposed models are generally deterministic in essence; average material data are used to produce approximate models for material behavior. In some cases, mean-field models work well, but we can’t expect them to be predictive for cases when properties exhibit statistical distributions or depend on microstructural features or environment. Being damage tolerant, the CMCs exhibit nonlinear deformations as a result of cracks that form in the matrix, in the interfaces, and in the fibers. The damage modes in woven CMCs are hierarchically organized, sequential, and stochastic as a result of microstructure and the respective properties of constituents. Being ceramic materials, the constituents are highly sensitive to microstructural flaws. The flaw populations govern matrix cracking and fiber failures, and strengths exhibit statistical distributions. Thus, a bottom-up multiscale approach based on micromechanics must account for the contribution of inherent fracture-inducing flaws, variability of constituent strengths, and associated size effects. Such approach is essential to understand the phenomena at low length scale that influence the behavior at macroscopic scale and to predict the mechanical behavior of CMCs. It can be the basis of an approach to composite and component design. The chapter deals with modeling of the stochastic processes of multiple fracture of the matrix and the fibers that govern damage and failure on fiber-reinforced ceramic matrix composites. The models permit determination of stresses at crack initiation from microstructural flaws and resulting crack pattern. Composite deformation is derived from the stress profile on fibers in the presence of matrix cracks and associated interface cracks. The models that are proposed in the present chapter have been used to predict the mechanical behavior of SiC/SiC and C/SiC ceramic matrix composites. An

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

957

analytical approach was applied to unidirectionally reinforced composite specimens under uniaxial tension, and a finite element-based approach was applied to 2D woven composite specimens under more complex loads. These results were reported in previous papers [1–6]. The statistical-probabilistic approaches to fracture are based on the weakest-link concept which assumes that fracture of the bulk specimen occurs instantaneously as soon as the crack initiated. Brittle material fails like a chain, the links of which would be specimen volume elements. The chain model can be extended to damage by progressive multiple cracking observed in multilayers and continuous fiber-reinforced composites. It must be stressed that care must be taken not to violate the weakest-link assumption and to use the probability equations properly. The “weakest-link” concept assumes that fracture of the bulk specimen is determined by the strength of its weakest volume element. This implies that the weakest-link concept is restricted to unstable fracture by direct extension of crack throughout the bulk of the specimen. Those materials that can be assimilated to structures including fiber-reinforced composite or multilayered materials are designed to develop mechanisms of crack arrest. In those structures made of an assembly of brittle elements like fibers, a matrix or multilayers, cracks initiate also from flaws, propagate through the elements, and are arrested at the closest interface. From this point of view, it may be considered that cracking of elements of the structure does not violate the weakest-link concept and that it may be approached using extreme value theory-based models. When the load is increased, the above mechanism of cracking repeats. Additional cracks can form from other flaws, which are less severe than the previous ones, but these flaws are the most severe in the population of remaining flaws. It is clear that models based on the extreme value theory do not apply when one considers the initial population of flaws and the whole structure. The fundamental approach based on the probability density function of flaws strengths (g(S)) can be extended by introducing the changes in the density function resulting from the elimination of flaws during the formation of previous cracks. This model requires an appropriate expression of g(S) with respect to the whole population of flaws that can be activated by stresses. Models based on the extreme value theory can be devised when a network of cracks appears which delineate fragments of material. Cracking is thus regarded as the brittle fracture of a fragment from the most severe flaw present in the fragment. It is generally assumed that the expression of flaw strength density function is independent of volume, so that the size effects induced by the formation of successive can be described.

2

Probabilistic Statistical Approaches to Brittle Failure

The fracture of brittle ceramics is caused by microstructural flaws that act as stress concentrators. The flaws are generally distributed randomly, and they exhibit wide variability in terms of severity, as a result of variability in shape, nature, size, and location with respect to the stress state. As a consequence, stress-induced fracture is a

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stochastic event, and fracture stresses measured on specimens with identical dimensions have a statistical distribution. There are two main types of probabilistic-statistical approaches to brittle fracture [1]: – The phenomenological and macroscopic approaches such as the Weibull model [7, 8] – The fundamental approaches that recognize the flaws as physical entities [9–21] The fundamental probabilistic approaches make better predictions and are more robust than the Weibull approach because of the underlying flaw strength density function. They are detailed in [1]. They are based on flaw severity distributions [1, 9–11, 15–21]. Flaw severity is defined either as elemental strength (i.e., the local stress that causes extension of flaw) [9, 14–20] or flaw size [8–10]. The failure probability is given by the following equation: 0

ð

ðX

1

P ¼ 1  exp@ dV gðXÞdXA V

(1)

0

where g(X) is the flaw density function, X measures flaw severity, and V is the stressed volume. X = S the elemental strength, or X = a the flaw size. The multiaxial elemental strength model [1, 19–21] is based on the distribution of multiaxial elemental strength gE(SE). For a unit volume element sphere, the number of flaws with a resistance between S and S + dS is given by the following formula, assuming a power law for gE(SE): π

1 gðSÞdS ¼ 2π

ð2 2ðπ

m mSm1 E S0 dSE cos ∅d∅dψ

(2)

0 0

Angles ∅ et ψ define the direction normal to flaw or crack plane. SE depends on the local stress state on the flaw. SE ¼ σ 1 f v ð∅, ψ, σ 2 =σ 1 , σ 3 =σ 1 Þ

(3)

SE ¼ 0 when σ 1 < 0,

(4)

σ 1, σ 2, and σ 3 are the principal stresses; σ 1 > σ 2 > σ 3. σ 2, and σ 3 can be negative (compressive components) when σ 1 and σ E > 0. The resulting failure probability is >0. Otherwise, failure probability = 0. The expression of failure probability is:   m   ð 1 σ1 σ2 σ3 P ¼ 1  exp  dV I V m, , V0 V S0 σ1 σ1

(5)

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . . π=2   ð 2ðπ σ2 σ3 1 With I v m, , fm ¼ v cos ∅d∅dψ 2π σ1 σ1 o

959

(6)

o

It is reduced to the following equation [1, 19–21] after integration for given stress state: 

 m  V σ ref P ¼ 1  exp  K V0 λ0

(7)

where λ0 is the scale factor and m the shape parameter of the power law. K is obtained by integrating the stress state over stressed volume V. K depends on the form of underlying flaw strength density function g(S). σ ref is a reference stress (peak stress) in the stressed volume V. V0 is the reference volume (V0 = 1 m3 when international units are used). By contrast, the Weibull approach is based on the density function of component failure stresses (macroscopic length scale), assumed to be a power law [7, 8]. The statistical distribution of failure strengths is given by the following equation:  ð  P ¼ 1  exp  ðσ=σ 0 Þm dV=V 0

(8)

where σ is the stress, σ 0 is the scale factor, m is the Weibull modulus, m reflects the scatter in data, and σ 0 is related to the mean value of strength. From Eq. 7, it is demonstrated that the Weibull equation of failure probability is a particular solution of Eq. 1 [1]. Thus, under a uniform uniaxial tension, K = 1, σ ref is the specimen tensile strength, so that Eq. 7 reduces to:     V σ m PW ¼ 1  exp  V0 σ0

(9)

where σ 0 is the scale factor. In this particular case, σ 0 = λ0. For single filaments under uniaxial tension, Eq. 9 defines the statistical distribution of flaw strengths in fibers. Equation 1 can be used to predict failure in several loading cases including multiaxial stress states [1, 19–21], for various formulae of flaw density function. Analytical expressions can be established for simple loading cases like uniform tension, for which closed form expressions of stress state are available. A number of postprocessors for calculation of failure probability were developed [1, 22]. They are based either on the Weibull model or on the Batdorf model [14] or on the multiaxial elemental strength model. The CERAM computer program which includes the multiaxial elemental strength model was used to calculate matrix cracking and mechanical behavior on 2D woven SiC/SiC [1–3]. The Weibull equation provides a sufficient approximation of failure probability for various simple uniaxial loading cases. It applies satisfactorily to filaments and matrix under uniaxial stress state.

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J. Lamon

Flaw Strength Distributions

The power law was found to be a satisfactory form of flaw strength density function. It allowed sound failure predictions for ceramics under various loading conditions [see 1 and references therein]. A few researchers have suggested a Gaussian distribution of strengths in the locality of flaws [23–25]. It has been shown recently that the distribution of flaw strengths in single filaments is a normal distribution [26]. Figure 1 defines fiber-reinforced composites from fracture initiation point of view, using flaw strength density functions for the matrix and the fibers. These composites are unidirectional as well as 2D woven with one direction of fibers in the loading direction. In Fig. 1a, the more severe flaws are located in the matrix. In Fig. 1b, the populations of flaws in the fiber and in the matrix exhibit similar severity. The respective positions of flaw strength distributions reflect the order of damage sequences under uniaxial tension. When Em ~ Ef, cracks initiate first in the matrix for the composite of Fig. 1a and in both for that of Fig. 1b. When Em < Ef, low stresses operate on the matrix, and cracks will appear first in the fibers of composite of Fig. 1b that will fragment; in the composite of Fig. 1a, the matrix and fibers may crack simultaneously. It is worth pointing out that the cracking sequence depends on stress state and size of constituent stressed volumes. It can be refined by computing failure probabilities using Eq. 5. Practically, fiber and matrix strength data exhibit a wide variation as a result of inherent variability in flaw severity. Small values of shape parameter or Weibull modulus m indicate a wide scatter. For instance, m = 5.3 (Table 1) for SiC filaments (Nicalon) and the scatter in strengths is very wide 500–3000 MPa [5, 26]; m = 6 for the CVI SiC matrix and strengths vary between 50 and 900 MPa depending on stressed volume (Fig. 2 and Table 2) [5, 27] (The strength values are indicated to show the magnitude of variation. They don’t have a reference value since they

a g(S)

Density function

Fig. 1 Schematic diagrams showing examples of flaw strength density functions for matrix and fibers: (a) more severe flaws in the matrix and (b) comparable severity of flaws in the matrix and in fibers [1]. The elemental strength is the flaw strength

gm(S)

gr(S)

Stmin

Smin

Ss St(αc)

S

b gm(S) gr(S)

g(S)

Smin

Ss St (αc)

Elemental strength (S)

S

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

961

Table 1 Statistical parameters of normal and Weibull distributions of flaw strengths for Nicalon filaments [26]

Test specimen 1 2 3

Normal distribution μ(%) S(%) 1.16 0.25 1.11 0.24 1.15 0.24

Number of filaments 487 986 924

Weibull distribution m e‘(%) e0(%) 5.30 1.25 0.01 5.23 1.20 0.01 5.43 1.23 0.01

σ 0(MPa) 23.4 21.1 25.7

V0 = 1 m3 Fig. 2 Statistical distributions of stresses at onset of matrix cracking on SiC/SiC minicomposite specimens: (a) probability versus strength, (b) log-log plot. Volume fractions of matrix: 27% batch A, 60% batch B, 70% batch C

a

1,2

Failure probability

1 0,8 0,6

Batch A Batch B

0,4

Batch C

0,2 0 0

b

200

400 600 800 Strength (MPa)

1000

2 1

Ln(-Ln(1-P))

0 -1 Batch A

-2

Batch B Batch C

-3 -4 -5 0

2

4 6 Ln (Strength (MPa))

8

depend on specimen volume size.). Uncertainties resulting from practical difficulties in the determination of fiber diameter (enhanced by diameter variation along fiber) and in the direct measurement of deformations may also contribute to variation in strengths. Practically much effort is done to improve accuracy in the determination of strengths, using up-to-date measurement techniques or by testing multifilament tow specimens, as discussed in a subsequent section. It is thus reasonable to consider that strength variability is dominated by the distribution of fracture-inducing flaws.

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Table 2 Estimates of CVI SiC matrix Weibull parameters obtained on microcomposites and minicomposites

m m σ 0 (MPa) σ 0 (MPa) V0 (m3)

2.2

Linear regression Maximum likelihood Linear regression Maximum likelihood

Minicomposites Vm = 61% [5] 6.2 (1.9) 6.1 (1.7) 11.4 (0.6) 10.5 (1.2) 1

Minicomposites Vm = 70% [5] 5.1 (1) 4.9 (0.7) 5.3 (0.2) 4.3 (0.2) 1

Microcomposites Vm = 67% [5, 27] 4.5 (1.6) 5.1 (1.8) 1.3 (0.1) 2.1 (0.1) 1

Determination of Statistical Parameters

2.2.1 Weibull Distribution Various methods of estimation of Weibull statistical parameters for filaments and matrix have been used in the literature. Filament strengths can be measured on batches of fiber specimens or on tows of several hundreds of filaments using tensile tests. Specimens made of CVI (chemical vapor infiltrated) matrix cannot be produced for obvious practical difficulty inherent to the CVI process. The CVI matrix is available only as coating on one filament (microcomposite) or a multifilament tow (minicomposite). Matrix strengths are obtained from tensile tests on micro- or minicomposites. The Weibull modulus, m, is obtained by fitting Eq. 9 to the distribution of experimental strength data or as the slope of the “Weibull plot” of ln (ln(1/1P)) versus ln σ. The scale factor σ 0 is determined from the value of ordinate at origin k: lnðlnð1=1  PÞÞ ¼ m ln σ þ k

(10)

Either graphical or curve fitting methods (like linear regression analysis, least squares method, or maximum likelihood estimator) can be used [1]. The Weibull plot is constructed, using an estimator for estimation of failure probabilities associated to failure data ranked in ascending order. Various estimators have been devised in the literature [1]. The estimator Pj = j/N can be used on very large sets of strength data. The estimator Pj = ( j0.5)/N is recommended for limited sample sizes (>30 specimens): j is the rank of filament strength, and N is the total number of data. An alternate method of determining fiber Weibull modulus uses strengths measured at various gauge lengths (for filaments when the cross-sectional area is constant) or volume sizes or strengths derived from fragmentation tests. The dependence of strength on size at given failure probability P is given by the equation: ln σ ðPÞ ¼ 1=m ln

X

þ k0

(11)

where  is specimen size (either gauge length (L ) or stressed volume (V )) and k0 is a constant. Thus, the slope of a log-log plot of strength versus gauge length is 1/m.

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

963

The tow testing technique is interesting because a significant set of filament strengths can be obtained using a single tensile tow test. Filament strengths are derived from forces and corresponding effective bundle sections as fibers are broken individually [28–32]. The number (Nj) of fibers broken at load Fj is derived from compliance Cj:   N j ¼ N t 1  C0 =Cj (12) where Nt is the initial number of intact fibers in the tow and C0 is the corresponding compliance. The estimator, P=Nj/Nt = 1  C0/Cj, can be used since the sample size is generally large, owing to the number of filaments present in a tow. A deformationbased analysis can be preferred which allows measurement of fiber sections to be alleviated. Then, deriving stresses on filaments from strains is straightforward when filament Young’s modulus is available. The values of Weibull modulus of most SiC-based fibers are rather small. They span a range from 2.3 to 9.9 [33]. A significant scatter in m estimates is generally observed for each fiber: for instance, m ffi 2.3–7.1 was reported for Nicalon fibers [5, 33–35]. The variation is inherent to the use of an estimator to construct the Weibull plot, to the use of limited sample size, and to the selection of sample [26]. Because of lack of uniformity in treatment of test data in the literature, values of scale factors were not summarized in this chapter. Detail can be found in [33]. Figure 2 shows plots of statistical distributions of stresses at onset of matrix cracking determined on SiC/SiC minicomposites tested in uniaxial tension [5]. Note the influence of matrix volume fraction. Comparable Weibull moduli were derived from these distributions and from those measured on microcomposites (Table 2) [27]. The statistical parameters were estimated by linear regression and maximum likelihood methods (Table 2). The Anderson-Darling test (A* < 0.5) indicates that the Weibull statistics properly describes the distribution of stresses at onset of matrix cracking. The lower values for A* were obtained for the statistical parameters determined using the maximum likelihood estimator. These parameters thus represent better estimates of statistical parameters. They were used in the computations in the foregoing section. Figure 3 illustrates results of a fragmentation tests on SiC/SiC microcomposite test specimens. It shows a logarithmic plot of fragment strength-size data measured using scanning electron microscopy during tensile tests. The best fit line follows the linearized equation of size effect for Weibull modulus m = 7.5: Ln σ Mi ¼ 

1 Ln V i þ constant 7:5

(13)

This SiC matrix Weibull modulus estimate compares satisfactorily with values extracted from the statistical distributions of matrix strengths at onset of cracking determined on three batches of SiC/SiC minicomposites [5]: 5.1–6.2 (standard deviation: 1.0 and 1.9). These results demonstrate that failure of fragments is described satisfactorily by extreme value theory-based model. This implies that the

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6

Ln(strength (MPa))

5,9

''

5,8 5,7

Predicted

5,6 5,5 5,4 5,3 5,2 5,1 5 0,01

0,1

1 Vi (mm3)

10

100

Fig. 3 Logarithmic plot of fragment strengths versus volumes during fragmentation of the SiC matrix of a unidirectionally reinforced SiC/SiC specimen. The solid line represents the best fit curve. The black triangles show predictions using Eq. 32

failure of a fragment is dictated by the most severe among the flaws present within the fragment and that the flaw density function is independent of fragment size. This latter hypothesis is implicit in the Weibull model. Note from Fig. 3 that the trend is observed down to small fragment volumes. The statistical parameters are important characteristics of failure. Sound estimates are required for failure predictions at given size and stress state, as well as information on associated V0. If V0 is not specified, the value of σ 0 is meaningless. It is important to stress that the estimate of σ 0 depends on dimension units [51]. For instance, the σ 0 estimates are substantially different if V0 is set to 1 m3 or 1 mm3 as shown by Eq. 14 derived from Eq. 9:         1=m   ¼ σ 0 mm3 109=m σ 0 m3 ¼ σ 0 mm3 V 0 mm3 =V 0 m3

(14)

where σ 0(m3) and σ 0(mm3) correspond, respectively, to V0(m3) = 1 m3 and V0(mm3) = 1 mm3. Some authors assume that the sectional area is constant. They use the following equation of failure probability: P ¼ 1  exp ½ðσ=σ L Þm L=L0 

(15)

σ L is a characteristic strength associated to characteristic length L. σ 0 is related to σ L by the following equation derived from Eqs. 14 and 15:        1=m σ 0 m3 ¼ σ L ðmmÞ V mm3 L0 ðmmÞ=LðmmÞV 0 m3 The volume V corresponds to L, and L0 = 1 mm.

(16)

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

965

2.2.2 Normal Distribution It has been demonstrated recently on SiC Nicalon fibers that flaw strengths follow normal distribution [26]. For the demonstration, large sets of 500 and 1000 filament strengths were obtained from tensile tests on multifilament tows. The analysis was based on strain to failure, which allowed elimination of fiber strength uncertainty due to diameter variation. Equations of Gaussian probability density function f(e) and normal distribution PN(Σ < e) are: " # 1 ðe  μÞ2 f ðeÞ ¼ pffiffiffiffiffi exp (17) 2S2 S 2π PN ðΣ  «Þ ¼

ðe 0

fð«Þd«

(18)

where Σ and e are strains, μ is the mean, and S is the standard deviation. The probability density function derived from acoustic emission monitoring is a bell curve described by a Gaussian function, symmetric about its mean as shown by the histogram of filament strengths (Fig. 4). Quite identical parameters μ and S were estimated on three bundles of Nicalon filaments (Table 1). The corresponding normal distribution coincides with the Weibull distribution function (Eq. 9) for the statistical parameters reported in Table 1 (Fig. 5). The scatter in Weibull scale factors is quite negligible. The shape parameter values (5.23–5.43; Table 1) show a very small variation when comparing to the data reported in the literature: 2.3–7.1 [5, 34, 35]. Referring to the limited scatter in Weibull parameters that was obtained, and to the large size of the data samples that were analyzed, it is considered that these are the true statistical parameters. The Gaussian probability density function shows the presence of a single population of fracture-inducing flaws, while volume- and surface-located flaws have been observed by SEM fractography [5]. It confirms previous results 200 175

AE events Normal distribution

150 AE enent count

Fig. 4 Histogram of filament strengths described by the distribution of AE events during a tensile test on a bundle of 1000 Nicalon SiC filaments [26]

125 100 75 50 25 0 0,0

0,5

1,0 Strain (%)

1,5

2,0

966

J. Lamon

1,0

Probability of failure

Fig. 5 Cumulative distribution functions of filament failure strains obtained on a Nicalon SiC fiber bundle: Normal distribution versus Weibull distribution (Eq. 9)

Normal distribution Weibull distribution (Eq. (x))

0,8 0,6 0,4 0,2 0,0 0,0

0,5

1,0

1,5

2,0

Strain (%)

which showed that the contributions of the populations of surface and volume located flaws cannot be separated [5].

3

Damage Modes by Successive Cracking in Continuous Fiber-Reinforced Composites

Damage modes by successive microcracks in continuous fiber-reinforced composites are shown in Fig. 6. The microcracks form in one of the constituents depending on respective constituent mechanical properties: the fibers of unidirectional polymer matrix composites, the ceramic matrix of unidirectional composites, the transverse plies of (0 /90 )n laminates, or the transverse tows of woven composites. The microcracks are arrested at an interface between the fiber and the matrix in unidirectionally reinforced composite, as well as between crossing plies and tows in 2D composites. As the load is increased, multiple cracks are created progressively as a result of the presence of fracture-inducing flaws with variable strength. The ultimate failure of composite material occurs when the population of fibers is no longer able to carry the applied load after saturation of matrix cracking (global load sharing). The population of fibers in plies or in bundles suffers damage prior to ultimate failure. Like the matrix, the fibers fail progressively as a result of flaw strength variation. Matrix cracking and fiber failure in composites can be modeled as a series system (matrix or fiber fragmentation) or a parallel system (damage of fiber bundles). The damage in series system displays the following characteristics: – The total volume under stress is constant during successive cracking. – The fragments (Fig. 6) are subject to stresses. – The whole population of flaws present in the total volume of material is able to induce fracture.

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

a

b

F

967

F

matrix fibre fragments

cracks F

F Fiber tows

c F

d F

F

F

Fig. 6 Diagrams of damage modes by successive failures (series systems): (a) fragmentation of the fiber in an organic matrix composite, (b) fragmentation of the matrix in a single fiber-reinforced ceramic matrix composite, (c) multiple cracking of transverse plies in a laminated composite, and (d) multiple cracking in a composite with woven plies

– Flaws are eliminated when new cracks form. Several flaws may be eliminated simultaneously when the stress state within fragments is affected by fiber-matrix debonding. By contrast, damage in parallel system of independent elements (Fig. 7) shows the following characteristics: – The total volume under stress decreases as elements break. – A broken element is unloaded after it fractured. – The fracture of an element is caused by the most severe flaw present in that element. The population of fracture-inducing flaws is restricted to the low strength extreme of the whole population of flaws. – The fracture of an element obeys the weakest-link concept and the statisticalprobabilistic models discussed above.

3.1

Stochastic Models of Matrix Fragmentation in 1D Composite and Minicomposite

In the series system-based approaches, the cracking stresses of matrix elements (chain-of-segment model [36]), or fragments (fragment dichotomy model [37]), are determined using the extreme value theory. Bayesian statistics are used for the

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Fig. 7 Damage by filament fractures to a dry tow of parallel and independent filaments (system of parallel elements) under uniaxial tensile forces (f) on filaments

f

filaments fractures

f

chain-of-element model [1, 36]. These approaches can be extended to 2D and 3D reinforced composites. The models are discussed in the following in the case of 1D composite.

3.1.1 The Chain-of-Segment Model The chain-of-segment model has been widely used in the Monte Carlo simulation of fragmentation of fibers in a polymer or metal matrix. The distribution of element strengths is constructed from the weakest-link failure of independent fibers. Fibers are assumed to follow the Poisson-Weibull model for strength where the average number of flaws per unit length with stresses less than or equal to stress σ is of the power form [38]: N ðσ Þ ¼

L1 o

 m σ σo

(19)

where σ o is the scale parameter relative to the test length Lo and ρ is a positive constant. As a consequence, fiber elements as well as a single fiber tension tested at arbitrary gauge length follow a similar Weibull distribution (extreme value statistics): 

   L σ m Pðσ, LÞ ¼ 1  exp  Lo σ o

(20)

where L designates the length of either the element or the fiber. In Monte Carlo simulations, a fiber length L is subdivided into Ne segments, each of length L/Ne  L. In some cases, Ne was selected arbitrarily as large as 2000 [39] or 104 [40]. In other cases, segment length was the final fragment size at saturation or was defined as twice the slip length at characteristic strength [38, 41]. Each segment is randomly assigned strength according to the Weibull cumulative probability distribution derived from Eqs. 19 and 20:

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

    m  L L σ P σ, ¼ 1  exp  Ne N e Lo σ o

969

(21)

A uniform stress σ f is gradually applied along the entire fiber length, and a break is formed at a segment when the stress on the segment equals the assigned strength of that segment. Stress on segment may be determined using various models for shear stress, or the slip zone is considered to be an exclusion zone. Comparison with available experimental fragmentation results is very sensitive to the statistical parameters introduced in Eq. 21 [40, 41]. Agreement was not obtained with those parameters derived from the failure of independent fibers with gauge length L [41, 42]. Drawbacks associated to fiber strength distribution dependence on length and statistical fiber strength characteristics have been addressed in [43]. Other experimental results showed that predictions are satisfactory only for low stresses quite far from the saturation of fragmentation [39, 41]. In summary, it can be concluded that the process of fragmentation is regarded as an in-parallel arrangement but it is treated as an in-series one: weakest-link statistics are used instead of Bayesian statistics, and the underlying flaw strength distribution refers to the weakest flaws instead of the whole population of flaws present in the material.

3.1.2 The Fragment Dichotomy Model [5, 36, 37] The fragment dichotomy model reproduces the process of fragment generation: fragments of decreasing size form from the brittle failure of parent fragments. Failure of a fragment is caused by the most severe flaw that it contains. The corresponding flaw strength is located at the lowest extreme of the distribution of flaw strengths for that fragment. From Eq. 7, it comes for fragment strengths:  σ iref ¼ λ0

 m  V0 1 Ln 1  Pi Vi K

(22)

where Vi is the volume of ith fragment and Pi is the failure probability of the fragment (with respect to the cumulative distribution of strengths of fragments with the same volume). The fragmentation process is characterized by a set of stresses σ iref . Calculation of the series of stresses σ iref requires the successive fragment volumes Vi and their failure probabilities Pi. In order to make a comprehensive presentation of the approach, let’s consider uniaxial tensile loading conditions and specimens with a uniform cross section (Fig. 6). When a crack is created in the matrix, it is perpendicular to the fiber axis, and it propagates through the matrix, leading to two new fragments with cross section Si. Thus, Vi = Si Li, Li is fragment length. The length of a fragment depends on the location of the critical flaw in the parent fragment. Since flaws have a random distribution in location and size, the length of successive fragments is a statistical variable. It can be derived from the spatial distribution of flaw strengths within the fragment using the following equation:

970

J. Lamon

ð P ð Li Þ ¼

hðxÞdx

(23)

Li1

where P(Li) is the probability that fragment length is Li, h(x) describes the spatial density of flaw strengths along the fragments, and x is the abscissa along fragment axis. Li is the length of the fragment generated from the failure of a parent fragment with length Li1. Fragment length is given by the reciprocal function: Li ¼ ½PðLi Þ1

(24)

Rupture occurs in that fragment having fracture stress smaller than the stress operating on fragments: σ  σ iref ðPi Þ

(25)

Analytical equations are developed in the following for a composite reinforced with parallel continuous fibers and subjected to a tensile load parallel to fiber axis. The stress state is affected by fiber/matrix debonding. The resulting shear lag is shown in Fig. 8. Introducing stress expressions into expression (8) gives K [1]. The strength of a matrix fragment of length 2li (or li at the ends of specimen) and constant cross-sectional area Sm is:  σ Mi ðLi , Pi Þ ¼

σ RM ðPÞ

V0M Vi

m1  m1 lei li

(26)

F

Poisson effect

Em Vm 0m mm

debond zone

Matrix

 = I0/Id  = 0 + k

Interphase

Fiber

F

0 1 0

Stress

Ef Vf 0f mf

u+l0

u

u+ld

(x)

x

0 Fiber ix

Matr

x

Fig. 8 Effect of matrix crack and associated fiber debonding on the stress state operating on the matrix and on the fiber. Also indicated are the matrix and fiber pertinent characteristics

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

971

where V0M is the volume of uncracked matrix, Vi is the volume of fragment (Vi = 2 li Sm), R reference matrix strength at probability P ( σ RM ðPÞ ¼ λ0 hσ M ðPÞ is the initial i V0 1 m according to Eq. 22), and lei is equivalent fragment length [36, V 0M K Ln 1P 37]:with   ldi m þ β when li > ldi lei ¼ li 1  li m þ 1

(27)

ðli  βldi Þmþ1 m when li  ldi ðm þ 1Þl m di ð1  βÞ

(28)

lo ld

(29)

lei ¼

β¼

lo is the length of the zone along which Poisson’s effect operates (there is no contact between the interface crack surfaces), and ldi is the debond length associated to matrix crack i. Vi, li, and lei are derived from the probability of location of a fracture-inducing flaw in a fragment. A uniform spatial distribution of fracture-inducing flaws was considered: hðxÞ ¼ constant ¼

1 2li  2ld

(30)

In [5], it was assumed that fragment fracture was restricted to the volume between debond crack tips. The probability of location of a fracture-inducing flaw in fragment is thus given by: PðxÞ ¼

x  ld ¼X 2ðli  ld Þ

(31)

where x is the distance to fragment edge and ld is debond length. Simpler analytical equations can be derived from the above general equations, for either Pi = 0.5 or X = 0.25 (cracks in the middle of fragments) or li > ldi. For the condition Pi = 0.5 and X = 0.25, satisfactory estimates of stresses and strains were obtained in [37]. For the condition Pi = 0.5, X = 0.25 and li > ldi, V0M/Vi = k þ 1 at step i so that the Eq. 27 is reduced to: σk ¼

σ RM L1=mm



L mþβ  2ld kþ1 mþ1

1=mm

(32)

σ k and σ RM are the strengths for Pi = 0.5. L is the gauge length, k is the number of cracks, and β accounts for Poisson’s effect in the interface crack (0  β  1). Note that the statistical parameters correspond to the brittle failure of initial volume of

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material. They can be derived from the strengths of a set of specimens. Matrix cracking saturation is obtained when the term between brackets is negative. The amount of cracks at saturation is thus, when β < 1, which means that some amount of load transfer is still effective through the interface: ns ¼

L mþ1 1 2ld m þ β

(33)

When β = 1, there is no more load transfer; ns is now ns ¼ 2lLd  1. The stress at matrix cracking saturation is then derived from Eq. 27, assuming that saturation is reached when li = ldi: σ sM

¼

σ RM ðn

þ 1Þ

1=mm

  1 þ mm 1=mm 1β

(34)

The σ k versus Vi relationship computed using Eq. 32 has been satisfactorily compared to experimental results in [6], as shown in Fig. 3. This model has been discussed in [6, 36, 37], and it has been shown to provide satisfactory distributions of fragment strengths for the SiC matrix in SiC or C fiberreinforced minicomposites, as well as the tensile stress-strain behavior of minicomposites [6, 36, 37, 44].

3.1.3 The Bayesian Model The Bayesian model of fragmentation is the exact solution when fragmentation is assimilated to progressive multiple failure of a chain of elements. Failure of a link in a chain, which can experience successive failures, involves conditional probabilities; it depends on the response of the other links. Probability of the kth failure is [1]: Pðk= k  1Þ ¼ 1 

1  Pk 1  Pk1

(35)

The subscript k  1 designates the event of nonfailure for the (k  1)th link and k the event of failure for link k. Pk1 is the probability of no failure of link k  1; Pk is the probability of kth failure. Equation 35 represents the operation of truncating the flaw strength distribution at probability Pk1, to obtain the pertinent distribution after the (k1)th failure. Pk1 and Pk pertain to the non-truncated flaw strength distribution. Pðk= k  1Þ corresponds to the truncated distribution after the (k1)th failure (Fig. 9). For simplicity, it will be referred to as Pk . h  p i It was assumed for simplicity that P1 ¼ 1  expXXX  VV0 SS0 for the first failure. The power form distribution that was selected in a first step may not be satisfactory because it is an extreme value theory-based equation. More accurate strength distributions can be used. The statistical parameters are designated p and S0

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

Fig. 9 Truncated flaw strength density functions g (S) after formation of (a) the (k1)th crack from flaw having elemental strength Sk1, (b) the kth crack from flaw having elemental strength Sk [1]

973

a

g(S) Density functions

30

S k-1

Sk

S

b

g(S)

Sk

S k+1

S

Elemental strength (S) because they characterize a wider strength distribution when compared to the low strength extreme Eq. 9. The following equation for the stress at formation of the kth crack was established in [36]:  m1 1 V0 Sk ¼ S0 k ðLnð1  Pα ÞÞm V 1 p

(36)

Pα may be taken to be 1%, 0.1%, or less, depending on the flaw population size. Pα may be also selected randomly, or k dependent, but it must remain smaller than a certain limit. To some extent, this limit depends on the number of flaws in the distribution. It is assumed that the number of remaining flaws is much larger than 1 so that statistics may apply. In [1] the Bayesian model was extended to nonuniform and multiaxial stress states and applied to 2D reinforced composites by using the multiaxial elemental strength.

3.1.4 Comparison of Models Good predictions of fragmentation strengths (Fig. 10) were obtained with the fragment dichotomy model and the Bayesian model [36]. By contrast, the fragmentation stresses were either overestimated (low stresses) or overestimated (high stresses) by the Monte Carlo simulation method. The upper and lower bounds show that predictions depend on the number of elements (Ne) and exhibit a significant uncertainty. Although it presents flaws in its foundation, the Monte Carlo simulation method is generally used for the treatment of multiple failures in a chain of links. It may provide a rough approximation of the low fragmentation stresses when Ne is similar to the actual number of fragments at saturation. However, it is difficult to

974

J. Lamon 1200 1100

Lower bound Ne=2000

Stress (MPa)

1000 900

Lower bound Ne=3000

800

Experiment

700

Bayesian model

600 500

Upper bound Ne=2000

400 300

Upper bound Ne=3000

200

Fragment model

100 0

0

500 1000 Number of cracks

1500

Fig. 10 Stress versus crack number diagrams for the SiC matrix of SiC/SiC minicomposite test specimens: comparison of experimental data with predictions using the Monte Carlo simulation method, the Bayesian model, and the fragment dichotomy model

meet this requirement for prediction purposes, since the actual number of fragments is unknown as object of prediction. Using larger amounts of elements leads to high variability in predictions, overestimation of most of fragmentation strengths, and misestimation of fragmentation stresses (Fig. 10).

4

Single Filament and Multifilament Tows

4.1

Single Filament and Multifilament Tows

Populations of single filaments such as multifilament tows represent a fundamental entity in composites. Tows comprise several hundreds or thousands of single filaments. They progressively carry the load as the matrix is damaged, and they control ultimate failure of 2D fiber-reinforced composites [45]. Although single fibers are brittle, multifilament tows are damage tolerant. They exhibit [46] nonlinear deformations and stable failure as a result of individual successive fiber breaks under controlled deformation rate (Fig. 11). Under certain loading conditions or in the presence of filament interactions, stable failure ends at maximum load (Fig. 11). The fraction of broken filaments at maximum force governs the failure of tow. The particular filament that fails at maximum load is referred to as the “critical fiber.” It has rank Nc, filament strengths being ordered from smallest to largest. The corresponding critical fraction of broken filaments is αc = Nc/Nt (Nt is the initial number of unbroken filaments).

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

Fig. 11 Typical force-strain behavior obtained under strain-controlled conditions for NLM 202 and Hi-Nicalon tows and curves predicted assuming equal load sharing [46]

975

160 Hi-Nicalon 140

Simulation

120

Force (N)

30

100 NLM 202 80 60 40 20 0

0

0.5

1

1.5

2

Strain (%)

4.2

The Bundle Model

Several authors have modeled relationships between tow tensile behavior and fiber properties. H.E. Daniels [47] considered tows containing parallel and non-contacting fibers. He demonstrated that tow strengths are described by a normal distribution for large numbers of fibers. Coleman [48] proposed a relationship between tow and fiber strengths and evidenced a significant drop in tow strength when comparing to mean fiber strength. S.L. Phoenix and H.M. Taylor [49, 50] introduced effects of nonuniform loading resulting from fiber misalignment and scatter in fiber lengths within tows. Calard and Lamon [46] introduced random load sharing. They predicted tow strength drops caused by filament interactions. The bundle models are based upon the following hypotheses [47, 48]: the bundle contains Nt identical and parallel independent filaments, and filament strengths exhibit a statistical distribution. When a fiber fails, the load becomes carried equally by all the surviving fibers, whereas the broken fiber no longer carries any load (equal load sharing). The fraction (α(σ)) of broken filaments is the failure probability of the Nth filament (filament strengths being ordered from smallest to largest) when Nt is large and when filament failures are equally probable events: αðσ Þ ¼

N ¼ Pð σ Þ Nt

(37)

σ is the tensile stress on filaments, and N is the number of broken filaments. The corresponding force F(σ) carried by the bundle is: Fðσ Þ ¼ N t ð1  αðσ ÞÞSf σ with Sf filament cross-sectional area.

(38)

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J. Lamon

According to Sect. 2.2, Weibull or normal distribution expressions for α(σ) can be introduced into Eq. 38. The maximum force Fmax is given by one of the following conditions, depending on the loading mode: – Stable failure under deformation-controlled conditions: dF dσ ¼ 0 – Unstable failure under load-controlled conditions: α = αc Both conditions yield the same expression: Fmax ¼ Fðαc Þ ¼ N t Sf σ F ð1  αc Þ

(39)

where σ F is the strength of critical filament. αc can be satisfactorily estimated using the following relationship: αc ¼ 1  expð1=mÞ

(40)

The expression of coefficient of variation of maximum force (Cv) [47, 51, 52] shows that it is small when the initial number of unbroken filaments Nt is large: SðFmax Þ CV ðFmax Þ ¼ ¼ EðFmax Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αðσ max Þ N t ð1  αðσ max ÞÞ

(41)

where E(Fmax) and S(Fmax) are the expectation and the variance of the maximum force Fmax. Nt  500 in most SiC tows. Therefore, the maximum force should show no variation. Some experimental results are at variance with theory [46]. When scatter in tow strengths was observed, it was attributed to imperfect local load sharing caused by fiber interactions (such as friction) or dynamic effects due to individual fiber breaks [46]. Imperfect load sharing induces a drop in tow strengths (Fig. 11). Fiber stress redistribution depends on the degree of interaction of the broken fiber with its neighbors. Taking into account, random local load sharing allowed sound predictions of tow strengths and associated scatter [46]. Although tow strength data may be influenced by extrinsic factors, testing of tows is an interesting technique to determine fiber properties [28–32].

4.3

Ultimate Failure of Single Filaments in the Presence of Fragmented Matrix

4.3.1 Fragment Dichotomy Model The characteristics of matrix multiple cracking are determined using the fragment dichotomy model. The approach to ultimate failure is based on the survival probabilities of fiber volume elements subjected to the stress state generated by the presence of matrix and interface cracks as shown in Fig. 8 [5]. According to the

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

977

weakest-link principle, the failure probability is obtained from the product of no failure probabilities of filament volume elements: P ¼ 1  π ð1  Pi Þ π ð1  PjÞ

(42)

  mf  σf f Pw ¼ 1  expXXX  Sf l ew σ of

(43)

t

k

with

w = i for those elements where ldi < li, and w = j for those where ldi > li, σ f is the f stress operating on fiber (equations are given in Appendix A), and lew is equivalent length. Failure probability of the filament is: "   # X f σ f mf X f Pf ¼ 1  expXXX  Sf ð l ei þ l ej Þ σ of t k

(44)

where lfei and lfej are equivalent lengths, respectively, for the k elements where ldi  li and the t elements where ldi > li.   Fiber strength can be derived in terms of reference strength σ Rf Pf , using the same method as above for matrix fragments: " #1=m     X l fei X l fei R σ f Pf ¼ σ f Pf þ L L t k

(45)

Then, for li  ldi, the following equation of fiber failure probability can be derived from Eq. 44, in the presence of n cracks in the matrix:   mf   σf 2nldi A VF 1 þ Pf ¼ 1  exp  σ of L

(46)

where ldi is the average debond length and VF is fiber volume.  A¼

" ! #  u 1  β ð1 þ aÞmf þ1  1 mf þ β ð 1 þ aÞ þ 1 ld mþ1 a

(47)

A is an increasing function of load sharing parameter a = EmVm/EfVf. After saturation of matrix cracking (li = ldi), Eq. 46 leads to the following equation of stress on fiber: " σ f ¼ σ of

1 Ln 1P f

#1=mf

V F ð1 þ AÞ

(48)

978

J. Lamon

4.3.2 Implications When the contribution of matrix to load sharing approaches 0, A ! 0 and Eq. 48 tends to:  σ f ¼ σ of

 1 =m 1 1 Ln VF 1  Pf

(49)

From the comparison of Eqs. 46, 47, and 49 at given value of Pf, it appears that σ f increases with decreasing sharing parameter a. Furthermore, Eq. 49 defines the failure stress of a filament subjected to a uniform tension, which represents the worst loading condition. It can be deduced that fibers should not fail prior to saturation of matrix cracking, unless extra loads or localized phenomena make fiber failure probability to increase. Ultimate failure of critical filament that dictates ultimate failure of tows and composite is derived from Eq. 49 for Pf = αc.

4.3.3 Curtin’s Model The approach to calculating 1D composite stress-strain curve as a function of matrix cracking that was discussed in [53] was motivated by the need for a detailed understanding of tensile failure in brittle matrix composites. It is based on Weibull forms for the number of flaws in the matrix and in the fiber. Curtin considered that, in a composite, there must be some characteristic matrix strength that is related to a characteristic volume of composite [53]. This characteristic volume is associated with slip around the matrix cracks. The number of flaws was written as a function of associated critical matrix cracking stress σ R and critical slip length δR. A main outcome of analyses in the literature is that at matrix crack saturation, there is a characteristic matrix crack spacing that is related to the characteristic slip length: δR ¼

rσ R τ

(50)

where r is fiber radius and τ is the interface shear stress. Using average crack spacing, the stress profile on fiber is integrated to yield composite strain. The stress-strain equation is a function of various stresses (thermal stress, minimum stress, debond stress, critical matrix cracking stress) and effective statistical parameters. All these parameters are not intrinsic by essence, and they show variation. Estimating these parameters is an issue. As for the matrix strength, it is considered that there is a characteristic length δC and an associated characteristic strength σ C that are relevant for the fibers in the composite. 1  m mþ1 rσ c σ 0 τL0 δc ¼ ¼ τ r

(51)

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

979

where σ 0 is here the fiber characteristic strength at a tested gauge length L0. m is the Weibull modulus. The ultimate strength is directly proportional to the characteristic strength:  σu ¼ Vf σc

2 mþ2

1  mþ1

mþ1 mþ2

 (52)

Equation 52 is based also on global load sharing when the fibers begin to fail and on the assumption that the fibers broken within a sliding length carry some load. It was found that it underestimates significantly the ultimate strength of SiC/SiC minicomposites. Furthermore, significant underestimations of composites strengths were also reported in [54]. Curtin indicated that this equation has no rigorous applicability to the composite performance, but it is often utilized nonetheless.

5

Application to the Prediction of Tensile Behavior

5.1

Unidirectionally Reinforced Composite

Figure 11 shows the nonlinear tensile curve for a SiC/SiC minicomposite specimen with a BN interphase that was predicted for the fragment dichotomy model of matrix cracking [4]. Predictions in good agreement with experimental results were also obtained on SiC/SiC minicomposites with PyC interphase [5–7] or with multilayered PyC/SiC interphases [6]. Pi was set to 0.5. Figure 12 shows that using random numbers for Pi does not influence significantly the predictions. A good approximation of behavior is obtained for Pi = 0.5 and X = 0.25 (cracks are placed at the center of fragments) (Fig. 13).

5.2

2D Woven Composite: Numerical Approach

Matrix damage can be simulated as a process of multiple failures in volume elements in the matrix and in the tows for 1D composite or in the longitudinal and the transverse tows in 2D composite. In [2, 3] a finite element analysis of failure probabilities has been conducted for the computation of matrix damage evolution in 2D woven SiC/SiC and C/C composites. Stresses were computed using a finite element commercial computer code and, then, failure probabilities using the postprocessor CERAM which uses the multiaxial elemental strength model [1, 19–21]. The finite element mesh reproduced the composite structure with different tow directions, interply matrix, and macropores. The cracking process involves crack initiation from flaws and extension element by element assuming that the elements contain a sufficient amount of flaws. The deformations were increased stepwise. Stress and failure probabilities were computed in mesh elements. Cracks were introduced in critical elements using a

980

J. Lamon 250

Length of minicomposite: 25.04 mm Interface shear stress = 84 MPa Vf = 0.4 Em = 400 Gpa Ef = 300 Gpa mm = 5.03 mf = 4.2 S0m = 5.7 Mpa S0f = 6 Mpa Srf = 0 Mpa Number of fibres : 500 Radius of fibres : 7 micrometers

Force (N)

200

150

100

experiment prediction

50

0

0

0.1

0.2

0.3

0.5

0.4

0.6

0.7

0.8

Strain (%) Fig. 12 Experimental and predicted force-strain behavior for a Hi-Nicalon/BN/SiC minicomposite test specimen with a BN interphase at fiber/matrix interface. The characteristics of constituents are indicated on the figure. They were estimated independently [4]

2500

Pi,X=random

2000 Stress s (MPa)

Fig. 13 Stress-strain curves for SiC/SiC single fiberreinforced composite predicted using the fragment dichotomy-based model for various values of matrix fragment failure probability (Pi) and crack location probability (X) and using uniform distribution of cracks (Vi = VOM/n)

Pi =0.5, uniform distribution 1500 Pi =0.5, X=random

1000

Pi =0.5, X=0.25

500

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Strain e (%)

node-splitting technique. A critical element is characterized by the failure probability of 1. When a critical element appears, successive computations are done at constant deformation, and the crack is increased element by element, while failure probability at crack tip is 1. Computations of crack locations and damage evolution under uniaxial tension compared quite well with in situ observations on 2D SiC/SiC and C/C composites

30

Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics. . .

981

Table 3 Comparison of observed and predicted damage modes in the matrix of 2D woven SiC/SiC composites under tensile load

Damage mode Cracking from macropores Matrix cracking in transverse tows Matrix cracking in longitudinal tows

In situ observations Deformation (%) Occurrence 0.035 Yes 0.085 Yes

Computations Deformation (%) 0.04 0.11

Occurrence Yes Yes

0.185

0.2

Yes

Yes

Fig. 14 Predicted crack pattern for a cell of 2D woven SiC/SiC made via CVI, under flexural load. The mesh reproduces the structure with the interply matrix, a macropore, and transverse and longitudinal tows

Young's modulus (GPa)

300 250 200 Simulation 1

150

Simulation 2

100

Simulation 3 Experiment

50 0

0

0,2

0,4 0,6 Strain (%)

0,8

1

Fig. 15 Young’s modulus versus applied strains for 2D woven SiC/SiC composite under tensile loading parallel to longitudinal fiber axis: comparison of results of experiments and computations for different ply arrangements: Simulation 1: nonparallel tow undulation, no fiber/matrix debonding. Simulation 2: nonparallel tow undulation, with fiber/matrix debonding. Simulation 3: parallel tow undulation, with fiber/matrix debonding

(Table 3). An example of crack pattern obtained under bending conditions is shown in Fig. 14. Figure 15 shows that predictions of composite Young’s modulus during damage compared satisfactorily with experimental results. This approach can be applied to 3D meshes.

982

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J. Lamon

Concluding Remarks

Probabilistic micromechanics allow modeling of damage of CMCs by multiple cracking of the matrix and population of fibers during loading, the cracks being generated by flaws having random distribution. The stress-strain behavior of composite is then derived from the stress state on fibers in the presence of cracks in the matrix and associated debonding. From the point of view of damage, multifilament tows can be assimilated to a parallel system with the failure of filaments that obeys extreme value theory. By contrast, the matrix can be assimilated to a series system. An extreme value theory-based approach and a Bayesian approach have been shown to be suitable for modeling multiple matrix cracking. The former one reproduces the process of formation of successive fragments as the load is increased and provides the evolution of the crack pattern as well as the series of stresses at crack formation. The latter describes the evolution of the distribution of flaw strengths as flaws are eliminated by the formation of matrix cracks and provides the corresponding set of stresses at crack formation. The physics-based approaches consider the flaws as physical entities. They are based on flaw density function. The Weibull approach is a particular solution of the elemental strength approach. It is suitable for uniaxial loading cases. The elemental strength approach is based on distribution of flaw strengths (flaw strength density function). The elemental stress is the stress that operates on a flaw. The multiaxial elemental strength model is suitable for numerical analysis of matrix damage of 2D or 3D reinforced composites and for determination of stress-strain curves. The statistical parameters that define the flaw strength distributions pertinent to the fibers can be estimated using tensile tests on filaments or multifilament tows. For those parameters pertinent to the matrix, micro- or minicomposites are required. Methods that provide large sets of strength data (like matrix fragmentation or tensile testing of tows) are preferred, in order to limit variation of estimated statistical parameters.

Appendix Stress on fiber in the presence of fragmented matrix: – In the vicinity of interfacial matrix cracks: u þ lo  x < ld σ m ðxÞ ¼ σ m

x  u  lo ld  lo

  x  u  lo σ f ðxÞ ¼ σ f 1 þ a  a ld  lo – In the rest of fragment: u þ ld < x < 2 li  (u þ ld)

(53) (54)

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983

σm ¼

σ a þ σ th m Vm 1 þ a

(55)

σf ¼

σ 1 þ σ th f Vf 1 þ a

(56)

where σ m and σ f are, respectively, the stresses operating on the fiber and the matrix; u, ld, and lo are defined in Fig. 8; 2li is length of fragment i; σ is the remote stress applied to the specimen; Vf and Vm are the volume fractions of fiber and matrix, respectively; a ¼ EEmf VV mf is the load sharing parameter; Ef and Em are fiber and matrix th Young’s moduli; and σ th m and σ f are the residual stresses, respectively, in the matrix and in the fiber.

References 1. Lamon J. Brittle fracture and damage of brittle materials and composites. London/Oxford: ISTE Press Ltd/Elsevier Ltd; 2016. 2. Lamon J, Thommeret B, Percevault C. Probabilistic-statistical approach to the matrix damage and stress-strain behavior of 2-D woven SiC/SiC ceramic matrix composites (CMCs). J Eur Ceram Soc. 1998;18:1797–808. 3. Guillaumat L, Lamon J. Probabilistic-statistical simulation of the nonlinear mechanical behavior of a woven SiC/SiC composite. Compos Sci Technol. 1996;56:803–8. 4. Pailler F, Lamon J. Micromechanics-based model of fatigue oxidation behaviour for SiC/SiC composites. Compos Sci Technol. 2005;65:369–74. 5. Lissart N, Lamon J. Statistical analysis of failure of SiC fibres in the presence of bimodal flaw populations. J Mater Sci. 1997;32:6107–17. 6. Bertrand S, Forio P, Pailler R, Lamon J. Hi-Nicalon/SiC minicomposites with (PyC/SiC)n nanoscale-multilayered interphases. J Am Ceram Soc. 1999;82(9):2465–73. 7. Weibull W. A statistical theory of the strength of materials, Ingeniorsvetenkapsakodemiens hadlinger NR 151. Stockholm: Generalstabens Litografiska Anstaltz Förlag; 1939. 8. Weibull W. A statistical distribution function of wide applicability. J Appl Mech. 1951; 18:293–7. 9. De Jayatilaka S, Trustrum K. Statistical approach to brittle fracture. J Mater Sci. 1977; 10:1426–30. 10. De Jayatilaka S, Trustrum K. Application of a statistical method to brittle fracture in biaxial loading systems. J Mater Sci. 1977;12:2043–8. 11. Poloniecki JD, Wilshaw TR. Determination of Surface Crack Size Densities in Glass. Nature. 1971;229:226. 12. Foray G, Descamps-Mandine A, R’Mili M, Lamon J. Statistical distributions for glass fibers: correlation between bundle test and AFM-derived flaw size density functions. Acta Mater. 2012;60:3711–8. 13. Hild F, Marquis D. A statistical approach to the rupture of brittle materials. Eur J Mech A Solids. 1992;11(6):753–65. 14. Batdorf SB, Crose JG. A statistical theory for the fracture of brittle structures subjected to nonuniform polyaxial stresses. J Appl Mech Trans ASME. 1974;41:459–64. 15. Argon AS, McClintock FA. Mechanical behavior of materials. Reading: Addison-Wesley; 1966. 16. Argon S. Surface cracks on glass. Proc R Soc London Ser A. 1959;250:482–92.

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17. Evans AG, Langdon TG. Structural ceramics. Prog Mater Sci. 1976;21:320–31. 18. Evans G. A general approach for the statistical analysis of multiaxial fracture. J Am Ceram Soc. 1978;61(7–8):281–376. 19. Lamon J. Ceramics reliability: statistical analysis of multiaxial failure using the Weibull approach and the multiaxial elemental strength model. J Am Ceram Soc. 1990;73(8):2204–12. 20. Lamon J, Evans AG. Structural analysis of bending strengths for brittle solids: a multiaxial fracture problem. J Am Ceram Soc. 1983;66(3):177–82. 21. Lamon J. Statistical approaches to failure for ceramic reliability assessment. J Am Ceram Soc. 1988;71(2):106–12. 22. Duffy SF, Janosik LL, Wereszczak AA, Schenk B, Suzuki A, Lamon J, Thomas DJ. Chapter 25, Life prediction of structural components. In: Van Roode M, Ferber MK, Richerson DW, editors. Ceramic gas turbine component development and characterization, vol. II. New York: ASME Press; 2003. p. 553–606. 23. Peirce FT. Mechanical properties of boron fibers. J Text Inst Trans. 1926;17:355. 24. Epstein B. Application of the theory of extreme values in fracture problems. Am Stat Assoc J. 1948;43(243):403–12. 25. Lu C, Danzer R, Fischer FD. Phys Rev E. 2002;65:067102:1. 26. R’Mili M, Godin N, Lamon J. Flaw strength distributions and statistical parameters for ceramic fibers: the normal distribution. Phys Rev E. 2012;85(5):1106–12. 27. Lamon J, Lissart N, Rechiniac C, Roach DM, Jouin JM. Micromechanical and statistical approach to the behavior of CMCs. In: Composites and advanced ceramics, Proceedings of the 17th annual conference and exposition, Cocoa Beach, Florida (USA), 10–15 janvier 1993, The American Ceramic Society, Ceramic Engineering and Science Proceedings, Sept–Oct, p. 1115–24. 1993. 28. Chi Z, Wei Chou T, Shen G. Determination of single fiber strength distribution from fiber bundle testing. J Mater Sci. 1984;19:3319–24. 29. Lissart N, Lamon J. Evaluation des propriétés de monofilaments à partir d’essais de traction sur mèches. In: Favre JP, Vautrin A, editors. Comptes-rendus des 9 Journées Nationales sur les Composites (JNC9), vol. 2. AMAC. Paris, France; 1994. p. 589–98. 30. R’Mili M, Murat M. Caractérisation des fibres par amélioration de l’essai sur mèche avec mesure directe de la déformation. C R Acad Sci. 1997;324(6):355–64. 31. R’Mili M, Bouchaour T, Merle P. Estimation of Weibull parameters from loose bundle tests. Compos Sci Technol. 1996;56:831–4. 32. (ENV1007-5). Advanced technical ceramic-ceramic composites – methods of test for reinforcements – Part 5: Determination of distribution of tensile strength and tensile strain to failure of filaments within a multifilament tow at ambient temperature. European Committee for Standardization, CEN TC 184 SC1, Brussel; 1997. 33. Lamon J, Mazerat S, R’Mili M. Reinforcement of ceramic matrix composites: properties of SiCbased filaments and tows. In: Bansal NP, Lamon J, editors. Ceramic matrix composites: materials, modeling and applications. Hoboken: Wiley; 2014. p. 3–26. ISBN 978-1-118-23116-6. 34. Eckel J, Bradt RC. Statistical analysis of failure of SiC fibres in the presence of bimodal flaw populations. J Am Ceram Soc. 1989;72(3):455–8. 35. Goda K, Fukunaga H. The evaluation of the strength distribution of silicon carbide andalumina fibres by a multi-modal Weibull distribution. J Mater Sci. 1986;21:4475–80. 36. Lamon J. Stochastic models of fragmentation of brittle fibers or matrix in composites. Compos Sci Technol. 2010;70:743–51. 37. Lamon J. Stochastic approach to multiple cracking in composite systems based on the extreme value theory. Compos Sci Technol. 2009;69:1607–14. 38. Hui C-Y, Phoenix SL, Ibnabdeljalil M, Smith RL. An exact closed form solution for fragmentation of Weibull fibers in single filament composite with applications to fiber-reinforced ceramics. J Mech Phys Solids. 1995;43(10):1551–85.

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39. Baxevanakis C, Jeulin D, Valentin D. Fracture statistics of single-fiber composite specimens. Compos Sci Technol. 1993;48:47–56. 40. Wagner HD. Stochastic concepts in the study of size effects in the mechanical strength of highly oriented polymeric materials. J Polym Sci B Polym Phys. 1989;27:115–49. 41. Okabe T, Takeda N, Komotori J, Shimizu M, Curtin WA. A new fracture mechanics model for multiple matrix cracks of SiC reinforced brittle-matrix composites. Acta Mater. 1999;47(17): 4299–309. 42. Rousset G, Lamon J, Martin E. In situ fiber strength determination in metal matrix composites. Compos Sci Technol. 2009;69(15–16):2580–6. 43. Glushko V, Kovalenko V, Mileiko S, Tvardovsky V. Evaluation of fiber strength characteristics using fiber fragmentation test. In: Mileiko S, Tvarkovsky V, editors. Composites: fracture mechanics and technology. Chernogolovska: Russian Composite Society; 1992. p. 66–76. 44. Pailler F, Lamon J. Micromechanics-based model of fatigue/oxidation for ceramic matrix composites. Compos Sci Technol. 2005;65:369–74. 45. Lamon J. A micromechanics-based approach to the mechanical behavior of brittle-matrix composites. Compos Sci Technol. 2001;61:2259–72. 46. Calard V, Lamon J. Failure of fibres bundles. Compos Sci Technol. 2004;64:701–10. 47. Daniels HE. The statistical theory of the strength of bundles of threads I. Proc R Soc. 1945; A183:405–35. 48. Coleman BD. On the strength of classical fibers and fibers bundle. J Mech Phys Solids. 1958;7:60–70. 49. Phoenix SL, Taylor HM. The asymptotic strength distribution of a general fiber bundle. Adv Appl Probab. 1973;5:200–16. 50. Phoenix SL. Probabilistic strength analysis of fiber bundles structures. Fiber Sci Technol. 1974;7:15–31. 51. McCartney LN, Smith RL. Statistical theory of the strength of fiber bundles. ASME J Appl Mech. 1983;105:601–8. 52. Gurvich M, Pipes R. Strength size effect of laminated composites. Compos Sci Technol. 1995;55:93–105. 53. Curtin WA. Stress-strain behavior of brittle matrix composites. In: Kelly A, Zweben C, editors. Comprehensive composites materials, Elsevier Science Ltd. Great Britain, vol. 4; 2000. p. 47–76. 54. Curtin WA. Ultimate strengths of fibre-reinforced ceramics and metals. Composites. 1993; 24(2):98–102.

Micromechanics of Polymeric Materials in Aggressive Environments

31

Xiaohong Chen

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Development of Coupled Multidisciplinary Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonlinear Hygro-thermo-viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nonlinear Chemo-thermo-viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Advancement in Computational Micromechanics for Aggressive Environment Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hygrothermal Environment Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Thermal Oxidative Environment Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Future Directions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

988 990 990 992 996 998 999 1000 1007 1008 1009

Abstract

The advantages of polymeric materials have led to substantial interests in a variety of engineering applications, such as high-speed aircrafts, ultra-deepwater operations, and biomedical devices. Polymeric materials used in these systems must be capable of withstanding aggressive environments and still maintain their long-term load-bearing capability. With the development of novel polymer blends and composites to meet physical, chemical, thermal, and mechanical requirements, constitutive relations, deformation and toughening mechanisms, and damage and fracture characteristics of these multiphase polymeric materials have drawn significant attention. Multidisciplinary principles are required for studying the behavior of polymeric materials subjected to combined thermomechanical

X. Chen (*) Aerostructures, UTC Aerospace Systems, Chula Vista, CA, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_65

987

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X. Chen

loading and physicochemically active environment exposure. The major challenge lies in the coupling of physical, chemical, thermal, and mechanical effects at multiple scales. It is necessary to establish generally applicable material models for implementation in robust design analysis tools. In this chapter, an overview on development of coupled multidisciplinary approach and advancement in computational micromechanics for polymeric materials in aggressive environments is provided with discussion on key methods, future directions, and open issues. Keywords

Micromechanics · Polymeric materials · Aggressive environments · Coupled multidisciplinary approach

1

Introduction

The term “polymer” has ancient Greek origin, referring to a large molecule consisting of multiple repeating units known as monomers. Since polymeric materials have many advantages over metallic materials, e.g., lightweight, easy manufacture, corrosion resistance, and cost-effectiveness, aerospace, transportation, packaging, building, and construction have greatly benefited from their usage. It is well-known that lightweight means less fuel consumption for aircrafts and trucks. Polymeric materials with time and temperature dependence can be classified as nonlinear viscoelastic solids, exhibiting both viscous and elastic characteristics. Schapery [1] reviewed constitutive equations, strength, and fracture models for nonlinear viscoelastic solids based on two types of nonequilibrium thermodynamics: functional thermodynamics and state-variable thermodynamics. In functional thermodynamics, the free energy is expressed as a functional of the histories of strain (stress), temperature, etc. with the employment of a laboratory or material time scale. In state-variable thermodynamics, the free energy is expressed as a function of current strain (stress), temperature, and other variables, including so-called internal state variables (ISVs). An individual polymer often cannot satisfy all the physical, mechanical, thermal, chemical, and optical requirements for a specific application. Thus particulatemodified polymers, rigid-rigid polymer blends, and fiber-reinforced, polymer-matrix composites (PMCs), consisting of two or more organic or inorganic constituents, have been developed to achieve better mechanical performance, thermal stability, and chemical resistance. Studies of the relationships between microscopic structure and macroscopic property are necessary for tailoring composite materials to meet specific design requirements. The evaluation of effective properties of heterogeneous material from those of its constituents is one of the classical problems in micromechanics (e.g., [2]). Micromechanical modeling forms a unified approach in the prediction of the effective properties of composite materials through analyzing individual constituents (e.g., viscoelastic, elastic-plastic, or nonlinear elastic) and enables us to investigate the influence of parameters such as volume fraction, aspect

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ratio, spacing, and layout of fibers or inclusions. It allows the analysis of a representative volume element (RVE) rather than the whole composite, on the assumption that a two-phase composite has a microscopic structure in which the reinforcing phase is arranged periodically. Bridging rules such as affine assumption, homogenization technique, and internal state-variable approach have been used for multi-scale formulations [3–7].Puglisi and Saccomandi [6] assessed multi-scale modeling of rubberlike materials and soft tissues, linking the macroscopic stretches with the network chain elongations via specific network cells (single chain, 4-chain tetrahedral structure, 8-chain cubic structure, etc.) from statistical mechanics analysis. Tong and Mei [7] studied mechanics of composites of multiple scales with the homogenization technique, providing the properties or response on the higher-scale level given the properties or response on the lower-scale level. The periodic solutions at the micro- or mesoscale are first solved by finite element analysis (FEA). These solutions are then used to establish the constitutive relations at the macroscale. Because FEA is used, complex geometry can be treated in a routine manner. The use of thermodynamically constrained ISVs that can be determined based upon microstructure-property relations is a top-down approach, indicating that the ISVs exist at the macroscale but reach down to various subscales to receive pertinent information. In-depth understanding of constitutive relations, deformation and toughening mechanisms, and damage and fracture characteristics of polymeric materials is crucial for material development, design validation, and life prediction. The addition of rubber particles as potent craze or shear yielding initiation sites is proved to be a very efficient way to enhance the toughness of many brittle or semi-brittle polymers such as polystyrene, polycarbonate, nylon, and epoxy. The dominant toughening mechanisms have been identified as rubber cavitation, matrix crazing, or shear yielding. The significance of rubber cavitation and the sequence of the associated toughening mechanisms have been the subject of many studies as summarized in [8].Rubber modification is an effective way to improve fracture toughness; however, unfortunately it also leads to reduction in modulus and strength. Conversely, inorganic fillers and reinforcements are widely used to enhance the strength and rigidity of ductile plastics but generally sacrifice toughness. In the development of a new generation of polymer blends with optimum combination of toughness, strength, and modulus for a wider range of engineering applications, the rigid-rigid polymer toughening concept originates from the idea of rubber toughening. PC/ABS, PA/PPO, PBT/PC, and PEI/PC blends with or without impact modifiers are typical examples of such blends. It is noteworthy that Chen and Mai proposed a face-centered cubic unit cell model for micromechanical modeling of rubber-modified polymers [9] and rigid-rigid polymer blends [10] and also developed a particle-crack tip interaction model to study the deformation and fracture behavior of rubber-modified polymers at different stress triaxiality using 3D elastoplastic FEA with sub-modeling technique [11]. Polymer degradation is often caused by the scission of polymer chains via hydrolysis under environmental factors such as heat, light, moisture, oxygen, and

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other chemical agents, resulting in change in dimension, weight, and properties [12–15]. The addition of reinforcements such as fibers or inclusions brings about additional complexity due to microstructure inhomogeneity and anisotropy. Absorption of moisture or water in PMCs is one of the primary environmental causes of their deterioration and loss of service life. Furthermore, degradation of PMCs with exposure to thermal oxidative environment is a major concern for high-performance propulsion applications. The length scales range from fibers or particles with sizes in microns, to individual plies with thicknesses in fractions of millimeters, to laminates with thicknesses in millimeters. The governing equations for diffusion, reaction, heat transfer, damage evolution, and viscoelastic deformation at different scales are controlled by multidisciplinary principles, including physics, chemistry, mechanics, thermodynamics, etc. Thus evaluation of environmental effects on the behavior of PMCs requires multidisciplinary approach and multi-scale modeling to define problems outside normal boxes. Due to lack of general material models and robust analysis tools for such complex problems, an industry-acceptable methodology is to use environmental knockdown factors for design of composite structures operating in aggressive environments. It is extremely costly and time-consuming to conduct a large variety of tests on statistically sufficient samples under various environmental conditions. Thus it is necessary to develop the ability to predict deformation, damage, and failure mechanisms of PMCs at multiple scales under combined mechanical loading and environmental attack. The objective of this chapter is to provide an overview on micromechanics of polymeric materials in aggressive environment. Sect. 2 recalls basic equations and fundamental aspects for development of coupled multidisciplinary approach. Sect. 3 reports recent advancement in computational micromechanics for hygrothermal or thermal oxidative environment application. The last section is devoted to future directions and open problems.

2

Development of Coupled Multidisciplinary Approach

A coupled multidisciplinary approach, as illustrated in Fig. 1, provides an effective way to study the behavior of polymeric materials in aggressive environments. The multiple disciplines should be coupled at the level of fundamental equations so as to accurately capture the multidisciplinary aspects of a complex problem.

2.1

Basic Equations

2.1.1 Mass Balance Equations Consider simultaneous diffusions and reactions in a polymeric material. Based on the conservation law of mass, the concentration of the kth constituent and the total mass must satisfy the following balance equations, also called diffusion-reaction equations:

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Thermomechanics

Physics

Fundamental Equations

Chemistry

Materials Science Fig. 1 Coupled multidisciplinary approach

dcðkÞ 1 dρ ¼ ρ∇  v ¼  ∇  jðmkÞ  r ðkÞ , ρ dt dt

(1)

with r ðkÞ ¼ 

N 1 X ðkÞ ðkÞ _ X νp M ξp , νðpkÞ MðkÞ ¼ 0 ρ p k¼1

(2)

where d/dt = (@/@t + v  ∇) is the material derivative operator, c(k) = ρ(k)/ρ is the N N P P mass fraction of the kth constituent, cðkÞ ¼ 1, ρ ¼ ρðkÞ is the total mass density, k¼1

ρ

(k)

is the density of the kth constituent, v ¼ u_ ¼

k¼1

N P

ρðkÞ vðkÞ =ρ is the mass center

k¼1

velocity, u is the mass center displacement, v(k) is the velocity of the kth constituent, N   P jðmkÞ ¼ 0, r(k) is the jðmkÞ ¼ ρðkÞ vðkÞ  v is the diffusion flux of the kth constituent, k¼1

consumption rate of the kth constituent, ξ_p is the rate of the pth reaction, ξp is ðk Þ the extent of the pth reaction, νp is the stoichiometric coefficient of the kth constituent in the pth reaction equation, and M(k) is the molar mass of the kth constituent.

2.1.2 Energy Balance Equation Based on the first principle of thermodynamics (conservation law of energy), the local energy balance equation is given by ρ

N1  ðk Þ X d^e ðN Þ ¼ ∇  jq þ σ : v∇ þ jðmkÞ  ^f  v_ ðkÞ  ^f þ v_ ðN Þ Þ dt k¼1

(3)

where ^e is the internal energy per unit mass, jq is the heat flux, and the condition N P jðmkÞ ¼ 0 is used to eliminate jðmNÞ .

k¼1

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2.1.3 Entropy Production Inequality Based on the second principle of thermodynamics, the entropy production inequality is given by d i^s d^s 1 ¼ þ ∇  js  0 dt ρ dt

(4)

where ^s is the entropy per unit mass and js is the entropy flux. The relationships between the thermodynamic fluxes (e.g., heat flux, diffusion flux, rate of reaction, rate of damage) and the thermodynamic forces (e.g., temperature gradient, chemical potential gradient, affinity of reaction, damage driving force) must satisfy the entropy production inequality.

2.1.4 Thermodynamic Potentials The following thermodynamic potentials are of particular importance: Helmholtz free energy per unit mass : ^h ¼ ^e  T^s

(5)

^ ¼ ^e  T^s  1 Σ:Ε Gibbs free energy per unit mass : Ω ρ0

(6)

where T is the absolute temperature, Σ = F1σFTρ0/ρ is the second Piola-Kirchhoff stress tensor, Ε = (FTF  I)/2 is the Green strain tensor, ρ0 is the initial mass density, σ is the Cauchy stress tensor, F = @x/@X is the deformation gradient, I is the unit tensor, x = x(X, t) is the current position, and X is the initial position. Knowledge of a free energy allows derivation of constitutive equations based on nonequilibrium thermodynamics, e.g., viscoelasticity or thermoviscoelasticity (see review article [1]). Fundamental aspects of nonlinear hygro-thermo-viscoelasticity and chemo-thermo-viscoelasticity will be summarized below.

2.2

Nonlinear Hygro-thermo-viscoelasticity

With increasing application of lightweight polymeric materials in aerospace and offshore structures, effects of temperature and fluid on deformation, strength, and durability have attracted more and more attention (e.g., [12–22]). Fluid sorption in polymeric materials causes swelling and weight gain. Swelling in the polymer matrix cannot develop freely due to significant mismatch in hygroscopic expansion between the matrix and the reinforcement, which causes internal stress buildup. Long-term aging and environmental degradation may also happen simultaneously, resulting in loss of strength and durability. Therefore, considerable efforts have been made to develop analytical models for studying the behavior of polymeric materials subjected to combined hygro-thermo-mechanical loadings. A major challenge for this class of problems is to understand the coupling and interaction

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among fluid absorption, heat transfer, microstructure damage, long-term aging, and environmental degradation. Weitsman [16, 17] did pioneering work in modeling stress-assisted diffusion in elastic and viscoelastic materials. Weitsman [18] also examined anomalous fluid sorption in polymeric composites and its relation to fluid-induced damage. Aboudi and Williams [19] conducted a coupled micro-macromechanical analysis of hygrothermoelastic composites. Roy and Xu [20] applied thermodynamic methodology to characterization of non-Fickian diffusion coefficients from moisture weight gain data in the presence of damage in polymer-matrix composites. Moreover, Rajagopal [21] studied diffusion through polymeric solids undergoing large deformations. To address important coupling effects for load-bearing polymeric materials in hygrothermal environment, Chen and Wang [22] presented a thermodynamic approach to coupled fluid transport, heat transfer, long-term deformation, and damage. The Gibbs free energy is taken as a functional of stress, temperature, and fluid concentration with damage being introduced as an ISV. Coupled hygrothermo-viscoelastic constitutive equations can be obtained from the Gibbs free energy functional expansion in an elegant form: Εij ¼

ðt 1

J ijkl ð0, t  λ, X; dÞ

ðt @Σkl ðλ, XÞ @T ðλ, XÞ dλ þ dλ αij ð0, t  λ, X; dÞ @λ @λ 1 ðt @cðf Þ ðλ, XÞ þ dλ þ L0ij βij ð0, t  λ, X; dÞ @λ 1 (7)

ðt

ð @Σij ðτ, XÞ 1 t @T ðτ, XÞ dτ dτ þ ρ^s ¼ αij ðt  τ, 0, X; dÞ Cðt  τ, 0, X; dÞ T @τ @τ 0 1 1 ðt @cðf Þ ðτ, XÞ þ dτ þ M0 ηðt  τ, 0, X; dÞ @τ 1 (8)   Ð @Σij ðτ, XÞ t dτ ρ μ ^ ðsÞ ^ μ ðf Þ ¼ 1 βijðtτ, 0, X; dÞ @τ ðt Ðt @T ðλ, XÞ @cðf Þ ðλ, XÞ þ 1ηð0, tλ, X; dÞ dλ dλþN 0 H ð0, tλ, X; dÞ @λ @λ 1

(9) ^ ðsÞ are chemical potentials of fluid and solid phases, d is the damage where μ ^ ðf Þ and μ tensor, and Jijkl(0, t  λ, X; d), αij(0, t  λ, X; d), βij(0, t  λ, X; d), C(t  τ, 0, X; d), η(0, t  λ, X; d), and H(0, t  λ, X; d) are appropriate thermodynamic property functions (i.e., compliances, coefficients of thermal expansion, coefficients of hygroscopic expansion, heat capacity, hygrothermal coefficient, and hygroscopic capacity).

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Equations (7, 8, and 9) are in general anisotropic form. The first terms on the right-hand side stand for mechanical contributions, the second terms for thermal contributions, the third terms for hygroscopic contributions, and the last term for values in the reference state. Energy can be converted from one form to another due to mechanical, thermal, and hygroscopic coupling, accompanied by intrinsic dissipation associated with mechanical, thermal, and hygroscopic hysteresis. The majority of the reported work in the literature on predicting the long-term response of polymeric materials from short-term data is limited to physical aging models based on the effective time theory [23, 24]. However, hygrothermal aging can be one of the primary environmental causes of life-limiting deterioration for polymeric materials [13–15]. For evaluating the long-term performance of polymeric materials in hygrothermal environment, with introduction of an intrinsic (material) time scale, ψ¼

ðt 0

h0  0   0   0 i 0 ata , T , cf , Σ t ; t0a , T t , cf t , Σ t dt

(10)

the long-term property functions may be obtained from momentary master curves to account for the effects of aging, temperature, fluid concentration, and stress, that is, h i J Mu J ijkl ð0, t, X; dÞ  J uijkl ðX; dÞ ¼ btaijkl, T , cf , Σ J M ð ψ, X; d Þ  J ð X; d Þ ijkl ijkl

(11a)

h i α Mu αij ð0, t, X; dÞ  αuij ðX; dÞ ¼ btaij, T , cf , Σ αM ð ψ, X; d Þ  α ð X; d Þ ij ij

(11b)

h i β Mu βij ð0, t, X; dÞ  βuij ðX; dÞ ¼ btaij, T , cf , Σ βM ð ψ, X; d Þ  β ð X; d Þ ij ij

(11c)

  Cð0, t, X; dÞ  Cu ðX; dÞ ¼ bCta , T , cf , Σ CM ðψ, X; dÞ  CMu ðX; dÞ

(11d)

  ηð0, t, X; dÞ  ηu ðX; dÞ ¼ bηta , T , cf , Σ ηM ðψ, X; dÞ  ηMu ðX; dÞ  M  Mu Hð0, t, X; dÞ  Hu ðX; dÞ ¼ bH ta , T , cf , Σ H ðψ, X; dÞ  H ðX; dÞ

(11e) (11f)

where t0a is aging time prior to loading; ta ¼ t þ t0a is the total aging time; ata , T , cf , Σ β J α is the total horizontal shift factor; btaijkl, T , cf , Σ, btaij, T , cf , Σ, btaij, T , cf , Σ, bCta , T , cf , Σ, bηta , T , cf , Σ, M M M and bH ta , T , cf , Σ are total vertical shift factors; J ijkl ðψ, X; dÞ, αij ðψ, X; dÞ, β ij ðψ, X; dÞ, M M M C (ψ, X; d), η (ψ, X; d), and H (ψ, X; d) are master functions; J uijkl ðX; dÞ, αuij ðX; dÞ, βuij ðX; dÞ, Cu(X; d), ηu(X; d), and Hu(X; d) are instantaneous coefficients; and Mu Mu Mu J Mu (X; d), ηMu(X; d), and HMu(X; d) are ijkl ðX; dÞ , αij ðX; dÞ , βij ðX; dÞ , C instantaneous master coefficients.

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To satisfy the entropy production inequality, the heat and diffusion fluxes are related to the temperature and chemical potential gradients with Onsager’s reciprocal relation, that is, " ! # ^f ðf Þ  ^f ðsÞ 1 μ ^ ðf Þ  μ ^ ðsÞ v_ ðsÞ ðf Þ qm jq ¼ L  ∇ þ L  ∇  v_  þ T T T T qq

jðmf Þ

¼L

mq



(12)

" ! # ^f ðf Þ  ^f ðsÞ 1 μ ^ ðf Þ  μ ^ ðsÞ v_ ðsÞ ðf Þ mm  v_   ∇ þ L  ∇ þ (13) T T T T Lqm is positive definite and LqqT = Lqq, LmqT = Lqm, LmmT = Lmm, Lmm

Lqq Lmq ^f ðf Þ , and ^f ðsÞ are body forces acting on fluid and solid phases. Traditional transport laws such as Fourier’s law of heat conduction and Fick’s law of mass transport as well as the Soret and Dufour effects can be taken as special cases. Effects of stress (pressure) and damage may be taken into account through their influence on the chemical potentials. Substitution of Eqs. (8, 9, 12, and 13) into Eqs. (1) and (3) with the use of (6) yields the coupled mass balance and heat transfer equations. Chen [25] further developed a consistent thermodynamic formulation of a coupled hygro-thermo-viscoelastic fracture theory, which laid a foundation for fracture characterization of load-carrying polymeric materials in hygrothermal environment. From Eq. (7), the total strain under combined mechanical, thermal, and hygroscopic loadings can be expressed as the sum of the mechanical (M), thermal (T ), hygroscopic (H ), and reference-state (R) strains: where

ðMÞ

Εij ¼ Εij

ðT Þ

ðH Þ

ðRÞ

þ Εij þ Εij þ Εij

(14)

Derrien and Gilormini [26] looked into the effect of moisture-induced swelling on the absorption capacity of transversely isotropic elastic polymer-matrix composites with the use of the Helmholtz free energy. The reduced expression of the molar chemical potential is obtained as μ ^ w ¼ μ w ðcÞ þ

3Mw η p ρp

(15)

where μw(c) is the molar chemical potential in the stress-free polymer, p is the hydrostatic pressure, η is the coefficient of moisture expansion, Mw is the molar mass of water, and ρp is the mass density of the dry polymer. Similar expression with the linear dependence of the chemical potential on the hydrostatic pressure was obtained previously, for instance, by Weitsman [16]. Later Sar et al. [27] used a thermodynamical approach for coupling the diffusion

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of moisture to the mechanical behavior of a polymer, based on the definition of the chemical potential of water.

2.3

Nonlinear Chemo-thermo-viscoelasticity

Carbon fiber/epoxy composites are capable of satisfying design requirements for aerospace components with prolonged service temperature up to 250  F/121  C. High-performance propulsion systems have pushed service temperature well beyond the capability of epoxies. Thus high-temperature polymer-matrix composites (HTPMCs) such as polyimides and bismaleimides (BMI) have been developed for aerospace industry. A major advantage of HTPMCs is high specific stiffness and strength. Experimental work has been conducted on thermal oxidation of hightemperature neat resins and their carbon fiber composites by such groups as Bowles et al. [28–30], Wang et al. [31, 32], Schoeppner et al. [33], Abdeljaoued [34, 35], and Colin et al. [36]. Nevertheless, limited research has been attempted to model mechanical property degradation associated with thermo-oxidation. Inherent difficulties involve strong anisotropy of composite microstructure, formation of a nonuniform skin-core structure, and complex roles of the fiber-matrix interphase. Reaction kinetics is the part of physical chemistry that studies the rate of reaction, which depends on various parameters such as concentrations, activation energy, temperature, and pressure. The oxygen consumption rate is approximated by Eq. (16) for PMR-15 after Abdeljaoued et al. [34, 35] or (17) for BMI after Colin et al. [36]:  1 r ¼ r0 1  1 þ β cðOxÞ  2β cðOxÞ β cðOxÞ ¼ r0 1 1 þ β cðOxÞ 2ð1 þ β cðOxÞ Þ ðOxÞ

r ðOxÞ



(16)

(17)

where c(Ox) is the oxygen concentration, r0 is the oxygen consumption rate at the saturation level with temperature dependence through the Arrhenius equation, and β is a reaction parameter. r0 and β can be determined from experimental weight loss data. Pochiraju and Tandon [37] and Tandon et al. [38] studied the thermo-oxidative behavior of high-temperature resin (PMR-15) with a diffusion-reaction model in which temperature, oxygen concentration, and weight loss effects were considered. McManus et al. [39] proposed a mechanism-based modeling of long-term degradation in polymer-matrix composites. Colin et al. [40] built a strategy for studying thermal oxidation of organic matrix composites, based on the kinetic model starting from a radical chain mechanism, initiated by hydroperoxide decomposition, and coupling the oxygen diffusion and consumption kinetics. Since proper material models with inclusion of important coupling effects are critical for the success of analytical prediction and life assessment, Chen et al. [41]

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established a thermodynamic formulation of high-temperature mechanics for PMCs with reactions and damage. Coupled chemo-thermo-viscoelastic constitutive equations can be obtained from the Gibbs free energy functional expansion in an elegant form: ðt @Σkl ðλ, XÞ @T ðλ, XÞ dλ þ dλ Εij ¼ J ijkl ð0, t  λ, X; dÞ αij ð0, t  λ, X; dÞ @λ @λ 1 1 ð X t @ξp ðλ, XÞ dλ þ L0ij  γ pij ð0, t  λ, X; dÞ @λ 1 p ðt

(18) ρ0^s ¼

ðt 1

αij ðt  τ, 0, X; dÞ

ðt

@Σij ðτ, XÞ 1 @T ðτ, XÞ dτ dτ þ Cðt  τ, 0, X; dÞ T 0 1 @τ @τ X ðt @ξp ðτ, XÞ dτ þ M0 þ κp ðt  τ, 0, X; dÞ @τ 1 p (19)

ρ0 Ap ¼  ρ

ðt 1

where Ap ¼ 

γ pij ðt  τ, 0, X; dÞ

X i

ðt @Σij ðτ, XÞ @T ðλ, XÞ dλ dτ þ κ p ð0, t  λ, X; dÞ @λ @τ 1 X ðt @ξq ðλ, XÞ dλ þ N p0  Bpq ð0, t  λ, X; dÞ @λ 1 q (20)

νðpiÞ MðiÞ μ ^ ðiÞ is the affinity for the pth reaction and Jijkl(0, t  λ,

X; d), αij(0, t  λ, X; d), C(t  τ, 0, X; d), γ pij ð0, t  λ, X; dÞ, κp(0, t  λ, X; d), and Bpq(0, t  λ, X; d) are appropriate thermodynamic property functions (i.e., compliance, coefficients of thermal expansion, heat capacity, coefficients of chemical shrinkage, thermochemical coefficients, and reaction capacity). Equations (18, 19, and 20) are in general anisotropic form. The first terms on right-hand sides stand for mechanical contributions, the second terms for thermal contributions, the third terms for chemical contributions, and the last term for the values in the reference state. Energy can be converted from one form to another due to mechanical, thermal, and chemical coupling, accompanied by intrinsic dissipation associated with mechanical, thermal, and chemical hysteresis. Similar to nonlinear hygro-thermo-viscoelasticity discussed in Sect. 2.2 the intrinsic time scale may be used to obtain long-term property functions from momentary master curves to account for the effects of aging, temperature, reaction, and stress for evaluating the long-term performance of polymeric materials in thermal oxidative environment. For reaction far from equilibrium, the Arrhenius-type kinetic equation is usually adopted:

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 Ea _ξp ¼ kp exp  p 1  exp  Ap RT RT

(21)

where Eap is the activation energy for the pth reaction, kp is a rate parameter, and R is the gas constant. From Eq. (18), the total strain under combined mechanical, thermal, and oxidative loadings can be expressed as the sum of the mechanical (M ), thermal (T ), oxidative (Ox), and reference-state (R) strains: ðMÞ

Εij ¼ Εij

ðT Þ

ðOxÞ

þ Εij þ Εij

ðRÞ

þ Εij

(22)

When a linear form is used to relate the diffusion flux to the concentration gradient, the normalized diffusion-reaction equation for isothermal oxidation (see [42]) is given by ðOxÞ

d~ c d~t

@ @ c~ðOxÞ ΩðkÞ ¼ @ x~1 @ x~1

!

@ @ c~ðOxÞ Ωð k Þ þ @ x~2 @ x~2

!

 ΦðkÞ r~ðOxÞ ðmÞ

ðmÞ

where ~t ¼ tD0 =φ2; x~i ¼ xi =φ; ΩðkÞ ¼ DðkÞ =D0 ; ΦðkÞ

@ @ c~ðOxÞ Ωð k Þ þ @ x~3 @ x~3

!

(23)    ðk Þ ðmÞ ¼ r 0 φ2 = D0 cs ; c~ðOxÞ 

¼ cðOxÞ =cs ; r~ðOxÞ ¼ r ðOxÞ =r 0 ; cs is the surface oxygen concentration; ϕ is the fiber ðmÞ diameter; D(k) is the oxygen diffusivity in the fibers, matrix, or interphase; D0 is ðk Þ the oxygen diffusivity in the non-oxidized matrix; and r 0 is the oxygen consumption rate at the saturation level in the fibers, matrix, or interphase. Typically, oxygen diffusion and oxidation reaction in the fibers are negligible in comparison with those in the matrix and interphase, that is, Ω( f )  0 and Φ( f )  0. With the use of Eq. (16), the oxidation-induced mass loss follows

 ~ dm 1 ¼ α Φ 1  d~t 1 þ β c~ðOxÞ

(24)

~ ¼ Δm=m0 is the percentage mass loss, α = cs(vMV  32)/ρ(m), ρ(m) is where m the density of PMR-15 matrix, ν is the stoichiometric coefficient, and MV is the molar mass of volatile products.

3

Advancement in Computational Micromechanics for Aggressive Environment Applications

Multi-scale composite analysis provides an integrated tool to assist both the materials scientists for developing advanced materials and the structural analysts for conducting realistic prediction of performance and life (e.g., [3–7]. The scale levels include the microscale (constituent level), mesoscale (ply level), and macroscale

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(laminate level). The existing micromechanics studies range from the simplest approximations (e.g., Voigt and Reuss) to more involved models (e.g., Eshelby’s equivalent inclusion method, Tanaka-Mori theorem, Hershey-Kröner-BudianskyHill self-consistent scheme, Hashin’s composite sphere model, Hashin-Shtrikman bounds), to fully numerical methods (e.g., finite element method, boundary element method, meshless local Petrov-Galerkin method, high-fidelity generalized method of cells) that are the most general and yet the most computationally intense. Each method has its realm of applicability and advantages. Nevertheless, most of them are unable to cover nonlinear, time-dependent response, thus limiting their ultimate usefulness. Computational micromechanics allows general userdefined material models for multiphysics analysis, which makes it feasible to be extended to aggressive environments. A summary on recent advancement in computational micromechanics for polymeric materials in hygrothermal or thermal oxidative environment is given below.

3.1

Hygrothermal Environment Application

Abhilash et al. [43] conducted micromechanical study on diffusion-induced damage evolution in reinforced polymers. In particular, FEA with commercial code ABAQUS was used to demonstrate the role of fiber distribution in the damage evolution due to fiber-matrix interfacial debonding under moisture ingress. Fick’s law was used for modeling transient moisture diffusion. To study the effect of degrading interfaces on the damage evolution, the cohesive strength was allowed to weaken with the local moisture concentration. Figure 2 compares the microscopic stress distributions in the RVE with different fiber arrangements corresponding to the time at which the RVE attains equilibrium concentration in the entire domain. The deformed profiles are scaled 50 times and fibers are removed for clarity. In analogy to thermal residual stresses, hygroscopic residual stresses are induced by hygroscopic mismatch between the fiber and matrix phases. The results indicate that microstructural heterogeneity strongly affects the moisture diffusion

Fig. 2 Microscopic stress distributions in the RVE with different fiber distributions when steadystate condition is reached (Reprinted from [43], with permission from Elsevier)

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Fig. 3 Different fiber cluster arrangements (Reprinted from [44], with permission from Elsevier)

characteristics that in turn hurt the overall load-carrying capacity of a composite due to aggravated damage. Jain et al. [44] further developed local micromechanical model of moisture diffusion in PMCs with an emphasis on the diffusion behavior of different filament clusters with variable densities (see Fig. 3) to serve as a clue for designing optimum filament geometry for impeding moisture ingress and improving durability of composites.

3.2

Thermal Oxidative Environment Application

Chen et al. [41] and Wang and Chen [42] developed a computational micromechanics approach based on the coupled high-temperature constitutive equations with reaction and damage so as to address oxidation-induced mass loss, chemical shrinkage, residual stresses, microstructure damage, and property degradation of PMCs in thermal oxidative environment. The computational micromechanics approach accounts for complex interactions among neighboring fibers and enables us to investigate the effects of the oxidized surface layer and active oxidation zone along the fiber-matrix interface on local stress concentrations. Numerical simulation was given for a commonly used G30-500/PMR-15 composite with the RVEs for transverse and axial oxidation as shown in Fig. 4. A fiber volume fraction of 0.65 and a fiber diameter of 7 μm were used for the case study. The isothermal oxidation was described by Eq. (23). The mass loss was calculated by Eq. (24). The mechanical, thermal, and oxidative contributions to the total strain were represented by Eq. (22). A homogenization technique was adopted in conjunction with finite element representation of material and oxidation details to determine the evolution

31

Micromechanics of Polymeric Materials in Aggressive Environments

1001

Fig. 4 RVEs for oxidizing fiber composite with staggered periodic structure (Reprinted from [41, 42], with permission from ICCM-14 and ASME)

of composite microstructure and property degradation. The microscopic stress and strain fields were used to determine the effective properties through volume averaging in the oxidizing composite. The oxidation-induced damage variables were calculated through the effective modulus reductions. The effective properties and damage variables can eventually be used for composite structure analysis at the macroscale. Thermal residual stresses are expected in the composite due to mismatch in the coefficients of thermal expansion for carbon fibers and the polyimide matrix.

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Due to the mismatch in the coefficients of chemical shrinkage for carbon fibers and the polyimide matrix as well as the formation of the nonuniform skin-core structure, the oxidation reaction introduces additional oxidative residual stresses in the composite. To illustrate quantitatively the effects of thermal oxidation on microscopic stresses, a comparison was made on the pure thermal stresses and the thermal oxidative stresses in the G30-500/PMR-15 composite subject to high-temperature exposure at T1 = 343oC/650oF in inert and oxidative environments, as shown in Fig. 5 for axial oxidation. The stress-free reference state was taken at the post-curing temperature T0 = 316oC/600oF. It can be seen that the oxidation-induced stress field is dilatational (tensile) in nature, acting as a driving force for matrix cracking and interface debonding, as observed in high-temperature experiments. The oxidation-induced surface microcracks and fiber-matrix interface debonding, in turn, provide additional pathways for oxygen diffusion and surface areas for oxidation reaction. The maximum oxidation-induced stress is found to be located at the fiber-matrix interface with the longest distance between the adjacent fibers (θ = 30o). The much needed, but not yet available, thermal, chemical, mechanical, and interphase properties for accurate prediction of the coupled reaction, diffusion, damage, and thermomechanical problems were also identified. Pochiraju et al. [45] studied the detailed evolution of stress and deformations in HTPMCs during thermo-oxidative aging. Three-dimensional FEA simulating the thermo-oxidative layer growth with time were used together with homogenization techniques to analyze lamina-scale behavior with the RVEs as shown in Fig. 6. The fibers are in a square arrangement with each fiber having a diameter of 10 μm and a spacing of 12.532 μm (center to center) between adjacent fibers. The RVE is taken to have a nominal fiber volume fraction of 50% but can be varied easily by changing the FEM mesh. The multi-fiber volume element consists of 10 unit cells. Thermo-oxidation-induced shrinkage was characterized from dimensional changes observed during aging in inert (argon) and oxidative (air) environments. Temperature-dependent macroscale (bulk) mechanical testing and nano-indentation techniques were used for characterizing the effect of oxidative aging on modulus change. Figure 7 shows the simulated oxidation state and von Mises stress distribution in a G30-500/PMR-15 composite after 10 h (a) and 200 h (b) of oxidative aging. It appears that stress concentrations occur at the fiber-matrix interface regions and in the interstitial regions between the fiber and matrix with the peak stress at the fiber-matrix interface and at the free edge. After 200 h of aging, the level of stresses induced due to the shrinkage mismatch is quite high (of the order of the matrix strength of 41 MPa at 288  C/ 550  F) and therefore leads to onset of damage at the fiber-matrix interface/interphase or decohesion of the matrix. Tandon and Pochiraju [46] further studied the interaction of oxidation and damage during high-temperature aging of polymeric matrix composites. Unidirectional carbon fiber-reinforced PMR-15 composites were used for both experimental characterization and numerical simulation. A discrete crack was placed in the ten fiber model with the crack plane between the 7th and the 8th unit cells, which allows determination of the region (as a function of number of fiber diameters) affected by the presence of oxidation. The crack is propagated at a rate of 1.3 μm/h,

31

Micromechanics of Polymeric Materials in Aggressive Environments

Fig. 5 Microscopic stresses in G30-500/PMR-15 composite subject to hightemperature (343  C/650  F) exposure in inert and oxidative environments (Reprinted from [41], with permission from ICCM-14)

s11

a

1003

VALUE –3.20E + 02 –1.00E - 16 +5.27E + 02 • 1.0.5E + 03 +1.58E + 03 • 2.11E + 0.3 +2.63E + 0.3 • 3.16E + 0.3 +3.69E + 0.3 • 4.21E + 0.3 +4.74E + 0.3 • 5.27E + 0.3

Thermal oxidative (radial) stress (psi)

b

6

0 degree (T+Ox) 30 degree (T+Ox) 0 degree (T) 30 degree (T)

5

srr (Ksi)

4

r

3 θ

2 1 0 -1 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

r/f

Radial stress (ksi) along the paths at q = 0º and 30º

c

6 5

σrr (T+Ox)

sij (Ksi)

4 3 σrz (T+Ox)

2 1

-1

σrθ (T+Ox)

σrθ (T)

0

σrr (T)

σrz (T) 0

15

30

45 q (degree)

60

75

90

Stresses (ksi) along the fiber-matrix interface

which is the average value estimated from experimental observations. The crack growth was simulated by setting the oxygen concentration at the crack face nodes equal to that of boundary sorption at the appropriate instant in time when the crack surfaces form. Figure 8 shows the oxidation growth around the propagating discrete crack in the composite. Consistent with experimental observations, the oxidation

1004 Fig. 6 Representative single fiber (axial oxidation modeling) and multi-fiber (both axial and transverse oxidation modeling) volume elements used in this analysis (Reprinted from [45], with permission from Springer Science and Business Media)

X. Chen

a Fiber

Exposed Surface

b

Interphase Fiber

Exposed Boundary

Exposed Boundary 125 x 125 x 12.5 µm

front is always seen ahead of the crack and the oxidation depth near the crack surface is substantially higher than the regions away from the crack. On the other hand, Gigliotti et al. [47] studied local shrinkage and stress induced by thermo-oxidation in the IM7/977-2 carbon/epoxy composite material at elevated temperatures. The oxygen concentration and mechanical fields were established through a coupled model constructed from a unified multiphysical approach and the thermodynamics of irreversible processes with internal state variables. The model was implemented in the ABAQUS finite element commercial code. The induced shrinkage and stress were simulated in polymer-matrix composite samples in thermo-oxidative (5 bars O2, 48 h, 150  C) and neutral environments (5 bars N2, 48 h, 150  C). The RVE is a simplified square-packed fiber/matrix arrangement with the elementary cell of 4 μm  4 μm and the fiber diameter of 6 μm. The instantaneous cooling phase from the stress-free cure temperature (210  C) to the test temperature (150  C) was simulated in one static step. Figure 9 shows the evolution of the von Mises stress at point P1 as a function of the simulation time and of the environment. Figure 10 presents the von Mises stress along the path P1–P2, that is, along the distance from the fiber/matrix interface, for the purpose of illustration, indicating that the fiber/matrix interface can effectively become the place for damage onset. Such an event may depend dramatically on the environmental conditions to which the composite sample is exposed. Figure 11

31

Micromechanics of Polymeric Materials in Aggressive Environments

a

1005

Stress Levels (MPa)

0.180

0.385

+2 .353e+00 +2 .157e+00 +1 .961e+00 +1 .765e+00 +1 .569e+00 +1 .373e+00 +1 .177e+00 +9 .813e–01 +7 .853e–01 +5 .893e–01 +3 .933e–01 +1 .973e–01 +1 .242e–03

Oxidation State 0.591 0.796

1.00

after 10 hours of oxidative aging

b

Stress Levels (MPa)

0.180

Oxidation State 0.385 0.590 0.795

+4 .731e+01 +4 .337e+01 +3 .943e+01 +3 .549e+01 +3 .155e+01 +2 .761e+01 +2 .367e+01 +1 .973e+01 +1 .579e+01 +1 .185e+01 +7 .913e+00 +3 .973e+00 +3 .353e–02 1.00

after 200 hours of oxidative aging Fig. 7 Oxidation state and von Mises stress distributions (MPa) after 10 h (a) and 200 h (b) of oxidative aging (Reprinted from [45], with permission from Springer Science and Business Media)

shows the average dimensionless (%) von Mises stress at the end of the simulation as a function of the angle along the fiber-matrix interface (free edge), since a criterion for damage onset could be given with comparison of a suitable average value of stress with local strength while taking into account the embrittlement effects due to thermo-oxidation for the polymer matrix.

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Fig. 8 Oxidation growth around a discrete crack propagating at 1.3 μm/h (Reprinted from [46], with permission from Elsevier)

Von Miscs stress point P1 (MPa)

35

5

30

N2 O2

4 3

25

P2 P1

2

20

1

15

0 0

5

10

15

20

10 5 0 0

10

20

30

40

50

Time (h)

Fig. 9 Evolution of von Mises stress of point P1 during the transient simulation of thermooxidation for a composite sample in oxidative and neutral environments (Reprinted from [47], with permission from Elsevier)

Roy and Singh [48] also presented mechanism-based modeling of damage evolution in HTPMCs under thermo-oxidative aging conditions using micromechanical modeling and homogenization. Overall, the reported micromechanical simulations are compatible with experimental observations and insightful for

31

Micromechanics of Polymeric Materials in Aggressive Environments P2

400 350 Von Mises (MPa)

1007

300

N2

250

O2

P1

200 150 100 50 0 0

0.5 1 1.5 2 2.5 Distance from the interface (mm)

P2

3 P1

Fig. 10 von Mises stress profile along the path P1–P2 for a composite sample in oxidative and neutral environments (Reprinted from [47], with permission from Elsevier)

Von Mises

160%

q x

140% N2 adimensional

120%

O2 adimensional

100% 0

10

20

30

40

50

60

70

80

90

angle (degrees)

Fig. 11 Average dimensionless (%) von Mises stress for a composite sample in an oxidative environment along the fiber-matrix interface (free edge) (Reprinted from [47], with permission from Elsevier)

material developments. Additional information on thermal oxidation of PMCs can be found in the review articles by Lafarie-Frenot et al. [49] and Cinquin et al. [50].

4

Concluding Remarks

A brief overview on micromechanics of polymeric materials in aggressive environment is given with a focus on multidisciplinary approach and computational micromechanics. Based on the fundamental principles of thermodynamics, the coupled effects are described by the theories of nonlinear hygro- or chemo-thermo-viscoelasticity at the constituent scale, which drive nonlinear responses at the ply, laminate, or structural scale. Significant efforts have been made to include coupled multidisciplinary approach at different scales in efficient

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X. Chen

algorithm and commercial software. Finite element method and homogenization technique are generally used to simulate the complicated processes of coupled diffusion, reaction, heat transfer, and damage evolution in fiber-reinforced, polymer-matrix composites with material anisotropy and inhomogeneity. Concentration, temperature, stress, and strain fields are tracked on the microstructural level and parameters characterizing homogenized behaviors are determined on higher-scale levels so as to assess mass gain or loss, dimensional expansion or shrinkage, and damage onset and propagation. It is also important to characterize diffusivity, strength, and toughness of the fiber-matrix interphase in hygrothermal or thermooxidative environments, which can effectively become the propitious place for damage initiation due to stress concentration. There is a concurrence that microstructural damage evolution is caused by the synergistic effects of environmental attacks and mechanical loads. The resulting thermal, hygroscopic, or chemical residual stresses combined with mechanical stresses may be sufficiently large enough to influence failure and thus should be accounted for in design analysis and life prediction. The investigation of physical, chemical, thermal, and mechanical mechanisms that are strongly coupled at various scales requires multiple skills in different fields, including physics, chemistry, thermomechanics, and materials science together with technological aspects related to engineering applications. The route to validate analytical prediction is not so linear, which requires continuous exchange of information between experiment and modeling at different scales. Numerical simulations accounting for various environmental factors can be used to guide designers to generate composite microstructures with optimized material properties and minimized stress concentrations, thereby leading to enhanced structural performance and prolonged service life.

5

Future Directions and Open Problems

The development of micromechanical modeling for polymeric materials in aggressive environments by means of coupled multidisciplinary approach helps to establish mechanism-based material models for describing complex problems, involving diffusion, reaction, heat transfer, damage evolution, long-term aging, and degradation. Even though modeling of hygrothermal and thermo-oxidative behaviors at the microscale reaches a satisfactory level, upscaling toward structural analysis is still a big challenge. In-depth understanding of the coupling of physical, chemical, thermal, and mechanical effects at multiple time and length scales is the key for realistic performance evaluation and life prediction. In the future, significant efforts should be made on generalizing multi-scale composite analysis methodology across multiple disciplines with inclusion of user-defined material models for all the involved mechanisms. Due to inherent complexity and novelty, many uncertainties still remain open. Research frontiers that require special attention involve robust multiscale composite analysis tool development, multidisciplinary design optimization, and macrocrack-microcrack interactions for aggressive environment applications. Correlation of prediction with measurement under operation conditions is vital for

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Micromechanics of Polymeric Materials in Aggressive Environments

1009

various engineering applications. The emergence of new classes of polymeric materials, e.g., magneto- or electro-active polymeric materials acting as sensors or actuators, brings about additional challenges. Electromagnetic effects and their coupling with other physical, chemical, thermal, and mechanical effects should be added for consideration.

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43. Abhilash AS, Joshi SP, Mukherjee A, Mishnaevsky L. Micromechanics of diffusion-induced damage evolution in reinforced polymers. Compos Sci Technol. 2011;71(3):333–42. 44. Jain D, Mukherjee A, Kwatra N. Local micromechanics of moisture diffusion in fiber reinforced polymer composites. Int J Heat Mass Transf. 2014;76:199–209. 45. Pochiraju KV, Tandon GP, Schoeppner GA. Evolution of stress and deformations in hightemperature polymer matrix composites during thermo-oxidative aging. Mech Time Depend Mater. 2008;12:45–68. 46. Pochiraju KV, Tandon GP. Interaction of oxidation and damage in high temperature polymeric matrix composites. Compos Part A. 2009;40:1931–40. 47. Gigliotti M, Olivier L, Vu DQ, Grandidier JC, Lafarie-Frenot MC. Local shrinkage and stress induced by thermo-oxidation in composite materials at high temperatures. J Mech Phys Solids. 2011;59:696–712. 48. Roy S, Singh S. Prediction of thermo-oxidative degradation in polymer matrix composites using micromechanical modeling and homogenization at the macro-scale. Mech Adv Mater Struct. 2012;19:68–78. 49. Lafarie-Frenot MC, Grandidier JC, Gigliotti M, Olivier L, Colin X, Verdu J, Cinquin J. Thermooxidation behaviour of composite materials at high temperatures: review of research activities carried out within the COMEDI program. Polym Degrad Stab. 2010;95:965–74. 50. Cinquin J, Colin X, Fayolle B, et al. Thermo-oxidation behaviour of organic matrix composite materials at high temperatures. Adv Aircr Spacecr Sci. 2016;3(2):171–95.

Crack Paths in Graded and Layered Structures

32

Ivar Reimanis

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Brittle Materials (Linear Elastic with No Plasticity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Thermal Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Toughness Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Weibull Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 T-Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Effects of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1014 1014 1017 1021 1022 1023 1026 1031 1031 1032 1032

Abstract

This chapter reviews the prediction of crack paths in materials with graded, and/ or layered, composition, microstructure, and/or properties. The relevant applications span a wide variety of technologies where structural integrity is important and includes composites, layered materials, coatings, and joints. For most cases, the prediction of failure requires a priori knowledge of the crack path. In many cases, the crack path may be sufficiently defined by considering just crack kinking, that is, a small or infinitesimal increment of crack deviation from its plane. For other cases, the full crack path needs to be determined, something usually done with remeshing techniques in numerical simulation. In linear elastic systems, the residual stress that arises from the coefficient of thermal expansion mismatch between constituents generally dominates the behavior, but not if the elastic mismatch is very large or if there are big toughness mismatches. For linear I. Reimanis (*) Colorado School of Mines, Golden, CO, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_67

1013

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I. Reimanis

elastic systems where plasticity occurs, but small-scale yielding still applies, the crack is usually drawn toward the softer material which also is generally tougher. For all cases, if the crack stress fields from all the sources are accurately defined, and materials properties known, it is possible to predict the crack path. Keywords

Crack kinking · Deflection · Numerical simulation · Joining · Residual stress

1

Introduction

Materials and structures with composition and/or microstructure gradients usually lead to property and/or residual stress gradients that significantly perturb crack growth. It is important to predict the crack path not only to be able to assess the overall structural integrity but also to predict the strength and fracture toughness. If the crack tip stress field and its evolution during crack growth are well-defined, and the toughness is known, then it is generally straightforward to predict the crack path. For the case when the crack and the applied loading are symmetric with respect to the gradient, then the crack continues on a symmetric path. However, if the crack and/or loading is asymmetric with respect to the gradient (Fig. 1), it is necessary to superpose the stress fields from the various sources and apply the appropriate fracture criterion. This superposition determines the local crack tip stress field that may be used to determine the stress intensity factor, K. Crack growth occurs when K obtains or exceeds KC.

2

Brittle Materials (Linear Elastic with No Plasticity)

Most of the theoretical work describing crack tip stress fields in graded materials is for linear elastic systems [1–10]. As Erdogan and others have shown, the crack tip stress fields in nonhomogeneous materials vary in a similar manner as they do in homogeneous materials [1, 4, 11]. In particular, the square root singularity of the stress field still applies, unlike the case for cracks at discrete interfaces. The stress field for mode I and II cracking may be given as 1 X KI K II 1= σ ij ¼ pffiffiffiffiffiffiffi f Iij þ pffiffiffiffiffiffiffi f IIij þ r j 2 Aj hj ðβ, θÞ 2πr 2πr j¼1

(1)

where the (r, θ) coordinate system is the standard one (see Fig. 2), and KI and KII are the mode I and mode II stress intensity factors. The angular variations in the crack tip stress fields, fij, are identical to the angular variations in a homogeneous solid. Higher-order terms in r comprise the last term, and Aj are the expansion series coefficients, also present in the case for a homogeneous solid. hj are functions of the grading parameter, β, and the angle, θ. These higher-order terms are typically

32

Crack Paths in Graded and Layered Structures

Fig. 1 Schematic showing a crack positioned asymmetrically with respect to the property gradients. Black and red arrows indicate applied opening and shear loading. Hypothetical variations in Young’s elastic modulus (E), yield strength (σ0), and coefficient of thermal expansion (CTE) are given. A crack kink is shown with a crack kink angle θm

1015

E

θm

CTE

Fig. 2 Schematic of crack tip region showing the initial crack (dotted line) with stress intensity factors KI and KII, and the kinked crack defined by new stress intensity factors, and K I and K II

Initial crack

KI, KII

σ0

y r

θ θm

x

Kinked crack K I* , K II*

neglected in homogeneous materials, but as described later, there are several instances in graded materials where they should be included. Except for the inclusion of the grading parameter, β, Eq. (1) is identical to that for a homogenous solid. In reality, discreteness usually appears on a local microstructural scale, and may significantly alter fracture processes, even though a global application of traditional fracture mechanics still applies. The main material parameters of interest in the linear elastic fracture studies on graded materials are the elastic moduli and the coefficients of thermal expansion, the latter being relevant since most materials are made at elevated temperature and then cooled to room temperature. A rule of mixtures description is in many cases sufficient to describe the variance of these parameters across a graded region [12–14], but more extensive micromechanics models may be used if a higher accuracy is needed [15]. It has been shown that the effect of Poisson’s ratio gradient on fracture is generally negligible and most elastic gradients need only consider the change in Young’s elastic modulus, E. The variance of elastic moduli has also been described in other ways which are more mathematically convenient for theoretical analyses. For example, Erdogan and others have described the variance of E exponentially with distance, x [1, 4, 5]. Some of their calculations indicate that the

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I. Reimanis

stability of any thermally or mechanically loaded crack depends more strongly on the ratio of end member elastic moduli, E2/E1, than on E(x) [1, 5], yet this remains to be verified experimentally and may not be true in all cases. In certain cases it has been verified that the type of elastic modulus variation strongly impacts mechanical behavior [16, 17]. The influence of the thickness of the graded region, t, has also been examined in linear elastic systems, and it has been found that the stress intensity factors depend mildly on t [3]. The direction of cracking (e.g., perpendicular or parallel to the gradient) is important and may determine the relative influence of E2/ E1, E(x), and t [1, 5]. Crack kinking may be described with the formulation set forth by Cotterell and Rice and [18]. A kinked crack may be defined by a set of local, crack tip stress intensity factors which are related to the main crack stress intensity factors. Figure 2 depicts a kinked crack. The relation between the kinked crack stress intensity factors and the main crack stress intensity factors may be given by [18] K I ¼ C11 ðθm ÞK I þ C12 ðθm ÞK II K II ¼ C21 ðθm ÞK I þ C22 ðθm ÞK II

(2)

Here the superscript * designates the kinked crack. Cij are functions dependent on the crack kink angle, θm (see Fig. 2), and are given by   1 θm 3θm 3 cos þ cos 4  2 2  3 θm 3θm sin þ sin ¼ 2 2 4 1 θm 3θm sin þ sin ¼ 4 2 2  1 θm 3θm cos þ cos ¼ 4 2 2

C11 ¼ C12 C21 C22

(2a)

If a maximum tangential stress criterion [19] is used, then the crack kink angle is such that the shear stress, τrθ = 0. Note that the maximum tangential stress is a principal stress. The maximum tangential stress corresponds to the well-known KII = 0 criterion where in this case, K II ¼ 0 (see Fig. 2). For most problems, it is the same as a maximum energy release rate criterion. For the case when K II ¼ 0, it is convenient to define an equivalent mode I stress intensity factor which then corresponds uniquely to the case when the kink angle is θm:   K EI ¼ K I when K II ¼ 0 and θ ¼ θm

(3)

Once the crack kinking direction is determined as described above, the condition for crack extension must be considered. The criterion for crack extension is K EI ¼ K IC

(4)

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Crack Paths in Graded and Layered Structures

1017

This approach has been used successfully to predict crack paths in graded epoxy/ glass composites [10, 20] and other materials as described below. It is shown next that a diverse set of crack paths is possible in the presence of residual stress and toughness gradients. Because of the large number of variables that describe the stress field, analytic solutions are frequently not practical or even possible, and thus, it is recommended that numerical solutions be developed. The next section describes a framework within which to build a numerical solution, and some specific examples are provided.

2.1

Thermal Residual Stresses

Thermal residual stresses are typically present in graded, layered materials because the materials are usually processed at elevated temperature and differences in thermal expansion lead to differential strains upon cooling. The stresses may be very high compared with the fracture strength. Furthermore, residual stresses may lead to mixed-mode crack tip stresses, as described below. The relationships between thermal residual stresses and the architecture of graded materials, particularly layered, graded materials, are relatively well-defined [12–14, 21–26]. The coupled influence of thermal stresses and elastic properties on the fracture behavior has also been examined in significant detail [1, 2, 27–32]. The presence of residual stresses may amplify the effects of elastic modulus [31, 32]. When the crack is positioned asymmetrically with respect to the applied load, it typically experiences a mixed-mode state of loading. The exception would be the case where the thermal residual stress perfectly balances the applied stress. The mode mixity, φ, which determines the angle at which the crack will kink, is given by φ ¼ tan -1

K II K Res þ K App II ¼ tan -1 IIRes KI K I þ K App I

(5)

where the superscripts Res and App indicate the contributions to the mode I and mode II stress intensity factors from the residual and applied loads. It is apparent from Eq. (5) that when a residual stress field is present, as the applied load is changed, ϕ changes. The end result is that, for highly brittle materials (low applied load to fracture), the effect of residual stress on the kink angle is much greater than for tough materials. Since crack extension occurs when Eq. (4) is satisfied, the kink angle depends on the toughness. K eI may be defined as K eI ¼ K I cos3

θm θm θm  3K II cos2 sin 2 2 2

(6)

With a maximum tangential stress criterion, the tangential stress ahead of the crack tip needs to be defined. The tensile stress may be defined as

1018

Fig. 3 (continued)

I. Reimanis

32

Crack Paths in Graded and Layered Structures

  1 θ 3 2θ K I cos  K II sin θ σ θ ¼ pffiffiffiffiffiffiffi cos 2 2 2 2πr

1019

(7)

where r is the distance from the crack tip. The angle at which the maximum tangential stress occurs (θm) may be found by employing an algorithm in which the region around the crack tip is examined. Because the maximum stress occurs where the shear stress is zero, the algorithm may be set up to identify the angle at which τrθ = 0; it is generally easier to identify where a curve intersects an axis than to find the location of a broad maximum. The above equations support a numerical modeling framework in which it is possible to establish how toughness and residual stress determine the crack path. It is noted that analytic approaches have been formulated for the case of mixed-mode cracks in layered solids, particularly for cracks at interfaces [33], and that those formulations provide a good foundation for crack path prediction in graded composites. The following example illustrates a specific case for a graded, layered Cu-W composite, a material of interest for plasma facing components in thermonuclear reactors [25]. Example Figure 3 provides an example numerical simulation result for a layered, graded system machined to a single-edge notched beam. The finite element method is used with material properties corresponding to Cu-W composites. W (E = 411 GPA) is elastically stiffer than Cu (E = 130 GPa). In addition to an elastic modulus mismatch, a thermal expansion mismatch is present between Cu and W. The thermal expansion coefficient, α, may be estimated with a rule of mixtures type relation:

ä Fig. 3 (a) Typical single edge notched in tension specimen geometry for the numerical modeling studies. The specimen is 25 mm long, 8 mm tall, and a 3 mm long crack exists in the center layer. The composition of the layers, their thickness, the total number of layers, and the location of the crack are mechanics-based parameters in the study. The composition determines the elastic modulus and the thermal expansion coefficient in the FE model. Load is applied at the ends of the specimen in a direction parallel to the gradient. (b) Schematic of crack tip showing the stress components in cylindrical coordinates. The crack tip displacement axes, u, are also shown. (c) Finite element predictions (elastic) of crack angle as a function of the crack tip driving force, or the equivalent stress intensity factor, for a crack in the configuration shown in Figure 3a; load is applied at specimen ends in a direction parallel to the gradient direction. When no residual stress is present, the crack kink angle is constant and arises entirely due to the elastic mismatch. In the presence of residual stresses that arise during cooling from elevated temperatures and are due to the thermal expansion coefficient mismatch between layers, the crack kink angle is a function of the crack tip driving force. (d) Finite element predictions (elastic) of crack angle as a function of the crack tip driving force, or the equivalent stress intensity factor, for a crack in the configuration shown in Figure 3a. The effect of employing different layer thicknesses is examined. The combined effect of elastic modulus and thermal residual stress results in a situation in which crack deflection is extremely sensitive to the layer thickness. The effect is pronounced for lower toughness materials (low KIe). (e) Finite element predictions (elastic) of crack angle as a function of the crack tip driving force, or the equivalent stress intensity factor, for a crack in the configuration shown in Figure 3a. The effect of number of layers is examined for a constant layer thickness of 3 mm. The deflection angle is extremely sensitive to the number of layers

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I. Reimanis

P αi V i K i α ¼ Pi iV i Ki

(8)

where αi is the thermal expansion coefficient of material i, Vi is the volume fraction, and Ki is the bulk modulus. Thermally induced residual stresses are modeled by simulated cooling from 300  C to 25  C. The notch is located in a layer comprising 40% by volume W, a composite in which it was determined that smallscale yielding is operative. However, to simplify the configuration for the present demonstration, plastic yielding is turned off in the simulation. The notched beam is loaded in tension, and the stress intensity factors are calculated by appropriate mesh modification to produce the square root singular behavior [34]. The angle of crack kinking was obtained by determining the direction of cracking in which τrθ = 0. This may be done by calculating the shear stress near the crack tip for angles from 90 to 90 (Fig. 3b). The distance from the tip where one chooses to calculate the tangential stress (r in Fig. 3b) is somewhat arbitrary but should be related to the relevant length scales for crack kinking; for example, this may be on the order of the grain size for a polycrystalline material. Without thermal residual stress, that is, if the coefficients of thermal expansion of the two materials are identical, the crack in this geometry is predicted to kink at an angle of 7 (toward the less compliant material – to the left in Fig. 3a). When thermally induced residual stress is present, the crack kink angle becomes positive (toward the more compliant material) and depends strongly on K eI. Figure 3c illustrates this result. Since the crack propagates only when K eI equals KIC, it is clear that the crack kink angle depends strongly on the toughness of the material. Two representative values of the fracture toughness are provided on the graph as reference points. The kink angle for the less tough material is about 54 , whereas that for the tougher material is about 24 . While this example only applies for a specific geometry and set of materials (as detailed in the Fig. 3c inset), it illustrates how the presence of residual stress has a major effect on the crack path. Note that residual stress is frequently present in graded materials due to thermal expansion mismatch, but it may also arise due to nonuniform sintering [35]. The geometry plays a strong role in governing not only the magnitude of the angle of crack deflection but also the sign, as demonstrated in Fig. 3d, e. Figure 3d considers the five-layer geometry shown in Fig. 3a but examines the effect of changing the layer thickness from 1 to 5 mm. In this case, when the layer thickness is 2 mm or less, then the crack deflection angle is positive (toward the right in Fig. 3a) for most values of toughness; the angle becomes exceedingly high for low toughness materials. When the layer thickness is 3 mm or greater, then the crack deflection angle is negative. For typical glasses and ceramics where the toughness varies between 0.5 and 7 MPa m1/2, the crack deflection angle is highly dependent on the toughness. Figure 3e shows the effect of varying the number of layers while keeping the layer thickness constant at 3 mm. In this case, the composition for each of the three layers is 0, 40, and 80 (in volume % W). That for the seven-layer specimens is 0,13,26,40,53,67, and 80. In all cases, the crack is embedded in the

32

Crack Paths in Graded and Layered Structures

1021

Fig. 4 (a) Schematics of the three specimen types referred to in Figure 4b. The four point bending (4PB) specimen is the same as that depicted in Figure 3a. Arrows indicate applied loading. (b) Calculations showing how the crack kink angle depends on the applied load as well as the specimen geometry. See text for more detail. Figure obtained from [36]

40% W layer. As with the previous cases, there are situations in which the sign and magnitude of the crack deflection angle are extremely sensitive to the geometric configuration. When the example above is applied to a different specimen loading geometry, a completely different behavior may arise. Figure 4 illustrates a result that compares the single-edge notched bar loaded in tension (SENT) geometry with a four-point bending (4 PB) geometry and a center-cracked plate loaded in tension (CCT) geometry perpendicular to the notch for the case when the layer thickness is 1 mm. The abscissa now is the applied load, σ a, which corresponds directly to K eI , making it easier to compare these three geometries from a mechanical testing perspective. The curves represent best fits of the individual simulations shown by the symbols. A drastically different kink angle is predicted for the CCT specimen, particularly when failure occurs at low applied load (this would correspond to high values of thermal residual stress or low toughness). However when the material fails at relatively high applied loads (corresponding to low residual stress or high toughness), then the difference between the geometries is much less. Recall that θm converges to 7 at very high loads for the four-point bending geometry (e.g., this is the same result as no residual stress – see Fig. 3c).

2.2

Toughness Gradient

Because the toughness determines the crack kink angle as described above, if a toughness gradient exists, the problem of predicting the crack path becomes

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I. Reimanis

significantly more complicated. Interestingly, it has been shown that the driving force for a crack propagating within a particular composition within a graded material is lower than that within an infinite medium of the same material [36]. Because the toughness and crack kink angle are coupled in a way that depends on the geometry and material properties, a general analytic solution may be challenging, and numerical simulation is probably the best approach. Based on results from the previous section, the presence of a toughness gradient may have a major impact on the crack stability when residual stresses are present. The stability of a crack has a major impact on material reliability. Consider the previous example with the Cu-W composite. For no residual stress, if a crack deflects (at an angle of 7º for the case described in Fig. 3c) into a material with higher toughness, then R-curve behavior should be active, crack stability is expected, and the crack trajectory may remain linear. Whether or not the crack path remains linear depends on the material gradient. When residual stress is present, the crack kinking model alone is not sufficient to predict failure in the presence of a toughness gradient, because the crack kink angle continually changes. However, in that case, it may be inferred from Fig. 3 that if a crack deflects into a material with higher toughness, the continued trajectory will be nonlinear, and the R-curve will be weaker, implying higher instability. One would conclude that the presence of residual stresses in graded structures inherently leads to lower strength reliability.

2.3

Weibull Statistics

The flaw population in a graded material may affect the crack path when crack growth occurs by the extension of flaws in the volume of material ahead of the crack tip. It has been noted that in these cases, a statistical approach is required to predict the crack path. Becker et al. have examined the statistical aspects of fracture in graded materials and found that the Weibull modulus, m, may be used as a parameter to predict crack path and driving force in graded materials [37]. When the crack is oriented perpendicular to the elastic modulus gradient, (Fig. 1), then the subsequent crack kinking that arises due to mixed-mode loading depends on m. m determines the distribution of damage ahead of the crack tip. If m is relatively low, the distributed damage ahead of the tip shields the crack tip, leading to higher effective toughness. When m is high, fracture is dominated by only the near strip stress fields. Since toughness influences the crack path, clearly the flaw population does as well. It is likely that a gradient in composition or microstructure also corresponds to a gradient in m, thereby making it more challenging to predict the crack path. Becher et al. showed that when a gradient in the Weibull scaling stress exists, there is a significant decrease in the initiation fracture toughness, but they did not examine the effects of gradients in m. The coupled effects of the various properties imply that numerical simulation is necessary to predict the crack path.

32

2.4

Crack Paths in Graded and Layered Structures

1023

T-Stresses

The first of the higher-order terms in Eq. (1) is called the T-stress. For mode I loading, Eq. (1) may be written  KI T σ xx ¼ pffiffiffiffiffiffiffi f Ixx ðθÞ þ 0 2πr

 pffiffi 0 þO r 0

(9)

where σ xx refers to the stress perpendicular to the plain of the main crack, and the last term represents the square root-order contribution to the stress field, negligible in nearly all cases. The zero-order constant term, T, represents a stress which acts parallel to the crack plane, referred to commonly as the “T-stress,” and has been noted to be significant for graded materials [38, 39] and generally important in many complex cracking situations. Under this mode I loading, KI dominates the stress field as r approaches zero, but many fracture events in advanced materials are associated with a process zone in the immediate vicinity of the crack tip, where the value of T may become significant compared with KI. Physically, the T-stress is related to the crack tip constraint: a positive T-stress increases the level of stress triaxiality at the crack tip, thereby leading to high crack tip constraint, while a negative value implies low stress triaxiality and the loss of crack tip constraint. The T-stress must be taken into account when conducting purely analytic calculations. It is “naturally” incorporated when conducting most numerical simulations. The difference with a graded compared to a homogeneous solid is that the stress intensity factors, KI and KII, depend not only on the loading and geometry of the solid but also on the material property gradients. It has been demonstrated that the presence of material inhomogeneities influences the magnitude and sign of the T-stress and that these are correlated with crack kinking angle [40]. However, it is again noted that the order of the stress singularity at the crack tip, r-1/2, is independent of the material gradients. A few studies on T-stresses have been performed in the context of examining the driving force and direction of cracks near interfaces. Some of these have been motivated by the need to understand cracking in joints, for example, within adhesive layers [41, 42], where interfaces frequently dominate cracking behavior. Generally, it is accepted that a two-parameter fracture model must be used for cracks in these heterogeneous situations: the two parameters are K and T for cracks in linear elastic materials [43]. However, there are relatively few studies that have modeled the crack tip behavior with the T-stress as an important parameter. In addition to affecting the crack path, the T-stress influences the crack driving force for nominally brittle materials that exhibit a process zone [44–47]. In the case of glass where heterogeneities exist at the nanoscale [46], the effect is indirect in that the T-stress influences the crack path which in turn influences the crack driving force, ultimately leading to a dependence between crack velocity and fracture surface roughness [48]. It is interesting that the length scales of the process zones vary from nanometer sized in the case of glass [46] to millimeter sized in the case of rocks [47].The importance of the T-stress spans several length scales.

1024 Fig. 5 Schematic illustrating (a) a directionally stable configuration, and (b) a directionally unstable configuration. S, the crack excursion size, should be related to the size of the microstructure. The figure is adapted after [46]

I. Reimanis

a S dθ

b S dθ

dφ ideal path

dφ ideal path

The relation between crack stability, crack path, and T-stress in heterogenous materials has not been examined in great detail. A crack is directionally stable if after a small, random perturbation, for example, due to its intersection with some microstructure entity such as a flaw, it returns to its expected, ideal path. Figure 5 shows a schematic of stable and unstable crack paths, depicting an infinitesimal crack kink angle as dθ and the subsequent infinitesimal kink as dϕ. A common mathematical definition of crack path directional stability is given by the crack deviation angles: by definition, if dϕ/dθ < 1, then the crack path is directionally unstable. While the analysis is infinitesimal, there is general consensus that it should involve some size scale for real materials. Cotterell and Rice are credited for establishing the fact that the directional stability of a crack depends on the Tstress; they further describe that cracks should be directionally stable when T < 0 and unstable when T > 0 [18].The general influence of the T-stress with respect to crack stability was even pointed out significantly earlier by Cotterell [49]. Since Cotterell and Rice’s well-cited paper, there has been significant work attempting to link crack path stability and the T-stress [41, 42, 50, 51]. Many studies have simply assumed validity of the original T-stress criterion (namely, the crack path is directionally unstable if T > 0), but there have been a few studies that suggest it is a necessary, but not sufficient condition [52, 53], and one study that states the sign of the T-stress has no bearing on the crack path stability [54]. A specific example illustrating the latter point exists for the standard compact tension test specimen: the crack in such a specimen is directionally stable for many materials, yet the T-stress is positive. Despite these inconsistencies, it is well-accepted that the T-stress is an independent parameter that directly affects events in the process zone around a crack tip. The relation between microstructure and the directional stability of the crack path is connected to the excursion parameter, S (Fig. 5). In order to apply this to real materials, S would be related directly to the microstructure size scale as compared with other geometric parameters in the problem. For example, S may be related to the particle spacing in graded, particulate-reinforced composites or to the scale of the residual stress field around a second phase particle. In a graded material, if S is substantially large, the excursion will transport the crack tip to a region that exhibits

32

Crack Paths in Graded and Layered Structures

1025

a different toughness. Therefore, the scale of the gradient is coupled with the scale of the excursion size in terms of predicting the crack path. Tong examined how the ratio of crack deviation angles, dϕ/dθ, depends on the excursion parameter, S [50]. By applying a maximum tangential stress criterion for crack growth, Tong showed that  1=2 "     1=2 # dϕ S 4 T A3 S ¼1 þ dθ a π A1 a A1

(10)

where a is the crack length, T is the T-stress, and A1 and A3 are the coefficients, Aj, from the stress field series expansion in Eq. (1). Figure 6 illustrates how sensitive dϕ/ dθ is to the excursion parameter S for a particular representative set of values of a and Ai, and a few values of T. Specific values of these can be readily calculated for specific specimen geometries. Values of dϕ/dθ less than 1 indicate a directionally unstable configuration. It is apparent from this that even for values of T < 0, there are directionally unstable scenarios (dϕ/dθ < 1). It is further apparent that in certain cases, the value of S alone alters the general condition for crack directional stability, though this does not appear to be a strong effect. For example, in Fig. 6, the curve for T = 4 is above 1 (directionally stable) for values of S less than 10 μm, but it is below 1 at larger values of S (Fig. 6). It seems in most cases, S influences the magnitude of the deviation but does not change the general tendency. There is an inherent limitation in this approach with regard to the values of S chosen. For example, when S approaches the size of geometric parameters such as crack length, the analysis is clearly not valid. Pook has proposed a criterion for directional stability that incorporates the T-stress as well as a crack tip length scale [53]. The criterion is based on a nondimensional parameter, TR, which implies directionally unstable crack growth when it exceeds a particular value: Fig. 6 df/dq, the tendency of crack deviation from the ideal path, is shown as a function of the magnitude of the crack excursion size, S, using equation (10). Figure 5 defines S. For dφ /dθ < 1, the crack is by definition unstable. Several values of the T-stress (in MPa) are given

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I. Reimanis

TR ¼

T T ¼ ffiffiffiffiffiffiffiffiffiffi σ xx KI p2πr

(11)

micr

where σ xx is the tensile stress ahead of the crack tip at a given r = rmicr. rmicr is related to a microstructure size scale. As an example, one can take rmicr = 15.9 μm (using MN-m units) and obtain the convenient expression [53] TR ¼

0:01T KI

(12)

The attractive aspect of this criterion is that there is a direct link to a crack tip size scale, and yet the expression is still nondimensional. Pook applied this for various situations and observed that the sign of TR did not correlate with directional stability but that the particular value of TR did. For example, for a biaxially loaded, center-cracked sheet of Waspaloy (a nickel-based material for gas turbines), the crack path was stable when TR < 0.0216. A similar study on poly (methyl methacrylate) indicated that a value of TR < 0.0122 led to a stable crack path [52]. Pook also examined the compact tension specimen for the same Waspaloy material as above and found that the value of TR was below 0.0216 for all specimens conditions. Since all of the conditions exhibited stable crack paths, the result suggests that the critical value of TR is fixed for a given material. Unfortunately, this result differs from the original postulate of Cotterell and Rice [18]; future work should try to clarify this. The above framework to predict crack stability and crack path based on T-stress has not been applied to graded materials. A comprehensive study that examined the effect of T-stress on crack kinking in graded materials [38] showed that the T-stress is generally higher in graded than in homogenous materials. Furthermore, when the kink length scale is about the same as the gradient dimensions, then the kink length predicted by the main two criteria (KII = 0 and the maximum energy release rate) diverge significantly; in homogeneous materials they only differ by at most about 3 degrees.

3

Effects of Plasticity

The effects of plasticity on crack paths in graded materials are separated into two categories here. In the first case, the crack tip is embedded in a linear, elastic material in which small-scale yielding may occur and for which plasticity may occur away from the crack tip; crack extension is mode I, occurring in a direction such that KII = 0, unless the crack is at an interface. For the second case, the crack is embedded in an elastic-plastic material, and elastic-plastic fracture mechanics must be applied and mixed-mode fracture is possible. As in the case for purely elastic materials described above, the focus here is on problems in which the crack plane is located asymmetrically within the gradient.

32

Crack Paths in Graded and Layered Structures

1027

Case 1 Small-Scale Yielding To consider the effects of plasticity on crack path, it is relevant to consider a layered geometry in which the crack is located near and parallel to an interface. There have been previous studies that describe the crack driving force for the case when a crack is located parallel to an interface, a small distance away from it [41, 55–61]. When the crack is very close, it has been shown that the Dundurs’ parameter is important in governing crack deflection [57, 58]. An analytic description exists for a crack growing parallel to an interface [62]. It has been used to show that when the distance away from the interface is smaller than the plastic zone size, then the crack will kink toward the interface just purely based on an energy balance argument. Tilbrook et al. [60] examined the relative competition between the increased K eI due to plasticity-induced compliance [63] and the decreased K eI due to crack tip shielding [64]. The following example illustrates this competition for a particular geometry and set of materials. Example Figure 7a shows an example model geometry where asymmetry is only present due to the position of the crack. Material 1 contains the crack and is linear Fig. 7 (a) An example specimen geometry used to derive the calculations presented in Figure 7b. The inner ductile layer is pure Cu and the outer layers are composed of a composite of 60 vol. % W and 40 vol. % Cu. Taken with permission from [61]. (b) Finite element results for a crack propagating parallel to an interface with a ductile layer (Figure 7a). The normalized SIF is the driving force for crack extension compared to that when the inner layer has no plasticity. The crack was situated 100 mm from the interface, and the ductile region thickness, w, is 200 μm (‘Narrow’) or 2000 μm (‘Wide’). Plasticity in the ductile region leads to crack tip shielding during crack growth. Without ductility (‘No yielding’) anti-shielding occurs. Taken with permission from [61]

a P/2

Sin= 10mm

P/2

w = 2mm Material 1

Material 1

h = 8mm

Sout= 20mm

b

a

x < 1mm Material 2

1028

I. Reimanis

elastic, but Material 2 exhibits plasticity during the growth of the crack. The volume of plasticity is highly dependent on the distance the crack is away from the interface as well as the elastic and plastic properties of both materials. Figure 7b illustrates a typical result. As the crack extends, the normalized stress intensity factor (K eI) initially drops when plasticity is present but then rises as the plasticity in Material 2 grows. Compared to the case of no yielding (open squares in Fig. 7b), the crack driving force is substantially reduced, meaning the effective toughness is increased. Figure 8 illustrates typical crack trajectories determined with numerical simulation for cracks positioned within a linear elastic material, near an interface. The materials modeled are pure Cu (elastic-plastic) as Material 2 and a composite comprising 60 vol % W and 40 vol. % Cu (linear elastic) as Material 2. The thermal residual stress plays a big role in governing crack path and even dictates whether a crack is drawn toward the interface or repelled from it. To highlight a specific case, consider a crack located 200 μm from the plastically deforming ductile layer (Fig. 8a). The crack is drawn into the ductile layer when no thermal residual stress is present (ΔT = 0). However, if one carries out the same simulation by cooling the layered system from 300  C (ΔT = 300) such that thermal residual stresses develop, then the crack is repelled from the layer. In cases where the crack deflects away from the interface, the crack path shows concavity toward the interface, a reflection of the decaying residual stress field. It is noted that the crack path under fatigue growth would be expected to be different from monotonic since the amount of plasticity in Material 2 would be different. Case 2 Elastic-Plastic There are only a few studies that have examined crack paths for the case when an elastic-plastic crack is oriented perpendicular to a gradient. The stress singularity is no longer described by Eq. (1), and the strength of the singularity depends on the power law exponent that describes the plasticity. It has been shown that the size of the singularity dominated zone is affected when the material gradients are steep [65]. This presents a challenging scenario to model because the fracture criterion may change as the crack enters different compositions (Fig. 1). Rashid and Tvergaard have examined the case of a near-interface crack next to a graded material in which plasticity and toughness varied but in the absence of elastic or thermal mismatch [66].They showed that the crack generally deviates away from the plastically softer region. None of these approaches have considered the case that the fracture criterion might change during crack propagation. However, Wang has developed a simulation procedure that handles varying fracture criteria [67]. The following provides a specific example that illustrates some of the coupled effects of plasticity and thermal residual stress. Example Chapa and Reimanis have performed numerical simulations for the case of layered, graded elastic-plastic composites when the crack is located within a plastically deforming material but limited to relatively low loads such that the

Fig. 8 Simulated crack trajectories for the geometry depicted in Figure 7a. The initial crack length was 3.2 mm and the initial crack position was (1) 200 μm and (b) 600 μm from the interface. Paths were predicted using finite element analysis with a range of assumed cooling temperatures. (a) and (b) are elastic-plastic, but (c) is elastic only. Taken with permission from [61]

32 Crack Paths in Graded and Layered Structures 1029

1030

I. Reimanis

plastic zone is less than the crack size and a continuum approach with Eq. (6) is valid [60]. Here, the plastic zone size is given by 

σ0 r p ¼ 2:5 e KI

2 (13)

where σ 0 is the yield stress. The model by Chapa and Reimanis utilized the finite element method with a crack tip mesh modification to model the singularity at the crack tip [33]. The J-integral approach may be utilized to find the stress intensity factors. Figure 9 shows an example result for the case of Fig. 3a where Cu is allowed to plastically deform. Figure 9 compares results with and without the presence of thermal residual stress. Without residual stress, the crack deflection angle, θm, is zero until the applied load is sufficiently high (σ a > 25 MPa) that plasticity starts to dominate. For 30 MPa and higher, the crack is drawn toward the plastically weaker material. In general, when the crack is contained within a material with low inherent toughness (failing at K eI less than about 4 MPa m1/2 for the specific example shown in Fig. 3a), the crack does not kink. However, for tougher materials, the crack path is significantly perturbed by the plasticity. The presence of thermal residual stress (Fig. 9) significantly rotates the crack tip stress field such that the crack kinks toward the material with the lower coefficient of thermal expansion (CTE). The lower CTE material is also the plastically stronger material, and, thus, the crack path is completely different from the case with no residual stress. The effect of thermal residual stress is countered by the plasticity as the applied load increases (increasing K eI ). The simulation in Fig. 9 was terminated at an applied stress of around 50 MPa because above this stress, the plastic zone size (Eq. 13) exceeds the specimen crack size, and a continuum approach is no longer valid. Fig. 9 Finite element simulations of the crack kink angle as a function of the equivalent stress intensity factor for the geometry shown in Figure 3a, allowing for plasticity, for the case of no residual stress and with residual stress. Residual stress modifies the sign and magnitude of crack kinking. Figure adapted from [36]

32

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Crack Paths in Graded and Layered Structures

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Conclusion

Numerical simulations provide the best approach to predict crack path when the crack extends into an asymmetric stress field gradient. It has been shown above that in virtually all cases, the thermal residual stress plays a significant role that cannot be ignored. A complex set of behaviors is possible due to the coupling of elastic, plastic, and thermal properties with the geometry and gradient architecture, including the overall geometric configuration, the strength of the gradient, and in the case of layered, graded structures, the layer thickness and number of layers. The complexity makes it difficult to generalize behaviors, but the examples provided illustrate their richness. The extreme sensitivity of the crack path to some of these variables has major implication on material reliability.

5

Future Directions

It has been demonstrated that crack paths in graded structures are inherently linked to material reliability, but the details have not yet been fully explored. For example, how could a Weibull approach be used to predict strength in a graded structure, considering the fact that the exact position of the flaws as well as the geometric and architectural parameters influence crack extension? Crack path prediction must be coupled with the Weibull description. The thermal residual stress probably would significantly influence the results. The effect of T-stresses on crack path for elastic-plastic graded systems has not been considered in detail. One of the challenges is to establish what the relevant length scale is for the excursion parameter. In fact, the approach on crack stability described by Tong has not been experimentally verified. It is not clear whether or not the relevant length scale could be known a priori or whether one has to perform some experiments or simulations. Currently, the understanding is descriptive, but a predictive model would necessitate a determination of the length scale. It is noted that there should be different relevant length scales within the same specimen. For example, in a particular composite, the T-stress is altered by the second phase particle size and arrangement. This effect is expected to manifest itself as a macroscopic change in crack path and/or effective toughness. Thus, while the Tstress is simply a field parameter and does not impact local toughness, if it influences the crack path at a small scale, it impacts the crack growth mechanism and driving force at the larger scale. The change in driving force would be measured as a change in effective toughness of the composite. This appears to be an unexplored area that may lead new designs for composites. The present chapter did not consider dynamic loading situations, but these present an area for future work in the context of crack paths in graded and layered materials. Particularly when plasticity is present, the crack tip constraint may vary in high loading rate regimes. The T-stress correlates with this variation, but predictive models for crack path do not exist.

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In electrochemical applications such as batteries and separation membranes, chemical expansion may lead to stresses that would have a similar effect on crack path as the thermal residual stress. The coupled chemical and thermal expansion on the crack path in graded and layered structures has not been examined in much detail. For some of the newest materials for batteries, the challenge is to obtain accurate material properties for numerical simulation models.

6

Cross-References

▶ Micromechanics of Hierarchical Materials: Modeling and Perspectives

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44. Giannakopoullos AE, M. Olsson M. Influence of the nonsingular stress terms on small-scale supercritical transformation toughness. J Am Ceram Soc. 1992;75(10):2761–4. 45. Fett T. Friction-induced bridging effects caused by the T-stress. Eng Fract Mech. 1998;59(5):599–606. 46. Fett T, Guin JP, Wiederhorn SM. Interpretation of effects at the static fatigue limit of soda-limesilicate glass. Eng Fract Mech. 2005;72:2774–91. 47. Ayatollahi MR, Aliha MRM. On the use of Brazilian disc specimen for calculating mixed mode fracture toughness of rock materials. Eng Fract Mech. 2008;75:4631–41. 48. Wiederhorn SM, Dretzke A, Rodel J. Crack growth in soda-lime-silicate glass near the static fatigue limit. J Am Ceram Soc. 2007;85(9):2287–92. 49. Cotterell TB. Notes on the paths and stability of cracks. Int J Fract Mech. 1966;2(3):526–33. 50. Tong J. T-stress and its implications for crack growth. Eng Fract Mech. 2002;69:1325–37. 51. Marur PR, Tippur HV. Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. Int J Solids Struct. 2000;37:5353–70. 52. Cotterell TB. On fracture path stability in the compact tension test. Int J Fract Mech. 1970;6(2):189–92. 53. Pook LP. Crack paths. Southhamton: WIT Press; 2002. 54. Melin S. The influence of the T-stress on the directional stability of cracks. Int J Fract. 2002;114:259–65. 55. Cao HC, Thouless MD, Evans AG. Residual stresses and cracking in brittle solids bonded with a thin ductile layer. Acta Metall. 1988;36(8):2037–46. 56. Hutchinson JW, Mear ME, Rice JR. Crack paralleling an interface between dissimilar materials. J Appl Mech. 1987;54:828–32. 57. Ritchie RO, Cannon RM, Dalgleish BK, Dauskardt RH, McNaney JM. Mechanics and mechanisms of crack growth at or near ceramic-metal interfaces: interface engineering strategies for promoting toughness. Mater Sci Eng. 1993;A166:221–35. 58. McNaney JM, Cannon RM, Ritchie RO. Near interface crack trajectories in metal-ceramic layered structures. Int J Fract. 1994;66(3):227–40. 59. Tilbrook MT, Reimanis IE, Hoffman M. Finite element simulations of cracks near interfaces: effects of thermal, elastic and plastic mismatch. J Am Ceram Soc. 2005;88(10):2833–8. 60. Tilbrook MT, Reimanis IE, Rozenburg K, Hoffman M. Effects of plastic yielding on crack propagation near ductile/brittle interfaces. Acta Mater. 2005;53:3935–49. 61. Tvergaard V. Theoretical investigation of the effect of plasticity on crack crowth along a functionally graded region between dissimilar elastic-plastic solids. Eng Fract Mech. 2002;69:1635–45. 62. Kolednik O. The yield stress gradient effect in inhomogeneous materials. Int J Solids Struct. 2000;37:781–808. 63. Kim AS, Suresh S, Shih CF. Plasticity effects on fracture normal to interfaces with homogeneous and graded compositions. Int J Solids Struct. 1997;34:3415–32. 64. Charalambides PG, Mataga PA, McMeeking RM, Evans AG. Steady-state mechanics of a growing crack paralleling an elastically constrained thin ductile layer. Appl Mech Rev. 1990;43(5):IIS267–70. 65. Zin Z-H. Effect of material non-homogeneities on the HRR dominance. Mech Res Commun. 2004;31:203–11. 66. Rashid MM, Tvergaard V. On the path of a crack near a graded interface under large scale yielding. Int J Solids Struct. 2003;40:2819–31. 67. Wang Z, Nakamura T. Simulations of crack propagation in elastic-plastic graded materials. Mech Mater. 2004;36:601–22.

Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics

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Chi-Hua Yu, Chang-Wei Huang, Chuin-Shan Chen, and Chun-Hway Hsueh

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Micromechanics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Finite Element Formulation of Viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Cohesive Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Computational Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Competing Mechanisms for Creep Fracture of Polycrystalline ZrB2 Materials . . . . . . . . . . 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Micromechanics Mechanism Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Benchmark Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1037 1040 1045 1046 1046 1048 1050 1050 1061 1062 1062 1063 1064 1065

C.-H. Yu (*) Division of Research and Development, CoreTech Corp. (Moldex3D), Hsinchu, Taiwan e-mail: [email protected] C.-W. Huang Department of Civil Engineering, Chung Yuan Christian University, Chung-Li, Taiwan e-mail: [email protected] C.-S. Chen Department of Civil Engineering, National Taiwan University, Taipei, Taiwan e-mail: [email protected] C.-H. Hsueh Department of Materials Science and Engineering, National Taiwan University, Taipei, Taiwan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_68

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4 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Effects of Cavity Nucleation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Effects of Grain Boundary Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Effect of Applied Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Polycrystalline Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Creep-Competing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Diffusional Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Grain Boundary Sliding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1067 1067 1070 1072 1075 1077 1080 1081 1082 1084 1087 1090

Abstract

Ultrahigh-temperature ceramics has great potential for applications as refractory materials at high temperatures. ZrB2-SiC has been considered as an excellent candidate of the ultrahigh-temperature ceramics due to its relatively low density and excellent refractory properties. However, the creep fracture of ZrB2-SiC limits its potential applications. Mitigation of the creep fracture is thus imperative. It has been concluded from several experiments that the creep resistance of ZrB2-SiC decreases with the increasing temperature, and there also exists a transition of creep mechanism for temperature above 1500
C. The effects of grain boundary heterogeneity on the creep resistance were studied. The creep resistance of an isotropic and homogeneous ZrB2 polycrystalline material is affected by the applied strain rate and the grain boundary properties. Grain boundary heterogeneity would initiate the microcrack and thus lead to fracture. An isotropic grain interior modeled by user-defined material properties (UMAT) subroutine along with the grain boundary simulated by a ratedependent cohesive zone modeling using user-defined element (UEL) subroutine was constructed to study the creep fracture of ZrB2-20% SiC composites. The model is accounted for nucleation, growth, and coalescence of cavities along the grain boundaries in a localized and inhomogeneous manner, link up of microcracks to form macrocracks, and grain boundary sliding. For ZrB2-20% SiC composites, a micromechanism shift form diffusional creep-control for temperatures below 1500
C to grain boundary sliding-control for temperatures above 1500
C was concluded from simulations. Also, the simulation results revealed the accommodation of grain rotation and grain boundary sliding by grain boundary cavitation for creep at temperatures above 1500
C which was in agreement with experimental observations. Keywords

Creep fracture · Micromechanics · Finite element analysis · Cavitation · Grain boundary sliding

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Introduction

It is envisioned that the next-generation engine or nozzle design would be made of high-strength, lightweight, and excellent creep resistance materials that could be operated over 2000
C [1]. Most of the metallic alloys will not be able to sustain such an extremely high-temperature environment even with a layer of thermal barrier coating. In the early 1960s, scientists and engineers already found the potential application of boride-based ceramics as an ideal candidate for refractory materials [2, 3]. Gifted by the high melting points, usually over 3000
C, and a relatively low density, ultrahigh-temperature ceramics (UHTC) materials can be exploited for extensive applications in aerospace, aircraft, and military industries [4]. Zirconium diboride (ZrB2) has been considered as one of the most useful UHTCs because of its extremely high melting temperature, approximately 3300
C that is higher than the basic requirement of structural materials for a container of most burnable fossil fuel during its operation process [5–7]. Along with the relative low density and superb thermal shock resistance, ZrB2 has become a potential candidate of structural materials for cutting-edge aerospace applications such as hypersonic aircraft or missile propulsion. Research efforts have been devoted to the studies of basic properties of ZrB2, including chemical synthesis and mechanical and thermal properties since the 1960s. A comprehensive review of ZrB2 has been carried out by Fahrenholtz et al. [8]. To extend the trustworthy application life circle of ZrB2 as an UHTC, basic mechanical properties and flexure strength of ZrB2 have been studied to provide guidelines for industrial applications. The microhardness of single crystal ZrB2 was investigated by Xuan et al. [9] which exhibited the temperature-dependent softening behavior from 1400
C to 1700
C. Also, the mechanical properties showed a loading rate-dependent behavior. Chamberlain et al. [10] reported that the values of Young’s modulus and room temperature flexure strength of ZrB2 with an average grain size of 2 μm were 489 GPa and 563 MPa, respectively. For ZrB2-SiC composites, the modulus did not vary significantly with the SiC content while the strength increased as SiC content increased. Monteverde et al. [11] also conducted a study to examine the microstructure and mechanical properties of ZrB2, and Young’s modulus and room temperature flexure strength were about 346 GPa and 350 MPa, respectively. They also indicated a temperature-dependent softening behavior at high temperatures. The toughness of ceramic materials is much lower than metals and alloys owing to the nature of atomic bonding. Also, their fine-grained structures, typical microstructural features of high-strength materials, are expected to suffer from creep deformations at high service temperatures since the dominant creep mechanism occurs at the grain boundary. It has been concluded that fracture of ceramics at high temperatures by creep can be classified into two categories: slow crack growth and creep fracture. While slow crack growth is a direct extension of a crack by diffusion of atoms through the crack tip, creep fracture is a process of nucleation,

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growth, and coalescence of cavities along the grain boundary in a localized, inhomogeneous manner. Generally, fracture of ceramics at high temperatures exhibits a transition from slow crack growth to creep fracture as the temperature increases [12, 13]. This transition also depends on stresses, such that low stresses are in favor of creep fracture [12, 13]. However, in both fracture mechanisms, surface diffusion along the crack surfaces and grain boundary diffusion to remove atoms from the crack tip are involved. While the surface diffusion is driven by the curvature of the crack surfaces, grain boundary diffusion is driven by the stress distribution along the grain boundary [14]. Creep fracture of polycrystalline ceramic, which occurs at elevated temperatures and low stresses, is considered in the present study. Observations of the creep fracture process have revealed that cavities nucleate mainly on grain boundaries that are normal to tensile loading. These cavities grow under the influence of hightemperature mechanisms such as grain boundary diffusion, grain boundary sliding, and creep. They then coalesce to form grain boundary microcracks, and finally intergranular fracture occurs when these microcracks link up to form a macroscopic crack. It should be noted that because cavitation occurs in an inhomogeneous and localized manner, it would induce localized stresses which, in turn, would modify the resultant stress field [14–16] and the stress analysis during creep fracture is a nontrivial problem. In general, the strength of ZrB2 suffers from creep fracture at elevated temperature (which is usually higher than 1500
C). This also restricts its application to a low life cycle. As a result, it is essential to understand the possible mechanisms of creep fracture. It has been concluded by several studies that intergranular fracture dominates the creep fracture process because of the nature of the fine crystalline structure of ZrB2 with certain small amount of impurities. Talmy et al. [17] investigated creep fracture of ZrB2 with SiC additive subjected to a four-point bending flexure test. The creep deformation showed a strong dependence on the applied stress and the temperature. Meléndez-Martınez et al. [18] performed a series of experiments to characterize the mechanical properties, such as Young’s modulus, microhardness, fracture toughness, and compressive strength of ZrB2 and the creep resistance of ZrB2 was found to be inversely proportional to the temperature from 1400
C to 1600
C. To broaden the potential applications of ZrB2, thorough understanding of the grain boundary properties and the creep mechanisms of ZrB2 in ultrahightemperature environment is much needed. A number of research efforts have been devoted into this realm since the 1960s. The mechanism of oxidation and the nature of protective coatings of ZrB2 at ultrahigh temperatures have been reported by Kaufman and Clougherty [2] in 1965. These pioneering works provided a guideline to determine the material behaviors of ZrB2. The mechanical properties of ZrB2 have been tested by the resonance frequency method, compact tension test, and Vickers indentation test for two different sintering processes [18]. It revealed that the properties of ZrB2 grain boundaries could be changed by certain additives, which resulted in significant improvements of the densification and flexure strength of ZrB2 ceramics. Fahrenholtz and coworkers [8] studied the densification of diboride

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ceramics, and their results showed that the strength of ZrB2 was inversely proportional to the grain size at room temperature. In addition, the flexure strength of ZrB2 in high-temperature environment was found to be strongly dependent on the microstructure resulting from different synthesis methods. It all comes from the porous structure of ZrB2 ceramics which would jeopardize the stability of materials during forming and serving as a refractory component. Several additives have been tested and examined during the sintering process to enhance the toughness and the flexure strength in high-temperature ambient. There are two purposes for additives: (1) to improve the microstructure by finer additives, e.g., SiC or ZrO2 powders, as a ceramic composite and (2) to modify certain properties of grain boundary, such as grain boundary diffusivity and viscosity, by adding a small amount of metallic particles. Particularly, sintering of ZrB2 with a certain amount of SiC would not only enhance the strength but also improve the oxidation resistance. The microstructure and flexural strength of ZrB2 with 10–30% SiC were characterized and measured by Zhang et al. [19]. The flexure strength was proportional to the value of SiC contents at room temperature because of the pores in ZrB2 microstructure was largely occupied by the finer grains of SiC. Furthermore, the flexural strength was also related to the grain size of ZrB2 and the SiC content in high-temperature oxidation atmosphere [20]. Researchers also found that the ZrB2-SiC composite exhibited a relatively low creep resistance at high temperatures. Talmy et al. [17] revealed that the creep resistant was reduced with an increase of SiC contents and the grain size of SiC. This phenomenon would limit the applications of ZrB2-SiC composites as ultrahigh-temperature ceramics. The reason of the deterioration in creep resistance is mainly caused by the competing mechanisms occurred at the grain boundaries. There existed a mechanism shift from grain boundary diffusion to grain boundary sliding by comparing the stress components with different volume fractions of SiC grains [21]. In addition, an oxidation layer would thicken the interface and introduced an extra oxygen concentration at the ZrB2-SiC grain boundary. Grain boundary heterogeneity is also a key factor that would change the dominating mechanism to sliding. Bird et al. [22] tested the creep resistance of ZrB2 with 20% SiC content in an argon atmosphere to prevent oxidation from 1400
C to 1820
C. Experimental results implied that there existed a significant change of the activation energy with respect to temperature region for temperature below and above 1500
C. In addition, the stress component, n, in the power-law creep was observed with a change from 1.0 to 2.2. The objective of this chapter is to study the competing creep mechanisms of grain boundary for ZrB2-SiC ceramics. In this work, we focus on: • To establish a robust modeling framework for the intergranular creep fracture for ultrahigh-temperature ceramics • To carry out the parametric study of the heterogeneity effect at grain boundaries • To link the micromechanical mechanisms to the macroscopic creep behavior • To examine the sliding-induced dislocation gliding for ultrahigh-temperature creep deformation

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An important aspect of this study in modeling UHTCs is to elucidate the competing creep mechanisms and to develop adequate computational frameworks suitable for intergranular fracture and creep predictions. Motivated by the results of previous studies mentioned above, further development of modeling creep deformation for the ultrahigh-temperature ceramics mainly caused by grain boundary diffusion and grain boundary sliding. The present computational approach can be extended to develop a modeling framework and computational procedures for intergranular creep fracture for various microstructures.

1.1

Micromechanics Model

In this section, an introduction of the micromechanics modeling accounting for elasticity and creep will be presented. There are two different approaches for describing the grain interior in this work. The first approach is a power-law type of Norton creep modeling for the isotropic, homogeneous grain interior that is derived from the contribution of diffusion creep. The second approach is a crystal plasticity model based on the Schmidt’s slip for crystalline materials. A rate-dependent cohesive model contributed from the cavity nucleation, growth, and coalescence on the grain boundaries, and grain boundary sliding has been developed previously [23]. The constitutive equations used in Ref. [23] are also adopted in the present study and are summarized in this chapter. However, instead of adopting a fictitious layer to simulate the grain boundary deformation in Ref. [23], a rate-dependent cohesive element is developed in the present study. This would not only greatly improve the numerical stability but also yield more accurate results and allow systematic studies of microstructural effects on the grain boundary deformation.

1.1.1 Norton-Type Creep Within the grains, the material is assumed to be homogeneous and isotropic and deformed by both elasticity and power-law creep. In this case, the total strain rate tensor is the sum of the elastic and the plastic strain rate tensors, such that e_ij ¼ e_eij þ e_cij

(1)

These two strain rate components can be calculated from
 1þν ν σ_ ij  σ_ kk δij ¼ E 1þν

n Q σ e 3 Sij e_pij ¼ e_0 exp RT σ0 2 σe

e_eij

(2)

(3)

where E and v are Young’s modulus and Poisson’s ratio, respectively; δij is the Kronecker delta; Q is the activation energy; R is the ideal gas constant; T is the

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Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics

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absolute temperature; e_0 and σ 0 are the reference strain rate and the reference stress, respectively; n is the creep exponent; Sij is the deviatoric stress tensor; and σ e is the von Mises stress given by rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Sij Sij σe ¼ (4) 2

1.1.2

Rate-Dependent Cohesive Zone Model

Grain Boundary Sliding The grain boundary sliding dominates creep deformation in the ultrahigh-temperature region. Grain boundary sliding has been considered as a complementary mechanism yet independent of the grain boundary diffusion and cavity nucleation. Nevertheless, it plays as a necessary mechanism to meet the compatible requirement during creep deformation for polycrystalline ceramics. For grain boundary sliding, it is assumed to be governed by Newtonian viscous flow, such that [24] u_ s ¼ w

τ η

(5)

where u_ s is the relative sliding velocity of adjacent grains resulting from the shear stress, τ, on the grain boundary, and w and η are the thickness and the viscosity of the grain boundary, respectively. The temperature-dependent function of viscosity for ZrB2 and SiC ceramics were suggested by Kislyi and Kuzenkova [25], such that 1 A Qs ¼ e RT η T

(6)

where T is the absolute temperature, A is a constant value, Qs is the activation energy for grain boundary viscosity, and R = 8.314 JK1mole1 is the idea gas constant. Grain Boundary Cavitation Grain boundary cavitation requires removal of materials from the cavity surface and deposition on the grain boundary. This results in the separation between two adjacent grains. However, it is a complex process to model a set of grain boundary cavities during creep fracture. To simplify the cavitation problem to a manageable manner in modeling, a smeared-out representation of the grain boundary cavitation was proposed [26], in which the distribution of cavities on each grain boundary (Fig. 1a) was replaced by a continuous varying separation as shown in Fig. 1b. Adopting this simplification, the separation of grain boundaries resulting from cavitation can be analyzed using the continuum mechanics approach. Also, the cavity on the grain boundary can generally be characterized by its radius a, half spacing b, and spherical-caps shape parameter h that is determined by the cavity tip angle ψ (see Fig. 1a). In this case, the cavity volume can be expressed as

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Fig. 1 Schematics showing (a) the geometry of cavities in the spherical-caps shape located at the grain boundary and (b) the smeared-out representation of grain boundary cavitation in terms of a continuous varying normal separation, un

4 V ¼ πa3 hðψ Þ 3

(7)

where the spherical-caps shape parameter is defined by 

1 ð1 þ cos ψ Þ  cos ψ 2 hð ψ Þ ¼ sin ψ 1

 (8)

Based on observations, ψ has a typical value of 75
during the cavity growth [27], and this value is adopted in the present study. The smeared-out representative separation of grain boundaries, as shown in Fig. 1b, is described by un ¼

V πb2

(9)

As a result, the rate of change of the normal separation is u_ n ¼

V_ 2V b_  πb2 πb3

(10)

_ is related to the strain rate of adjacent grains The rate of void spacing change, b, and the cavity nucleation, such that [28] 1 N_ b_ 1 ¼ ðe_I þ e_II Þ  2N b 2

(11)

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Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics

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where e_I and e_II are in-plane logarithmic strain rate at the grain boundary and N is the cavity density, which is defined as the number of cavities per unit undeformed grain boundary to describe the nucleation of new cavities. Cavity Nucleation Cavity nucleation occurs at the atomic scale and depends on both the grain boundary microstructure and the local stress. Several studies on the mechanism of the cavity nucleation, which took place at grain boundary at high temperature, have been proposed for decades. However, there exists no unified theory to describe the complex process of cavity nucleation. A phenomenological model proposed by Tvergaard [29] is hence adopted in this work. According to their model, the cavity nucleation rate is proportional to the effective creep strain rate, e_ce, and normal stress, σ n, on the grain boundary, such that
2 σn _ N ¼ Fn P e_ce for σ n > 0

(12)

0

where Fn is a material parameter and Σ0 is a normalization factor. However, experimental evidences often show that cavity nucleation does not occur right from the initial stage of plastic deformation. To account for this nucleation threshold, a parameter that combines stress and accumulated strain has been proposed, such that [28]
S¼

σ Pn 0

2 ece for σ n > 0

(13)

The parameter, S, must reach a threshold value, Sthr, to trigger the cavity nucleation. This threshold value, Sthr, can be defined in terms of the initial cavity density, Ni, through Sthr ¼

Ni Fn

(14)

Cavity Growth Cavity growth results from both diffusion of atoms from cavity surfaces to the grain boundary layer and the creep deformation of the surrounding grains. Hence, the _ is volumetric cavity growth rate, V, V_ ¼ V_ 1 þ V_ 2

(15)

where V_ 1 and V_ 2 represent the contributions of atom diffusion and creep, respectively, and have been derived elsewhere, such that [28]

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C.-H. Yu et al.

σ n  ð1  f Þσ s V_ 1 ¼ 4πD
 1 1 ln  ð3  f Þð1  f Þ f 2 8 h in σm > < 2π_ce a3 hðψÞ αn j σσme j þ βn , for  >1 σe V_ 2 ¼ σ σ m m > : 2π_ce a3 hðψÞ½αn þ βn n , forj j < 1 σe σe

(16)

(17)

with (

2 ) a2
a f ¼ max , b a þ 1:5L   σ e 1=3 L¼ D c _e

(19)

3 2n

(20)

ðn  1Þðn þ 0:4319Þ n2

(21)

αn ¼ βn ¼

(18)

where L is a stress- and temperature-dependent length scale factor introduced by Needleman and Rice [30] and D is the grain boundary diffusion parameter. It should be noted that the effective stress σ e and the mean stress σ m are obtained from the adjacent grains instead of the cavity. The sintering stress σ s in Eq. 16 is usually neglected owning to its small value [28]. In general, L indicates the competition between two mechanisms: diffusion and plastic creep flow effect. This characteristic length, L, is usually a function of temperature and stress. When L has a relative high value compared with the void radius, a, or the half void spacing, b, the plastic flow effect could be omitted. It also implies a low stress and low-temperature condition under this circumstance. On contrary, when L is relative small compared with a and b for a high-temperature and high-stress environment, the void growth rate would increase tremendously with the ratio of a/L. The effects of grain boundary diffusion and plastic creep contributed to the void growth simultaneously. The cavity growth is dominated by creep for large values of a/L, while the atom diffusion is more important for small values of a/L (e.g., a/L < 0.1). Also, it is worth noting that the traction on the grain boundary decreases drastically prior to cavity coalescence (i.e., when a/b ! 1) which, in turn, would result in numerical instability. To circumvent this numerical instability, cavity coalescence was assumed to occur when a/b reached 0.7 in the existing work [23], and simulation of cavity growth from a/b = 0.7 to 1 was not required. However, with the improved numerical stability in our rate-dependent cohesive element, the entire process of cavity growth (i.e., for a/b up to 1) can be simulated.

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1.2

Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics

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Discussion

Modeling of intergranular creep fracture requires consideration at all relevant length scales. At the atomic/molecular level, cavity nucleation can start at the triple points, the grain boundary ledges, or the second-phase particles located at the grain boundary where stress concentrations exist. However, it remains a great challenge to quantify the nucleation directly from atomistic simulations, and cavity nucleation is often described by a phenomenological model. At the microscopic level, the cavity growth results from both the diffusion of atoms from the cavity surfaces into the grain boundary and the creep deformation of the surrounding grain materials [30, 31]. In addition, grain boundary sliding occurs in the presence of shear stress along the grain boundary, the material inside the grains is subjected to both elastic and creep deformations, and coalescence of cavities can lead to a microcrack on the grain boundary. At the mesoscopic level, grain aggregate is considered, and the microcracks can link up to form a macroscopic crack to result in fracture. While grain boundary sliding is dictated by a sliding velocity versus shear stress relation, deposition of materials from cavities to grain boundary due to grain boundary cavitation can be modeled by a grain boundary thickening rate versus normal stress relation. To model the viscoelastic constitutive behavior of the grain boundary, a fictitious layer of linear elastic spring against shearing and opening was introduced [23, 32] for the numerical convenience to approximately describe the cavitation process on the grain boundary. Sufficiently large values of the elastic stiffness of the normal and shearing spring layers were adopted, such that the results derived from the viscoelastic constitutive equations were independent of the stiffness of this fictitious elastic layer [28]. In this case, the relationship between traction and separation on the grain boundary is biased, and numerical errors are accumulated as the cavity grows during simulations. To overcome this inherent weakness in assuming a fictitious layer and to gain insights into the microstructural effects on the grain boundary deformation, it is essential to derive a cohesive law that explicitly specifies the relations between the normal/tangential displacement rates and the corresponding tractions, and the present study is aimed at deriving this cohesive law. Cohesive zone model has often been exploited in the numerical simulation of void nucleation and fracture in materials subjected to various kinds of loading conditions. Also, the cohesive element in finite element analysis has been investigated extensively. As the commercial finite element program ABAQUS [33] is well equipped with high robustness in solving a wide range of nonlinear problems, current model is to leverage this capability and to develop a robust user-defined subroutine of rate-dependent cohesive element in ABAQUS to simulate the intergranular creep fracture problem. The detail of implementation will be introduced in the following sections.

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Numerical Implementation

A nonlinear finite element model based on the commercial package ABAQUS [33], which is very robust in solving numerical instabilities during the time integration in transient analyses, is adopted in the present study. A rate-dependent cohesive element is developed to simulate the effects of sliding and cavitation on the grain boundary through the user-defined element (UEL) subroutine in ABAQUS. A userdefined material properties (UMAT) subroutine is also implemented in ABAQUS to simulate both the elastic deformation and the power-law creep in the grain. The tangent stiffness matrix for the proposed cohesive element is described below. For simplicity, only the plane-strain case is considered in the following.

2.1

Finite Element Formulation of Viscoplasticity

The formulation of rate-dependent viscoplasticity was extended by finite element method. Unlike linear elastic problem, the major point to be solved is the numerical integration to integrate the time-dependent plastic constitutive equation. In the present study, the infinitesimal deformation assumption was adopted. The superposition of strain rate can be written as the linear combination of elastic strain rate and creep strain rate. e_ij ¼ e_eij þ e_cij

(22)

The constitutive relationship for elastic strain rate and the plastic strain was given by: e_eij


 1þν ν σ_ ij  σ_ kk δij ¼ E 1þν e_cij ¼ e_eff

3 Sij 2 σe

(23) (24)

where Sij is the deviatoric stress tensor, σ e is the von Mises stress, and e_eff is the effective strain that can be evaluated from the creeping law e_eff


n σe ¼ e_0 σ0

(25)

Solution of this time-dependent problem is contingent upon the satisfaction of Eqs. 22, 23, 24, and 25 simultaneously. Generally, a virtual work formulation is made as a replacement for the equilibrium condition in the finite element analysis. The virtual work of viscoplasticity material system is given by ð σ ij Δekl Ω

ð ð @N a dV  bi N a dV  ti N a dA ¼ 0 @xj Ω

@Ω

(26)

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Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics

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where Na is the interpolation function of each element, bi is the body force, ti is surface traction, and the superscript a denotes the node number. The weak form of viscoplasticity could be obtained by combining Eqs. 22, 23, 24, 25, and 26. The essence of time-dependent finite element analysis is to obtain the stress state corresponding to an incremental total strain Δeij applied to the object during the incremental time period Δt. A robust implicit integration technique was employed in the present study to solve the stress state. At current time state t = tn, the stress state ðnÞ σ ij and accumulated plastic strain eðenÞ are given. For the next time increment tn þ 1 = tn þ Δt, the deviatoric strain increments Δeij,can be computed Δeij ¼ Δeij  Δekk

δij 3

(27)

The crux of this implicit integration method to assure the robustness is a predictor-corrector scheme. The elastic stress predictor was computed as a general elastic problem at the end of the time increment; i.e., E Δekk 3ð1  2νÞ

(28)

 E E  ðnÞ Δeeij ¼ Sij þ Δeij  Δepij 1þν 1þν

(29)

pðnþ1Þ ¼ pðnÞ þ ðnþ1Þ

Sij

ðnÞ

¼ Sij þ

for the hydrostatic stress p(n þ 1) and the deviatoric stress S(n þ 1). To achieve numerical stability, we calculate the plastic strain increment at the next time incremental, and the strain increment evaluated from Eq. 24 can be expressed as ðnþ1Þ

Δepij ¼ Δee

3 Sij 2 σ ðenþ1Þ

(30)

where Δee ¼ Δte_0

ðnþ1Þ

σe σ0

!n (31)

The predictor of deviatoric stress is constructed based on the absence of the plastic strain, that is ðnþ1Þ

Sij

ðnÞ

¼ Sij þ

E Δeij 1þν

(32)

The real stress state is assumed proportional to the predictor in Eq. 32. The scale factorβ is introduced for calculation and the expression of actual stress state becomes S(n þ 1) = βS(n þ 1). By substituting Eqs. 27, 28, 29, 30, and 31 into Eq. 32, we have

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C.-H. Yu et al.

ðnþ1Þ Sij

¼

ðnþ1Þ

By eliminating Sij time increment is

ðnþ1Þ βSij

¼

ðnþ1Þ βSij

ðnþ1Þ

3 E Sij  Δee 2 1 þ ν σ eðnþ1Þ

(33)

and combining Eqs. 32 and 33, the stress state at the end of "

ðnþ1Þ Sij

¼

ðnþ1Þ Sij

¼ 1

3E

#
Δee ðnþ1Þ

2ð1 þ νÞσ e

ðnÞ

Sij þ

E Δeij 1þν

 (34)

where Δσe* is the predictor of the effective stress. It is worth noting that the implicit algorithm or the backward-Euler method can be shown to be unconditionally stable. Even though the scheme is more complicated than the forward-Euler method, the advantages far outweigh the additional complexity.

2.2

Cohesive Element

The weak interfaces in a solid may separate under a sufficiently high stress. This interface separation behavior is usually described by the cohesive laws that relate the normal and tangential displacements to the corresponding tractions. There are a variety of possible methods for implementing cohesive laws using ABAQUS, and the development of cohesive elements is the most versatile one. The cohesive element is made up of two linear line elements as shown in Fig. 2, which connect the faces of adjacent plane elements during the fracture process. The two lines of the interface element initially lie together in the unstressed state with zero thickness and separate as the adjacent elements deform. The relative displacement of the element faces induces both the normal and the tangential displacements which, in turn, would result in element stresses depending on the constitutive equations. Description of these constitutive equations have been shown in the previous section which takes into account the grain boundary sliding, cavity nucleation, growth, and coalescence on the grain boundary. In the tangential direction, the relationship between the shear stress and the tangential displacement is governed by Eq. 5, in which the Newtonian viscous Fig. 2 The grain boundary is simulated by the cohesive element that is made up of two linear line elements and four nodal points

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Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics

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deformation is assumed. The partial derivative of the incremental shear stress with respect to the incremental tangential displacement is: @Δτ η ¼ @Δus wΔt

(35)

where Δt is the time step in the time integration scheme. In the normal direction, the normal stress is related to the normal displacement which is governed by the cavity nucleation, growth, and coalescence on the grain boundary. Substituting Eqs. 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 into Eq. 10, the normal displacement rate can be expressed as a function of the normal stress in a quadratic form, such that Δu_ n ¼ Aσ 2n þ Bσ n þ C

(36)

where A¼ B¼

1 π b2

C¼

V 1 Fn c e_ π b2 N Σ20 e

(37)

4πD
 1 1 ln  ð3  f Þð1  f Þ f 2 V_ 2 V  ðe_I þ e_II Þ π b2 π b2

(38)

(39)

It is worth noting that A, B, and C signify, respectively, the contributions of cavity nucleation, atom diffusion, and creep to the normal separation. The partial derivative of the incremental normal stress with respect to the incremental normal displacement can be derived from Eq. 36, such that @Δσ n 1 ¼ @Δun ð2Aσ n þ BÞΔt

(40)

Combining Eqs. 35 and 40, the material Jacobian (tangent stiffness) matrix can be obtained, such that

Δσ n Δτ



2

1 6 ð2Aσ n þ BÞΔt ¼4 0

3

0 7 Δu

n η 5 Δus wΔt

(41)

It is worth emphasizing that the normal separation and the shear separation on the grain boundary are decoupled since only diagonal terms exist in the material Jacobian matrix. In addition, all components in the material Jacobian matrix are derived in an explicit form and are implemented in the user subroutine UEL in ABAQUS.

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Computational Framework

Based on Eqs. 13 and 17, it is necessary for the scheme to access the information of state variables, including the stress and the strain, of adjacent grains in calculating the normal separation on each grain boundary. In other words, to calculate the normal displacement on the grain boundary, the analysis requires not only the knowledge of the state-dependent variables of the grain boundary element at Gaussian integration points from UEL user subroutine but also the variables in the surrounding grain elements adjacent to the grain boundary element. Although ABAQUS provides a default material model of the power-law creep to analyze the state variables of all grains due to elasticity and creep, these state variables of grain elements cannot be accessed directly from the UEL subroutine during the analysis process. Therefore, a user subroutine UMAT in ABAQUS, based on the elasticity and power-law creep, is developed to perform the creep analyses for grain elements. To achieve the stability of numerical integration, an implicit stress integration scheme is chosen for UMAT [34], while a semi-implicit integration scheme is chosen for UEL for the force recovery. The semi-implicit scheme involves updating all state variables at the end of the current step by adding the current increment to the values of the current step. For example, to update the normal stress on the cohesive element at the end of the k-th time step; i.e., at t = t(k þ 1), Eq. 36 can be rewritten as:  2   u_ ðnkþ1Þ ¼ A σ ðnkÞ þ Δσ ðnkÞ þ B σ ðnkÞ þ Δσ ðnkÞ þ C

(42)

where the superscript in the parenthesis represents the time step. Equation 42 can be extended and reformulated as:       2  2 ðk Þ ðkÞ ðk Þ ðkÞ ðk Þ A Δσ n þ 2Aσ n þ B Δσ n þ A σ n þ Bσ n þ C  u_ ðnkþ1Þ ¼ 0 (43) In this case, Eq. 43 is a quadratic function of Δσ ðnkÞ which is an unknown at the current time step. Since all the stress components at the beginning of the current time step and the displacement rate in the normal direction at the current time step are known, the unknown, Δσ ðnkÞ , can be solved by the Newton-Raphson method. In this case, the normal stress can be updated, such that σ ðnkþ1Þ ¼ σ ðnkÞ þ Δσ ðnkÞ

(44)

The computational procedure of UEL is shown in Fig. 3.

2.4

Numerical Examples

A bi-crystal with a single grain boundary is simulated first to verify its accuracy and numerical stability. Polycrystalline ceramics with typical hexagonal grains and

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Fig. 3 The flowchart of the computational procedure of the user-defined element (UEL) in our computational framework

multiple grain boundaries are then simulated to study the effects of grain boundary heterogeneities. Data of many material parameters are required in the simulation. To obtain the essential trends of the dependence of creep fracture on the material

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parameters, the material parameters were normalized by the initial half grain facet length, Ri, Young’s Modulus of the grain element, E, the applied strain rate, e_app, and the reference cavity density, Nr (=1/πR2i ), in a previous study [28]. Some material properties pertinent to ZrB2 high-temperature ceramics [17] and the normalized material parameters used in [28] are adopted in the present study. Specifically, using Ri = 5 μm, E = 400 GPa, ν = 0.15, and e_app = 1  106 s1, all other parameters are normalized by Ri, E, e_app , and Nr. The normalized reference stress, σ 0/E, is set at 0.025, the normalized reference strain rate, e_0 =e_app, is 1  102, and the power-law exponent for creep, n, is 1.1 for the grain element. For the grain boundary element, the initial cavity radius a is set at 0.67  103Ri, the half cavity spacing b is 0.16Ri, and the initial cavity density of the undeformed grain boundary Ni is directly related to the initial cavity spacing and is 40Nr. The material parameter in the cavity nucleation law, Fn, is set at 5.3  104Nr for brittle material, and the normalization factor, Σ0, in Eq. 12 is 2.5  103E. The time scale, t, is normalized by the applied strain rate, e_app, such that t* = te_app. Also, the ratio of the viscosity η to the width of grain boundary w, is normalized by the applied strain rate, e_app, and Young’s modulus E, such that  η  w

¼

η Ri e_app w E

(45)

The normalized diffusion parameter, D*, is defined by D ¼

DE e_app R3i

(46)

In the simulation, (η/w)* = 1.25  105, and D is determined by a fixed L adopted in simulations. For L/Ri = 0.19 (i.e., a/L = 3.36  103), the corresponding D* is 8  105. In this case, diffusional cavity growth is much faster than the creep deformation, and the fracture behavior is more brittle.

2.4.1 Bi-crystal with Simple Boundary Value Problem The first example of our implementation is a bi-crystal that consists of two square gains with facet length of 2Ri and a single grain boundary as depicted in Fig. 4 [33, 35]. This example is conducted to study the numerical stability and accuracy of the proposed implementation and to gain insights into the microstructure effects on the grain boundary deformation based on the cohesive law described in the present study. Specifically, the two grains are modeled by the plane-strain element (CPE4 in ABAQUS), and the grain boundary is modeled using the rate-dependent cohesive element described in the previous section. The system is loaded by displacing the top and bottom facets of the system at a constant vertical velocity V0, while the side facets are free. The velocity V0 is set at a value such that the applied strain rate, e_app, is 106 s1. Because the normal separation is uniform along the grain boundary in this case, the number of cohesive elements used for the grain boundary, Ngb, makes no difference in the simulation results. Also, it should be noted that because e_app is used

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Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics

Fig. 4 Schematics showing a bi-crystal with a single grain boundary used in our creep fracture simulations

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V0

E,v

2Ri

E,v

2Ri

2Ri V0

as a normalization factor in the present simulations, the following normalization results are independent of e_app for a fixed L defined by Eq. 19. However, hightemperature creep tests of ceramics have generally been performed at a strain rate of 106 to 108 s1 [32]. The initial normalized time step Δt* is set at 108, and the following time steps are adjusted automatically by ABAQUS to maintain the desirable convergence with efficiency. In the present example, no shear sliding occurs along the grain boundary between the two adjacent grains. The relationships between the normal traction and the normal separation on the grain boundary versus time are plotted, respectively, in Fig. 5a, b. From Fig. 5a, one can observe that the normal traction increases almost linearly until it reaches the maximum stress and then decreases rapidly afterward. On the other hand, the normal separation remains almost zero and increases rapidly after the normal traction reaches the maximum stress, and its increasement then becomes slower as the stress rapidly decreases. This implies that the bi-crystal behaves like a brittle material until it yields by grain boundary cavitation, the deformation of the system is attributed to grain deformation initially, and grain boundary separation prevails when yielding occurs. Combining Fig. 5a, b, the rate-dependent cohesive behavior of the grain boundary is shown in Fig. 5c. The brittle behavior prior to cavitation yielding reflected by Fig. 5a–c is attributed to the small value of a/L used in the simulation. Figure 6a–c show the relationships of the cavity radius a, half spacing b, and a/b, respectively, as functions of time. In Fig. 6a, the increase of the cavity radius, a, with time represents cavity growth on the grain boundary. On the other hand, the half spacing, b, decreases with time representing cavity nucleation. The rapid decrease of b with time reflects the high cavity nucleation rate used in the simulation. For t* > 0.007, b remains almost constant; i.e., b_ ! 0. This is mainly due to the large number of cavities and small value of b on the grain boundary (see Eq. 11). Near the end (at t* ~ 0.01), b attains a minimum constant. This corresponds

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Normalized normal traction, sn/E

a 0.0016

0.0012

0.0008

0.0004

0

Normalized normal separation, un/Ri

b

0

0.002

0.004 0.006 0.008 Normalized time, teapp

0.01

0

0.002

0.004 0.006 0.008 Normalized time, teapp

0.01

0

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Fig. 5 (a) The normalized normal traction and (b) the normalized normal separation versus normalized time relationships and (c) the ratedependent cohesive behavior of the grain boundary for a bicrystal subjected to creep fracture

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Normalized cavity radius, a/Ri

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b Normalized half cavity spacing, b/Ri

Fig. 6 (a) The normalized cavity radius a/Ri, (b) the normalized half cavity spacing b/Ri, and (c) the a/b ratio versus normalized time relationships for a bi-crystal subjected to creep fracture showing both cavity growth and nucleation

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to the rapid decrease of the grain boundary traction after reaching its maximum, and the corresponding cavity nucleation rate is almost zero. Finally, the a/b ratio increases monotonically and reaches the upper limit of 1 which represents the complete separation of the grain boundary. The result shown in Fig. 6c demonstrates the numerical stability of our rate-dependent cohesive element, such that the entire cavity growth process can be simulated without artificially setting a limit of a/b to represent cavity coalescence elsewhere [28].

2.4.2 Polycrystal with Regular Hexagonal Grains A polycrystal model presents a potential application of the cohesive element to the actual microstructure which considers the cavity nucleation/growth according to micromechanics. A configuration of regular hexagonal grains is shown in Fig. 7. The initial facet length of each grain is set at 10 μm. The grains are modeled by threenodes triangle elements (CPE3 element in ABAQUS), and the grain boundaries are modeled by cohesive (grain boundary) elements. At the grain boundary, the base of the triangle element from the grain is attached to one cohesive element of the grain boundary. Material properties of the grains, grain boundaries, and other parameters in microstructures adopted in simulations are the same as those used in the example for bi-crystal. The model is subjected to a constant vertical velocity at the top and the bottom boundaries, while the periodic boundary condition (PBC) is set at the right and the left boundaries; i.e., u2 ¼ V 0 t and u1 ¼ 0 on Γtop

(47)

u2 ¼ V 0 t and u1 ¼ 0 on Γbottom

(48)

PBC on Γright and Γleft

(49)

The velocity V0 is also set at a value such that the corresponding applied strain rate, e_app , is 106 s1. Because of the presence of sintering additives, impurities, and grain orientations, it is likely to have different properties among the grain boundaries which lead to grain boundary heterogeneities. Three different cases are considered in this example to investigate the effects of grain boundary heterogeneities on creep fracture. The first one is without any heterogeneity, the second one considers heterogeneity at the center of the model (shown by the dashed-line in Fig. 7 marked as heterogeneity I), and the third one considers heterogeneity at an oblique facet of the central grain of the model (shown by the dotted line in Fig. 7 marked as heterogeneity II). In the present study, the heterogeneity is taken into account by increasing the nucleation rate parameter, Fn, to 10.6  104Nr on the heterogeneous facet which is twice of the value on the other facets. Other types of heterogeneities can also be simulated accordingly. To ensure the accuracy of simulations, a convergence study was performed to examine the convergence of this polycrystal model, and a regular polycrystal model

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Fig. 7 Schematic drawing of polycrystalline ceramics with typical hexagonal grains and multiple grain boundaries. Three cases are considered in simulations: no heterogeneity, heterogeneity I located at the center of the model, and heterogeneity II located at an oblique facet of the central grain of the model

Fig. 8 Convergence study for polycrystal without heterogeneity showing the normalized normal separation versus time relationships with different numbers of cohesive elements, Ngb, for each grain boundary

without heterogeneity was chosen for this purpose. Each grain boundary was divided into 1, 2, 4, 8, or 16 cohesive elements, and the corresponding models would, respectively, have 6, 24, 96, 384, or 1536 triangle elements in each grain. The separation-time curve for the node at the geometrical center of the entire model was chosen for comparison among models with different numbers of elements, and the result is shown in Fig. 8. It can be observed that the solution converges when eight cohesive elements are used for each grain boundary. Hence, each grain boundary is divided into eight cohesive elements, as shown in Fig. 9, in the following simulations. Figure 10a–c demonstrates the evolutions of the map of the stress along the loading direction, σ II, for polycrystalline grains without heterogeneity. It should be noted that the grain boundary separation is enlarged by ten times in the figures for the ease of visualization and the stress is normalized by E. Before cracking, Fig. 10a shows relatively uniform stress distribution. At t* = 0.005395, Fig. 10b shows microcracks in all the grain boundaries normal to the loading direction, and

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Fig. 9 The mesh adopted in the simulation, Ngb = 8, for each hexagonal grain

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the load-carrying capacity decreases on the cavitated grain boundaries as the cracks grow. Also, because the stress transfers from the crack region to its neighboring, the stress intensity around the crack region can be observed. This would promote cavitation and cracking of the neighboring grain boundaries. Finally, these microcracks along grain boundaries link up (i.e., a/b reaches 1) to form macrocracks, and the load-carrying capacity is lost for the fully cavitated grain boundary as shown in Fig. 10c. The degree of cavitation and cracking on the grain boundary is reflected by its load carry capacity. The evolutions of σ II stress map for polycrystalline grains with heterogeneity at the central facet, depicted as heterogeneity I in Fig. 7, are shown in Fig. 11a–c. With a higher cavitation rate, the central facet’s load-carrying capacity is smaller, and stress concentration exists in the neighboring of the heterogeneity facet as shown in

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Fig. 11 Snapshots of simulation results of the map of the normalized stress along the loading direction, σ II/E, at (a) t* = 0.004095, (b) t* = 0.004695, and (c) t* = 0.006895 for polycrystal with a heterogeneity located at the center of the model subjected to creep fracture

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Fig. 11a. In Fig. 11b, one can observe that the first microcrack occurs on the facet with heterogeneity, and cavity nucleation and growth also start to occur in the neighboring facets. Compared to the case shown in Fig. 11, the first microcrack occurs earlier in polycrystal with heterogeneity than that without any heterogeneity. At t* = 0.006895, Fig. 11c shows that the central facet is fully cavitated, and microcracks along grain boundaries also start to link up to form macrocracks that would lead to failure eventually. The evolutions of σ II stress map for polycrystalline grains with heterogeneity at an oblique facet of the central grain, depicted as the heterogeneity II in Fig. 7, are shown in Fig. 12a–c. Although the oblique facet with heterogeneity has a higher cavitation rate, the applied loading is not in favor of its cavitation because of the orientation. In this case, the oblique facet with

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Fig. 12 Snapshots of simulation results of the map of the normalized stress along the loading direction, σ II/E, at (a) t* = 0.004095, (b) t* = 0.005395, and (c) t* = 0.006895 for polycrystal with a heterogeneity located at an oblique facet of the central grain of the model subjected to creep fracture

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Normalized normal traction, σn/E

0.002 No heterogeneity Heterogeneity I Heterogeneity II

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Normalized normal separation, un/ Ri Fig. 13 The comparison of normalized traction-separation curve at the center node among the three cases: polycrystals without heterogeneity, with heterogeneity I, and with heterogeneity II

heterogeneity and the horizontal facets start to crack about the same time. Compared to cracking for the case shown in Fig. 11, cracking for the case shown in Fig. 12 is slower. The comparison of normalized traction-separation curve at the center node among the above three cases is shown in Fig. 13. For heterogeneity I case, cavitation is easier and the maximum traction that the grain boundary can sustain is lower than the other two cases.

2.5

Discussion and Summary

Considering grain boundary sliding, cavity nucleation/growth, and creep, a robust rate-dependent cohesive model is developed in the present study to simulate creep fracture of polycrystalline ceramics at high temperatures. Using the commercial finite element software ABAQUS, we have successfully developed a user-defined element (UEL) subroutine to analyze the rate-dependent cohesive behavior of the grain boundary and a user-defined material properties (UMAT) subroutine to simulate the elastic and creep deformations of the grain. Both the explicit form of the Jacobian matrix for the grain boundary and the stress update algorithm are derived, and the numerical stability is improved by our proposed framework. It also allows systematic studies of microstructural effects on the grain boundary deformation. The developed methodology is applied to a bi-crystal with a single grain boundary first to verify its accuracy and numerical stability. Polycrystalline ceramics with typical hexagonal grains and multiple grain boundaries are then simulated to study the effects of grain boundary heterogeneities. Numerical results demonstrate that the stress distribution becomes localized in the presence of grain boundary heterogeneities which, in turn, results in localized grain boundary cavitation. The subsequent

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growth and coalescence of cavities lead to microcracking along grain boundaries. These microcracks then link up to form a macrocrack and result in fracture. It should be noted that the cavitation model in this work is based on the Tvergaard-van der Giessen model, which by itself was established by analyzing the cavitation problem in a three-dimensional case. Therefore, our constitutive law is not restricted to two dimension. However, our finite element simulations are performed in a twodimensional plane-strain case, which is idealized from real materials, to save the computation time. Nevertheless, the result is useful for a quick examination of the essential trends of the dependence of creep fracture on the material parameters. Also, the evolution of the cavity radius to spacing ratio, a/b, provides an index for evaluating the status of fracture based on a microscope point of view. A number of material parameters are involved in the model. However, some of them are difficult to acquire directly from experiments. In order to have extensive applications, atomistic simulations would be fruitful to obtain parameters such as grain boundary diffusivity, viscosity, and cavity nucleation parameter, Fn. Also, the authors aim to provide a viable numerical model for intergranular creep fracture. Therefore, the grains are treated as a homogeneous, isotropic material in an average sense. In reality, the anisotropy could arise from the crystallographic orientation dependence of the creep flow in the grains which, in turn, would affect the grain boundary decohesion process.

3

Competing Mechanisms for Creep Fracture of Polycrystalline ZrB2 Materials

A micromechanics model was developed to simulate creep fracture of ceramics at high temperatures, and material properties pertinent to zirconium diboride were adopted in the simulation. Creep fracture is a process of nucleation, growth, and coalescence of cavities along the grain boundary in a localized and inhomogeneous manner. Based on the grain boundary cavitation process, creep fracture can be categorized into cavity nucleation-controlled and cavity growth-controlled processes. On the other hand, based on the deformation mechanism, the separation between two adjacent grain boundaries can be categorized into diffusion-controlled and creep-controlled mechanisms. In this section, a parametric study was performed to examine the effects of applied stress, grain boundary diffusivity, cavity nucleation parameter, and applied strain rate on cavity nucleation-controlled versus cavity growth-controlled process as well as diffusion-controlled versus creep-controlled mechanism during creep fracture of ZrB2.

3.1

Background

In this section, we aimed to provide a simulation framework to systematically study the creep fracture of ZrB2 ceramic at elevated temperatures. A user-defined element (UEL) subroutine and a user-defined material properties (UMAT) subroutine have

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been developed to analyze the rate-dependent cohesive behavior of the grain boundary and the creep deformation of the grain, respectively, and implemented them in ABAQUS in our previous study [36] to simulate creep fracture of ceramics at high temperatures. However, there existed many unknown parameters in the simulations. Those parameters were determined in the present study by either using existing data reported for ZrB2 or comparing the simulation results with existing measurements.

3.2

Micromechanics Mechanism Study

The separation between two adjacent grain boundaries is dictated by grain boundary cavitation that involves cavity nucleation and growth, and both are functions of stresses. However, while the cavity nucleation depends on the creep strain rate, the cavity growth depends on not only creep but also diffusion. Hence, based on the cavitation process, creep fracture can be categorized into cavity nucleationcontrolled and cavity growth-controlled processes. On the other hand, from the point of the deformation mechanism, the separation between two adjacent grain boundaries can be categorized into creep-controlled and diffusion-controlled mechanisms. It should be noted that cavity nucleation and growth can be characterized, respectively, by the half cavity spacing, b, and the cavity radius, a, as functions of time. As a result, the study of nucleation-controlled versus growth-controlled cavitation process can be achieved by examining the evolutions of b and a during the cavitation process. Also, the derivation of the grain boundary separation rate shows three components describing the contributions from cavity nucleation, atom diffusion, and creep, respectively. While the second component is controlled by grain boundary diffusion, both the first and the third terms (i.e., cavity nucleation and creep) are controlled by the creep strain rate. Hence, the study of diffusioncontrolled versus creep-controlled mechanism can be achieved by examining the evolutions of the second term and the sum of the first and the third terms during creep fracture. It should be noted that data of many material parameters are required in the simulation; however, some of them (e.g., diffusion parameter, D, nucleation parameter, Fn, and viscosity, η) are not readily available. Fortunately, creep fracture of ZrB2 has been studied extensively and the stress-strain rate curves can be found in the literature [17, 18]. By adjusting certain material parameters to yield a fair agreement of the creep stress-stain rate curves between the simulations and the measurements, the selected parameters can be used as the benchmark data for further parametric studies. In this study, these material parameters are firstly determined by simulation of creep fracture under a constant loading stress and comparison with experimental measurements conducted by Talmy et al. [17]. Then, using these benchmark data, parametric studies are performed to examine the effects of (i) grain boundary diffusivity, (ii) cavity nucleation rate, and (iii) applied strain rate on creep fracture. Both the nucleation-controlled versus growth-controlled cavitation process and the creep-controlled versus diffusion-controlled mechanism are characterized.

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Benchmark Data

A polycrystal model consisting of hexagonal grains shown in Fig. 14 is adopted for simulation. Each grain is modeled by 384 three-nodes triangle elements (CPE3 element in ABAQUS) and is attached to 48 cohesive (grain boundary) elements. The grain elements and the grain boundary elements are constructed, respectively, by UMAT and UEL subroutines in ABAQUS. The material properties adopted in simulations are listed in Tables 1 and 2, respectively, for the grain elements and grain boundary elements. Among these parameters, Young’s modulus, Poisson’s ratio, and atomic volume, Ω, are material properties pertinent to high-temperature ceramics ZrB2 [37]; the reference stress, Σ0, of nucleation law is suggested by Onck Fig. 14 The mesh adopted in the simulation for each hexagonal grain

Table 1 Materials properties for the grain element used in simulations

Parameter e0

Value 1.0  108 s1

σ0 n

10 GPa 1.1

Table 2 Benchmark data of the materials properties for the grain boundary element used in simulations

Parameter T E ν Ω k R Q D0δ0 η Σ0 Fn

Value 1673 K 495 GPa 0.15 18.55 cm3.mole1 1.38(13)  1023 J.K1 8.314(75) J.K1.mole1 130 kJ.mole1 1.35  1020 mm3.s1 5.0  1010 MPa.s 10 GPa 5.6  1016 mm2

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et al. [28] for creep fracture of polycrystalline materials; and reference grain boundary diffusion coefficient, D0δ0; nucleation parameters, Fn; and grain boundary viscosity, η, are determined by fitting the simulation results to Talmy et al.’s experimental data [17]. These parameters are fitted and optimized by constructed a response surface [38]. Response surface function of stress in terms of the diffusion parameter, D, and nucleation parameter, Fn, are established, and then an optimization method [39] is carried out to determine values of these two parameters. These obtained parameter values are the optimal solutions which are closest to the experiment data [17]. To emphasize the nature of rate dependence of parameters, all the micromechanics quantities presented in follow sections were normalized. Stress quantity was normalized by Young’s modulus (T* = T/E); time was normalized by the reference creep strain rate (t ¼ te_0); and all the length quantities, such as cavity radius and cavity spacing, were normalized by initial half facet length (a* = a/Ri, b* = b/Ri).

3.4

Numerical Examination

By adjusting D0δ0, Fn, and η to fit simulation results to Talmy et al.’s experimental data on creep fracture of ZrB2 [17] at different loading stresses, good agreement was obtained as shown in Fig. 15 for the steady-state strain rate versus stress relation when D0δ0 = 1.35  1020 mm3/s, Fn = 8.66  1016 mm2, and η = 5  1010 MPa.s were adopted. The material properties used in simulations are listed in Tables 1 and 2. During creep fracture, the stress and deformation are nonuniform in the simulation domain. In order to study the micromechanics variables during the creep fracture process, data were extracted from the grain boundary located at the center of model (cross mark) for comparison. The evolution of the normalized traction is shown in Fig. 16a at different loading stresses. At t = 0, the traction at the center of

Fig. 15 The comparison of the steady-state strain rate versus applied stress relation between Talmy et al.’s measurements [21] and the present simulation results

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Fig. 16 (a) The normalized normal traction and (b) the normalized normal separation versus normalized time relationships, (c) the rate-dependent cohesive behavior of the grain boundary, (d) the normalized cavity radius a/Ri, (e) the normalized half cavity spacing b/Ri, and (f) the a/b ratio versus normalized time relationships for creep fracture of ZrB2 for applied stresses of 50, 75, and 100 MPa

the model equals to the loading stress. This traction increases quickly with time, reaches a maximum, and then decreases to zero when cracking occurs. The maximum normal traction increases with the applied stress. In addition, the normal traction decreases faster after reaching the maximum and reaches zero earlier, signifying earlier cracking, for a higher loading stress. The corresponding evolution of the normalized separation is shown in Fig. 16b. It is apparent that the high loading stress not only shortens the grain boundary cracking time but also reduces the

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maximum grain boundary separation. Combining Fig. 16a, b, the normalized traction-separation relation is shown in Fig. 16c. The traction increases quickly to reach its maximum leading to brittle fracture. The capability of grain boundaries to resist fracture can be observed from the traction-separation curve. It is interestingly to note that the cohesive energy increases with the applied loading stress. Rate-dependent effect would be examined by several micromechanics quantities, such as cavity radius, a, half cavity spacing, b, and the ratio of a/b in Fig. 16d–f. While cavity growth is signified by increasing a, cavity nucleation is signified by decreasing b. In Fig. 16d–f, the end of the curve denotes complete cracking of the brain boundary; i.e., a/b = 1. It can be observed from Fig. 16d, e that both the cavity growth and nucleation rates increase with the applied stress. Also, upon complete cracking of the grain boundary, the ending values of a and b are smaller for higher applied stress. This implies that there are more small cavities on the grain boundary for higher applied stress and the cavitation process is controlled by cavity nucleation at high loading stresses. Combining Fig. 16d, e, a/b as a function of the normalized time is shown in Fig. 16f. The fracture index, a/b, increases faster and reaches unity sooner for a higher applied stress. To determine whether the fracture process is controlled by diffusion or creep, the normal displacement rate, the component resulting from diffusion, and the component resulting from creep during the creep fracture process are shown in Fig. 17a–c for loading stresses of 50, 75, and 100 MPa, respectively. It can be see that the fracture process is controlled by diffusion for the three loading stresses which, in turn, leads to brittle fracture behavior and is in agreement with the results shown in Fig. 16c. However, the contribution of creep increases with increasing loading stress.

4

Parametric Study

The strength of grain boundary is governed by some vital micromechanical properties. For instance, the cavity growth is dominated by the grain boundary diffusion parameter and the cavity nucleation is controlled by the grain boundary cavity nucleation parameter. These parameters could be altered by the additive added into ZrB2 powders during the densification process. The impurities could cause the variations of grain boundary diffusion and cavity nucleation parameters as well. Besides those two micromechanics parameters, a different applied strain rate would also affects the cohesive behavior due to the nature of rate-dependent. Hence, effects of grain boundary diffusivity, cavity nucleation, and applied strain rate on creep fracture of ZrB2 are studied in the following.

4.1

Effects of Cavity Nucleation Rate

To study the effects of cavity nucleation rate, the cavity nucleation parameter, Fn, is arbitrarily set at ten times and one-tenth of the parameter shown in Table 2 (i.e., 5.6  1017 and 5.6  1015). In this case, a constant applied stress of 100 MPa is

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Fig. 17 The normal separation rate and the components resulting from diffusion and creep during the creep fracture process for applied stress of (a) 50, (b) 75, and (c) 100 MPa

assumed. The normalized normal traction and separation on the grain boundary as functions of normalized time are plotted, respectively, in Fig. 18a, b. It can be observed that the grain boundary with a lower cavity nucleation rate can withstand

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Fig. 18 (a) The normalized normal traction and (b) the normalized normal separation versus normalized time relationships, (c) the rate-dependent cohesive behavior of the grain boundary, (d) the normalized cavity radius a/Ri, (e) the normalized half cavity spacing b/Ri, and (f) the a/b ratio versus normalized time relationships for creep fracture of ZrB2 for cavity nucleation parameters of one-tenth and ten times of the benchmark data

a higher stress (Fig. 18a) and allows more grain boundary separation before full cracking of the grain boundary (Fig. 18b). The corresponding traction-separation curve of grain boundary is shown in Fig. 18c. There exists a significant increase of cohesive energy with the decreasing grain boundary cavity nucleation. The cavity radius a, half cavity spacing b, and a/b as functions of time are shown, respectively, in Fig. 18d, e, f. For a higher cavity nucleation rate, a faster

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decreasing rate of b (Fig. 18e) is accompanied by a slower increasing rate of a (Fig. 18d). The rapid decrease of b with time reflects the high cavity nucleation rate. For the late stage of grain boundary nucleation, i.e., for t* > 0.6  104 for the ten times of Fn, b remains almost constant; i.e., b_ ! 0. This is mainly due to the large number of cavities and small value of b on the grain boundary (see Eq. 18). At the end (i.e., at t* ~ 1.5  104), b attains a minimum constant. This corresponds to the rapid decrease of the grain boundary traction after reaching its maximum, and the corresponding cavity nucleation rate is almost zero. It is worth mentioning that cavity nucleation is the quadric term in Eq. 36 and it is a strong function of the grain boundary traction. The a/b ratio shown in Fig. 18f increases monotonically and reaches the upper limit of unity which represents the complete separation of the grain boundary. Also, at a fixed time, a/b (i.e., the relative cavitation region on the grain boundary) increases with the cavity nucleation rate. As mentioned previously, the fracture process can be categorized into creep and diffusion controlled. The contributions of diffusion and creep, which includes cavity nucleation at grain boundary and creep effects from adjacent grains, to creep fracture are shown in Fig. 19a, b, respectively, for Fn = 5.6  1017 and 5.6  1015. It can be observed from Fig. 18a that the separation rate increases rapidly initially and reaches a plateau for t* > 1  104 with the negligible contribution from creep. This is in agreement with the fast cavity growth rate and negligible cavity nucleation rate for t* > 1  104 as shown in Fig. 18d, e. Compared with Fig. 19a, b exhibits the peak of the contribution from creep and the plateau of the separation rate at a later stage. Fig. 19a, b shows the diffusion-controlled creep fracture. However, the contribution from creep increases as Fn increases.

4.2

Effects of Grain Boundary Diffusivity

To study the effect of grain boundary diffusivity, the grain boundary diffusivity parameter, D, is arbitrarily set at twice and half of the parameter shown in Table 2 (i.e., 1.77  1017 and 4.425  1018). A constant applied stress of 100 MPa is assumed. The normalized normal traction and separation on the grain boundary as functions of normalized time are plotted, respectively, in Fig. 20a, b. As can be seen in Fig. 21a, the normal traction reaches a higher peak value and a longer traction duration for a smaller diffusion parameter. It is shown in Fig. 20b that the grain boundary with a lower value of diffusion parameter exhibits a stronger resistance to opening and is able to carry the load for a longer time. The corresponding traction-separation behavior of grain boundary is shown in Fig. 20c. The cohesive energy stored in the grain boundary increases with the decrease in the diffusion parameter. The normalized cavity radius and half cavity spacing and a/b as functions of normalized time are shown, respectively, in Fig. 20d–f. When the diffusion parameter increases, the cavity growth rate increases (Fig. 20d). This would decrease the load-carrying capacity of the grain boundary, and hence the

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Fig. 19 The normal separation rate and the components resulting from diffusion and creep during the creep fracture process for cavity nucleation parameter of (a) one-tenth and (b) ten times of the benchmark data

grain boundary traction decreases (Fig. 20a). Also, because the cavity nucleation rate is a strong function of the grain boundary traction, the decrease of the grain boundary traction would lead to a significant decrease of the cavity nucleation rate and, hence, an increase of the cavity spacing (Fig. 20d). Depending upon the balance of the increases of both a and b with increasing diffusion parameter at a fixed time, Fig. 20f shows that a/b is relatively insensitive to the diffusion parameter for a constant applied stress. The normal displacement rate, the component resulting from diffusion, and the component resulting from creep during the creep fracture process are shown in Fig. 21a, b, respectively, for diffusion parameter of 1.77  1017 and 4.425  1018 at a constant loading stresses of 100 MPa. The fracture process is controlled by diffusion in both Fig. 21a, b. The contribution of creep goes through a maximum during the creep fracture process. This maximum is insensitive to the diffusion parameter; however, it occurs at a longer time for a smaller diffusion parameter.

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Fig. 20 (a) The normalized normal traction and (b) the normalized normal separation versus normalized time relationships, (c) the rate-dependent cohesive behavior of the grain boundary, (d) the normalized cavity radius a/Ri, (e) the normalized half cavity spacing b/Ri, and (f) the a/b ratio versus normalized time relationships for creep fracture of ZrB2 for grain boundary diffusivity of half and twice of the benchmark data

4.3

Effect of Applied Strain Rate

In order to examine the effects of applied strain rate, different applied strain rates, 2  106 s1, 106 s1, and 5  107 s1, are assumed in the present study, and the corresponding results are shown in Fig. 22a–e. Under a constant applied strain rate, there exists a surge of the normal traction at t = 0. It is shown in Fig. 22a that the

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Fig. 21 The normal separation rate and the components resulting from diffusion and creep during the creep fracture process for grain boundary diffusivity of (a) half and (b) twice of the benchmark data

normal traction has a higher surge and relaxes faster afterward for a higher applied strain rate. Also, for a higher applied strain rate, the grain boundary allows a smaller separation before full separation (Fig. 22b). The corresponding traction-separation relation is shown in Fig. 22c, and the cohesive energy increases with the increase in the applied strain rate. The evolutions of cavity growth and cavity nucleation are shown in Fig. 22d, e, respectively. Because of the surge of the normal traction at t = 0, cavity nucleates and grows rapidly initially. With the rapid decrease of the normal traction for t > 0, the cavity growth rate decreases, and the cavity nucleation rate decreases rapidly to almost zero because of both small normal traction and small cavity spacing. The fracture index, a/b, is plotted in Fig. 22f, and it increases monotonically and reaches the upper bound of unity which represents the complete separation of the grain boundary.

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Fig. 22 (a) The normalized normal traction and (b) the normalized normal separation versus normalized time relationships, (c) the rate-dependent cohesive behavior of the grain boundary, (d) the normalized cavity radius a/Ri, (e) the normalized half cavity spacing b/Ri, and (f) the a/b ratio versus normalized time relationships for creep fracture of ZrB2 for applied strain rates of half and twice of the benchmark data

The normal displacement rate, the component resulting from diffusion, and the component resulting from creep during the creep fracture process are shown in Fig. 25a, b, respectively, for the applied strain rates of 2  106 s1 and 5  107 s1. In response to the surge of the normal traction at t = 0, the separation rate also exhibits a surge initially. Although the creep fracture is controlled by diffusion in both Fig. 23a, b, the contribution of creep increases with the increase in the applied strain rate.

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Fig. 23 The normal separation rate and the components resulting from diffusion and creep during the creep fracture process for applied strain rate of (a) half and (b) twice of the benchmark data

4.4

Discussion and Summary

Creep fracture is a process of nucleation, growth, and coalescence of cavities along the grain boundary in a localized and inhomogeneous manner of polycrystalline ceramics at high temperatures. A micromechanics model was developed to simulate creep fracture of zirconium diboride. While many material parameters are involved in the simulation, some of them (e.g., grain boundary diffusivity, cavity nucleation parameter, and grain boundary viscosity) are not readily available. By adjusting the unknown material parameters to yield a good agreement of the creep stress-stain rate curves between the simulations and the measurements, the selected parameters were used as the benchmark data for parametric studies. Based on the grain boundary cavitation process, creep fracture can be categorized into cavity nucleation-controlled and cavity growth-controlled processes. On the other hand, based on the deformation mechanism, the separation between two

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adjacent grain boundaries can be categorized into diffusion-controlled and creepcontrolled mechanisms. In the present study, the parametric study was performed to examine the effects of applied stress, grain boundary diffusivity, cavity nucleation parameter, and applied strain rate on cavity nucleation-controlled versus cavity growth-controlled process as well as diffusion-controlled versus creep-controlled mechanism during creep fracture of ZrB2, and the results are summarized as follows.

4.4.1 Effects of Applied Stress Both the cavity growth and nucleation rates increase with the applied stress. Also, there are more number of small cavities on the grain boundary prior to fracture for higher applied stress, and the cavity nucleation contributes more to the cavitation process at high loading stresses. Creep fracture is controlled by diffusion; however, the contribution of creep increases with the increase in the applied stress. 4.4.2 Effects of Grain Boundary Diffusivity The cavity growth rate increases, while the cavity nucleation rate decreases with the increase in the grain boundary diffusivity. There are less number of large cavities on the grain boundary prior to fracture for higher grain boundary diffusivity. Creep fracture is controlled by diffusion. The contribution of creep goes through a maximum during the creep fracture process. This maximum is insensitive to the grain boundary diffusivity; however, it occurs at a longer time for a smaller grain boundary diffusivity. 4.4.3 Effects of Cavity Nucleation Parameter The cavity growth rate decreases, while the cavity nucleation rate increases with the increase in the cavity nucleation parameter. There are more number of small cavities on the grain boundary prior to fracture for higher cavity nucleation parameter. Creep fracture is controlled by diffusion; however, the contribution of creep increases with the increase in the cavity nucleation parameter. 4.4.4 Effects of Applied Strain Rate Both the cavity growth and nucleation rates increase with the applied strain rate. There exists a surge in the normal traction on the grain boundary upon loading with constant applied strain rate. As a result, cavity nucleates and grows rapidly initially and then slowly afterward. There are more small cavities on the grain boundary prior to fracture for higher applied strain rate. Creep fracture is controlled by diffusion; however, the contribution of creep increases with the increase in the applied strain rate. Creep Mechanism Study of ZrB2-SiC Composites at High Temperatures The creep deformation of the ultrahigh-temperature ceramic composite ZrB2-20% SiC at temperatures from 1400
C to 1700
C was studied by a micromechanical model, and a real microstructure was adopted in finite element simulations. Based on the experiment results of the change of activation energy with respect to the temperature, a mechanism shift form diffusional creep-control for temperatures

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below 1500
C to grain boundary sliding-control for temperatures above 1500
C was concluded from simulations. Also, the simulation results revealed the accommodation of grain rotation and grain boundary sliding by grain boundary cavitation for creep at temperatures above 1500
C which was in agreement with experimental observations.

5

Polycrystalline Modeling

To elucidate mechanisms that could affect the high-temperature creep behavior, simulations of ZrB2 with 20% SiC were performed using ABAQUS 6.12 [33]. The mechanisms for grains and grain boundaries described in pervious sections were considered in our simulations. Also, the experimentally observed change of activation energy resulting from different mechanisms of grain boundaries deformation at high temperatures was investigated in the present study. The creep deformation of ZrB2-20% SiC composites is significantly dependent on the properties of the grain boundary. In order to capture the nature of microstructure, an SEM image of the actual morphology of ZrB2-SiC composite was acquired from the specimen on the tension side after crept test as shown in Fig. 24. Based on this image, a finite element model was constructed in the present study. This SEM image was processed to accentuate the grain boundaries and then transformed into an eligible finite element model for the numerical convenience as shown in Fig. 25. All the microcracks and cavities at grain boundaries were eliminated at the initial setup of the simulation. The finite element mesh shown in Fig. 26 contains 20 grains and more than 38,000 elements. The grain was simulated by plane-strain triangle elements, CPE3, while the grain boundary was simulated by the user-defined cohesive elements. The boundary conditions of current finite element model are schematically shown in Fig. 27. The top and bottom surfaces are subjected to Fig. 24 SEM image after the high-temperature creep test on the tension side

1µm Mag = 25.00 K X

EHT = 16.00 KV WD = 4mm

Signal A = Inlens Date : 10 Jan 2013 Photo No = 7925 Time : 13:54:00

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Fig. 25 Demonstration of preprocessing procedure, (i) MATLAB image processing, (ii) ABAQUS python to establish a model without grain boundary element, and (iii) add grain boundary elements using MATLAB parser. SiC grains are in yellow

Fig. 26 ABAQUS mesh, SiC presets in yellow

constant tension loading with a magnitude of 75 MPa, while the right and left surfaces are traction free without any constraints. To examine the mechanisms, which dominate the creep process at different temperatures, the cavity nucleation ability on the grain boundaries was assumed to be controlled by two different kinds of simulations based on the experimental observation of creep results; i.e., negligible grain boundary cavitation for temperatures below 1500
C and some cavitations at ZrB2-SiC grain boundaries for temperatures above 1500
C. Accordingly, the first simulation was to neglect all the grain boundary nucleation probabilities, while the other one was to manipulate the ability of cavity nucleation only on the ZrB2-SiC grain boundaries.

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Fig. 27 Illustration of boundary condition and model setup

Table 3 Parameters for zirconium diboride grains Properties E ν σ0 n Q

Value 495 GPA 0.15 10 GPa 1.1 470 kJ/mole

Table 4 Parameters for silicon carbide grains Properties E ν σ0 n Q

Value 445 GPA 0.12 10 GPa 1.1 440 kJ/mole

It is rather difficult to obtain all grain boundary properties at elevated temperatures from literature or experiments. Some of them could be obtained by in situ measurements under ultrahigh-temperature surrounding, e.g., grain boundary diffusion parameter and grain boundary viscosity. Some of those material properties were adopted from our previous work pertinent to ZrB2, and the other parameters for SiC grains and grain boundaries were adopted from literature. The nucleation parameters were determined by fitting the creep data to experimental results from the work conducted by Bird et al. [22]. The parameters used for the simulations of all temperature regions are listed in Tables 3, 4, 5, and 6.

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Table 5 Parameters for ZrB2-ZrB2 grain boundary Properties Ω k R Q D0b0 A Qs w

Value 18.55 cm3mole1 1.38(13)  1023JK1 8.314(75) JK1mole1 530 kJ mole1 1.35  1020 mm3 s1 5.37  1011 MPa.s 660 kJ mole1 1.0 nm

Table 6 Parameters for ZrB2-SiC grain boundary Properties Ω k R Q D0b0 A Qs Fn Σ0

Value 18.55 cm3mole1 1.38(13)  1023JK1 8.314(75) JK1mole1 530 kJ mole1 4.57  1021 mm3 s1 3.9  1018 MPa.s 440 kJ mole1 8.65  1016 mm2 10 GPa

Fig. 28 The plot of steadystate strain rate with temperature denotes the change of activation energy

5.1

Results and Discussion

The logarithm of the total strain rate versus the reciprocal of temperature is shown in Fig. 28. The change of the slope implies the change of activation energy for the macroscopic creep deformation. When cavity nucleation was forbidden artificially for all temperatures, the activation energy remained nearly constant at temperatures

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from 1400
C to 1700
C. On the other hand, the activation energy exhibits a significant change once cavity nucleation is introduced at the ZrB2-SiC grain boundaries. The simulation result for the switch mechanism of allowing cavity nucleation when the temperature is above 1500
C shows a good agreement with the experimental data reported by Bird et al. [22] The ability of cavity nucleation at ZrB2-SiC grain boundary apparently affects the creep deformation in the hightemperature region. The apparent activation energy transition with cavity nucleation, which was artificially embedded in our simulation, appears as a strong softening mechanism operating in sequence with grain boundary sliding which is consistent with experimental findings in Ref. [37].

5.2

Creep-Competing Mechanism

The total creep strain rate could be decomposed by the grain creep component e_cgrain, the grain boundary diffusion component e_cdiff , and the grain boundary cavitation component e_ccav . The mean value of separation among all grain boundaries was projected onto the loading direction to emphasize the contribution to the total creep deformation during our simulation. As can be seen in Fig. 29, more than 99% of the total creep results from the grain creep component for all simulation results. The contribution from grain boundary cavitation to the total creep deformation is very small. However, it will be shown later that this small amount of grain boundary cavitation is essential to accommodate grain boundary sliding and grain rotation for creep at temperatures above 1500
C. To clarify the actual contribution of grain boundary cavitation at temperatures above 1500
C, the creep deformation behavior of each grain is fruitful to determine the dominant mechanism. Based on experimental observations [22], SiC grains act as deformation anchors due to a relatively strong resistance to grain boundary sliding [25]. The rotation angle and creep deformation of each grain are reported in Table 7 with the grain identification number labeled in Fig. 30. From Table 7, it can be concluded that the creep deformation is dominated by grain diffusional creep for temperatures below 1500
C since all the grain rotation angles are nearly zero. Additionally, the calculated activation energies validate the findings of Bird et al. [37] and Talmy et al. [21] for fine-grained ZrB2 composites containing ~20% SiC particulates below 1500
C. In addition to the grain boundary separation, there is a small amount of grain sliding. The grain elongation shows a good agreement with the total creep strain. When grain boundary cavitation is allowed at 1600
C, significant increase of grain rotation is observed for grains 5, 6, and 7. Taking grain 7 for instance, the magnitude of rotation increases from 0.02
to 0.51
when cavitation is allowed at 1600
C. A similar trend can be observed at 1700
C where the rotation angle of grain 7 increases dramatically from 0.035
to 2.67
. These significant grain rotations suggest a creep mechanism transition from diffusion-controlled creep to a grain boundary sliding (and rotation)-controlled deformation above 1500
C.

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Fig. 29 Creep deformation as a function of time showing small contribution from grain boundary to the total creep deformation at (a) 1400
C, (b) 1500
C, (c) 1600
C, and (d) 1700
C without grain boundary cavitation and (e) 1600
C and (f) 1700
C with grain boundary cavitation

5.3

Diffusional Creep

The diffusion mechanism dominates the creep deformation for temperatures below 1500
C. ZrB2 grains were observed to elongate along the loading direction. Even though the grain boundary separation contributes slightly to the total deformation for temperatures below 1500
C, the heterogeneity of grain boundary properties would

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Table 7 Rotation angle of each grain Grain ID 1 2 3 4 5 6 7 8 9 10 11 12 a

1400 1.59E-03 1.98E-03 1.58E-03 7.81E-04 2.51E-03 1.00E-03 5.33E-03 2.43E-03 5.53E-03 1.75E-04 2.74E-04 2.07E-03

1500 1.64E-03 2.19E-03 1.81E-03 8.05E-04 2.73E-03 1.23E-03 5.55E-03 2.58E-03 5.94E-03 1.84E-04 2.99E-04 2.25E-03

1600 1.47E-03 2.15E-03 1.54E-03 2.09E-03 7.63E-03 1.73E-03 1.97E-02 4.86E-03 1.76E-02 2.48E-03 1.08E-03 7.76E-03

1600a 0.01 0.02 0.03 0.01 0.29 0.13 0.51 0.02 0.04 0.01 0.00 0.02

1700 4.03E-03 8.22E-03 1.15E-02 4.03E-03 1.53E-02 6.66E-03 3.51E-02 7.87E-03 2.98E-02 4.07E-03 2.86E-03 1.41E-02

1700a 0.02 0.17 0.37 0.09 2.14 1.37 2.67 0.13 0.22 0.01 0.04 0.12

With grain boundary cavity nucleation introduced

Fig. 30 The simulated microstructure with grain identification numbers

cause stress localization as can be seen in Figs. 31 and 32, in which the von Mises stress contours are shown at 1400
C and 1500
C, respectively. One can observe the stress nonuniformity as a result of the stiffness and creep rate differences between ZrB2 and SiC grains resulting in maximum grain boundary stress concentrations of 350 MPa at t = 1 s (Fig. 31a). With increasing creep time, the stress redistribution appears to be concentrated within the ZrB2 grains at t = 1200 s as shown in Fig. 31b. Similar stress contour evolution can be observed in Fig. 32a, b at 1500
C. The regions of strain relaxation in the absence of cavitation qualitatively agree with the observed creep microstructures shown in Fig. 33 [22]. It is worth noting that the stress concentration near the ZrB2-SiC grain boundary is caused by a mismatch between ZrB2-ZrB2 and the ZrB2-SiC grain boundary diffusivity as predicted in our previous study [40] and suggested by Bird et al. [22].

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Fig. 31 The stress contour at 1400
C for (a) t = 1 s and (b) t = 1200 s. The diffusion creep-controlled mechanism shows a slow dissipation of stress concentration after creeping time of 1000 s

5.4

Grain Boundary Sliding

Compared to creep fracture below 1500
C, creep fracture above 1500
C revealed more cavities at ZrB2-SiC grain boundaries [22]. This grain boundary cavitation appears to be essential for grain boundary sliding to occur, and it poses as an accommodation mechanism in high-temperature region (above 1500
C). The comparison of the results at 1600
C with and without cavity nucleation allowed at grain boundaries is shown in Fig. 34. The stress contours at the initial step t = 1 s shown in Fig. 34a, c have similar stress distributions. This phenomenon implies that grain boundary cavitation does not influence the creep deformation at the initial state of the creep test. The stress heterogeneity at this step is mainly caused by the stiffness mismatch between ZrB2 and SiC grains. When t = 1000 s, the stress distributions become different from each other as shown in Fig. 34b, d. The stress concentration disappears because cavity nucleation on grain boundary plays a role as a softening mechanism.

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Micromechanics Modeling of Creep Fracture of High-Temperature Ceramics

Fig. 32 The stress contour at 1500
C for (a) t = 1 s and (b) t = 1200 s. The diffusion creep-controlled mechanism shows a slow dissipation of stress concentration after creeping time of 1000 s

Fig. 33 Mechanically polished crept microstructure at 1500
C and 97 MPa for t = 43,200 s showing negligible cavitation along ZrB2-SiC multi-grain junctions. White arrows highlight regions of SiC pullout from polishing, and the presence of B4C has negligible effect on creep behavior [22]

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Fig. 34 von Mises stress at 1600
C without cavity nucleation assigned at grain boundary at (a) t = 1 s, (b) t = 1000 s and with cavity nucleation allowed at ZrB2-SiC grain boundary at (c) t = 1 s, (d) t = 1000 s

Volume defects are visible after creeping for 1000 s at 1700
C as shown in Fig. 35. The stress concentration almost vanishes on ZrB2-SiC grain boundaries. Compared with Fig. 35a, b shows that the localized stress redistribution is dissipated by grain rotation and cavity nucleation around multi-grain junctions which provides space for grain rotation. It is worth noting that the capability of cavity nucleation described by Eq. 1.17 represents a tendency of opening in the direction normal to the grain boundary. When a grain rotates toward adjacent grains, geometric incompatibilities are observed at the triple junction which would result in new deformation gradients across the grain boundaries. The cavities nucleated around grain 7 in Fig. 36 accommodate the grain sliding and rotation event to relieve the majority of the geometric incompatibility. However, realistically any grain overlapping would be accommodated by other additional creep mechanisms such as diffusion or dislocation mechanisms. Experimental evidence has shown dislocation creep along zones adjacent to grain boundaries which can accommodate the sliding event [41] with ~10% cavitation contributing toward the total creep strain [22]. The present analysis includes only cavitation creep in the absence of dislocation creep mechanism. This may explain a lower predicted creep strain rate than those experimentally

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Fig. 35 von Mises stress at 1700
C (a) without nucleation assigned at grain boundary at t = 1000 s (b) cavity nucleation allowed at ZrB2-SiC grain boundary at t = 1000 s

observed in Ref. [22] as shown in Fig. 28. However, the distribution of cavities along ZrB2-SiC grain boundaries in the present analysis is similar to those observed for polished crept specimens shown in Fig. 37.

6

Summary

Creep deformation of ZrB2 with 20% SiC was investigated at temperatures from 1400
C to 1700
C using a temperature- and time-dependent finite element model and an actual ZrB2-SiC microstructure. Based on the experiment results of the change of activation energy with respect to the temperature, our simulations conclude the mechanism shift form diffusional creep-control for temperatures below 1500
C to grain boundary sliding-control for temperatures above 1500
C.

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Fig. 36 Cavitation as an accommodation mechanism for grain rotation. Solid ellipse highlights geometric incompatibilities as a result of grain rotation

Fig. 37 Mechanically polished crept microstructure of the tensile outer-fiber region at 1700
C and 97 MPa for t = 1210 s showing ZrB2SiC multi-grain junction cavity nucleation (black arrows) [37]. White arrows highlight regions of SiC pullout from polishing

A reasonable estimation of cavity nucleation was made by a curve fitting with experimental results. The simulation adequately predicted the observed steady strain rate including the mechanism transition at 1500
C. SiC additions to ZrB2 ceramics enhance the heterogeneity effect on grain boundaries for the stress distribution. Even though the creep effect is mostly attributed to grain diffusional creep, the grain boundary properties could also introduce the stress redistribution along grain boundaries. The dominant mechanisms of creep fracture at both low temperatures (below 1500
C) and high temperatures (above 1500
C) were examined. In the lowtemperature region, creep fracture is dominated by the diffusional creep with less

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than 1% total strain contribution from grain boundary cavitation. In the hightemperature regime, cavitation accommodates the grain boundary sliding event. We show that grain boundary cavity nucleation is necessary to accommodate grain boundary sliding, in the absence of dislocation creep mechanism, as a consequence of grain rotation and grain boundary sliding. In the present study, we link the micro-mechanisms to the macroscopic creep deformation. A numerical study of UHTC was conducted by combining micromechanical model and finite element analysis. Numerical results were validated with the experimental data. The main achievements of this dissertation are summarized below: • A robust model has been developed to capture grain boundary cavitation, grain boundary sliding, cavity growth, and creep deformation of grains. The model has been implemented via ABAQUS. The model has been tested and verified by two examples in this study, i.e., a bi-crystal model and a regular hexagonal grain model. The results show the robustness and scalability of the current model. • Grain boundary heterogeneity affects the macroscopic creep behavior. The stress distribution becomes localized due to the incompatibility of neighboring grains. Hence, the creep resistance for a strong or weak surface will be jeopardized once the grain boundary heterogeneity is generated. The nature of heterogeneity often occurs due to the presence of a second phase for high-temperature ceramic composites and thus results in reduction of creep resistance. • The change of microscopic creep mechanisms at the elevated temperature for ZrB2-20% SiC has been studied by numerical study in the present work. We concluded that grain creep and grain boundary diffusion dominate creep fracture at a low-temperature region (below 1500
C). Alternatively, grain boundary sliding with accommodation cavitation at ZrB2-SiC interface will monopolize the high-temperature realm. Numerical study reveals that the cavity nucleation is a necessary accommodation for grain boundary sliding. Some directions for future research are conceived. The micromechanics model implemented in the present study could be applied to study intergranular fracture behavior. However, owing to the convergence requirement for the rate-dependent cohesive element and computing capability, the modeling size is limited to a small area. To retain the numerical stability and extend the potential utility, more work is needed. To overcome the numerical disadvantages, a field equation for the initial crack would be applied. An asymptotic stress and strain field has been derived by Hui and Riedel [42]. The current model could be extended to mode I or mode II fracture simulation with an initial crack in the middle. Along with our current model, a reliable prediction of future material design could be delivered. Acknowledgments The research was jointly supported by AFOSR Grant No. FA9550-09- 0200 and the Ministry of Science and Technology, Taiwan, under Contract no. MOST 102-222 E-002063-MY3. We are grateful to the National Center for High-Performance Computing and Simutech Solution Corporation for providing the computational resources.

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Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

34

Chun-Hway Hsueh

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 ASTM Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Roark’s Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hsueh et al.’s Rigorous Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hsueh et al.’s Simplified Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Zirconia Monolayered Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Porcelain/Zirconia Bilayered Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 SOFC Trilayered Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Solutions for Bilayered Discs Subjected to Thermal Stresses . . . . . . . . . . . . . . . . . . . . . . . 4.2 Finite Element Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1094 1095 1096 1097 1099 1104 1107 1107 1108 1109 1111 1111 1113 1113 1120 1121 1123

Abstract

Although standard test methods for biaxial strength measurements of ceramics have been established and the corresponding formulas for relating the biaxial strength to the fracture load have been approved by the American Society for Testing and Materials (ASTM) and International Organization for Standardization, respectively, they are limited to the case of monolayered discs. Despite the increasing applications of multilayered ceramics, characterization of their strengths using biaxial flexure tests has been difficult because the analytical C.-H. Hsueh (*) Department of Materials Science and Engineering, National Taiwan University, Taipei, Taiwan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_69

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C.-H. Hsueh

description of the relation between the strength and the fracture load for multilayers subjected to biaxial flexure tests is unavailable until recently. Using ringon-ring tests as an example, the closed-form solutions for stresses in (i) monolayered discs based on ASTM formulas, (ii) bilayered discs based on Roark’s formulas, (iii) multilayered discs based on Hsueh et al.’s rigorous formulas, and (iv) multilayered discs based on Hsueh et al.’s simplified formulas are reviewed in this chapter. Finite element results for ring-on-ring tests performed on (i) zirconia monolayered discs, (ii) dental crown materials of porcelain/zirconia bilayered discs, and (iii) solid oxide fuel cell trilayered discs are also presented to validate the closed-form solutions. Finally, a case study of layer thickness effects in bilayered dental ceramics subjected to both thermal stresses and ring-on-ring tests is presented. Keywords

Modeling · Multilayer · Mechanics · Biaxial Strength

1

Introduction

Multilayered structures have extensive applications as microelectronic, optical, structural, dental, and biological components [1]. To maintain the functionality and reliability of multilayered systems, it is essential to understand the geometrical and material factors that influence their strengths. Uniaxial strength tests, such as threeor four-point bending of bars, have been used extensively in the past to determine the strength of brittle materials such as ceramics [2–4]. However, the measured strength depends on both the surface and the edge conditions, and it is very difficult to eliminate undesirable edge failures [5, 6]. On the other hand, biaxial flexure tests involve supporting a thin plate on three or more points near its periphery and equidistant from its center and loading a central portion. The area of maximum tensile stress thus falls at the center of the plate surface, and the measured strength is independent of the condition of the plate edges. Also, real material components are generally subjected to multiaxial loading during service applications, and the biaxial strength data are more useful than the uniaxial strength data for the material design. As a result, biaxial flexure tests, such as ball-on-ring (or ball-on-three-ball), ring-onring, and piston-on-three-ball tests, are becoming increasingly popular as a means of measuring the strength of brittle materials [7–28]. Stress analyses for biaxial flexure tests are very complex even for monolithic elastic materials. A method of handling a thin monolayered disc supported at several points along its boundary and subjected to normal loading symmetrically distributed over a concentric circular area was first developed by Nadai [29]. The relation between the transverse displacement of the disc and the load intensity was described by a biharmonic equation, and its general solution could be obtained using Muskhelishvili’s complex variable method [30]. Solving the biharmonic equation with the essential equilibrium conditions and boundary conditions is a formidable task, and a complex series solution has been obtained by Bassali [31]. However,

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Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

1095

because of the complex formulation, application of Bassali’s solution is quite difficult. As a result, Bassali’s solution was subsequently simplified to an expression that could be easily adapted to the design and analysis of thin monolayered discs [32, 33]. Specifically, Vitman and Pukh [32] formulated the equation for ring-on-ring tests, and Kirstein and Woolley [33] formulated the equation for piston-on-ring tests. While ASTM has selected ring-on-ring tests to establish C1499 for advanced ceramics [34], the International Organization for Standardization has selected piston-on-three-ball tests to establish ISO 6872 for dentistry-ceramic materials [35]. Both ASTM and ISO formulas are simplifications from those series solutions; however, they are applicable only when the disc is monolithic. While all-ceramic dental crowns have been used extensively for clinical restorations, they are usually fabricated into bilayered structures with esthetic but relatively weak veneer porcelains on stiff and strong ceramic supporting cores. To evaluate the biaxial strength of bilayered dental structures, researchers in the dental community often rely on Roark’s formulas [36], in which bilayered discs subjected to biaxial moment loading are considered and stresses are predicted only on the top and the bottom surfaces of the discs. Recently, the closed-form solutions for the stress distribution in multilayered discs subjected to biaxial flexure tests have been derived by Hsueh et al. [37–43], and the analytical results have been validated by the finite element analysis (FEA). With Hsueh et al.’s formulas, the biaxial strength of multilayers can be readily evaluated using biaxial flexure tests. The purpose of this chapter is to review the closed-form solutions for stresses in multilayered discs subjected to biaxial flexure tests. The specific example of ring-onring tests is considered in this chapter, and the closed-form solutions for other biaxial flexure tests can be obtained accordingly based on their loading configurations. First, ASTM formulas for monolayered discs [34] and Roark’s formulas [36] for bilayered discs are reviewed, respectively. It should be noted that Roark’s formulas are valid for a simply supported disc at its edge. To account for the overhang region of the disc for the actual loading configuration, modification of Roark’s formulas is required which is also shown in this chapter. Then, the derivations of Hsueh et al.’s rigorous and simplified formulas are presented. Also, specific results are calculated for (i) zirconia monolayered discs, (ii) dental crown materials of porcelain/zirconia bilayered discs, and (iii) solid oxide fuel cell (SOFC) trilayered discs subjected to ring-on-ring tests. Comparison between the closed-form solutions and the finite element results is made to validate the formulas. Finally, a case study of layer thickness effects in bilayered dental ceramics subjected to both thermal stresses and ring-on-ring tests is presented.

2

Modeling

A diametrical section through the axis of symmetry of the multilayered disc is shown schematically in Fig. 1a. The disc consists of n layers with individual thickness, ti, where the subscript, i, denotes the layer number with layer 1 being at the bottom of the disc. The cylindrical coordinates, r, θ, and z, are used, and the bottom surface of

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C.-H. Hsueh

Fig. 1 Schematics of the diametrical section through the axis of symmetry of a multilayered disc showing (a) the coordinate system used in analyses and the loading and the supporting surfaces during biaxial flexure tests and (b) a multilayered disc subjected to ring-on-ring tests

layer 1 is located at z = 0. The disc is subjected to biaxial flexure tests, and the interfaces between layers are assumed to remain bonded during tests. For biaxial flexure tests, ring-on-ring, ball-on-ring, and piston-on-ring biaxial flexural tests have been conventionally used. Derivation of the biaxial stress distribution in the disc during loading is contingent upon the determination of the following two relations: (i) the biaxial stress-biaxial moment relation and (ii) the moment-load relation. While the first relation is valid for all biaxial flexure tests, the second relation depends on the individual loading configuration. The ring-on-ring loading is considered in this chapter as an example. For the other biaxial flexure tests, the results can be obtained by replacing the corresponding moment-load relation. The schematic drawing for ring-on-ring tests is shown in Fig. 1b. The disc with radius R is supported by an outer support ring with radius a and loaded with a smaller coaxial inner ring with radius b. The load P is applied on the disc through the inner ring.

2.1

ASTM Formulas

For biaxial flexure tests, the ASTM formulas are available only for monolayered discs, and the guidelines for testing of discs have been provided by Salem and Powers [44]. In this case, the biaxial stress, σ, in the disc is proportional to both the

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

1097

distance from the neutral surface and the moment but inversely proportional to the flexure rigidity, such that [45] σ¼

Eðz  zn ÞM , ð1  ν2 ÞD

(1)

where E is Young’s modulus, ν is Poisson’s ratio, M is the biaxial moment per unit length, z = zn denotes the position of the neutral surface, and D is the flexural rigidity. For monolayered discs, the neutral surface is located at the midplane of the disc, and zn and D are given by [45]

D¼

t zn ¼ , 2

(2a)

Et3 , 12ð1  ν2 Þ

(2b)

where t is the thickness of the monolayered disc. For ring-on-ring tests, the central region bounded by the inner ring is subjected to uniform biaxial stresses. In a simplified form, the biaxial stress on the tensile surface is [6, 7, 32, 34] " #
a ð1  νÞa2  b2  3P σ¼ 2ð1 þ νÞln þ 4πt2 b R2

ðat z ¼ 0 and r  bÞ:

(3)

Considering the linear stress distribution through the disc thickness, the variation of biaxial stresses through the thickness is described by " #
a ð1  νÞa2  b2  Eðz  zn ÞP 2ð1 þ νÞln σ¼ þ 8π ð1  ν2 ÞD b R2

ðfor r  bÞ,

(4)

where zn and D are defined by Eqs. (2a) and (2b), respectively. In this case, the moment-load relation is described by " #
a ð1  νÞa2  b2  P 2ð1 þ νÞln M¼ þ 8π b R2

2.2

ðfor r  bÞ:

(5)

Roark’s Formulas

Bilayered discs are considered in Roark’s formulas. When the bilayered disc is subjected to a biaxial moment, the stresses at the top and the bottom surfaces of the disc are given by [36]

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C.-H. Hsueh

" # 6M ð1  ν2 2 Þð1 þ t1 =t2 Þð1 þ E2 t2 =E1 t1 Þ σ¼ 2 1þ t2 K p ð1 þ E2 t2 =E1 t1 Þ2  ðν2 þ ν1 E2 t2 =E1 t1 Þ2

ðat z ¼ t1 þ t2 Þ, (6a)

" # 6M E1 t1 ð1  ν2 2 Þ t2 ð1  ν2 2 Þð1 þ t1 =t2 Þð1 þ E2 t2 =E1 t1 Þ þ σ¼ 2 ðat z ¼ 0Þ, t2 K p E2 t2 ð1  ν1 2 Þ t1 ð1 þ E2 t2 =E1 t1 Þ2  ðν2 þ ν1 E2 t2 =E1 t1 Þ2

(6b)

where Kp ¼ 1 þ

E1 t 1 3 ð 1  ν 2 2 Þ 3ð1  ν2 2 Þð1 þ t1 =t2 Þ2 ð1 þ E2 t2 =E1 t1 Þ þ : E2 t2 3 ð1  ν1 2 Þ ð1 þ E2 t2 =E1 t1 Þ2  ðν2 þ ν1 E2 t2 =E1 t1 Þ2

(7)

In this case, t1 + t2 is the thickness of the bilayered disc. It should be noted that instead of the actual ring-on-ring tests, the following loading configuration is considered in Roark’s formulas. The disc is simply supported at its edge, i.e., R = a in Fig. 1b, and it is subjected to a uniform annular line loading at r = b. In this case, the biaxial moment is related to the load, P, by [36] " #
a ð1  ν Þa2  b2  P e 2ð1 þ νe Þln M¼ þ 8π b a2

ðfor r  bÞ,

(8)

where νe is the equivalent Poisson’s ratio of the bilayer and is given by νe ¼ ν2

Kq , Kp

(9)

and Kq ¼ 1 þ

E1 ν1 t1 3 ð1  ν2 2 Þ 3ð1  ν2 2 Þð1 þ t1 =t2 Þ2 ð1 þ ν1 E2 t2 =ν2 E1 t1 Þ þ : (10) E2 ν 2 t 2 3 ð 1  ν 1 2 Þ ð1 þ E2 t2 =E1 t1 Þ2  ðν2 þ ν1 E2 t2 =E1 t1 Þ2

The moment-load relation has different signs between Eqs. (5) and (8) because of different conventions in defining the moment. Roark’s formulas have been adopted by the researchers in the dental community to predict stresses for bilayered dental ceramics subjected to ring-on-ring tests. However, because the specimen radius is required to be greater than the support ring radius in actual ring-on-ring tests, it is impractical to use the condition of R = a in Roark’s formulas, and modification of Roark’s formulas is required. This modification can be obtained by comparing Roark’s formulas with ASTM formulas for the monolayered case which is shown as follows [42]. When E1 = E2 = E and ν1 = ν2 = ν, the bilayer becomes a monolayer, and Roark’s formulas can be simplified to

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

 # ð1  νÞ a2  b2 σ¼ 2ð1 þ νÞln þ (11a) b a2 4π ðt1 þ t2 Þ2 ðat z ¼ t1 þ t2 and r  bÞ, " #
a ð1  νÞa2  b2  3P 2ð1 þ νÞln þ ðat z ¼ 0 and r  bÞ: σ¼ b a2 4π ðt1 þ t2 Þ2 3P

"

1099


a

(11b) Comparison between Roark’s and ASTM formulas shows that Eqs. (5) and (11b) become identical for the monolayered case when R = a, which is the condition considered in Roark’s formulas. Hence, for R > a in ring-on-ring loading, the moment-load relation in Roark’s formulas should be modified, such that Eq. (8) becomes " #
a ð1  ν Þa2  b2  P e 2ð1 þ νe Þln M¼ þ 8π b R2

ðfor r  bÞ:

(12)

The modified Roark’s formulas, which are pertinent to ring-on-ring tests, can be obtained by combining the biaxial stress-biaxial moment and the biaxial momentload relations, i.e., Eqs. (6a), (6b), and (12).

2.3

Hsueh et al.’s Rigorous Formulas

Multilayered discs were considered in Hsueh et al.’s analyses [37–39]. In this case, the continuity conditions at the interfaces between layers are required in analyses, and the solutions are very difficult if not impossible to derive. Instead of solving directly the problem of multilayered discs subjected to biaxial flexure tests, the essence of Hsueh et al.’s analyses is to find the correlation between monolayered and multilayered discs subjected to biaxial moment. Then, utilizing this correlation, the existing solutions for monolayered discs can be converted to the solutions for multilayered discs.

2.3.1 Correlation It is shown in Fig. 1a that the interface between layers i and i þ 1 is located at hi and the top surface of layer n is located at z = hn. With these definitions, hn is the thickness of the disc, and the relation between hi and ti is described by hi ¼

i X

tj

ði ¼ 1 to nÞ:

(13)

j¼1

The interfaces between layers are assumed to remain bonded during biaxial flexure tests. The stress distributions in the multilayered disc can be determined

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C.-H. Hsueh

from the force and moment equilibrium conditions coupled with the existing solutions for monolayered discs, and the analytical procedures are described in the following. For small deflection (e.g., less than one-quarter of the disc thickness), the strains are proportional to the distance from the neutral surface and inversely proportional to the radius of curvature [45]. For a monolayered disc, the neutral surface coincides with the midplane of the disc. However, for a multilayered disc, the neutral surface would deviate from the midplane because of the nonuniform elastic properties through the disc thickness. The radial and the tangential strains, er and eθ, in the system are [45] er ¼

z  znr rr

ð0  z  hn Þ,

(14a)

eθ ¼

z  znθ rθ

ð0  z  hn Þ,

(14b)

where z = znr and z = znθ dictate the positions of neutral surfaces for bending in the radial and the tangential directions, respectively, and rr and rθ are the corresponding radii of curvature. It should be noted that by using Eqs. (14a) and (14b) to describe the strain distribution in the system, the strain continuity condition at the interfaces is automatically satisfied. For a thin disc, the stress normal to the disc is negligible, and the radial and the tangential stresses, σ r and σ θ, are related to strains by σ ri ¼

Ei ðer þ νi eθ Þ 1  ν2i

ði ¼ 1 to nÞ,

(15a)

σ θi ¼

Ei ðeθ þ νi er Þ 1  ν2i

ði ¼ 1 to nÞ,

(15b)

where E and ν are Young’s modulus and Poisson’s ratio, respectively; the subscript, i, denotes the layer number; and Ei/(1–νi2) has the physical meaning of the planestrain modulus of layer i. There is no in-plane applied force, and the force equilibrium condition requires n ð hi X i¼1

n ð hi X i¼1

σ ri dz ¼ 0,

(16a)

σ θi dz ¼ 0:

(16b)

hi1

hi1

When i = 1, hi-1 (i.e., h0) in Eqs. (16a) and (16b) is defined as zero. The positions of the two neutral planes can be obtained by substituting Eqs. (14) and (15) into Eqs. (16a) and (16b). It is found that solutions of znr and znθ are complex and they are functions of the two radii of curvature. However, solutions are greatly simplified if

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Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

1101

the difference between νi (i = 1 to n) is ignored in the parentheses in Eqs. (15a) and (15b). Using this simplification [i.e., letting νi = ν in the parentheses in Eqs. (15a) and (15b)], znr and znθ become independent of the two radii of curvature and ν, and their solutions are znr ¼ znθ ¼ zn  ¼

n X i¼1

n Ei ti
ti . X Ei ti h þ : i1 1  νi 2 2 1  νi 2 i¼1

(17)

In this case, the two neutral surfaces become one, and it is redefined as zn*. It should be noted that the simplification of Poisson’s ratio νi in the plane-strain modulus, Ei/(1 – νi2), in Eq. (15a) and (15b) is not required in order to obtain Eq. (17). Hence, Ei/(1 – νi2) should be regarded as one term without the simplification in νi. The moment equilibrium condition requires n ð hi X i¼1

n ð hi X i¼1

σ ri zdz ¼ Mr ,

(18a)

σ θi zdz ¼ Mθ ,

(18b)

hi1

hi1

where Mr and Mθ are bending moments per unit length. Solutions of Eqs. (18a) and (18b) yield Mr ¼ D



M θ ¼ D



 1 1 þν , rr rθ  1 1 þν , rθ rr

(19a)



(19b)

where D* has the physical meaning of the flexural rigidity of the multilayer and is D ¼

n X i¼1

 Ei ti ti 2
ti   2  h h þ h t þ þ , z i1 i1 i i1 n 1  νi 2 3 2

(20)

where zn* is given by Eq. (17). For a monolayered disc with uniform material properties, Ei = E and νi = ν, Eqs. (17) and (20) recover the neutral surface and the flexural rigidity of a monolayered disc shown in Eqs. (2a) and (2b). Combining Eq. (15a) and (15b) with Eqs. (19a) and (19b), the stresses can be related to bending moments by Ei ðz  zn  ÞMr  σ ri ¼  1  ν2i D

ði ¼ 1 to nÞ,

(21a)

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C.-H. Hsueh

Ei ðz  zn  ÞMθ  σ θi ¼  1  ν2i D

ði ¼ 1 to nÞ:

(21b)

It is worth noting that the above results for multilayers were derived under the condition that all layers have the similar Poisson’s ratio. When the layers have very different Poisson’s ratios, a correction factor is required, and the derivation of this correction factor is shown as follows. For a thin multilayered disc, the stress normal to the disc can be ignored, and the radial and the tangential stresses, σ r and σ θ, are related to the radial and the tangential strains, er and eθ, by σ ri ¼

Ei ðer þ νi eθ Þ 1  νi 2

ði ¼ 1 to nÞ,

(22a)

σ θi ¼

Ei ðeθ þ νi er Þ ði ¼ 1 to nÞ: 1  νi 2

(22b)

In order to obtain Eqs. (17) and (20), the difference in Poisson’s ratios νi (i = 1 to n) in the parentheses in Eqs. (15a) and (15b) is ignored, such that an average Poisson’s ratio, ν, is considered, and it is given by ν¼

n 1 X νi ti : hn i¼1

(23)

However, the simplification of Poisson’s ratio νi in the plane-strain modulus, Ei/(1 – νi2), in Eqs. 21a and (21b) is not required in deriving Eqs. (17) and (20). Under the biaxial stress condition, er = eθ = e, and Eqs. (22a) and (22b) become σ ri ¼ σ θi ¼ σ i ¼

Ei e 1  νi

ði ¼ 1 to nÞ:

(24)

The simplification of νi = ν used in the parentheses in Eqs. (15a) and (15b) corresponds to σ ri ¼ σ θi ¼ σ i ¼

ð1 þ νÞEi e 1  νi 2

ði ¼ 1 to nÞ:

(25)

While Eq. (24) is exact, Eq. (25) is approximate which leads to the solutions described by Eqs. (17), (20), and (21). However, if the stress described by Eq. (25) is multiplied by a factor of (1 þ νi)/(1 þ ν), it becomes identical to Eq. (24). Hence, the accuracy of the closed-form solutions for the stress distributions given by Eq. (21) can be improved by multiplying a correction factor of (1 þ νi)/(1 þ ν), such that σi ¼

Ei ðz  zn  ÞM ð1  νi Þð1 þ νÞD

ði ¼ 1 to nÞ:

(26)

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Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

1103

The above stress-moment relation, i.e., Eq. (26), is valid for any biaxial flexure test in correlating monolayer solutions to multilayer solutions.

2.3.2 Conversion For the moment-load relation, Eq. (5) shows that it depends on the loading configuration and Poisson’s ratio of the disc but is independent of Young’s modulus for the monolayered case. Compared to the monolayered case, Eq. (12) shows that the moment-load relation for bilayer can be obtained by replacing Poisson’s ratio of monolayer, ν, with equivalent Poisson’s ratio of bilayer, νe. Hence, Eq. (5) can also be used as the moment-load relation for multilayer by considering ν as the average Poisson’s ratio of multilayer. Combining Eq. (5) with Eq. (26), the solutions for the multilayered discs subjected to ring-on-ring tests become " #
a ð1  νÞa2  b2  Ei ðz  zn  ÞP σi ¼ 2ln þ 8π ð1  νi ÞD b ð1 þ νÞR2

ðfor r  b and i ¼ 1 to nÞ, (27)

where zn* and D* are defined by Eqs. (17) and (20), respectively, and ν is given by Eq. (23). Depending upon the comparison between the strength of the individual layer and the stress distribution through the thickness of the multilayer during tests, failure is expected to initiate from the layer in which its tensile strength is first exceeded by the stress due to loading. Eq. (27) can be used readily to evaluate the biaxial strength of multilayered discs using ring-on-ring tests.

2.3.3 Bilayered Systems For the special case of bilayered discs, i.e., n = 2, Hsueh et al.’s rigorous formulas become " #
a ð1  νÞa2  b2  E1 ðz  zn  ÞP 2ln σ1 ¼ þ 8π ð1  ν1 ÞD b ð1 þ νÞR2

ðfor 0  z  t1 and r  bÞ, (28a)

σ2 ¼

" #
a ð1  νÞa2  b2  E2 ðz  zn  ÞP 2ln þ 8π ð1  ν2 ÞD b ð1 þ νÞR2 ðfor t1  z  t1 þ t2 and r  bÞ,

(28b)

where zn*, D*, and ν are given by

zn 

E1 t1 2 E2 t2 2 E2 t1 t2 þ þ 2 2ð1  ν1 Þ 2ð1  ν2 2 Þ 1  ν2 2 ¼ , E1 t1 E2 t2 þ 1  ν1 2 1  ν2 2

(29a)

1104

C.-H. Hsueh

D ¼

E1 t 1 3 E2 t 2 3 E2 t1 t2 ðt1 þ t2 Þ þ þ 2 2 1  ν2 2 3ð1  ν1 Þ 3ð1  ν2 Þ h i 2 E1 t1 2 E2 t2 2 E2 t1 t2 2ð1ν1 2 Þ þ 2ð1ν2 2 Þ þ 1ν2 2  , E1 t1 E2 t2 þ 1  ν1 2 1  ν2 2 ν¼

ν1 t1 þ ν2 t2 : t1 þ t2

(29b)

(29c)

Equations (28a) and (28b) can be compared with Roark’s formulas. However, while Hsueh et al.’s formulas give the stress distribution through the disc thickness, Roark’s formulas give only stresses on the top and the bottom surfaces of the disc. For the special case of monolayered discs, i.e., n = 1, Hsueh et al.’s formulas become identical to ASTM formulas.

2.4

Hsueh et al.’s Simplified Formulas

Although the above rigorous formulas are closed-form solutions, the expressions are lengthy which may deter the engineers, who are less adept in mechanics and mathematics, from using them. Hence, it would be fruitful to derive simplified formulas, such that they can be used readily by engineers to evaluate the biaxial strength of multilayered dental ceramics using biaxial flexure tests. The derivations of the simplified formulas are shown as follows [46]. Under the condition of small deflection, the (in-plane) strain, e, varies linearly through the disc thickness, such that e ¼ αz þ β,

(30)

where α and β are unknown constants to be solved. For biaxial moment loading, the biaxial stress/strain relation is described by σi ¼

Ei e 1  νi

ði ¼ 1 to nÞ:

(31)

Strictly speaking, Eq. (31) is valid for monolayer or when all the layers in the multilayered system have the same Poison’s ratio [47]. For a general multilayered system, the stress state in the disc is not equibiaxial because of different Poisson’s ratios between layers, and Eq. (31) is satisfied in an approximate sense [47]. Based on Eqs. (30) and (31), the solutions of the biaxial stress/ strain distributions in the multilayered disc are contingent upon the determination of α and β. The biaxial stresses at the top and the bottom surfaces of the multilayered disc, σ T and σ B, can be defined from Eqs. (30) and (31), such that

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

σT ¼

En ðαhn þ βÞ 1  νn

σB ¼

E1 β 1  ν1

ðat z ¼ hn Þ, ðat z ¼ 0Þ:

1105

(32a) (32b)

Solutions of Eqs. (32a) and (32b) yield α¼

E1 ð1  νn Þσ T  En ð1  ν1 Þσ B , E1 En h n

(33a)

ð1  ν1 Þσ B : E1

(33b)

β¼

Because α and β can be expressed in terms of σ T and σ B, the biaxial stress/strain through the disc thickness can be obtained from Eqs. (30), (31), and (33) if σ T and σ B are known. Hence, if σ T and σ B can be measured during testing, the biaxial stress/ strain through the thickness of the multilayered disc can be determined using Eqs. (30), (31), and (33). However, if σ T and σ B are unattainable experimentally, they still can be derived analytically as shown in the following. When the disc is subjected to biaxial bending moment, satisfactions of force and moment equilibrium conditions require n ð hi X i¼1

n ð hi X i¼1

σ i dz ¼ 0,

(34a)

σ i zdz ¼ M:

(34b)

hi1

hi1

Solutions of Eqs. (30), (31), and (34) yield n P Ei t i i¼1 1  νi α¼ n  n   , n   2 P Ei ti P P Ei t i ti 2 Ei ti ti 2 h þ  h þ h t þ i1 i1 i1 i 1νi 2 3 i¼1 i¼1 1  νi i¼1 1  νi (35a) n P Ei ti
ti  M hi1 þ 2 i¼1 1  νi β¼ 2  n  n   : n  P Ei ti P E i ti P Ei ti  ti 2 ti 2  hi1 þ hi1 ti þ 1νi hi1 þ 2 3 i¼1 i¼1 1  νi i¼1 1  νi (35b)

M

With α and β given by Eqs. (35a) and (35b), σ T and σ B can be determined from Eqs. (32a) and (32b), and the solutions for the biaxial stress/strain through the disc

1106

C.-H. Hsueh

thickness are completed. For ring-on-ring, the moment-load relation is given by Eq. (5), in which ν is the average Poisson’s ratio of multilayer, and the biaxial stress/strain distribution can hence be expressed in terms of the load. For the special case of bilayered discs, i.e., n = 2, α and β given by Eqs. (35a) and (35b) become 

 E1 t 1 E2 t2 þ 1  ν1 1  ν2 α¼h , i 2 4E1 E2 t1 t2 ðt1 2 þ t1 t2 þ t2 2 Þ E1 t1 2 E2 t2 2 þ þ ð1ν1 Þ ð1ν2 Þ ð1  ν1 Þð1  ν2 Þ  E1 t1 2 E2 t 2 2 2E2 t1 t2 6M þ þ 1  ν1 1  ν2 1  ν2 β¼h i2 4E E t t ðt 2 þ t t þ t 2 Þ : 1 2 1 2 1 1 2 2 E1 t1 2 E2 t2 2 þ ð1ν1 Þ þ ð1ν2 Þ ð1  ν1 Þð1  ν2 Þ 12M

(36a)

(36b)

The stresses at the top and the bottom surfaces of the disc, σ T and σ B, become  6E2 M E1 t1 2 E2 t2 2 2E1 t1 t2 þ þ 1  ν2 1  ν1 1  ν2 1  ν1 σT ¼ h i2 4E E t t ðt 2 þ t t þ t 2 Þ , 1 2 1 2 1 1 2 2 E1 t1 2 E2 t2 2 þ þ ð1ν1 Þ ð1ν2 Þ ð1  ν1 Þð1  ν2 Þ  6E1 M E1 t1 2 E2 t 2 2 2E2 t1 t2 þ þ 1  ν1 1  ν1 1  ν2 1  ν2 σB ¼ h i2 4E E t t ðt 2 þ t t þ t 2 Þ : 1 2 1 2 1 1 2 2 E1 t1 2 E2 t2 2 þ ð1ν1 Þ þ ð1ν2 Þ ð1  ν1 Þð1  ν2 Þ

(37a)

(37b)

The stresses at the interface, i.e., σ 1 and σ 2 at z = t1, become σ1 ¼

E1 ð1  ν2 Þt1 σ T t2 σ B þ E2 ð1  ν1 Þðt1 þ t2 Þ t1 þ t2

ðat z ¼ t1 Þ,

(38a)

σ2 ¼

t1 σ T E2 ð1  ν1 Þt2 σ B þ t1 þ t2 E1 ð1  ν2 Þðt1 þ t2 Þ

ðat z ¼ t1 Þ:

(38b)

Equations (38a) and (38b) can also be applied to Roark’s formulas to obtain the stresses at the interface of the bilayer because both σ T and σ B are given in Roark’s formulas. It should be noted that assuming ν1 = ν2 and a linear strain distribution through the thickness of a bilayered disc subjected to biaxial moment loading, equations equivalent to Eqs. (38a) and (38b) have also been derived by Rosenstiel et al. [48, 49]. Furthermore, when the system becomes a monolayer, i.e., E1 = E2 and ν1 = ν2, Eqs. (37) and (38) become

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

σ T ¼ σ B ¼ σ1 ¼ σ2 ¼

6M ðt1 þ t2 Þ2

t1 σ T þ t2 σ B t1 þ t2

,

ðat z ¼ t1 Þ:

1107

(39) (40)

Equation (39) recovers the classical solution of the biaxial stress on the surfaces of monolayer subjected to biaxial moment [45].

3

Results

Specific analytical and FEA results are calculated using material properties pertinent to (i) zirconia monolayered, (ii) porcelain/zirconia bilayered, and (iii) SOFC trilayered discs subjected to ring-on-ring tests in order to validate the closed-form solutions. Although the following calculated results are for some specific loads, the results for other loads can be obtained by scaling with the load because of the linear elasticity used in analyses. However, ASTM standards for biaxial flexure tests [34] should be followed. For example, deflections should be less than one-quarter of the disc thickness during the test. Also, it should be noted that although the FEA provides a powerful means for analyzing stress fields in complex multilayered systems, it suffers from the drawback that it is a case-by-case study, and computation needs to be performed for each change in geometrical parameters and material properties.

3.1

Zirconia Monolayered Discs

Zirconia has been used as the supporting core in all-ceramic dental crowns [24]. A zirconia (E = 220 GPa, ν = 0.33) disc of 1.6 mm thickness is considered. The disc has a radius, R, of 7 mm. The supporting and the loading rings have radii of a = 6 mm and b = 2.5 mm, respectively, and P = 1000 N is loaded on the disc through the loading ring. The FEA results show that the stresses are fairly biaxial and uniform within r  b. However, a small peak stress exists on the tensile surface opposite and just inside the load ring position which is greater than the stress at r = 0 by ~6% on the tensile surface. This small peak stress results from the contact stress and is a function of the size of the overhang region. Specifically, the small peak stress diminishes as the overhang region becomes smaller [39]. For FEA, the simulated biaxial stress at r = 0 and through the thickness is shown in Fig. 2. Fig. 2 shows that the stress is linear through the thickness and the neutral surface is located at the midplane of the disc. While ASTM and Hsueh et al.’s rigorous formulas are identical for the monolayered case, they are also in excellent agreement with the FEA results. At the top and the bottom surfaces of the plate, the predicted stresses from Roark’s

1108

C.-H. Hsueh

Fig. 2 The biaxial stress through the thickness of a zirconia monolayered disc subjected to ring-on-ring loading of P = 1000 N showing the comparison among ASTM formulas, Roark’s formulas, modified Roark’s formulas, Hsueh et al.’s rigorous formulas, and FEA results

formulas deviate from the FEA results. However, after corrections for the overhang region of the disc using Eq. (12) instead of Eq. (8), the stresses predicted from modified Roark’s formulas agree with the FEA results.

3.2

Porcelain/Zirconia Bilayered Discs

The porcelain/zirconia bilayered system has significant applications in all-ceramic dental crowns. Both the porcelain layer (E = 64 GPa, ν = 0.21) and the zirconia layer (E = 220 GPa, ν = 0.33) are 0.8 mm thick [24]. The disc has a radius, R, of 7 mm. The supporting and the loading rings have radii of a = 6 mm and b = 2.5 mm, respectively, and P = 1000 N is loaded on the disc through the loading ring. The biaxial stresses through the disc thickness are shown in Fig. 3a, b, respectively, for zirconia and porcelain being the top layer. Compared to the FEA results, excellent agreement is obtained for Hsueh et al.’s rigorous formulas, Eq. (28), while a slight deviation is observed for the modified Roark’s formulas, Eqs. (6) and (12). Also, despite the simplicity of Hsueh et al.’s simplified formulas, Eqs. (37) and (38), they agree well with both the rigorous solutions and FEA results. However, it should be noted that, if the same Poisson’s ratio is used for both layers (i.e., ν1 = ν2), the solutions from the three sets of equations become the same (not shown in figures). It is worth mentioning that for a bonded interface, continuities of the displacement, the shear stress, and the stress normal to the interface are required. However, unless the material properties change continuously across the interface, the in-plane stress is not continuous at the interface. For bilayered discs, the stress is linear through the thicknesses in each individual layer due to bending; however, because of different elastic properties between the two layers, the (in-plane) stress is discontinuous at the interface and the stress gradients are different. When zirconia is the top layer, Fig. 3a shows a peak compressive stress at the loading (zirconia) surface and two peak tensile stresses, one each at the bottom surfaces of the zirconia and the porcelain layers. The

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

1109

Fig. 3 The biaxial stress through the thickness of porcelain/zirconia bilayered discs subjected to ring-on-ring loading of P = 1000 N with (a) zirconia and (b) porcelain as the top (loading) layers showing the comparison among Hsueh et al.’s rigorous and simplified formulas, modified Roark’s formulas, and FEA results

neutral surface (i.e., zero stress) is located at z = ~1 mm (i.e., within the zirconia layer). When the porcelain layer is on the top, Fig. 3b shows a peak tensile stress on the bottom surface of the zirconia and two peak compressive stresses, one at the loading (porcelain) surface and another at the top surface of the zirconia. The neutral surface is located at z = ~0.6 mm (i.e., within the zirconia layer). The maximum tension predicted in Fig. 2 for monolayered discs is very different from those predicted in Fig. 3a, b for bilayered discs. Hence, the existing solution for monolayers, i.e., Eq. (3), cannot be used to predict the results for bilayers (or multilayers).

3.3

SOFC Trilayered Discs

The SOFC trilayer consists of La0.75Sr0.2MnO3 (LSM) cathode layer (E = 35 GPa, ν = 0.25) of 0.025 mm thickness, YSZ electrolyte layer (E = 215 GPa, ν = 0.32) of 0.15 mm thickness, and NiO-YSZ anode layer (E = 55 GPa, ν = 0.17) of 0.025 mm

1110

C.-H. Hsueh

thickness [21]. The disc has a radius, R, of 25 mm. The supporting and the loading rings have radii of a = 19 mm and b = 4.75 mm, respectively, and P = 10 N is loaded on the disc through the loading ring. For this LSM/YSZ/NiO-YSZ trilayer, either the NiO-YSZ layer or the LSM layer can be the top loading layer during tests. The biaxial stresses through the disc thickness are shown in Fig. 4a, b, respectively, for NiO-YSZ and LSM being the top layers. Excellent agreement is obtained between Hsueh et al.’s rigorous formulas and FEA results. Fig. 4a, b is similar. This is because (i) the middle (i.e., YSZ) layer is much stiffer and thicker than the two outer layers, and the load is mainly shared by this middle layer, and (ii) the difference in stiffness between the two outer layers is not much, and the difference between which layer being at the top is small. Fig. 4a, b shows that the greatest tension does not occur at the tensile surface of the specimen. Instead, it exists at the lower surface of the middle layer. This is because YSZ is much stiffer than both LSM and NiO-YSZ. Depending upon the comparison between the strength and the Fig. 4 The biaxial stress through the thickness of LSM/YSZ/NiO-YSZ trilayered discs subjected to ring-on-ring loading of P = 10 N with (a) NiO-YSZ and (b) LSM as the top loading layers showing the comparison between Hsueh et al.’s rigorous formulas and FEA results

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

1111

Fig. 5 The biaxial stress through the thickness of a YSZ monolayered disc subjected to ring-on-ring loading of P = 10 N showing the comparison among ASTM formulas, Roark’s formulas, modified Roark’s formulas, Hsueh et al.’s rigorous formulas, and FEA results

maximum tension in each layer, cracking can initiate in the inner layer instead of at the tensile surface of the specimen. When the trilayered disc becomes YSZ monolayer of 0.2 mm thickness, the results are shown in Fig. 5 for comparison with Fig. 4a, b.

4

Case Study

The purpose of the case study is to analyze the stress distribution through the thickness of bilayered dental ceramics subjected to both thermal stresses and ringon-ring tests and to systematically examine how the individual layer thickness influences this stress distribution and the failure origin [41]. In this case, ring-onring tests are performed on In-Ceram Alumina/Vitadur Alpha porcelain bilayered discs with porcelain in the tensile side, and In-Ceram Alumina to porcelain layer thickness ratios of 1:2, 1:1, and 2:1 are used to characterize whether failure originated at the surface or the interface. Based on (i) the thermomechanical properties and thickness of each layer, (ii) the difference between the test temperature and the glass transition temperature, and (iii) the ring-on-ring loading configuration, the stress distribution through the thickness of the bilayer is calculated using the closed-form solutions. Finite element analyses were also performed to verify the analytical results. While the solutions for bilayered discs subjected to ring-on-ring tests are shown in Sect. 2.3.3, the solutions for bilayered discs subjected to thermal stresses are shown as follows.

4.1

Solutions for Bilayered Discs Subjected to Thermal Stresses

For simplicity, the average thermal stress in each layer has been considered by many researchers. With this simplification, the thermal stress is constant in each layer, and

1112

C.-H. Hsueh

the layer is subjected to either pure tension or pure compression. However, deflection of the bilayer occurs in order to balance the moment which, in turn, would result in a thermal stress gradient in each layer. Because of this stress gradient, the individual layer can be subjected to combined tension/compression depending upon the thickness ratio as well as the elastic modulus ratio between layers. Hence, to examine the effects of thermal stresses on failure of bilayers, consideration of the thermal stress distribution through the layer thickness is essential. While the general solutions for multilayered discs subjected to thermal stresses [50, 51] and ring-on-ring tests have been derived, respectively, the solutions for a bilayered disc subjected to both thermal stresses and ring-on-ring tests can be obtained by (i) using 2 as the layer number in the general solutions and (ii) superposing the solutions obtained from thermal mismatch and ring-on-ring loading. The results for thermal stresses are summarized as follows [50, 51]. The coefficients of thermal expansion (CTEs) of layers 1 and 2 are α1 and α2, respectively. With a temperature change ΔT from the stress-free temperature, thermal stresses develop due to the mismatch in thermal strains. For the bilayered systems, the in-plane thermal stresses in layers 1 and 2, σ 1 and σ 2, are biaxial which have been derived, such that [50, 51] ( σ1 ¼

E01

" #) E02 t2 ðα2  α1 ÞΔT E01 t1 2 þ E02 t2 2 þ 2E02 t1 t2   þκ z E01 t1 þ E02 t2 2 E01 t1 þ E02 t2

ðfor 0  z  t1 Þ,

(41a) (

σ2 ¼

E02

" #) E01 t1 ðα1  α2 ÞΔT E01 t1 2 þ E02 t2 2 þ 2E02 t1 t2   þκ z E01 t1 þ E02 t2 2 E01 t1 þ E02 t2 ðfor t1  z  t1 þ t2 Þ,

(41b)

where E0 = E/(1–ν) is the biaxial modulus with E and ν being Young’s modulus and Poisson’s ratio, respectively, and κ is the curvature of the bilayer which is given by [50, 51] κ¼

E01 2 t1 4

6E01 E02 t1 t2 ðt1 þ t2 Þðα2  α1 ÞΔT : þ E02 2 t2 4 þ 2E01 E02 t1 t2 ð2t1 2 þ 2t2 2 þ 3t1 t2 Þ

(42)

The above solutions, Eqs. (41a), (41b), and (42), are exact for positions away from the edge of the bilayered disc. Some limitations of the analytical solutions are discussed here. For the epitaxial growth of one layer on another, different lattice parameters between layers would induce the lattice mismatch strain, which is not considered here. The phenomenon of glass transition is also not considered. In this case, the glass transition temperature is a function of the cooling rate, and a strain component is induced by the free volume of the glass. In the presence of lattice mismatch between layers and free volume of the glass, the corresponding lattice mismatch strain and/or free volume-induced strain should be superposed on the

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

1113

thermal strain. When the temperature dependence of CTE is considered, the thermal strain component in equations should be replaced by an integral of the CTE with respect to the temperature range. Also, uniform temperature distribution in the system is assumed in formulating above equations, and the nonelastic deformation is not considered.

4.2

Finite Element Analyses

The discs were modeled using higher-order 8-node quadratic axisymmetric elements (plane 82) in the finite element code ANSYS version 9.0. The interface between layers was assumed to remain perfectly bonded at all stages of computation. After performing detailed mesh sensitivity analysis, the selected mesh distribution contained 166,811 elements and 501,766 nodes. These elements were meshed to be 5 μm in size within the inner ring, while significantly coarser mesh was utilized to model the disc overhang region. The loading was simulated as pressure applied over a small ring width of 25 μm, while the support ring was modeled as rollers to permit free radial displacement.

4.3

Results

The thermomechanical properties of In-Ceram Alumina and Vitadur Alpha porcelain [52] are listed in Table 1. Using the geometry shown in Fig. 1 for bilayer, Vitadur Alpha porcelain is layer 1 and In-Ceram Alumina is layer 2. The total thickness of the bilayered disc t1 + t2 is kept at 1.8 mm, and three thickness combinations of bilayered discs with the thickness ratios t2:t1 = 1:2, 1:1, and 2:1 are used in stress analyses. Also, the discs have a radius, R, of 8 mm. First, the biaxial thermal stresses away from the disc edge are analyzed using Eqs. (41a) and (41b). Then, the biaxial stresses in the central region bounded by the inner ring and through the disc thickness are analyzed using Eqs. (28a) and (28b). Finally, the total stresses resulting from both the thermal mismatch and ring-on-ring loading are presented. In all cases, the FEA is also performed to compare with the analytical results.

4.3.1 Thermal Stresses Veneer porcelain is coated on In-Ceram Alumina at 950  C; however, the glass transition temperature of this porcelain is 570  C [53, 54]. Hence, the temperature Table 1 Thermomechanical properties of In-Ceram Alumina/Vitadur Alpha porcelain bilayered dental ceramics Materials In-Ceram alumina Vitadur Alpha porcelain52 a

VITA product information

E (GPa) 280a 64

ν 0.23a 0.19

α (106/ C) 7.4a 7.1a

1114

C.-H. Hsueh

change ΔT = 545  C for cooling to room temperature (25  C) is considered for thermal stress development in the bilayer, and the possible stress relaxation by nonelastic deformation at temperatures below the glass transition temperature is not considered. The (in-plane) biaxial stress distributions through the thickness are shown in Fig. 6a–c, respectively, for the thickness ratios of 1:2, 1:1, and 2:1. Excellent agreement is obtained between analytical and FEA results in Fig. 6, the stress is linear through the thickness in each layer, but the stress is discontinuous at the interface because of the different CTEs between the two layers. Also, the stress gradients are different because of the different elastic properties between the two layers. Fig. 6a, b shows that for 1:2 and 1:1 thickness ratios, both In-Ceram Alumina and porcelain are subjected to combined tension/compression because of the deflection of the system. Specifically, the porcelain surface is subjected to tension at z = 0. However, a net compressive force exists in porcelain, while a net tensile force exists in In-Ceram Alumina because In-Ceram Alumina has a higher CTE than porcelain. When the thickness ratio changes from 1:2 to 1:1 and to 2:1, the tensile stress at the porcelain surface decreases and then becomes compressive. Because the observed failures of bilayered dental ceramics in the study always initiate in the veneer porcelain layer, it is beneficial to have compressive residual stresses in the porcelain layer to discourage its failure. Although the porcelain layer is subjected to a net compressive force from the thermal mismatch, Fig. 6 shows that the porcelain surface can be subjected to a tensile thermal stress depending upon the relative layer thickness ratio. Hence, it is of interest to examine the condition for the existence of compressive thermal stresses through the entire thickness of the porcelain layer. This can be achieved by finding the critical relation for a stress-free porcelain surface, i.e., σ 1 = 0 at z = 0, and this critical relation derived from Eq. (41a) is   E02 t2 3 3t2  2¼0 E01 t1 t1

ðfor σ 1 ¼ 0 at z ¼ 0Þ:

(43)

A critical ratio for t2/t1 as a function of E0 2/E0 1 can be obtained from Eq. (43), such that the porcelain layer is subjected to combined tension/compression and pure compression, respectively, when t2/t1 is below and above this critical ratio. For example, when E0 2 = E0 1, the critical thickness ratio is 2. For In-Ceram Alumina/ Vitadur Alpha porcelain bilayered discs, E0 2/E0 1 = 4.6 and t2/t1 > 1.04 are required in order to have the entire porcelain layer subjected to compressive thermal stresses.

4.3.2 Stresses Due to Ring-on-Ring Loading For ring-on-ring tests, the supporting and the loading rings have radii of a = 5.5 mm and b = 2.5 mm, respectively. For each thickness ratio, 30 specimens were used to measure the failure loads. Both surface and interface failures were observed, and they are indicated, respectively, by red circle and blue square in Fig. 7a–c for the thickness ratios of 1:2, 1:1, and 2:1, respectively. The mean failure loads are 568.5, 555.2, and 688.0 N, respectively, for specimens with relative thickness ratios of 1:2,

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

Fig. 6 The biaxial thermal stress through the thickness of In-Ceram Alumina/Vitadur Alpha porcelain bilayered discs for ΔT = 545  C and layer thickness ratios of (a) 1:2, (b) 1:1, and (c) 2:1 showing both the analytical and FEA results

1115

1116 Fig. 7 The measured failure loads for 30 In-Ceram Alumina/Vitadur Alpha porcelain bilayered discs subjected to ring-on-ring tests and layer thickness ratios of (a) 1:2, (b) 1:1, and (c) 2:1 showing red•, surface failure; blue■, interface failure; and ––––, mean failure load

C.-H. Hsueh

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

1117

1:1, and 2:1. The above loads are hence used accordingly for each thickness ratio to calculate stresses resulting from the failure loads. It is instructive to compare ring-on-ring tests performed on monolayered discs and bilayered discs. For monolayered discs, the stress distribution given by Eq. (4) is independent of Young’s modulus. Using ν = 0.19 (for porcelain), t = 1.8 mm, and P = 555.2 N, the stress distribution is calculated. The FEA results show that the stresses are fairly biaxial and uniform within r  b with the stress just inside the loading ring greater than the stress at r = 0 by 5.3% on the tensile surface. The biaxial stress at the disc center and through the thickness is shown in Fig. 8, in which excellent agreement between the analytical and FEA results is obtained. For bilayered discs, the biaxial stresses (due to the failure load) at the disc center and through the thickness are shown in Fig. 9a–c, respectively, for the thickness ratios of 1:2, 1:1, and 2:1. Excellent agreement is obtained between Hsueh et al.’s rigorous formulas and FEA results in Fig. 9. The difference in stress distributions between monolayers and bilayers subjected to ring-on-ring tests can be observed by comparing Fig. 8 with Fig. 9b. For bilayered discs, Fig. 9a–c shows that porcelain layers are subjected to tension, while In-Ceram Alumina layers are subjected to combined tension/compression. There are two peak tensile stresses in the stress distribution through the disc thickness, one on the bottom surface of porcelain and another on the bottom surface of In-Ceram Alumina. For 1:1 and 2:1 thickness ratios, the peak tension on the bottom surface of In-Ceram Alumina is greater than the peak tension on the bottom surface of porcelain.

4.3.3

Total Stresses Due to both Thermal Mismatch and Ring-on-Ring Loading In the presence of both the thermal mismatch and ring-on-ring loading, the biaxial stresses at the disc center and through the thickness are shown in Fig. 10a–c, respectively, for the thickness ratios of 1:2, 1:1, and 2:1, and excellent agreement is obtained between the analytical and FEA results. The peak tension on the bottom Fig. 8 The biaxial stress at the center through the thickness of a monolayered porcelain disc subjected to ring-on-ring tests with P = 555.2 N

1118 Fig. 9 The biaxial stress at the center through the thickness of In-Ceram Alumina/Vitadur Alpha porcelain bilayered discs subjected to ring-on-ring tests for layer thickness ratios of (a) 1:2 with P = 568.5 N, (b) 1:1 with P = 555.2 N, and (c) 2:1 with P = 688.0 N showing both Hsueh et al.’s rigorous formulas and FEA results

C.-H. Hsueh

34

Modeling of Multilayered Disc Subjected to Biaxial Flexure Tests

Fig. 10 The biaxial stress at the center through the thickness of In-Ceram Alumina/Vitadur Alpha porcelain bilayered discs subjected to both thermal mismatch with ΔT = 545  C and ring-onring loading for layer thickness ratios of (a) 1:2 with P = 568.5 N, (b) 1:1 with P = 555.2 N, and (c) 2:1 with P = 688.0 N showing both Hsueh et al.’s and FEA results

1119

1120

C.-H. Hsueh

sr core

veneer

sq core

veneer

MPa –183

–100.5 –141.75

–18 –59.25

64.5 23.25

147 105.75

Fig. 11 FEA simulated stress fields, σ r and σ θ, for In-Ceram Alumina/Vitadur Alpha porcelain bilayered discs with thickness ratio 2:1 subjected to both thermal mismatch with ΔT = 545  C and ring-on-ring loading P = 688.0 N

surface of the In-Ceram Alumina layer is greater than the peak tension on the bottom surface of the porcelain layer for both 1:1 and 2:1 thickness ratios. An example of the radial and the tangential stresses, σ r and σ θ, obtained from FEA for the 2:1 bilayer disc specimen is shown in Fig. 11 [41]. The characteristics of biaxial stresses (i.e., σ r = σ θ) and uniformity of these biaxial stresses with respect to r in the central region bounded by the inner ring can be observed in Fig. 11.

4.4

Discussion

Although Fig. 10b, c shows that maximum tension, 91 MPa and 146 MPa, occurs on the bottom surface of the In-Ceram Alumina layer, respectively, for 1:1 and 2:1 thickness ratios, cracking is unlikely to initiate there because of the high tensile strength of In-Ceram Alumina (400 to 600 MPa). However, it should be noted that an atomically sharp and smooth interface is assumed for the bilayered systems in modeling. In this case, the material properties are discontinuous at the interface which, in turn, would result in discontinuity of the in-plane biaxial stress at the interface. Experimentally, an interphase and/or roughness might exist at the interface. In the presence of an interfacial interphase (e.g., interaction between the two layers at the interface to form a thin third layer), the material properties would have a continuous variation at the interface which, in turn, would smooth the stress discontinuity at the interface. In the presence of interfacial roughness, the mechanical interlocking at the interface would result in the stress transfer of the biaxial stresses within the two layers at the interface which would also smooth the stress discontinuity at the interface. In both cases, the stress at the interface is expected to

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have a value in between the stresses at the bottom surface of layer 2 and at the top surface of layer 1. Observations using scanning electron microscope showed no evidence of the interphase at the interface. However, the In-Ceram Alumina discs were air abraded to remove the excess glass before veneering. Based on experimental results, it has been concluded that this abrasion increases the surface roughness which, in turn, would enhance the bond strength between the abraded and the veneer layers through mechanical interlocking. Our surface roughness measurements showed Ra of 1.2 to 1.5 μm and Rt of 9 to 12 μm for the In-Ceram Alumina discs. This surface roughness would result in interface roughness and defects in the porcelain layer after veneering. It is believed that while the interfacial roughness could transfer the high tensile stress from the In-Ceram Alumina layer to the porcelain layer at the interface, the interface defects would lower the strength of porcelain at the interface, and both would promote failure of porcelain at the interface. The increasing probability of interface failure for the layer thickness ratios from 1:2 to 1:1 to 2:1 shown in Fig. 7 corresponds to the shift of the maximum tension within the porcelain layer from the surface to the interface. The reported strengths for porcelains are 60 to 70 MPa [52]. The predicted maximum stresses on the tensile surfaces of porcelain layers at failure loads shown in Figs. 9 and 10 depend on the input material properties; however, they are in the vicinity of reported strengths. Finally, it should be noted that a common strategy for achieving natural esthetics of all-ceramic prostheses is to minimize the core to veneer thickness ratio to provide more depth of translucency. The results of the present study show that such a practice creates a relatively high tensile stress at the free surface of the veneering porcelain (Fig. 10) which is an undesirable situation.

5

Summary

For biaxial strength measurements, ASTM has selected ring-on-ring tests to establish C1499 for advanced ceramics [34], and the International Organization for Standardization has selected piston-on-three-ball tests to establish ISO 6872 for dentistryceramic materials [35]. However, monolayered specimens are considered in tests for both standards. Although Roark’s formulas [36] for bimetallic circular plates subjected to biaxial moment loading have been used to evaluate the biaxial strength of bilayered structures, it is not commonly recognized that Roark’s formulas are valid for a simply supported disc at its edge. Hence, to account for the overhang region of the disc for the actual loading configuration, modification of Roark’s formulas is required. This modification is given in this chapter. Also, Roark’s formulas give stresses only for the two outer surfaces of bilayer. Recently, Hsueh et al. showed that the rigorous closed-form solutions for multilayered discs subjected to biaxial flexure tests could be obtained by the following procedures. First, derive the correlation between monolayered and multilayered discs subjected to biaxial moments. Then, utilizing this correlation, the existing solutions for monolayers subjected to biaxial flexure tests can be converted to the solutions for multilayers. The rigorous

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closed-form solutions for multilayers subjected to ring-on-ring tests are presented in this chapter and validated by the FEA results, and the solutions for multilayers subjected to other biaxial flexure tests can be obtained using the same methodology. Specifically, formulas pertinent to multilayered discs subjected to piston-on-ring tests as well as comparison between results obtained from formulas and FEA for both monolayered and bilayered discs can be found in a previous publication [38]. Although Hsueh et al.’s rigorous formulas are closed-form solutions, the expressions are lengthy which may deter the engineers, who are less adept in mechanics and mathematics, from using them. Hence, Hsueh et al. also derived the simplified formulas [46], such that they could be used readily by engineers to evaluate the biaxial strength of multilayered dental ceramics using biaxial flexure tests. Despite the simplicity in deriving the simplified formulas, it is sufficiently accurate in comparing with the rigorous formulas and finite element results. Also, if the biaxial stresses on the top and the bottom surfaces of the disc can be measured during testing, the biaxial stress/strain through the entire thickness of the multilayered disc can be determined using Eqs. (30), (31), and (33) in the simplified model. However, if measurements of biaxial stresses on the top and the bottom surfaces of the disc are unattainable experimentally, they still can be derived by satisfying the force and moment equilibrium conditions, and the solutions are given by Eq. (35). A case study is also presented in this chapter to analyze the stress distribution through the thickness of In-Ceram Alumina/Vitadur Alpha porcelain bilayered dental ceramics subjected to both thermal stresses and ring-on-ring tests and to systematically examine how the individual layer thickness influences this stress distribution and the failure origin [41]. In this case, porcelain was placed in the tensile side for ring-on-ring tests. The calculated stress distributions show that the location of maximum tension during testing shifts from the porcelain surface to the In-Ceram Alumina/Vitadur Alpha porcelain interface depending upon the layer thickness ratio. This trend is in agreement with the experimental observations of the failure origins. The analytical results are also verified by the finite element analyses. It should be noted that only the flexural stress but not the contact stress is considered in Hsueh et al.’s analytical modeling. Although the loading mode of biaxial flexure tests is mainly the flexural stress, the contact stress is induced between the loading fixture and the disc. This contact stress influences the stress field at locations close to the contacts but has only little effects on the stress field in the central region of the disc, and compressive stress concentrations can be observed at the contacts in the simulated stress fields obtained from FEA. However, because only the maximum tension in each layer is required to evaluate the biaxial strength of the disc from tests, the inaccuracy of the analytical solutions due to neglect of the contact stress is not of concern. Acknowledgments The work was performed mainly at Oak Ridge National Laboratory and supported by US Department of Energy, Division of Materials Sciences and Engineering, Office of Basic Energy, under Contract No. DE-AC05-00OR22725 with UT-Battelle, LLC. This chapter was compiled at National Taiwan University and supported by the Ministry of Science and Technology, Taiwan, under Contract No. MOST 103-2221-E-002-076-MY3.

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Micromechanics of Dual-Phase Steels: Deformation, Damage, and Fatigue

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Behnam Anbarlooie, Javad Kadkhodapour, Hossein Hosseini Toudeshky, and Siegfried Schmauder

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Micro-deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Microstructure and Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Metallographical Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Interrupted SEM Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 In Situ SEM Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Micro-damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Shear Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Interface Debonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 X-Ray Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Micro-fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Low Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 High Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B. Anbarlooie · H. Hosseini Toudeshky Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran J. Kadkhodapour (*) Shahid Rajaee Teacher Training University, Tehran, Iran e-mail: [email protected] S. Schmauder Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_70

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Abstract

Ferritic–martensitic dual-phase (DP) steels are increasingly being used in various automotive components because of their favorable material behavior for lightweight and crash-safe designs. DP steels deform with strong strain and stress partitioning at the microscale. Deformation pattern of ferrite and martensite phases under tensile loading condition is the most important issue which can be effective on the prediction of mechanical behavior of DP steel. Deformation pattern and strain localization play an important role in the process of damage initiation and final fracture. Failure in DP steels is a phenomenon that has been extensively investigated in the last decade through experimental tests and simulation methods. Experimental procedures have shown that failure has a ductile pattern and that shear failure of ferrite matrix is dominant in these materials. On the other hand, experimental findings have shown that failure due to fatigue loading occurred with different pattern comparing to the other loading conditions. To understand and improve DP steels, it is important to identify connections between the microstructural parameters and the mechanical behavior of these materials at macroscale. This work provides a detailed micromechanical investigation of DP steels focusing on micro-deformation, micro-damage, and microfatigue analysis of DP steels based on experimental and numerical approaches to highlight the current and future directions and open problems about these materials. Keywords

Dual-phase steels · Deformation · Damage · Fatigue · Microscale

1

Introduction

Dual-phase (DP) steels were developed in the late 1970s and early 1980s in response to the demand for high-strength, highly formable steels [1–3]. In recent years, there has been resurgence of interest in dual-phase steels because the applications of these materials have made significant inroads, particularly in the automotive sector [4–6]. The automotive industries explore a material solution for lightweight and crash-safe designs. High-strength steels such as dual-phase steel and the transformationinduced plasticity (TRIP) steel have been produced to attain this aim. Hot rolling and cold rolling with subsequent annealing are common ways to produce DP and TRIP steels in the industry. DP steels require single-stage heat treatment in intercritical temperature range for ferrite formation followed by fast cooling to form the martensite phase, while TRIP steels can be obtained using a twostage heat treatment, namely, for ferrite formation and bainite formation. DP steels offer a two-phase microstructure, in which the hard martensite islands are located in the soft ferrite matrix [7, 8]. The microstructure and the final amount of ferrite and martensite in DP steel can be controlled by the holding time, intercritical temperature, and the cooling rate in the heat treatment cycle.

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Recently, researchers examined the microstructure of deformation in dual-phase steels under monotonic and complex loadings in order to better understand their macroscopic behavior. The key microstructural characteristics for DP steels are the amount, strength, and distribution of the martensite islands. Metallographical imaging, SEM analysis, and in situ SEM testing were recently carried out to observe the deformation field in dual-phase steels. In situ SEM investigations are valuable in assessing the strain partitioning between ferrite and martensite phases. There have been also numerous experimental investigations on the deformation and fracture behaviors of DP steels, and analytical models have been proposed for prediction of macro- and micromechanical properties of DP steels [7, 9, 10]. Some investigations performed finite element modeling using the real microstructure of DP steels to predict the deformations pattern and global mechanical response. For a successful modeling in microscale which leads to reasonable predicted results, an appropriate selection of representative volume element (RVE) plays an important role. In recent years, the obtained real microstructure from metallographic images containing ferrite and martensite phases with determined volume fraction has been used to generate the required RVEs for finite element analyses. 2D and 3D analyses have been performed to predict the stress–strain behavior of uniaxial tensile tests based on the plane stress conditions, and the results have been reported by Paul et al. [5] for different selected RVEs. Cohesive zone modeling (CZM), the brittle failure modeling using extended finite element method (XFEM), elastic–plastic analyses, and localization pattern are the main techniques and procedures which have been used to predict the mechanical behavior and damage in each phase. Uthaisangsuk et al. [11] applied the GursonTvergaard-Needleman (GTN) damage model and cohesive zone modeling to the RVE in order to describe damage propagation in the microstructure. In another attempt Vajragupta et al. [12] used XFEM and damage concept for martensite and ferrite modeling, respectively, and created a model coupled with the GTN damage model for ferrite and used a traction–separation law as the criterion for XFEM analyses in order to simulate the brittle fracture of the martensite. Also, Ramazani et al. [13] performed a quantitative identification of parameters for XFEM modeling of martensite cracking. In addition to cohesive zone and brittle damage modeling using XFEM method, numerical investigation based on localization pattern has been done by others. Recently, it has been found that failure or damage mode depends on the stress state in the material [14]. Stress and strain localizations lead to failure and could be a proper criterion for understanding damage mode in dual-phase steel [15]. Strain localization occurs in the ferrite matrix between tips of two closely martensite islands. With respect to the deformation localization, Katani et al. [16] showed that damage occurrence between ferrite-ferrite or ferrite-martensite grains caused void nucleation and growth. On the other hand, micromechanical modeling using the crystal plasticity approach has been extensively employed in simulating the mechanical response of materials. The crystal plasticity finite element method has now become a regular tool for studying the microscopic heterogeneity associated with the plastic deformation in

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metals. Crystal plasticity simulations were presented by many authors to predict the fracture and failure of alloys. The microstructure was analyzed, and damage mechanisms were experimentally derived as a function of strain. Void nucleation, growth, and coalescence were also characterized [17–19]. Various experimental fatigue behavior analyses of DP steels are also available in the literature. Sperle [20] showed that strain hardening and bake hardening increase the fatigue strength of DP steels. Some of the investigations also concentrated on the influence of martensite content volume fraction on fatigue behavior of DP steels. Wännman and Melander [21] reported the increased number of cyclic stress with increase in the martensite content according to the results of strain-controlled fatigue tests. The behavior of micro-cracks under fatigue loading differs substantially from the behavior of long cracks [22]. Crack initiation becomes important in high cycle fatigue, as it can amount to more than 90% of component life cycles [23], and the effect of microstructure on fatigue crack initiation and fatigue behavior of DP steels is an important issue that should be noticed to optimize microstructure morphology for extended fatigue lives. Even though numerous journal articles and conference papers exist in the literature on the effect of microstructure on mechanical properties of DP steels, it is still a favorite topic to the scientific community. This study is undertaken in the context of understanding the role of DP steel microstructure on the micro-deformation, microdamage, and micro-fatigue behavior of material via the experimental and numerical methods to concentrate on some of the pending key challenges in this field. The next section of this chapter contains the reviewing of microstructural deformation patterns of DP steels using metallographic, interrupted SEM, and in situ SEM analysis, followed by micro-damage investigation with respect to the localization, shear bands, interface debonding, and X-ray tomography. Finally, it is closed with a discussion on the micro-fatigue behavior of DP steels.

2

Micro-deformation

The local deformation field and its effect on the damage pattern have been the subject of many works. Even though dual-phase steels exhibit a macroscopically uniform and homogeneous deformation mode, from a micromechanical perspective, its plastic deformation is inherently inhomogeneous due to its nature of the grainlevel inhomogeneity [9–11].

2.1

Microstructure and Mechanical Properties

Appropriate proportion of the martensite and ferrite phases is the key factor for the favorable mechanical properties of DP steels. Martensite volume fraction, grain shape, grain size, carbon content, martensite–ferrite morphology, ferrite grain size, and ferrite texture are the dominant factors influencing the ductility of DP steels. Tremendous efforts have been performed to investigate the effects of these

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Fig. 1 Microstructure of the DP600 steel in two different magnifications [27]: (a) 200x and (b) 500x magnification

parameters on the mechanical behavior of DP steels. The effect of volume fraction of the martensite phase on the mechanical properties of DP steels has been investigated by different authors [24–26]. Increasing the volume fraction of the martensite phase was found to increase the yield and ultimate strengths of the aggregate. On the other hand, the size and dispersion of ferrite and martensite phases become more critical. Finer ferrite grain size and higher carbon content in martensite principally increase the strength of DP steels. However, these factors deteriorate their ductility. As a popular type of DP steels, the microstructure of DP600 steel provided by metallography is shown in Fig. 1. The pattern of the microstructure shows martensite islands of 4 μm average sizes which are evenly dispersed in the ferrite matrix [27]. Some martensite islands of larger size can be also observed in some locations in the two-phase microstructure. As shown in Fig. 1, the martensite islands are distributed in the ferrite matrix, and martensite volume fraction can be determined from the metallographical images. Rodriguez and Gutierrez [28, 29] reported the stress–strain relation for each phase, and nanohardness experiments have been carried out in order to estimate the accuracy of the predicted stress–strain behaviors given in (1). It has been already shown that the accuracy of the results is satisfactory and this method can be used to define the mechanical behaviors of ferrite and martensite phases. The stress–strain relation is expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1  expðM  k  ep Þ σ ¼ σ 0 þ Δσ þ α  M  μ  b kL σ 0 ¼ 77 þ 80%Mn þ 750%P þ 60%Si þ 80%Cu þ 45%Ni þ 60%Cr þ 11%Mo þ 5000Nss

(1)

(2)

where α is a constant (α = 0.33), M is the Taylor factor (M = 3), μ is the value of shear modulus (μ = 80 GPa), and b is the value of Burgers vector (b = 2.5  10e-10 m). L is the dislocation mean free path, k is the indication of recovery rate, and ep is the

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plastic strain. Δσ is the additional strengthening due to the precipitation and carbon in solution. The value of σ 0 takes into account the contribution of the lattice friction and the elements in solid solution. The expression for Δσ and the values for α, L, and k for different phases are listed in Table 1. In this table, dα is the ferrite mean linear intercept in meter, Css is the carbon in solution, and [C] is the martensite carbon content.

2.2

Experimental Works

The steel sheet samples with DP microstructures can be obtained through laboratory heat treatment. Before annealing, the steel sheets were pickled by a solution of 20% HCl in water at the temperature of 80  C. The sheet thickness was then reduced by cold rolling in a twin rolling mill. The thickness decrease was approximately according to a cold-rolled deformation degree of 65%. Subsequently, sheet specimens for tensile test were prepared parallel to the rolling direction [30]. The metallographic examination of the as-received steel showed a ferrite–pearlite microstructure with a grain size of about 13 μm, as illustrated in Fig. 2. A laboratory heat treatment should be performed for the prepared specimens in order to produce defined DP microstructures. The intercritical annealing process at different temperatures could be carried out for obtaining steel samples with DP microstructures containing different martensite phase fraction and corresponding ferrite grain sizes. The heat treatment experiments must be conducted in a salt bath. Table 1 Parameters defined in Eq. 1 Ferrite Martensite

Δσ 5000Css 3065[C]-161

Fig. 2 Microstructure of the as-received steel sheet [30]

α 0.33 0.33

L(m) dα 3.8  108

k 105/dα 41

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Intercritical annealing temperature

750

AC3 α+γ ACI(723)

α

0.09 25 0.0168

Temperature (°C)

Temperature (°C)

830

735-800°C,5 min

AC1 As-received Cold-rolled t= 1.2 mm

0.6327 Carbon content (wt%)

Water quenching

Time

Fig. 3 Overview of the applied intercritical treatment process [30]

The sheet samples were immersed in the salt bath at five different temperatures of 750, 770, 790, 810, and 830  C. These temperatures lay between the Ac1 and Ac3 temperature of the investigated steel so that the samples just had an austenitic–ferritic structure. The holding time after reaching each intercritical temperature was 5 min. Finally, the samples were directly quenched in water to room temperature, in which the phase transformation from austenite to martensite occurred. A route overview of the performed intercritical annealing process was presented with the Fe-C diagram in Fig. 3. The produced DP microstructures should exhibit the martensite phase fraction (MPF) of 25%, 35%, 45%, 60%, and 90% with regard to the defined intercritical annealing temperatures of 750, 770, 790, 810, and 830  C, respectively [30]. Some investigations on simple tension tests were carried out by researchers [18, 31–35] on dual-phase steels. After each tension test, Tasan et al. [36] performed indentation tests on the cross section of the specimen. They expected to observe a sudden drop in the hardness curve due to the damage near the failure surface. However, no noticeable drop in either nanohardness or microhardness was observed. They relate this phenomenon partly due to other mechanisms that are acting in parallel to the simple damage mechanism, such as strain hardening, grain shape change, texture development, residual stress, and indentation pileup.

2.3

Metallographical Images

Figure 4 shows the details of microstructures for deformed specimens. Figure 4a, b shows the microstructure of the specimen loaded up to the necking and different places in the necking area. Figure 4c, d also shows the microstructure of specimen at two different places in the specimen loaded very close to the failure point. Comparison of this figure with the microstructure in Fig. 1 shows that the dominant deformations in the microstructural level in these two figures are in the ferrite matrix. Also some minor deformations in the martensite grains can be observed. It has to be mentioned that in these two stages of deformations, a very limited number of voids are observed. It can be due to the standard production and quality of the commercial steel. If the steel was

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Fig. 4 Deformation pattern in the microstructure; (a) and (b) at strain value of 20% and different places, (c) and (d) at strain value of 28% and different places [27]

produced by a laboratory process [30], the rearrangement of grain boundaries could not be tolerated by the two phases only, and the void initiation in the grain boundaries might be observed. On the other hand, due to the severe deformation of phases and the rearrangement of grain boundaries, the microstructure tolerates a lot of deformation energy that causes the final failure of the material.

2.4

Interrupted SEM Analysis

Scanning electron microscopy (SEM) images of the specimens can be used to depict the void growth in the microstructure of DP steels. SEM microstructure analyses of DP steels [37, 38] revealed three distinctive mechanisms of void nucleation: cracking of martensite, decohesion at ferrite and martensite interfaces, and separation of

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Fig. 5 Investigation of failure process in tensile specimen of DP steel using SEM images, (a) diffuse necking, (b) void initiation, (c) void growth, and (d) ferrite and martensite interface debonding [9]

adjacent martensite regions. The majority of voids formed at ferrite and martensite interfaces were located between closely spaced martensite particles. The SEM analysis of void nucleation mechanisms were carried out on the DP steels by Kadkhodapour et al. [9]. Figure 5a–d shows some detailed inspection of void initiation. Figure 5a shows the top view of the specimen which is deformed to the diffuse necking point, and Fig. 5b–d is the SEM images from a section of that specimen. Figure 5b shows that some of the voids are nucleated in the ferrite–ferrite grain boundaries. This type of void initiation seems to take place always in the direct neighborhood of a martensite particle. Thus, stress concentration or deformation mismatch might be a reason. In ferrite–martensite grain boundary, two patterns of void initiation can be observed. The first pattern which is shown by arrow in Fig. 5c is similar to crack propagation. It seems that in this region the initial void forms in ferrite–ferrite grain boundary and then propagates to the ferrite–martensite grain boundary like a crack. The second pattern, which is indicated by the rectangle in Fig. 5c and magnified in Fig. 5d, can be named normal separation of ferrite–martensite grain boundary.

2.5

In Situ SEM Analysis

The in situ tensile testing is an almost accurate method to investigate the strain localization in ferrite and martensite phases. Avramovic-Cingara et al. [37]

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Fig. 6 In situ SEM: void growth events at localized necking. The arrows indicate shear bands: (a) void initiation and (b) void growth [37]

performed quantitative measurements to determine the rate of ductile damage accumulation during in situ SEM tests (Fig. 6). They reported that rate of development of the void population with strain was initially low and then increased at large strains. They observed that voids mostly initiated on the interfaces normal to the tensile axis and grew along ferrite grain boundaries. The development of localized deformation in the specimen was analyzed from the in situ SEM images obtained during the tensile test. The plastic strain partitioning between the two phases and the difference in local deformation among the ferrite grains has been the subject of ongoing research and discussion [19, 39, 40]. According to the tensile in situ observations in the literatures [41], local slip bands could be observed in three different sites in the ferrite phase: (1) in the center region of large ferrite grains, (2) between the martensite islands, and (3) near the cracked martensite grains. It has been shown that before the final failure, most of the applied deformations are imposed into the deformation bands. In this condition, the microscopic deformational features of the microstructure are dominantly controlled by growth and coalescence of deformation bands. This phenomenon occurs in high deformational fields inside the necked zone of the specimen. In contrast, the effect of dislocation density and martensite–ferrite grain size is more pronounced in relatively low strains [33]. The ferrite strain localization between the martensite islands was reported using the real microstructure of Fig. 7. The applied strain was 13%, as determined from the experimental analysis [42]. Figure 7a shows the undeformed microstructure. The local martensite volume fraction was 43%, and the microstructure contained both equiaxed and elongated martensite grains. No crack or void was evident in the ferrite phase. Figure 7b shows the deformed microstructure before the macroscopic necking, where the applied strain was 13% in vertical direction. The distribution of microscopic strain in the loading direction was calculated using the image processing computational code, and the strain maps were superimposed onto the SEM images of the deformed microstructure. The strain localization in the ferrite phase

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Fig. 7 Ferrite strain localization between the martensite islands: (a) initial microstructure, (b) deformed shape of microstructure (applied strain is 13%), and (c) superimposed microscopic strain map onto the deformed microstructure [42]

trapped among the martensite grains could be observed to be considerably higher than that in the other regions. For example, the microscopic strain value in the region A (110%) was almost eight times higher than the applied macroscopic strain (13%). There are also some areas with very low strain values in the microstructure. From Fig. 7c it can be seen that most of the low strain regions are located in parallel with elongated vertical martensite island (correspondent to the loading direction). This could be attributed to the difference in the amount of elongation in the two phases of the material (martensite and ferrite) under the uniaxial tensile loading.

2.6

Numerical Investigations

In micromechanical modeling, chemical composition and volume fraction and morphology of phases and grain size are taken into account. Different modeling scales and methods for micromechanical modeling of dual-phase steels are summarized by Tasan et al. [43]. The representative volume element (RVE) is a popular technique for micromechanical modeling of materials. A basic RVE can be defined using a simple cell model or it can be more complicated using an actual microstructure. A finite element model for the DP steels consisting of martensite particles with two different sizes dispersed in a ferrite matrix has been developed by Al-Abbasi and Nemes (Fig. 8). They assumed different particle sizes and distinguishes two mechanisms of deformations taking place side by side in the process of tensile straining [44]. The trend of increasing the strength and decreasing the uniform strain with increasing the volume fraction of martensite was also reported by them. The local failure mode was closely related to the stress state in the material. 2D and 3D RVEs can also be generated based on real microstructures. In the case of dual-phase steels, martensite and ferrite can be clearly distinguished due to the

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Fig. 8 Axisymmetric twoparticle model RVE idealization, showing the homogeneous dispersion of small and large particles [44]

a

Martensite

Ferrite

b

Rolling direction

Ferrite Martensite Rolling direction

52 μm Y Z

X

Fig. 9 (a) An actual microstructure of DP 980 and (b) the corresponding finite element model used for the RVE of DP 980 [45]

color contrast between ferrite matrix and martensite grains in a micrograph. Sun et al. [45] modeled the DP steel under plane stress conditions with different form of boundary conditions and showed that the shear dominant failure mode or split mode perpendicular to the loading direction is the final failure of 2D RVE (Fig. 9). A good agreement between the predicted failure mode by FEM and experimental results was also observed for the uniaxial tensile specimen. The simplest assumption for generating a 3D RVE of a two-phase material is based on Mori–Tanaka’s approach [46]. In this approach, the second-phase particles are considered as inclusions that are distributed in a matrix representing the first phase. As can be seen in Fig. 10, Uthaisangsuk et al. [11], Paul [47], and

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a

RVE

10% Martensite

b

90% Ferrite

c

Ferrite Femte Martensite

Martensite

Fig. 10 3D RVEs based on real microstructures generated by (a) Uthaisangsuk et al. [11], (b) Paul [47], and (c) Ramazani et al. [48]

Fig. 11 RVEs based on real microstructures and the martensite islands were modeled as ellipsoids: (a) RVE including 11 martensite particles and (b) RVE including 26 martensite particles [49]

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Ramazani et al. [48] generated cubic RVEs of dual-phase steels. They considered ferrite as the matrix and martensite as inclusions. The numbers of ferrite and martensite cubes were determined according to their volume fraction in the dualphase steels. In another work [49], both the volume fraction and the morphology of martensite were considered to generate a 3D RVE that is more representative of the actual microstructure. Martensite islands were modeled as ellipsoids, and the aspect ratio and size distribution of martensite islands were determined according to the quantitative metallography results. 3D RVEs with oval martensite islands are depicted in Fig. 11.

3

Micro-damage

Damage mechanisms are classically used to explain the failure mechanism in metals and more specifically in steel. For ferritic steel, in which the martensite grains do not exist, the process of failure is explained in the following way. Ceramic inclusions are often weakly bonded and initiate voids easily. As these voids grow, straining of the inter-void material becomes intense, which may lead to breakage or decohesion of the strong and well-bonded carbide particles. However, in modern steels the volume fraction of ceramic inclusions has decreased significantly so that their effect on the final failure is considered to be very low [50–52]. By micromechanical modeling of DP steel, different possibilities for describing damage and failure in microstructure should be considered, as several failure mechanisms occur simultaneously. The ductile failure of ferrite, brittle fracture of martensite, and interface debonding between phases could be observed regularly when the failure behavior of DP steel was studied [12]. Therefore, an appropriate damage modeling technique should be implemented.

3.1

Localization

Strain localization can be considered as the early stage of failure and fracture. Determination of the strain localization is quite helpful to understand the fracture modes in DP steels. The prediction of ductile failure of DP steels in the form of plastic strain localization resulting from the incompatible deformation between the harder martensite phase and the softer ferrite matrix was reported. The strain localization occurs between the tips of two closely located martensite particles. The stresses in the sandwiched ferritic phase are larger than the other region stresses in the uniaxial loading; thus there is more chance of localization in this area. The relative deformation of martensite grains causes high deformation localization in the ferrite matrix and plays an important role in the final failure of the material. The failure of boundaries between the ferrite–ferrite or ferrite–martensite grains initiates the void nucleation and growth. The slip of the martensite grains on each other causes severe deformation of the boundaries between them, which results in the accelerated failure of the material [9]. The stress and strain localization is seen in Fig. 12.

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Micromechanics of Dual-Phase Steels: Deformation, Damage, and Fatigue

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a

9.47 4.00 3.60 3.20 2.80 2.40 2.00 1.60 1.20 0.80 0.40 0.00 –0.40 –0.80 –0.84

b

2424 2000 1667 1333 1000 667 333 0 –333 –667 –1000 –1333 –1667 –2000 –3348

Fig. 12 Stress and strain distribution in ferrite matrix: (a) strain localization and (b) stress localization [9]

3.2

Shear Bands

Propagation of ductile cracks occurred by the coalescence of voids along shear bands [37]. Shear bands were formed inside ferrite grains, while martensite was hardly deformed, and the deformation was localized in the ferrite matrix [10]. A more detailed pattern of shear band propagation is shown in Fig. 13. As depicted in this figure, shear cracks propagated parallel to shear band directions. It is shown in Fig. 13 that the crack propagation occurred in three important regions as follows [53]: • Micro-crack initiation started between the tips of two closely located martensite particles. The stress in the sandwiched ferritic phase was usually higher than in other regions in the uniaxial loading case, and there was a higher chance of

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a

b

c

Fig. 13 Micro-crack (and void) nucleation and propagation during load increase: (a) ε = 12%, (b) ε = 15%, and (c) ε = 18% [53]

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localization in these regions. As can be observed in Fig. 13a, the strain localization started in the ferrite matrix that was sandwiched by two martensite islands. • Martensite underwent plastic deformation when development of shear bands happened in the ferrite phase. Micro-cracks propagate in martensite islands in localized plastic strain. Localized shear caused splitting of martensite grains in regions with less thickness. In Fig. 13b, it is observed that shear failure was caused by localization fracture of the martensite grains in the narrow regions. High deformation created further relative displacement for martensite grains, which led to additional plastic strain localization. Due to changes in relative position of martensite grains, the ferrite matrix experienced severe deformation at some points. These points were incapable of load bearing in final stages of loading and had a rupture-like failure. • Shear failure and primary voids occurred in the ferrite matrix in the vicinity of martensite islands (Fig. 13b, c). Propagation of cracks occurred by coalescence of these voids. The voids coalesced along the shear bands and led to final fracture.

3.3

Interface Debonding

According to the performed investigations, the void nucleation could occur at the interfaces between ferrite and martensite phases [54]. On the other hand, using the cohesive elements in the finite element analysis provides the capability to predict the voids and micro-crack locations along the ferrite and martensite interfaces in a 2D finite element model with respect to the separation and stress values in the cohesive elements. Therefore, the finite element meshes should be compatible with the real microstructure because of the ferrite and martensite interface importance. Initiation of voids creation can be predicted by the damage initiation and propagation capability of CZM interface elements. Figure 14 shows the damage and debonding occurrence at ferrite and martensite interfaces. Comparison of the predicted results with the observed voids from the experiments shows that the void initiation at ferrite and martensite interfaces has been acceptably well predicted.

3.4

X-Ray Tomography

X-ray microtomography can now be routinely used for the imaging and quantification of damage in ductile materials [55–57]. Recently, Maire et al. [58] carried out in situ tensile tests on a dual-phase steel specimen. In addition, using X-ray tomography, he followed and quantified the evolution of damage nondestructively in three dimensions. He concluded from micrographs and through other types of observation that damage is never observed in the ferrite phase. Also he observed

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Fig. 14 Damage and debonding predictions at grain boundaries and compatibility with voids in the experimental results: (a) SEM image and (b) finite element result [54]

both martensitic fracture and ferrite/martensite decohesion during the test. By X-ray tomography, he carried out some measurements on the process of void initiation and void growth in the material. Hence, there is need for the development of proper modeling tools to account for the evolution of damage. The existing models for damage evolution were mainly developed to predict the growth of cavities in a ductile matrix. The simplest and oldest one is the uncoupled Rice and Tracey (RT) approach [59]. It has been shown by different authors [56, 60] that X-ray absorption microtomography is the most reliable way to obtain quantitative three-dimensional (3D) information on damage. Using an in situ tensile test during X-ray tomography experiments on a highstrength steel, it has been shown by Maire et al. [58] that it is possible to qualify and quantify damage in three dimensions in the bulk of steels. The quantification of the observations led to the determination of some important parameters for damage. It was possible to measure not only the evolution of the density and size of the cavities but also the local strain and triaxiality. The main finding of this quantification is the quasi-stagnation of the value of the average equivalent diameter of the cavities. Figure 15a, b shows the typical evolution of a reconstructed vertical section of the sample. The sections are extracted from the tomographic 3D block at tensile states just after necking (a) and just before fracture (b), respectively. They are oriented parallel to the tensile axis, which is vertical in the figure. The figure shows clearly that a high number of cavities nucleate, grow, and sometimes coalesce. Figure 16a, b shows a 3D rendering of the outer surface of the cavities for the initial state (a) and the state just before fracture (b, same as in Fig. 15b). In this figure, the matrix is transparent, and only the outer surface of the sample is visualized in a semitransparent mode. As is often the case for this 3D rendering mode, adding in the same view the total population inside the volume of the sample makes the amount of damage appear to be enormous.

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Fig. 15 Tomographic slices parallel to the tensile axis at the steps: (a) just after necking and (b) just before fracture [58]

Fig. 16 3D representation of the population of cavities inside the deforming sample in its initial state (a) and just before fracture (b, same as in Fig. 14b) [58]

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4

Micro-fatigue

4.1

Low Cycle Fatigue

The case of low cycle fatigue (LCF) occurs when the loading is large enough to produce plastic strain of the order of magnitude of the elastic strain. This means that the number of cycles to failure is relatively low. An order of magnitude is less than 10,000 cycles. For mechanical components subjected to cyclic loading leading to plastic strain, most materials exhibit the phenomenon of either mean stress relaxation or strain ratchetting, or a combination of the two, depending on the applied load and structure geometry (Fig. 17) [61]. Cyclic ratcheting occurs if the strain cycle is open, e.g., in stress-controlled cyclic deformation. It is the result of net plastic strain accumulation in a given direction with each cycle. Large strain accumulation leads to the ductile failure of the material by void growth and coalescence, when the total accumulated plastic strain reaches a critical value that is comparable to the strain to failure in monotonic tension [62]. Mean stress relaxation under strain-controlled cyclic loading with nonzero mean strain is the observation of the same mechanisms as ratcheting. Mean stress can strongly influence the life for crack initiation in fatigue loading of engineering materials. In addition to the significance of mean stress in fatigue life estimation, the instability of mean stress is a key factor to calculate the fatigue life of mechanical components subjected to high degree of plasticity during cyclic loading. When a cyclic-biased deformation is applied in a material, mean stress tends to relax to a stable value. The rate of relaxation, the amount of relaxation, and the stable values

Fig. 17 Elastoplastic deformation behavior subjected to (a) constant cyclic strains, (b) constant cyclic stresses, and (c) remote constant nominal cyclic stresses [61]

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Micromechanics of Dual-Phase Steels: Deformation, Damage, and Fatigue

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are all dependent on the loading conditions, the component geometry, and the material [63].

4.1.1 Dual-Phase Steel Under Low Cycle Fatigue One major difference between DP microstructures and other steel grades is the mixture of hard and soft phases. It has been shown in low cycle fatigue that strain partitioning can occur, where strain is concentrated in the softer phase [64]. Cyclic plastic deformation is an interesting consideration for multiphase steels such as DP, as the accommodations of the plastic strain between phases influence the fatigue properties [65]. Influence of microstructure and annealing conditions on straincontrolled LCF response of DP steels has been studied, and it has been found that LCF behavior of these materials was sensitive to the microstructure of the materials [66]. Figure 18a shows measured LCF life of the specimens. Each curve refers to the response of a single specimen subjected to the half of the total strain range given. In all of the specimens regardless of strain amplitude, hardening occurs for most of the fatigue life followed by a rapid drop in cyclic stress prior to failure. The cyclic response curves show three stages, firstly a fast hardening during the first few cycles. This is followed by a gradual hardening for most of the fatigue life and finally a rapid drop in stress carrying capability prior to failure. Figure 18b shows the effect of cyclic hardening on rising stress amplitude with increasing number of cycles at the first 50 cycles of each specimen. This figure shows that the rate of hardening at the first stage depends strongly on the level of the applied total strain range [67]. Figure 19a, b shows photographs taken by TEM from interrupted (at 20% of their life) LCF specimens at high and low strain amplitudes (0.1 and 0.002), respectively. These figures show damage initiation by fractured martensite particles at the high strain amplitude and by separation of ferrite and martensite at their interface for the low strain amplitude. Examination of the gauge surfaces of the interrupted LCF specimens indicates that the damage almost always started from martensite phase or at its vicinity. During LCF tests, cyclic deformation plays an important role in crack nucleation and growth. Irreversible deformation of the materials is strongly influenced by variables such as alloy composition, microstructure, heat treatment, and loading conditions. Damage nucleation occurs at second-phase particles, inclusions, and inhomogeneities and is controlled by a stress or strain property which has to be exceeded in order to form damages. During cyclic straining, plastic deformation occurs in the ferrite areas, and hard martensite particles resist plastic deformation, leading to inhomogeneous deformation. The final failure of the material occurs by linking the cracks or voids and coalescence.

4.2

High Cycle Fatigue

When a material is loaded with lower values for stress, the plastic strain at the mesolevel remains small and is often negligible. It may be high at some points at the

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Fig. 18 (a) LCF response and fatigue life of the specimens and (b) cyclic hardening behavior of the specimens under different strain amplitudes at the first 50 cycles [67]

microlevel where transgranular micro-cracking occurs only on some planes and most often at the surface of the specimen by the mechanism of intrusion–extrusion. The number of cycles to failure may be very large (more than 10,000 cycles). Note that for brittle damage and high cycle fatigue (HCF) damage, a stress–strain curve obtained from a classical tension–compression test at the mesoscale usually does not represent the true behavior for strain and damage because the space localization induces microplastic and damage zones much smaller than those of the specimens. Nevertheless, it is used because mechanical experiments at the microscale are difficult to perform. The microhardness test may help to characterize a micro-volume as it involves a size of the order of microns, but its state of stress is complex [68].

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Micromechanics of Dual-Phase Steels: Deformation, Damage, and Fatigue

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Fig. 19 (a) TEM photograph taken from a specimen tested at strain amplitude of 0.1 and interrupted at 20% of its lifetime and (b) TEM photograph taken from a specimen tested at strain amplitude of 0.002 and interrupted at 20% of its lifetime [67]

Subcritical crack growth can occur at stress intensity K levels generally far less than the fracture toughness Kc in any metallic alloy when cyclic loading is applied. In simplified concept, it is the accumulation of damage from the cyclic plastic deformation in the plastic zone at the crack tip that accounts for the intrinsic mechanism of fatigue crack growth at K levels below Kc. The process of fatigue failure itself consists of several distinct processes involving initial cyclic damage (cyclic hardening or softening), formation of an initial fatal flaw (crack initiation), macroscopic propagation of this flaw (crack growth), and final catastrophic failure or instability [69]. The general nature of fatigue crack growth in metallic materials and its description using fracture mechanics can be briefly summarized by the schematic diagram in Fig. 20 showing the variation in fatigue crack growth rates with the nominal stressintensity range [70].

4.2.1 Dual-Phase Steel Under High Cycle Fatigue The concept of microplasticity within individual grains is introduced as a key driving force parameter for fatigue crack formation. From a top-down perspective, the methodology of quantifying the effects of microstructure morphology on the distribution of fatigue response is a logical component of microstructure-sensitive design and prognosis strategies. Figure 21 provides a philosophical construct for using computational micromechanics to relate variation of microstructure to variability in fatigue response (or properties) [71]. Crack initiation and the subsequent propagation behavior are strongly affected by the microstructure of materials. In ductile materials, cracks initiate along slip bands in a grain, or at grain boundaries on the surface. Crack growth immediately after initiation is often blocked at the grain boundaries so that propagation rates of short cracks vary between rather high and very low values [72]. Because of this complex

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Fig. 20 Schematic illustration of the typical variation in fatigue crack growth rates da/dN, as a function of the applied stress-intensity range ΔK in metallic materials [70]

Fig. 21 Fatigue microstructure-sensitive simulation for desired failure probability [71]

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Fig. 22 Population of crack initiation sites [75]

behavior, linear elastic fracture mechanics is of only limited usefulness for small cracks [73]. Experimental studies on small cracks are usually based on surface observation. Small cracks, however, propagate in depth direction as well as on the surface, so subsurface observations are also necessary to fully understand the small crack behavior [74]. Numerous cracks were observed on the dual-phase steel specimen surface caused by fatigue loading. Crack initiation occurred in three different types of sites: in ferrite grains, at martensite–ferrite interphase boundaries, and at ferrite–ferrite grain boundaries. No cracks were initiated in any of the martensite grains, but some cracks grew into martensites after initiating at the above sites. Figure 22 shows that the most common initiation site was in the ferrite grains; the second was at the martensite–ferrite boundaries. Figure 23 shows development of the number of crack initiation points at each site. After 5000 cycles, most cracks were initiated at the martensite–ferrite boundaries, whereas the transgranular crack initiation in the ferrite grains was just about to start. The density of the transgranular initiation sites increased gradually until the end of the test, while the increase of initiation at the other two sites slowed down after 10,000 cycles. Initiation occurred rarely at martensite–ferrite boundaries and ferrite–ferrite boundaries after 50,000 cycles [75].

5

Summary

In this work, microstructure-based studies are used to investigate the deformation, damage, and fatigue behavior of DP steels at the microscale. Martensite volume fraction, grain shape, grain size, carbon content, martensite–ferrite morphology, ferrite grain size, and ferrite texture are the dominant factors influencing the ductility

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Fig. 23 Development of the density of cracks initiated in a ferrite grain, at a martensite–ferrite phase boundary, or at a ferrite–ferrite grain boundary [75]

of DP steels. SEM images of the specimens can be used to depict the deformation pattern and void growth in the microstructure of DP steels. SEM microstructural analyses of DP steels revealed different mechanisms of void nucleation. The in situ tensile testing is an almost accurate method to investigate the strain localization in ferrite and martensite phases. In micromechanical modeling, results can be affected by chemical composition, volume fraction, and morphology of phases and grain size. 2D and 3D RVEs can also be generated based on real microstructures. By micromechanical modeling of DP steel, different possibilities for describing damage and failure in microstructure should be considered, as several failure mechanisms occur simultaneously. The ductile failure of ferrite, brittle fracture of martensite, and interface debonding between phases could be observed regularly when the failure behavior of DP steel was studied. Strain localization can be considered as the early stage of failure and fracture. Determination of the strain localization is quite helpful to understand the fracture modes in DP steels. Propagation of ductile cracks occurred by the coalescence of voids along shear bands. Shear bands were formed inside ferrite matrix, while martensite grains were hardly deformed, and the deformation was localized in the ferrite matrix. X-ray microtomography can now be routinely used for the imaging and quantification of damage in ductile materials. Moreover, it has been shown in low cycle fatigue that strain partitioning can occur, where strain is concentrated in the softer phase. During cyclic straining, plastic deformation occurs in the ferrite areas, and hard martensite particles resist plastic deformation, leading to inhomogeneous deformation. The final failure of the material occurs by linking the cracks or voids and coalescence. Also, numerous cracks were observed on the dual-phase steel specimen surface caused by high cycle fatigue loading. Crack initiation occurred in three different types of sites: in ferrite grains, at martensite–ferrite interphase

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boundaries, and at ferrite–ferrite grain boundaries. No cracks were initiated in any of the martensite grains, but some cracks grew into martensite grains after initiating at the above sites.

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Defect Accumulation in Nanoporous Wear-Resistant Coatings Under Collective Recrystallization: Simulation by Hybrid Cellular Automaton Method

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Dmitry D. Moiseenko, Pavel V. Maksimov, Sergey V. Panin, Dmitriy S. Babich, and Victor E. Panin

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Cellular Automata Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Heat Transfer Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Estimation of Heat Transfer Simulation Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Modeling of Heat Transfer in Porous Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Mechanical Energy Transfer Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Rotational Deformation Modes and Defect Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D. D. Moiseenko (*) · P. V. Maksimov Laboratory of Physical Mesomechanics and Non-destructive testing, Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, Russia e-mail: [email protected]; [email protected] S. V. Panin National Research Tomsk Polytechnic University, Tomsk, Russia e-mail: [email protected] D. S. Babich Tomsk State University, Tomsk, Russia e-mail: [email protected] V. E. Panin Laboratory of Physical Mesomechanics and Non-destructive testing, Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, Russia National Research Tomsk Polytechnic University, Tomsk, Russia National Research Tomsk State University, Tomsk, Russia e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_72

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Abstract

A modification of a multiscale hybrid discrete-continual approach of excitable cellular automata is developed. The new version of the method is accomplished by considering the porosity and nanocrystalline structure of a material and the algorithms of calculation of local force moments and angular velocities of microscale rotations. The excitable cellular automata method was used to carry out numerical experiment (NE) for heating of continuous and nanoporous specimens with nanocrystalline TiAlC coatings. The numerical experiments have shown that nanoporosity allows to substantially reduce the rate of collective crystallization. In so doing the nanoporosity slowed down propagation of the heat front in the specimens. This fact can play both positive and negative roles at deposition of the coatings and their further use. On the one hand, by slowing the heat front propagation, one can significantly reduce the level of thermal stresses in deeper layers of the material. On the other hand, such deceleration in case of the high value of the thermal expansion coefficient can give rise to the formation of large gradients of thermal stress, which initiate nucleation and rapid growth of a main crack.

1

Introduction

Currently, composite coatings on the basis of MAX phases with multifunctional properties (tribological, heat-proof, etc.) become more and more relevant [1–7]. This new class of materials is characterized by unique performance combining the benefits of both metallic alloys and ceramic materials. On the one hand, such compounds demonstrate high heat conductivity and thermal resistance. On the other hand, they feature high elastic moduli, low thermal expansion coefficient, high heat resistance, and extreme high-temperature strength. Such materials are highly resistant to cycle loads and high temperatures even above 1000
C. Also, in this temperature range, their resistance to cycle loads leaves behind the majority of known heat-proof and heat-stable materials, including the intermetallide-based alloys [3]. The practical interest to new materials of this kind is primarily related to new requirements to critical units of aircrafts. In combination with low density of the composite, the unique characteristics of MAX phases unveil a wide range of their application in space and aircraft engineering. For instance, the implementation of heat-stable Ti-Al-based material with protective coating makes it possible to increase up to 50% of the lift force-to-weight ratio in comparison with the best Ni-based analogues [6]. However, the design of new classes of wear-resistant materials operating under extremely high-temperature gradients (~1000 K) demands for detailed study of both initial microstructure of newly developed coatings and understanding of their structure evolution under thermal and mechanical loads. For example, the melting temperature of γ-TiAl phase is about 1750 K, which limits the application of such intermetallides under extreme thermal cycling.

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Recent years have witnessed the works on modeling of either behavior of wearresistant and heat-proof coatings under mechanical loading [8, 9] or the process of their design [10, 11]. The majority of them are dedicated to the investigation of titanium-aluminum alloys. However, to the best of our knowledge, the results of detailed studies of the mesostructure changing of the Ti-Al-C coatings under thermal cycling at temperatures close to melting ones are not available. The capability of collective crystallization as well as its impact on redistribution of internal thermal stresses within the mesoscopic structure is also far from being studied thoroughly. In this concern the aim of the paper was to carry out computer modeling to study collective crystallization of nanostructured highly porous Ti-Al-C coating, as well as the effect of dynamic structure rearrangement on distribution of thermal stresses. In the present work, the authors refer to the principles of physical mesomechanics providing that under scale invariability assumption, a solid underloading might be treated as a hierarchically organized system [12]. Such 3D system represents the combination of interfaces. The latter are independent 2D systems where rotary nature wave fluxes of mass and energy propagate [13]. Figure 1 depicts the scheme that characterizes interactions of 2D planar subsystem (all interface and surface layer) and 3D subsystem (the crystalline lattice). In the physical mesomechanics, it is postulated that a solid (being a multilevel hierarchical structure) subjected to extreme loading conditions (mechanical, thermal, radiative, and other forces) undergoes structural phase transformations at each scale which result from the influence of fluxes and flows of various nature. These fluxes and flows can be divided into two groups: (i) fluxes of substances (particles of matter) and (ii) flows of energy. The first group includes mass transfer fluxes (diffusion, convective and advective transfer, etc.). The second group is broader and more diverse; it comprises of mechanical energy transfer in an acoustic wave, heat transfer, electric current, etc. As was shown in [14], earlier theoretically predicted in [15–17], and experimentally found in [18–22], the localized waves of plastic deformation result from complex self-organization of energy flows and local structural transformations in the hierarchy of structural scales. It should be emphasized that according to the modern quantum-mechanical concept of matter, each elementary particle is a carrier of energy quantum. Fig. 1 Scheme of interaction between 2D planar subsystem (interfaces) and 3D subsystem (crystalline lattice)

3D-crystal •

E high energy region

defects flows 3D-crystal

2D-interface with generated defects region with high density of defects

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Consequently, the above-stated grouping of fluxes and flows is quite relevant, and the choice of types that govern a given flow process depends on the specificity of the problem stated and scale level of the process. The construction of an adequate mathematical model of the transfer process, which would take into account the hierarchy of scales in structural phase transformations, is an acute problem in solid-state theory. The approach of this kind is developed in physical mesomechanics [12]. In doing so, it considers deformation of a loaded solid as a multilevel process associated with shear stability loss at different structural scales. An essential role in the multilevel self-organization of structural phase transformations belongs to all types of interfaces in a loaded solid, where normal and tangential tensile and compressive stresses are distributed in a chessboard-like manner. Further development within physical mesomechanics allowed researchers to describe (i) the formation of the chessboard-like distribution of opposite-sign normal and tangential stresses on surfaces and interfaces in a loaded solid, (ii) propagation of single and double spiral waves along the surface, (iii) quasi-periodic cracking and delamination of protective and thermal barrier coatings, etc. A fundamental result of physical mesomechanics is the theoretically and experimentally proven concept stated that nonlinear waves of localized plastic flow arise due to self-organization of flows of nanoscale local structural transformations which are channeled by shear banding at the mesoscale and by accommodation kink bands at the macroscale [14, 15]. The concept provides a methodological basis for describing the self-organization of local structural transformations in a solid underloading at various structural scales as well as for constructing multiscale models of deformation and fracture of solids. Continuum mechanics, being a good tool for the solution of applied problems in the elasticity theory at the macroscale, was not initially developed for the description of a media possessing the structure. There are a number of factors that complicate the application of continuum mechanics methods for computer-aided simulation of the dynamics of internal structure evolution/degradation. One of the reasons is related to the fact that in such problems, one must consider both (i) the presence of internal structure affecting energy transfer and (ii) dynamic rearrangement of the structure and interfaces. As a rule, currently available methods of continuum mechanics take into account the internal structure by specifying the so-called nonlinearity of properties, i.e., the scatter of parameters of the simulated medium over the space [23]. However, when solving physical mesomechanics problems, it is necessary not only to explicitly account for the internal structure of the material by assigning various properties to individual elements (as it is conventionally done in numerical methods of continuum mechanics) but also to account for a hierarchical interaction between different structural elements and for the transformation of the initial structure during plastic deformation of a solid. Classical continuum mechanics does not inherently consider the internal structure of a medium. Consequently, it employs artificial and often very complicated mathematical tricks to imitate the presence of structure.

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Since it is impossible to explicitly define the specific character of interaction between individual elements of a medium, continuum mechanics employs the variation laws of the element properties which are, as a rule, derived from experimental data on the macroscopic material behavior. For example, the law of stress variation depending on the strain of each local element of a medium is specified by introducing a response function being analogous to the macroscopic “σ–ε”diagram, while such parameters as yield stress (used in a model by taking into account strain hardening) are derived from macroscopic statistical laws (e.g., the Hall-Petch law). Therefore, to simulate deformation of materials with heterogeneous structure (tending to multiscale structural transformation) along with classical continuum mechanics methods, one should use approaches that allow explicitly specifying both (i) the structure of the medium under simulation and (ii) laws of interaction between structural elements. These are so-called discrete methods, among which molecular dynamics and cellular automata methods are the most well-known [24–32].

2

The Cellular Automata Approach

Numerical simulation on the basis of cellular automata methods is performed in the following steps: 1. Specification of the internal structure geometry of a specimen 2. Specification of the distribution of the material’s local properties for each automaton over the entire volume (e.g., spread in values of strength, elastic moduli, degradation rate of the material, dislocation density, and so on) 3. Specification of relations between elements 4. Specification of a set of possible states of elements and formulation of laws of their switching 5. Numerical implementation of the procedures to simulate redistribution and propagation dynamics of disturbances between elements under the action of external forces In the stochastic excitable cellular automata (SECA) method, the specimen under simulation is divided into elementary volumes when each one is represented by the active element of the cellular automaton. Every active element can run through the set of its states defined by certain logical rules. Apart from the state, each element has parameters corresponding to the simulated volume of the space. Cellular automata are divided into three main types: bistable, excitable, and self-oscillating. The active element of the bistable automaton can take one of two possible states. The active element of the excitable automaton may sequentially switch its states under applying external energy flux. The active elements of the self-oscillating automata also can take a number of possible states and change them when external energy flux is not applied. Based on the above-stated peculiarities of simulated processes that imply thermal and mechanical energy distribution in the material, the excitable

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cellular automaton was chosen as a simulation tool. Each automaton of this type has a number of neighbors within the first coordination sphere as well as numerical parameters corresponding to the material contained in the simulated volume of space, such as elastic modulus, density, shear modulus, dislocation density, thermal conductivity, specific heat capacity, thermal expansion coefficient, etc. During the interaction with neighboring automata, the thermal and mechanical components of energy and hence related physical parameters (temperature, entropy, stress, strain, density, etc.) may change. Depending on the specificity of the simulated process, such parameters of an automaton are chosen whose change may lead to switching of its state. For example, in the case of thermal processes, such states can be responsible for aggregate states of matter characterized by particular temperature intervals and values of heat capacity, heat conductivity, density, etc. When simulating processes associated with the inflow and redistribution of mechanical energy, it is worthwhile to relate the automaton state with the stages being passed through at loading of a solid: elastic deformation, inelastic deformation, prefracture, and fracture. In doing so, every stage can be defined by particular values of such automaton parameters: density, elastic modulus, shear modulus, etc. The switching between states can be deterministic or stochastic. Depending on the type of switching, the automata are divided into deterministic and stochastic ones. In the proposed SECA method, a cellular automaton is stochastic, i.e., a set of probability values for switching to a corresponding state corresponds to each value of a studied parameter. An important factor at simulation is related to the type of the cellular automata packing. Each packing type has its own set of energy propagation directions and number of neighbors in each coordination sphere. Generally speaking, the type of the automata packing does not necessarily coincide with the type of the atomic packing in the crystal lattice of the simulated material. In the model under consideration, either a simple cubic or hexagonal close packing of automata is to be chosen. The packing pattern depends on the peculiarities of the simulated process. For example, when solving an indentation problem, it is reasonable to choose a hexagonal close packing, since it provides a more accurate simulation of the propagation of a hemispherical front of disturbances. Under simulation of thermal and mechanical energy flows being directed deep into the specimen from plane energy flux front and uniformly loaded boundaries, one should choose a simple cubic packing whose structure is more appropriate for plane deformation and temperature fronts. By varying these two types of automata packings, one may check the correctness of a physical process modeling: if results obtained for different types of packings are close enough, the model under construction is correct. Within the proposed approach, a cellular automaton has the following parameters being related to the material contained in the simulated volume: elastic modulus, shear modulus, density, thermal expansion coefficient, specific heat capacity, thermal conductivity, total energy of defects, maximum rate of material response at applying external mechanical force, total energy, mechanical energy, thermal energy, and temperature.

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Apart from these parameters, much like was mentioned above, an important characteristic of a cellular automaton is the automata packing type. In the framework of the proposed method, simple cubic and hexagonal close packings are used to construct the network. In the case of the cubic packing, an automaton has 6 neighbors in the first coordination sphere (a unit cell is a cube), and in the case of hexagonal packing, it has 12 ones (a unit cell is a dodecahedron). Since under mechanical loading, simulation account is taken only at energy flows from automata of the first sphere, the above difference suggests that the simulated material has a larger number of energy propagation directions from the local mesovolume than in the case of the cubic packing pattern. However, in many cases, e.g., in simulating plane fronts of mechanical energy fluxes, the cubic packing has an advantage over the hexagonal one since it offers a higher accuracy and computational speed of an algorithm for the calculation of energy flows in the directions parallel and perpendicular to the front.

3

Heat Transfer Simulation Method

To determine the temperature of the elements, it was necessary to introduce a discrete model of heat transfer of a heterogeneous medium. At zero time step of the numerical experiment, for each element, the initial values of temperature, deformation, heat conductivity, heat capacity, and coefficient of linear thermal expansion were assigned. Then, at each n-th time step, a new temperature value is calculated for an element with due consideration of heat fluxes from each adjacent element. T ni ¼ T n1 þ i

N 1 X Qn , ci ρi V k¼1 ik

(1)

n where T n1 i and T i are temperatures of the i-th element at n-1 and n-th time steps, ci is heat capacity of the i-th element, ρi is the density of the i-th element, V is element volume, Qnik is the flux of heat energy from neighboring k-th element into considered i-th at the n-th time step, and N is the number of neighbors. The change of heat energy Qnik is calculated using the Fourier law:

Qnik ¼

 λik Ω  n1 T k  T n1 Δτ: i l

(2)

Here λik is the coefficient of mutual heat conductivity, l is the distance between the centers of the elements under consideration, Ω is the area of adjacent plane, and Δτ is time step. The coefficient of mutual heat conductivity is calculated as follows: λik ¼ ðλi þ λk Þ=2:

(3)

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Here λi and λk are heat conductivity coefficients of the i-th and k-th elements, with λ j eφj θik ðj ¼ i, kÞ, λj ¼ e

(4)

where eλ j is the coefficient of heat conductivity of the j-th automaton, φj is the coefficient determining the fraction of impact of phonons on the material heat conductivity, and θik is angle (in degrees) determined by the difference of crystal lattice orientations of i-th and k-th automata (in current program version θik = |ψi  ψk|). After the calculation of new temperature values T ni using Eq. (1), for each i-th element, the following values are calculated: (a) Change in deformation due to thermal expansion:   ΔQ εni ¼ αi  T ni  T n1 i

(5)

(b) Corresponding changes in stress and mechanical energy Ani :   n  Yi , ΔQ σni ¼ αi  T ni  T n1 i ΔQ Ani ¼

Y ni  V CA 2     2    n1 2   εi  sign εn1 þ ΔQ εni  εn1 þ ΔQ εni  sign εn1 ; i i i

(6)

(7)

(c) New values of heat energy Qni :    ΔQ Ani ; þ ci  T ni  T n1 Qni ¼ Qn1 i i

(8)

(d) According to (8), a correction is made for a new temperature value:

T ni ¼ T ni 

ΔQ Ani : ci

(9)

Thus, the fraction of element heat energy variation is consumed for the change in its mechanical energy due to the process of thermal expansion. As a result, internal thermal stresses are accumulated, which are transformed into fluxes of mechanical energy distributed over the network of elements.

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4

Estimation of Heat Transfer Simulation Accuracy

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With the use of the SECA method, a number of numerical experiments on the thermal impact on the single-layer rubber specimen were conducted. The specimen was simulated using the cellular automaton with the FCC packing pattern of the elements whose size made 0.1 μm. The specimens were parallelepiped shaped with dimension of 2  2  6 mm. Initial temperature of all elements was equal to 293 K. The boundary conditions were the following: the constant temperature T = 423 K was applied at the top and the bottom of the mesovolume faces; the other faces were heat-insulated. Total time of numerical experiment made 120 s; in doing so the number of time steps was equal to 100,000. The time step was equal to 1.2 μs. The terms of conducted numerical experiment are similar to the problem of heat propagation in an infinite plate which is formulated as follows. An infinite plate of 2R thickness is given. Suppose that the initial temperature of the material inside the plate is set to T0, the temperature of surrounding surfaces is a constant one throughout the entire process of heat propagation being equal to Tc. In this concern the temperature distribution over the plate at any time step should be found. The analytical solution of this problem is related to solving the differential equation: @T ðx, τÞ @ 2 T ðx, τÞ ¼a ðτ > 0,  R < x < RÞ @τ @x2

(10)

With the following initial and boundary conditions: T ðx, 0Þ ¼ T 0 , T ðR, τÞ ¼ T c :

(11)

Here a is the thermal diffusivity of simulated material that might be expressed as a = λ/сρ, where λ is the thermal conductivity, с is the specific heat, and ρ is the material density. The general solution of Eq. (10) might be attained by separation of variables: T ðx, τÞ ¼ T c þ ðT 0  T c Þ

1 X n¼1



 xi   π 2 aτ  h π 4ð1Þ cos  þ πn exp   þ πn : 2 R 2 π ð2n  1Þ R2 nþ1

(12)

The following parameters were set for the analytical solution: T0 = 293 K, Tc = 423 K, R = 3 mm, and thermal conductivity value being equal to one of the rubber. In order to find the sum of the series, a computational program that makes possible to calculate the sum of its members at any preset accuracy was developed. The calculation results allowed us to graph the dependencies of the temperature distribution through the plate thickness at various time steps. Similar curves were

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drawn for the analytical solution. Comparison of these curves for each specified time step might be seen in Fig. 2. The graphs depicted in Fig. 2 show that the numerical and analytical solutions are just slightly different. The following algorithm was applied for a mathematical estimation of the difference between analytical and computational results. For each layer of cellular automaton elements, being parallel to the heated faces, the element being closest to the x-axis and passing through the specimen center was selected. For each aforementioned element with the coordinate x at time τ, the value of the relative error δ (x, τ) of the numerical solution obtained by the SECA method is calculated with further comparison to the analytical one T (x, τ):   ^ ðx, τÞ T c  T   δðx, τÞ ¼   1: T c  T ðx, τÞ

(13)

The calculation results show that the average relative error δ (x, τ) decreases over time: (i) it is less than 2% after 10,000-th time step (12 s); (ii) it makes 0.5% after 50,000-th time step (60 s); it is equal to 0.1% after 100,000-th time step (120 s). Then, the numerical experiment on the uniform thermal loading on a copper specimen was conducted with the help of the SECA method. Cellular automaton with the FCC packing pattern was employed. The element size made 1 μm. The mesovolume was rectangular shaped with dimension of 60  50  40 mm. The initial temperature of all components was equal to T = 300 K. The follows boundary conditions were set: the constant temperature of 1000 K was set at all faces; the overall time of the numerical experiment made 100 μs; the number of time steps was equal to 100,000; the time step was equal to 1 ns.

Fig. 2 Temperature distribution through the rubber plate depth for numerical and analytical solutions at various time steps

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The numerical experiment parameters were similar to the problem of heat propagation in a rectangular parallelepiped. In doing so, the parallelepiped with the dimensions 2R1, 2R2, and 2R3 is given. Let us take that the initial temperature of the material inside the parallelepiped is T0, while the temperature of its faces is constant Tc throughout the heat loading. In this concern the temperature distribution at any time step inside the parallelepiped is to be found. One should find an analytical solution for the following differential equation: @T ðx, y, z, τÞ @τ2

@ T @2T @2T þ þ ðτ > 0,  R1 < x < R1 ,  R2 < y < R2 ,  R3 < z < R3 Þ ¼a @x2 @y2 @z2 (14)

with the following initial and boundary conditions: T ðx, y, z, 0Þ ¼ T 0 , T ðR1 , y, z, τÞ ¼ T ðx,  R2 , z, τÞ ¼ T ðx, y,  R3 , τÞ ¼ T c :

(15)

Here a is the thermal diffusivity of simulated material being expressed as a = λ/сρ, where λ is the thermal conductivity, с is the specific heat, and ρ is the density of material. The general solution of this equation might be attained by means of the variable separation: T ðx, y, z, τÞ ¼ T c þ þðT 0  T c Þ  2

 1 X 1 X 1 X x y z μ μ2 μ2  An Am Ak cos μn cos μm cos μk exp aτ n2 þ m2 þ k2 : R1 R2 R3 R1 R2 R3 n¼1 m¼1 k¼1

(16a) 2 2 2 , Am ¼ ð1Þmþ1 , Ak ¼ ð1Þkþ1 ; μn μm μk π π π μn ¼ ð2n  1Þ , μm ¼ ð2m  1Þ , μk ¼ ð2k  1Þ : 2 2 2

An ¼ ð1Þnþ1

(16b)

Assuming that T0 = 300 K, Tc = 1000 K, R1 = 30 μm, R2 = 25 μm, and R3 = 20 μm and taking the calculating parameter a as one corresponding to the copper, the analytical solution of the problem similar to one from the numerical experiment can be obtained. The calculation results at various time steps allowed us to graph the curves of temperature distribution along the z-axis, i.e., at crossing the centers of the top and bottom faces through the bulk specimen. Similar curves were drawn for the analytical solution. Comparison of these curves for each specified time step might be seen in Fig. 3. The graphs in Fig. 3 show that the numerical and analytical solutions differ but slightly. For the mathematical-based estimation of the difference, the same algorithm as in the case of the infinite plate was applied. In this case, the calculation of the

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Fig. 3 Temperature distribution through the depth of copper parallelepiped obtained under numerical and analytical solutions

relative error δ (x, y, z, τ) of the numerical solution at the (x, y, z) point at τ time step was conducted by the following relation:   ^ ðx, y, z, τÞ T c  T   δðx, y, z, τÞ ¼   1: T c  T ðx, y, z, τÞ

(17)

These calculation results have shown that the average relative error δ(x, y, z, τ) decreases over time, and it does not exceed (i) 2.2% after 1000-th time step (1 μs); (ii) 0.36% after 2000-th time step (2 μs); (iii) 0.13%, after 3000-th time step (3 μs); 0.04%, after 4000-th time step (4 μs); and 0.01% after 5000-th time step (5 μs).

5

The Modeling of Heat Transfer in Porous Body

The modeling of heat transfer in porous body should account the transfer of heat energy by radiation. In the framework of the method under consideration, a state of an element is introduced that indicates the fact of its belonging to a pore. If the considered i-th automaton element on the first coordination sphere has an element in pore state, then the heat inflow into the i-th element is calculated from the assumption of the cubic packing of elements. If this takes place, the pore boundaries are reflective, i.e., radiated and reflected rays propagate perpendicularly to the boundary. In this case, the neighboring element is the elements in material state that is the closest on the way of propagation of rays from the i-th element. Let this element have index k. The change in heat energy of the i-th element after the interaction with the k-th element at current n-th time step Qnik is determined as follows.

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In the case of radiant heat exchange, the i-th element, in addition to its generated radiation Eni, partially reflects the incident energy from the k-th element. The sum of energy of and reflected radiation is the effective radiation of the i-th  generated  element Eni eff . For the calculation of the radiant heat exchange between elements, it is of importance to determine resulting radiation, which is the difference between the radiant flux received by an element and radiant flux which it radiates. To determine the flux density of the resulting radiation qnik , an energy balance equation is composed:     qnik ¼ Eni eff  Enk eff :

(18)

On the other hand, the resulting radiation is the difference between the flux radiated by an element and the fraction of absorbed energy:   qnik ¼ Eni  μi Enk eff ,

(19)

where μi  [0; 1] is the absorption coefficient of the i-th element. Taking into account (18) and (19) yields: 

Eni

 eff



1 En ¼ qnik 1  þ i: μi μi

(20)

To calculate the heat energy Qnik transferred from the k-th element to the i-th one, a model of heat flux is used in a system of paralleled reflecting plates (Fig. 4). In the frame of the model, using the principles of the law of radiant heat exchange, the equation for calculating the radiant heat exchange between elements can be derived. Let us derive the equation for determining heat flux qnik from the k-th element to the i-th one. This value is the difference between effective fluxes from each of the elements: qnik ¼ Enk  Eni :

(21)

Thus, according to (20) and taking into account that under stabilized regime, the resulting fluxes for each of the elements have equal value and opposite signs  n  qik ¼ qnki : Fig. 4 The scheme of heat radiation model

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 n 1 En n Ek eff ¼ qik 1  þ k, μk μk

n  n 1 E Ei eff ¼ qnik 1  þ i, μi μi

(22)

where μi is the absorption coefficient of material of the i-th element. Substituting (22) into (21) yields the expression for heat flux qnik : qnik

 n  μi μk Ek Eni ¼  : μi þ μk  μi μk μk μi

(23)

According to the laws of Stefan-Boltzmann and Kirchhoff, the relation of the flux density of heat energy Eni – radiated by the surface of heated i-th element-containing material – to its absorption coefficient is calculated as follows: n 4 Eni Ti ¼η : μi 100

(24)

Here η = 5.67 W/(m2K4) is the radiation coefficient of a gray body, μi is the absorption coefficient of material of the i-th element, and T ni is the temperature of the i-th element at previous time step. Obviously, the heat flux Qnik over the time step Δτ through the element plane area Ω is expressed as: Qnik ¼ qnik Ω Δτ:

(25)

Taking into account (23) and (24) yields the expression for heat flux Qnik : Qnik

"

n 4 # μi μk T nk 4 Ti ¼ η  Ω Δτ: μi þ μk  μi μk 100 100

(26)

Here, μi, μk  [0; 1] are absorption coefficients of the i-th and k-th elements. Thus, the modeling of radiant heat exchange using (2) requires adding expression (26): Qnik

 λik Ω  n1 μi μk T k  T n1 þ ¼ η i l μi þ μk  μi μk

"

T nk 100

4



T ni  100

4 # Ω Δτ:

(27)

Taking into account that the amount of radiated energy is known to exceed the amount of energy absorbed by the walls of pores due to dissipation (related to the absorption and dissipation coefficients), the energy dissipation during the heat transfer through pores is expressed using optical thickness of gaseous filler of pores. Hence, μi and μk coefficients should be selected with the regard for the heat energy dissipation in the filler gas in pores. The chemical composition of the gas can be determined experimentally.

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Further, due to heterogeneous thermal expansion, in different structural elements, internal thermal stresses accumulate and transform into fluxes of mechanical energy redistributed over the network of a cellular automaton. The method explicitly accounted grain boundaries, their curvature, and grain boundary angles of the crystal lattice. The created algorithms allowed calculating the value of local force moments, vorticity tensor, and dissipation of torsion energy conditioned by the formation of new defect structures.

6

Mechanical Energy Transfer Simulation Method

In the method of excitable cellular automata [33–35, 40, 41], the polycrystalline structure of the material, i.e., the presence of internal interfaces, is taken into account in explicit form. A network of active elements that make up a specimen is divided into clusters when every one of them models a grain with its own lattice orientation characterized by the Eulerian angles ψ, φ, and η. The Eulerian orientation angles of neighboring grains are used as a basis for determining misorientation angles at the interface. The misorientation angle is one of the important parameters which govern the energy characteristics of the interface. The developed algorithm for the simulation of mechanical energy transfer was supplemented with the procedure of calculating the vectors of moments of forces acting on a local mesovolume. The SECA approach described below firstly was published by Pleiades Publishing Ltd. in [34, 40]. An essential difference of the proposed approach from most of the discrete methods (e.g., molecular dynamics or movable cellular automata) is that an active element describes not a discrete mesovolume of the material but a fixed region of space through which the material flow takes place. In terms of hydrodynamics, a SECA element models a “reference volume,” while classical approaches deal with material volumes. This concept opens up new avenues for the numerical simulation of flows of matter and energy of different origin. One of the principal equations of the method is the modified Turnbull equation which is used to calculate the velocity of the flux of matter through the interface between active elements vnik under the action of stress σn-1ik at this interface, approximated in terms of pairwise interactions: vnik ¼ mik  σ n1 ik

(28)

where mobility mnik is determined as

Q mik ¼ ðm0 Þik  exp  ik kB T ik

(29)

while (m0)ik is the maximum mobility value written in the SECA method in the following form:

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ðm0 Þik ¼

c ðYiYYY k Þ e ik YiYk

2

(30)

where Yi and Yk are the elastic modulus values of the i-th and k-th elements, c is the effective response rate of the medium to an external mechanical action, kB is the Boltzmann constant, and Qik is the energy of the boundary between the i-th element and its k-th neighbor on the (n – 1)-th time step. The boundary energy value is determined according to the Read-Shockley approximation: Qik ¼ γ HAGB



jθk  θi j jθ k  θ i j 1  ln θHAGB θHAGB

(31)

where γHAGB is the maximum boundary energy corresponding to the maximum misorientation angle of the lattice, θHAGB is the maximum misorientation angle, θi and θk are the effective orientation angles of the lattice of the i-th and k-th elements, and Tik is the temperature value on the considered boundary which depends on the element temperatures Ti and Tk. Crystal lattice and its misorientation between grains in the framework of SECA approach are illustrated in Fig. 5. Figure 6 demonstrates schematically the triple grain junction in the polycrystalline structure under simulation. The volume fraction of the material that moved to the neighboring element, Δζik, and mechanical energy variation of the i-th element as a result of interaction with the k-th neighbor ΔAnik are defined as follows: ΔV nik vn1 Δt ¼ Δζ nik ¼ ik lc Vc

(32)

ΔAnik ¼ Δζ nik σ n1 ik V c

(33)

Here, Δt is the time step value, lc is the element size, and Vc is the element volume. As has been said above, it is easy to show that in the limiting case (uniaxial loading of a homogeneous material), the transfer equation in the SECA method degenerates into a linear Hooke’s law. In the case of a homogeneous medium Yi = Yk = Y, Qik = 0 and expression (28) with regard to (29) and (30) is transformed in the following way: vnik ¼

c n1 σ Y ik

(34)

On the other hand, the flux velocity vikn = Δl/ Δt, where Δl is the distance to which the material penetrates from one neighboring element to another for time Δt. By the definition of linear deformation Δl = εiknl0, where l0 is the linear size of the element and εikn is the strain value for the material of the i-th element during interaction with the k-th. Thus, we come to the linear stress-strain dependence:

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Fig. 5 Schematic images of a grain with lattice (a) and neighboring grains with different lattice orientations (b)

enik Y ¼ σ n1 ik

cΔt l0

(35)

If с is chosen in the following way, cΔt/l0 = 1, this dependence coincides with the classical Hooke’s law. Note that mobility in the Turnbull equation can be broadly interpreted depending on the specificity of the process under consideration. For example, mobility is the quantity inversely proportional to viscosity when describing particle penetration into a solid at impact interaction or vortex formation in hydrodynamics. According to the concept of Prof. G. Sih, the physical world cannot be described adequately without the hypothetical existence of the rate of change of volume with surface [36]. It actually means in the framework of his theory describing dissipative processes of mass and energy pulsation that there exist additional “body fluxes.” With body fluxes in locally open dynamic systems (among which are interfaces), Newton’s third law is never fulfilled, and there is always an addition to the classical Newtonian reaction force responsible for the action of these body fluxes. The developed SECA method is in complete agreement with the above concept. In the method of excitable cellular automata, mobility is considered as a quantity

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Fig. 6 Schematic of triple grain junction in polycrystalline structure

reciprocal to the (volume) specific impulse of response of the material contained in the neighboring element. The situation is schematically illustrated in Fig. 7. In this case, the Turnbull velocity in Eq. (28) is nothing but the rate of the flux of matter with regard to the “body flux” from the surrounding material (Fig. 7). This makes it possible to numerically simulate curvature formation through energy flows along grain boundaries with regard to the influence of the surrounding three-dimensional crystal structure. Owing to the introduction of the modified Turnbull equation, the developed approach explicitly accounts for the nonequilibrium character of simulated processes. On the other hand, the nonequilibrium pattern of the processes at the interface is closely related to curvature formation which is hardly to be described without taking into account local moments of forces along interfaces. The majority of solved problems in classical transport theories are reduced to a onedimensional case where an interface degenerates into a point. The properties of a barrier that must be overcome are assigned to each point (e.g., in the case of passage of a dislocation through the grain boundary). Clearly, in this problem statement, moments of forces are identically equal to zero, and the notion of curvature loses its meaning. However, as is known from theory and experimental results [12–16], interfaces are independent planar subsystems with non-zero thickness for which their own transfer equations must be written and solved. Consequently, in a threedimensional statement, two problems must be solved simultaneously: these are transfer through the interface and transfer along the interface. As is shown in [37],

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Fig. 7 Schematic interaction of active elements in the SECA method (a); mass exchange of an active element with the surrounding material as a result of action of body fluxes (b)

the grain boundary is a system of clusters surrounded by a quasiamorphous layer; in so doing, the mechanism of grain boundary sliding is described by an equation similar to the Turnbull one. It is thus implicitly stated that mass transfer in the grain boundary is nothing but an atom-by-atom rearrangement of the cluster structure of the interface. The given concept correlates well with the SECA method applied in the present paper. On the other hand, in paper [38] it is shown that a two-dimensional model of a liquid crystal consisting of clusters surrounded by a quasiamorphous phase is described by the Klein-Gordon equation for the oscillation of tangential stresses which gives rise to local rotations of particles. Below, we will demonstrate that the distribution of local moments of forces along grain boundaries in the method of excitable cellular automata determines a modulated vortex-like character of disturbance propagation along the interface. An obvious solution of the problem is to develop “evolutionary algorithms” in the form of simple numerical schemes for calculating the most important parameters of the system state. One of such parameters is the vorticity of mass transfer flow along the interface which arises due to curvature of the interface.

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Rotational Deformation Modes and Defect Accumulation

Rotary wave fluxes are particularly distinguished under dynamic loading, for instance, under friction when wear particles are formed. Most distinctively the microrotations are demonstrated in experiments on fragmentation during propagation of a shock wave [39]. Rotation modes of deformation occur after the impact of local moment of force. Angular velocity of the i-th element in Fig. 8 under the flux of matter acting through the boundary of the k-th and l-th elements (each k-th element lies on the first coordination sphere of the i-th element; each l-th element lies on the intersection of first coordination spheres of the i-th and corresponding k-th elements) is calculated as follows: ! ! ! ω ikl ¼ r ikl  v kl  2 !   r ikl  :

(36)

The velocity of matter flux at the boundary of the k-th and l-th elements is calculated using the Eq. (28). The total angular velocity of the i-th element is determined as the following sum: ! ωi ¼

K X L X

! ω ikl :

(37)

k¼1 l¼1

Here K is the number of elements on the first coordination sphere of the i-th element, and L is the number of elements at the intersection of the first coordination spheres of the i-th element and each k-th neighbor. Fig. 8 Scheme for calculation of angular velocity of material rotation in cellular automaton element

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1177

Three-dimensional rotation angle of the i-th element over time τ ( Δ! γ i ) is proportional to the total angular velocity: Δ! γi¼! ω i τ:

(38) !

Change of the force moment of the i-th element over time τ (ΔMi) is calculated as follows: !

ΔMi ¼

Gπ r 3c Δ! γ i Gπ r 3c ! ω iτ ¼ : 2 2

(39)

Here G is shear modulus of material, contained in the i-th element, and rc is element radius. Vortex vector is determined as follows: ! ! ω i ¼ rot v :

(40)

Components of vortex vector are expressed as follows: ω1 ¼ 2Ω23 , ω2 ¼ 2Ω31 , ω3 ¼ 2Ω12 ,

(41)

where Ω is vorticity tensor, Ωij ¼



1 @vi @vj  : 2 @xj @xi

(42)

The meaning of angular velocity calculated using expressions (20), (21), and (22) is similar to that of vortex vector. Taking into account Eq. (24), the following expressions for vortex vector components are yielded: 2M1 , Gπ r 3c τ 2M2 , ω2 ¼ Gπ r 3c τ 2M3 : ω3 ¼ Gπ r 3c τ

ω1 ¼

(43)

Thus, vorticity tensor Ω can be expressed as follows: 0

0

B B B M3 Ω¼B B  Gπ r 3 τ c B @ M2 Gπ r 3c τ

M3 Gπ r 3c τ 0 

M1 Gπ r 3c τ

1 M2 Gπ r 3c τ C C M1 C C: Gπ r 3c τ C C A 0



(44)

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The member responsible for accumulation of latent energy of defects is expressed in the following way: h

! in ΔAd i

  ktors Gi Δ! γ ni πr 3c !n Δγ i: ¼ 2

(45)

Here, ktors is the rate of accumulation of defects (can be determined experimentally). It should be noted that expression (45) has general sense and, depending on formulated rules of switching of cellular automaton, can have the meaning of energy required for reversible structure and phase transformation. Here the ktors parameter represents the fraction of the elastic energy, which is expended to form the defects within the volume of the active element per unit time. Thus the latter expression was incorporated directly into the numeric model as an integral term accounting for the defect microstructure in the first-order approximation. During each step of numerical simulation, the part of the elastic energy is irrecoverably transferred into the energy of defects, which reduces the shear stability of the material underloading. The increment of the local moment of force leads to curvature formation in the material. Curvature, in its turn, initiates defect fluxes to propagate along interfaces at different scale levels. The action of the force moment causes torsion of the material in an active element by a certain angle. The material rotation gives rise to lattice distortion through the formation of multiple slip lines. Moreover, by determining the angle of torsion of the material, we may quantitatively estimate the variation of the volume fraction of defects. To do this, an approximation of a model of torsion of a rod can be used (Fig. 9a).

Fig. 9 Computational model of a rod of circular cross section under torsion (a) and calculation of defect density variation (b)

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The sought angle γ is found as follows: γ¼

ðCC1  BB1 Þ ½r ðϕ þ dϕÞ  rϕ rdϕ dϕ ¼ ¼ ¼ rθ, θ ¼ BC dx dx dx

(46)

where θ is the relative angle of torsion. Let us employ the abovementioned model. The angle of element rotation γ is calculated by the formula γ ¼ rθ

(47)

where r is the element radius. The relative angle of torsion θ is defined as follows: θ¼

2M Gπ r 4

(48)

where M = |M| and G is the shear modulus. Thus, γ¼

2M Gπ r 3

(49)

The change of the number of defects in the element under its rotation by angle γ is determined depending on the number of elementary lattice cells that enter into the sector corresponding to this angle (Fig. 7b): ΔN d ¼ k

Ssect ðγ Þ Scell

(50)

where Ssect(γ) is the area of the sector corresponding to angle γ, Scell is the crosssectional area of the elementary lattice cell intersected by the lattice plane passing through the center of the cell, and k is the proportionality coefficient depending on the type of the atomic packing and material. It is known from the “first principles” that each material has its own energy of interatomic interaction, and hence coefficient k for different materials is different. However, it depends not only on the energy value but also on many other factors. The sector area Ssect(γ) is determined by the expression Ssect ðγ Þ ¼

γ R2 2

(51)

Substituting expressions (49) and (51) into (50), we have: ΔN d ¼

kM Gπ r Scell

(52)

The defect density variation in the element Δρd is defined as the ratio of ΔNd to the cross-sectional area of a cellular automaton element SCA:

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Δρd ¼

ΔN d kM ¼ Gπ r Scell SCA SCA

(53)

The derived relation allows calculating defect density variation in a material at curvature formation under the action of the local force moment.

8

Numerical Experiment

Theoretical study was performed in order to evaluate the character of structure and phase transformations in nanocrystalline and nanoporous specimens under thermal cycling. The hybrid method of excitable and bistable cellular automata was used to simulate heat transfer with due account for the process of collective recrystallization. The numerical experiments described below firstly were presented in ECCOMAS 2016 Congress [41]. Using the method of stochastic excitable cellular automata, a numerical experiment (NE) was conducted including the heat up of solid and nanoporous specimens consisting of nanocrystalline TiAlC with high (12106 K1) and low (7106 K1) values of the thermal expansion coefficient (TEC). The NE comprised of the simulation of collective crystallization and thermal expansion of materials, as well as the explicit account for thermal transition of mechanical energy into its latent fraction during local rotations. In doing so at every time step, the mechanical energy of each element of a cellular automaton was decreasing for a value being proportional to the square of absolute value of accumulated three-dimensional angle of rotation of this element. Each specimen was simulated by means of a cellular automaton with FCC arrangement of elements and the dimensions of 60 nm. The size of each element was 2.88  2.4  1.6 μm; the dimensions of grain and pore were set to 60 nm (Fig. 10). Initial temperature of each element was 300 K; initial values of deformation and stress were set to 0. The value of time step was 0.02 ns. Thermal loading consisted of two time steps of 1000 μs duration: at the first stage, heating to the upper

Fig. 10 Grain structure of specimens (in three-dimensional form and at the front face) at the initial moment of time: (a) without pores; (b) with pores

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limit of 1500 K was simulated; at the second one, cooling down to 300 K was computed (Fig. 11). Figure 12 illustrates the images of grain structure of specimens (in three-dimensional form and at the front face), formed after collective crystallization by specific time step. Figures 13, 14, 15, and 16 depict the time dependencies of mean values of effective stress, specific latent mechanical energy, relative latent mechanical energy, and modulus of force moment at the stage of heating of the specimens. The study performed by direct numerical experiment has elucidated that introduction of the nanoporosity allows appreciably reducing the rate of collective crystallization. It should be noted that the nanoporosity was slowing down the propagation of the heat front in modeled specimens under the studied. This fact can play both positive and negative roles in the process of deposition and further exploitation of a coating. On the one hand, the slowdown of heat front can considerably reduce the level of thermal stresses in deep layers of a material. On the other hand, the slowdown of heat propagation front in the case of high TEC of a material

Fig. 11 Scheme of thermal load application (a) and time dependence of temperature at upper face (b)

Fig. 12 Images of grain structure of specimens (in three-dimensional form and at the front face), formed after collective crystallization by specific time step: (a) without pores with high TEC (t = 300 ns); (b) without pores with low TEC (t = 300 ns); (c) porous with high TEC (t = 600 ns); and (d) porous with low TEC (t = 600 ns)

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Fig. 13 Time plots of mean effective stress for specimens without pores/with high TEC (black), without pores/with low TEC (red), porous/with high TEC (green), and porous/with low TEC (blue)

Fig. 14 Time plots of mean specific latent mechanical energy for specimens without pores/with high TEC (black), without pores/with low TEC (red), porous/with high TEC (green), and porous/ with low TEC (blue)

can result in the formation of sharp gradients of thermal stresses that initiate the origination and rapid propagation of a main crack. Note that in porous specimens, the rate of collective crystallization was substantially lower than that in nanocrystalline specimens. The results of the study demonstrate that the value of the thermal expansion coefficient exerts no impact on the rate of recrystallization in nonporous specimens. This means that the driving force in the

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Fig. 15 Time plots of mean relative latent mechanical energy for specimens without pores/with high TEC (black), without pores/with low TEC (red), porous/with high TEC (green), and porous/ with low TEC (blue)

Fig. 16 Time plots of mean modulus of force moment for specimens without pores/with high TEC (black), without pores/with low TEC (red), porous/with high TEC (green), and porous/with low TEC (blue)

recrystallization process in this case is primarily the temperature gradient, rather than mechanical stresses. Figures 17, 18, 19, and 20 represent the images of grain structure, distribution of effective stress, modulus of force moment, and relative latent mechanical energy in studied specimens at different time steps.

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Fig. 17 Images to illustrate grain structure, distribution of effective stress, modulus of force moment, and relative latent mechanical energy in specimen without pores and with high TEC at different time steps

Comparison of the dynamics of defect structure accumulation (Figs. 17, 18, 19, and 20) and the analysis of plots of relative specific torsion energies (Fig. 15) allows concluding that in nonporous nanocrystalline material, with low TEC, the rate of defect structure accumulation increases. This might be explained by the fact that for high values of thermal expansion coefficient, material has higher capability to relax

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Fig. 18 Images to illustrate grain structure, distribution of effective stress, modulus of force moment, and relative latent mechanical energy in specimen without pores and with low TEC at different time steps

thermal stresses via three-dimensional deformation without disturbing the translational invariability of the crystal lattice but increasing the interatomic distance of the latter. In the case of the low TEC, the major amount of thermal stresses transfer into local force moments, initiating curvature and torsion of crystal lattice, which unavoidably is accompanied by origination of defect structures at microscale level.

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Fig. 19 Images to illustrate grain structure, distribution of effective stress, modulus of force moment, and relative latent mechanical energy in porous specimen with high TEC at different time steps

This makes particularly indicative the plots of time dependencies of the total modulus of force moment (Fig. 16). It is evident that averaging the modulus of force moment over the entire specimen results at mesoscale level in substantial oscillations of the force moment in specimens with high TEC. This testifies elastic rotations of whole conglomerates of grains, which, nevertheless, does not lead to appreciable

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Fig. 20 Images to illustrate grain structure, distribution of effective stress, modulus of force moment, and relative latent mechanical energy in porous specimen with low TEC at different time steps

formation of defect structures in them. The plots of the time dependencies of force moments in specimens with low TEC, conversely, are smooth and possess the values of two to four times lower. The comparison of images of the defect structure energy distribution both in porous and nonporous specimens shows its more dense and uniform distribution in

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material with low TEC, while in material with high values of thermal expansion coefficient, the torsion energy is distributed as large regions of its high and low concentration (Figs. 17, 18, 19, and 20). This confirms the conclusion about material with low thermal expansion coefficient achieving larger curvature of crystal lattice at microscale, accompanied by bigger rate of defect structure formation. Such systems rapidly drop stresses into moment forces at microscale impeding the self-organization of rotation moment in large conglomerates of grains.

9

Conclusions

A new modification of the multiscale hybrid discrete continuum approach of stochastic excitable cellular automata was developed. The new method explicitly accounts for the porosity and nanocrystalline structure of material, the algorithm for computing local force moments, and angular velocities of microrotations occurring in structurally nonuniform medium with due account for energy dissipation. Using the direct numerical experiment, it was demonstrated that in porous specimens, the rate of collective crystallization was substantially lower than that in nanocrystalline specimens. The results of the study demonstrate that the value of thermal expansion coefficient has no impact on the rate of recrystallization in nonporous specimens. On the other hand, the value of thermal expansion coefficient of nanoporous and nanocrystalline specimens substantially impacts the distribution character and rate of defect structure accumulation. For instance, it was shown that the material with low thermal expansion coefficient reaches higher curvature of crystal lattice at microscale. The formation of curvature and torsion in the lattice is accompanied by the formation of defect structures. Such systems are more capable of relaxing moment stresses at microscale, which interferes the self-organization of rotation moment in conglomerates of grains at mesoscale. Thus, in addition to accounting the local rotational modes of structural elements and thermal radiation by pore walls, it is necessary to study the ability of a material to reversible structural transformations in the material of pore walls. The reversible structural transformations will accumulate peak values of energy in elastic rotational stress fields that are knowingly inertial, which will allow preventing catastrophic growth of main cracks.

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29. Smolin AY, Shilko EV, Astafurov SV, Konovalenko IS, Buyakova SP, Psakhie SG. Modeling mechanical behaviors of composites with various ratios of matrix–inclusion properties using movable cellular automaton method. Defence Technology. 2015;11:18–34. 30. Kroc J. Application of cellular automata simulations to modelling of dynamic recrystallization. Lect Notes Comput Sci. 2002;2329:773–82. 31. Godara A, Raabe D. Mesoscale simulation of the kinetics and topology of spherulite growth during crystallization of isotactic polypropylene (iPP) by using a cellular automaton. Model Simul Mater Sci Eng. 2005;13:733–51. 32. Humphreys FJ, Hatherly M. Recrystallization and related annealing phenomena. New York: Pergamon; 1995. 33. Moiseenko DD, Panin VE, Maksimov PV, Panin SV, Berto F. Material fragmentation as dissipative process of micro rotation sequence formation: hybrid model of excitable cellular automata. AIP Conf Proc. 2014;1623:427–30. 34. Moiseenko DD, Panin VE, Elsukova TF. Role of local curvature in grain boundary sliding in a deformed polycrystal. Phys Mesomech. 2013;16:335–47. 35. Panin VE, Egorushkin VE, Moiseenko DD, et al. Functional role of polycrystal grain boundaries and interfaces in micromechanics of metal ceramic composites under loading. Comput Mater Sci. 2016;116:74–81. 36. Sih GC. Mesomechanics of energy and mass interaction for dissipative systems. Phys Mesomech. 2010;13:233–44. 37. Mott NF. Slip at grain boundaries and grain growth in metals. Proc Phys Soc. 1948;60:391–4. 38. Sadovskii VM, Sadovskaya OV. On the acoustic approximation of thermomechanical description of a liquid crystal. Phys Mesomech. 2013;16:310–6. 39. Meshcheryakov YI, Khantuleva TA. Nonequilibrium processes in condensed media: part 1. Experimental studies in light of nonlocal transport theory. Phys Mesomech. 2015;18:228–43. 40. Moiseenko DD, Pochivalov YI, Maksimov PV, Panin VE. Rotational deformation modes in near-boundary regions of grain structure in a loaded polycrystal. Phys Mesomech. 2013;16:248–58. 41. Moiseenko DD, Maksimov PV, Panin SV, Panin VE. Defect accumulation in Nanoporous wearresistant coatings under collective recrystallization. Simulation by hybrid cellular automaton method. In: Papadrakakis M, Papadopoulos V, Stefanou G, Plevris V, editors. Proceedings of VII European congress on computational methods in applied sciences and engineering. Published on-line https://eccomas2016.org/proceedings/pdf/10631.pdf.

Multiscale Fatigue Crack Growth Modeling for Welded Stiffened Panels

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Ž. Božić, Siegfried Schmauder, M. Mlikota, and M. Hummel

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Molecular Dynamics (MD) Simulation of Dislocation Development in Iron . . . . . . . . . . . . 2.1 Methods and Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Microstructural Crack Nucleation and Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Modeling and Simulation of Crack Propagation in Welded Stiffened Panels . . . . . . . . . . . . 4.1 Specimen’s Geometry and Loading Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modeling of Welding Residual Stresses in a Stiffened Panel by Using FEM . . . . . . 4.3 Stress Intensity Factors and Fatigue Crack Growth Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1192 1195 1195 1196 1197 1200 1200 1202 1202 1208 1209

Abstract

The influence of welding residual stresses in stiffened panels on effective stress intensity factor values and fatigue crack growth rate is studied in this paper. Interpretation of relevant effects on different length scales such as dislocation appearance and microstructural crack nucleation and propagation is taken into account using molecular dynamics (MD) simulations as well as a Tanaka-Mura approach for the analysis of the problem. Mode I stress intensity factors (SIFs),

Ž. Božić (*) Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia e-mail: [email protected] S. Schmauder · M. Mlikota · M. Hummel Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_73

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KI, were calculated by the finite element method (FEM) using shell elements and the crack tip displacement extrapolation technique. The total SIF value, Ktot, is derived by a part due to the applied load, Kappl, and by a part due to welding residual stresses, Kres. Fatigue crack propagation simulations based on power law models showed that high tensile residual stresses in the vicinity of a stiffener significantly increase the crack growth rate, which is in good agreement with experimental results. Keywords

Dislocation · Microstructurally small cracks · Fatigue crack growth rate · Residual stress

1

Introduction

Structural health monitoring and damage detection in aircraft, ship, offshore, and other structures are highly important for their fitness for service assessment. Under cyclic loading fatigue cracks may initiate at sites of stress concentration and further propagate, which can eventually result in unstable fracture and structural failure. In aircraft, ships, and other thin-walled structures, stiffened panels are widely used owing to their light weight and high strength and stiffness. Welded stiffened panels are mostly implemented in the deck and side structure of a ship. The crack growth rate in welded stiffened panels can be significantly affected by the residual stresses which are introduced by the welding process. The high heat input from the welding process causes tensile residual stresses in the vicinity of a stiffener. These tensile stresses are equilibrated by compressive stresses in the region between the stiffeners. Residual stresses should be taken into account for a proper fatigue life assessment of welded stiffened panels under cyclic tension loading. From a physics point of view, the fatigue phenomenon involves multiple length scales due to the presence of micro-cracks or inclusions that are small compared to the large size of structural components. Therefore, it is necessary to consider the fatigue process at all scales. A scale-dependent physics-based model is required for accurate simulation and understanding of material behavior in various operational environments to assess fatigue life of a structure [1–5]. The process of fatigue failure of mechanical components may be divided into the following stages: (1) crack nucleation, (2) small crack growth, (3) long crack growth, and (4) occurrence of final failure. In engineering applications, the first two stages are usually termed as the “crack initiation or small crack formation period,” while long crack growth is termed as the “crack propagation period.” In pure metals and some alloys without pores or inclusions, irreversible dislocations glide under cyclic loading. This leads to the development of persistent slip bands, extrusions, and intrusions in surface grains that are optimally oriented for slip. Dislocation development can be simulated by using the molecular dynamics (MD) simulation code IMD [6]. To analyze dislocation development, atomistic scale simulation methods are implemented [7–10]. With continued strain cycling, a fatigue crack can be nucleated at an extrusion or intrusion within a persistent slip band [11–18]. Nonmetallic inclusions, which are present in commercial materials as

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a result of the production process, can also act as potential sites for fatigue crack nucleation. In the high-cycle regime, fatigue cracks initiate from inclusions and defects on the surface of a specimen or component. For very high-cycle fatigue, fatigue cracks initiate from defects located under the surface of the specimen [19, 20]. Micro-crack nucleation can be analyzed by using the Tanaka-Mura model or some of its modifications [11, 21–24]. Fatigue crack growth prediction models based on fracture mechanics have been developed to support the damage tolerance concepts in metallic structures [25]. A well-known method for predicting fatigue crack propagation under constant stress range is a power law described by Paris and Erdogan [26]. Dexter et al. [27] and Mahmoud and Dexter [28] analyzed the growth of long fatigue cracks in stiffened panels and simulated the crack propagation in box girders with welded stiffeners. They conducted cyclic tension fatigue tests on approximately half-scale welded stiffened panels to study propagation of large cracks as they interact with the stiffeners. Measured welding residual stresses were introduced in the finite element model and crack propagation life was simulated. Sumi et al. [29] studied the fatigue growth of long cracks in stiffened panels of a ship deck structure under cyclic tension loading. For that purpose fatigue tests were carried out on welded stiffened panel specimens damaged with a single crack or an array of collinear cracks. In order to analyze the total fatigue life of a structural component or a test specimen, from crack initiation through cyclic slip mechanism up to long crack propagation and final failure, a multiscale approach is needed. Fig. 1 shows a schematic description of bridging between the three considered scales: nano, micro, and macro. The relevant material’s property parameter from atomistic scale needed for the micromechanics modeling is the critical resolved shear stress (CRSS). The CRSS is inferred from MD simulation and it is the input parameter for a modified Tanaka-Mura model. The modified Tanaka-Mura micromechanics model provides information on the number of loading cycles to initiate a small crack and its size. This information is further an input for the macroscale fatigue crack growth model based on power law equations, by means of which then a total fatigue life up to fracture as a final event can be assessed. Based on the presented procedure, the fatigue behavior of a material can be simulated, and new materials can be modeled and analyzed. In this way Wöhler curves can be obtained for new materials without experiments. This paper presents a study of the influence of welding residual stresses in stiffened panels on effective stress intensity factor values and fatigue crack growth rate. A total SIF value, Ktot, was obtained by a linear superposition of the SIF values due to the applied load, Kappl, and due to welding residual stresses, Kres. The effective SIF value, Keff, was considered as a crack growth driving force in a power law model [30, 31]. Mode I SIF values, KI, were calculated by the FE software package ANSYS using shell elements and the crack tip displacement extrapolation method in an automatic post-processing procedure [32]. Simulated fatigue crack propagation life was compared with the experimental results as obtained by Sumi et al. [29]. The molecular dynamics (MD) simulation was implemented to analyze dislocation development in an iron cuboid model with a triangular notch tip. Numerical simulations of the fatigue crack initiation and growth for martensitic steel, based on modified Tanaka-Mura, were carried out.

Micro crack growth

Microstructure (Meso)

(f 1, m 1, d 1, ...)

(f 2, m 2, d 2, ...)

FEM

1.000.000

2.000.000

experiments comput. model-total comput. model-initiation

Specimen (Macro)

Cycles to failure, Nf

Nf

0 100 101 102 103 104 105 106 107 108 number of cycles

5

10

15

20

25

Finite Element Method (FEM) Fatigue Crack Analysis (Fatigue Crack Propagation)

S-N curve

400 100.000

500

600

700

Atomistic scale (Nano)

Fracture toughness Kc, Jc, CTODc

Final failure

Microstructural FEM

Crack growth papameters K, J, CTOD

Crack growth period

Macro crack growth

Nf

N = 194000

a macro-crack is formed

crack length [mm]

MD

Microstructural features

Initiation period

Crack nucleation

da/dN, Nini=Âng =1 Ng , aini

p (1 – u)d(Δ – 2CRSS)2

8µWc

Fig. 1 A schematic description of bridging between the three considered scales

Cyclic slip

CRSS

Ng =

Finite Element Method (FEM) Crack initiation (Nucleation and initial growth)

max [MPa]

Molecular Dynamics (MD) Local Slipping (Plastic deformation)

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Molecular Dynamics (MD) Simulation of Dislocation Development in Iron

2.1

Methods and Model

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Taking a close look on dislocation development leads to the necessity of atomistic scale simulation methods. Therefore, we used for the present work the molecular dynamics (MD) simulation code IMD [6]. It was developed at the Institute of Theoretical and Applied Physics (ITAP) belonging to the University of Stuttgart. ! In MD the atoms are seen as mass m points at the position r for which the elementary Newton’s equations of motion
 ! ! F r , t ¼ m  @2 r

(1) @2t
 ! are solved in every time step. The force F r , t is given by the derivative of the
 ! interatomic embedded atom method (EAM) [7] pair potential U r , t (2):

 ! ! F r , t ¼ ∇ U r , t

(2)

The system we investigated contains about half a million iron atoms (see Fig. 2). They form a cuboid of the size 286  143  143 Å3 where a notch (dimensions 15  90 Å2) with a triangular notch tip was inserted along the (110) plane. Cyclic deformation of the simulation box was applied in the [001]-direction. Therefore, the z-component of the simulation cell was elongated with a constant rate of 5  107 at each time step. The value 5  107 represents a factor by which z-components of the system are multiplied at each time step; therefore, it is without units. The strain rate could be calculated as 2.5  108 s1. Such high strain rates are typical for MD simulations but very high for experimentalists. After reaching a strain of 7%, we applied pressure at the same rate until we reach 7% of strain in compression. Seven percent tend to be at the upper end of the elastic elongation regime. In order to Fig. 2 Bcc iron cuboid 286  143  143 Å3 with a 15  90 Å2 notch on a (110) plane. The 486,000 atoms are color coded via von Mises stress (red = ^ high stress, blue = ^ low stress). Image by MegaMol™ [8]. Full periodic boundary conditions are applied. The loading direction is marked by the arrows

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reduce computation time and still observe changes in the structure already in the very first cycles, one has to apply these high values of maximum strain. Still, this value should be realistic. For that reason the authors used an input strain value of 7%. There have been other simulations with lower maximum strain, where the formation of dislocations occurs after a higher number of cycles. The loading and unloading were repeated continuously. The temperature was chosen to be room temperature (300 K). The time step was fixed to 2 fs. Periodic boundary conditions were used in every direction.

2.2

Results and Discussion

During the continuous cyclic change from elongation to compression, different stages of the system appeared (see Fig. 3). • • • •

Stage I: Configuration under no pressure. Stage II: Initiation of reversible local restructuring under tensile loading. Stage III: Formation of one continuous plane with face centered cubic (fcc) structure. Stage IV: Compression leads to a resolution of the deformation introduced in the previous steps into the structure and to bending of the middle of the notch surfaces toward each other up to a minimum distance of 6.8 Å. • Stage V: During the fourth loading cycle, dislocations are initiated. The dislocation extraction algorithm (DXA) [9] detects “defect surfaces.” “The defect surface consists of those parts of the interface mesh, which have not been swept by elastic Burgers circuits” [10].

Fig. 3 Stress [MPa] in z-direction in terms of the time [ns] during cyclic loading of a nanostructure of a notched iron cuboid. System configurations at different times are depicted: blue is according to DXA [9] “defect surfaces,” and red represents dislocations. The view is from lower left

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• Stage VI: Dislocations still remain in the structure even though no pressure remains in the system.

The MD simulations presented here illustrate the development of the system from reversible plastic deformation to irreversible plastic deformation including dislocation nucleation, propagation, back-propagation, and multiplication. The quite high stress level of the system in the regime of 5.5 GPa during dislocation nucleation is attributed to the fact that the underlying model is assumed to be an internally defect-free single crystal, except for the external notch, and to a minor degree to the high strain rate which is applied in this simulation. Strain rate effects in MD simulations have been previously discussed in literature, e.g., [33], and will thus not be taken into further account here. The stress to propagate an artificially introduced dislocation in an MD model is frequently obtained via pure shear simulations in literature [34] and at the IMWF institute as demonstrated, e.g., in Kohler et al. [35] and in Molnar et al. [36] where typical values of 84 MPa were obtained for 0 K in close agreement to values of 81 MPa for 10 K obtained by Kumar et al. [37] with the Fe-Fe potential presented by Bonny et al. in [7]. In the here presented simulations, the intention was to force dislocation nucleation and to study such naturally formed dislocations as a realistic basis for dislocation movement analyses rather than to use artificially implanted dislocations in the model. In the first two peaks, a reversible bcc-fcc Bain transition from α- to γ-Fe takes place (Stage III, Fig. 3). The oscillation of the stress-time curve (Stage III, Fig. 3) is explained by the formation of stacking faults inside the transition fcc phase. In contrast to Farkas et al. [38], where it was believed that the emission of Shockley partial dislocations is relevant for the stacking fault formation, no dislocations are observed in our simulation at this early stage. Dislocation nucleation takes place as the irreversible deformation begins. The small spikes in the stress-time curve are identified as single dislocation movement. The height of the spike, which is related to the CRSS to move a dislocation, is calculated to a value of 293 MPa, taking into account that this value is higher than the CRSS due to the angle of the dislocation glide plane with respect to the loading axis and the, thus, involved Schmid factor which amounts to 0.40 in the present case: The CRSS value obtained from the present simulation is calculated according to Schmid’s law to be τ = 293 MPa cos 26.5 cos 63.5 = 117 MPa which is in very good agreement to pure shear simulations mentioned above [33, 36] and to an even better degree with an experimental value of 108 MPa which has been previously used for microscale modeling of fatigue by Jezernik et al. [24] for the present material.

3

Microstructural Crack Nucleation and Propagation

To solve problems of crack nucleation, the Tanaka-Mura model [11, 21–24] is frequently used. In the two articles [21, 22], Tanaka and Mura proposed dislocation models for treating fatigue crack nucleation at slip bands. Tanaka and Mura envisioned that fatigue crack nucleation occurs by the accumulation of dislocation

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dipoles in a single grain during strain cycling. In the theory of fatigue crack nucleation in slip bands, the forward and reverse plastic flows within slip bands are caused by dislocations with different signs moving on two closely located layers. It is assumed that their movements are irreversible. Based on the TanakaMura model, the monotonic buildup of dislocation dipoles is systematically derived from the theory of continuously distributed dislocations. The number of stress cycles up to the nucleation of a crack about one grain diameter in length is reached when the self-strain energy of the accumulated dislocation dipoles reaches a critical value. The number of stress cycles Ng required for crack nucleation in a single grain can be determined as follows: Ng ¼

8GW c π ð1  νÞd ðΔτ  2CRSSÞ2

(3)

Equation (3) presumes that cracks form along slip bands within grains, depending on slip band length d and average shear stress range Δτ on the slip band. Slip band length represents the distance along the slip band between grain boundaries in a single grain. Other material constants (shear modulus G, specific fracture energy per unit area Wc, Poisson’s ratio ν, and frictional stress of dislocations on the slip plane, i.e., critical resolved shear stress CRSS) can be found in the specialized literature [23] or calculated by means of MD (CRSS). According to Nakai [39] the initiation conditions of small fatigue cracks still have not been clarified enough, since no method for successive, direct, and quantitative observation of the process had been devised. In the presented model, cracks nucleate sequentially but on the segmental level. In each simulating iteration, just one segment of a particular grain is broken. It means that in one iteration a segment belonging to one grain brakes, while in the following iteration, a segment of some other grain can break. In the presented case, the number of segments in each individual grain is equal to four. The total number of stress cycles Nini needed for the initiation of a small crack is calculated by summing the cycles spent to nucleate all cracks, including those coalesced to form the final small crack. Jezernik et al. [24] used the Tanaka-Mura model to numerically simulate the small crack formation process. Three improvements were added to this model: (a) multiple slip bands inside each crystal grain as potential sites for crack nucleation, (b) crack coalescence between two grains, and (c) segmented crack generation inside one grain. A numerical model was directed at simulating fatigue properties of thermally cut steel. The authors took into account accompanying residual stresses in order to simulate the properties of the thermally cut edge as faithfully as possible. As residual stresses are not taken into account in the original Tanaka-Mura model according to eq. (3), in the present study, they are imposed as additional external loading. The superpositioning principle in linear elastic micromechanics analysis has been applied by taking residual stresses as part of the total load into account which leads to the shear stress distribution shown in Fig. 4. Therefore, the residual stresses are implicitly

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Fig. 4 Micro-crack nucleation and subsequent coalescence [24]

evaluated in the Tanaka-Mura equation through the average shear stress range on the slip band Δτ. Figure 4 shows the shear stress distribution and nucleated cracks for a typical high-cycle fatigue regime load level (450 MPa). In the beginning, cracks tended to occur scattered in the model and form in larger grains that are favorably oriented and show higher shear stresses. But after a while, existing single grain cracks started coalescing, causing local stress concentrations and amplifying the likelihood of new cracks forming near already coalesced cracks. When calculating cycles required for crack initiation, no cycles were attributed to crack coalescences (it is simulated as being instantaneous). The total number of cycles of crack initiation equals the sum of cycles needed for each micro-crack to nucleate. When the crack depth is less than a critical value, the crack growth behavior has been found to be highly dependent upon the microstructure [40–42]. With increasing length, the growing cracks leave the originally 45 -oriented slip planes and tend to propagate perpendicular to the external stress axis. The change of the crack plane from the active slip plane to a non-crystallographic plane perpendicular to the stress axis is called the transition from Stage I (crystallographic propagation) to Stage II (non-crystallographic propagation) or transition from crack initiation to crack propagation. In Stage II of fatigue crack propagation, only one crack usually propagates, while most of other cracks usually stop within Stage I [15]. Stage II of fatigue crack propagation is simulated by using the power law crack propagation models based on the linear elastic fracture mechanics (LEFM).

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4

Modeling and Simulation of Crack Propagation in Welded Stiffened Panels

It is well-known that the residual stress in a welded stiffened panel is tensile along a welded stiffener and compressive in between the stiffeners. Residual stresses may significantly influence the stress intensity factor (SIF) values and fatigue crack growth rate. A total SIF value, Ktot, is contributed by the part due to the applied load, Kappl, and by the part due to welding residual stresses, Kres, as given by eq. (4): K tot ¼ K appl þ K res

(4)

The so-called residual stress intensity factor, Kres, is required in the prediction of fatigue crack growth rates. The considered analysis method is based on the superposition rule of LEFM. The finite element method has been widely employed for calculating SIFs. In the FE software package ANSYS [32], the command INISTATE is used for defining the initial stress conditions. For evaluating Kres, it is important to input correct initial stress conditions to numerical models in order to characterize residual stresses [28, 43]. The effective SIF range ΔKeff was considered in crack growth models in order to take crack closure effects on fatigue crack growth rate into account. The Elber [30] and the Donahue [31] crack growth models are employed to simulate fatigue lifetime for welded stiffened panel specimens. In the Donahue model, the effective SIF range values are calculated based on the applied load, without taking welding residual stresses into account. The Elber model takes into account both the applied load and welding residual stresses, and the effective SIF range values are calculated based on the effective SIF ratio Reff, which depends on the Ktot,max and Ktot,min values. It is important to determine the total SIF values Ktot accurately, to model effects of residual stresses on crack propagation rate.

4.1

Specimen’s Geometry and Loading Conditions

Fatigue tests with constant stress range and frequency were carried out on a stiffened panel specimen with a central crack [29]. The specimen geometry is shown in Fig. 5. The material properties and chemical composition of the used mild steel for welding are given in Table 1 [44]. Table 2 shows the fatigue test conditions applied in the experiment. The crosssectional area of the intact section and the average stress range away from the notch are denoted as A0 and Δσ, respectively. The force range and the stress ratio are denoted by ΔF = Fmax  Fmin and R = Fmin/Fmax, respectively. The applied stress range was Δσ = 80 MPa. The initial notch length was 2a = 8 mm and the loading frequency was 3 Hz. In the experiment the crack lengths were measured by using so-called crack gauges. The crack gauges are bonded to a specimen’s surface in front

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Fig. 5 Stiffened panel specimen Table 1 Material properties Mechanical properties E – Young’s modulus 206,000 MPa Chemical composition (%) Cmax Simax 0.18 0.35

ν – Poisson’s coefficient 0.3

σ o – Yield strength 235 MPa

Mnmin 0.70

Pmax 0.035

Smax 0.035

Table 2 Fatigue test conditions A0 [mm2] 1200

ΔF [N] 96,000

Δσ [MPa] 80

R 0.0204

of a crack tip in order to measure length of a growing crack with respect to applied number of loading cycles. Different from usual strain gauges, the grid of crack gauges is cut along with crack development, resulting in resistance change.

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4.2

Modeling of Welding Residual Stresses in a Stiffened Panel by Using FEM

The residual stress distribution implemented in present simulations follows Faulkner’s model [45] in which the tensile regions around the stiffeners are modeled as rectangular shapes with a base width proportional to the plate thickness. For ship structures the rectangular width typically ranges from 3.5 to 4 times the plate thickness. Dexter et al. [27] performed fatigue tests on half-scale welded stiffened panel specimens which model a part of ship deck structure. The authors measured welding residual stresses in stiffened panel specimens and observed that the measured stresses vary mostly between a rectangular and a triangular shape [27, 28]. Subsequent crack growth simulations showed that the residual stress distribution can be well described by a rectangular shape, where the maximum tensile stress equals to yield stress of the considered material and compressive stresses between the stiffeners provide the equilibrium of internal forces [28]. In this study the distribution of welding residual stresses in the stiffened panel specimen is taken into account in a similar manner as in the model developed by Mahmoud and Dexter [28]. Residual stress distribution depicted in Fig. 6 was utilized for the considered specimen. This model presents the tensile regions around the stiffeners as rectangular shapes with a base width equal to 10 mm and with a stress level equal to yield strength σ o = 235 MPa. Estimated base width is in good agreement with Dexter’s model [28], where approximately ¼ of the span between the two stiffeners is exposed to tension. Compared with Faulkner’s model, the base width used in this study is slightly shorter. The compressive residual stresses between the stiffeners and on the stiffeners were applied in the model to satisfy equilibrium. To evaluate the SIF value contributed by the residual stresses, Kres, it is important to input correct initial stress conditions in the numerical model. Experimental measurements [27, 28] showed that the profiles of welding residual stresses are almost identical along the axis parallel to the weld line. Correspondingly, in the FE model, elements with the same horizontal coordinate should have the same initial stress condition. For that purpose regular 1 mm size FE mesh was used, so that elements can be selected in columns and associated with the corresponding initial stress level as given in Fig. 6. Applying initial stresses for the used eight node shell elements is based on the element integration point. These initial stresses are equilibrated in the first analysis step. Due to the symmetry of specimen’s geometry and loading conditions, it was sufficient to model only one quarter of the specimen. In ANSYS software package, the command INISTATE was implemented to define assumed initial stress conditions [32]. Figure 7 shows the σ y component of the welding residual stresses in the stiffened panel specimen obtained for the implemented stress distribution as given in Fig. 6 (the σ y stress component acts parallel to the loading axis).

4.3

Stress Intensity Factors and Fatigue Crack Growth Rate

For evaluating SIFs by FEM, the crack tip displacement extrapolation method was implemented [32]. The SIFs were determined in a linear elastic FE analysis. In the

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Fig. 6 Welding residual stress distribution

FEM modeling, the crack tip region was meshed by singular elements. The procedure for the calculation of SIFs is based on the application of well-known “quarterpoint” elements introduced by Barsoum [46] and by Henshell and Shaw [47]. The near crack tip displacements and stresses of LEFM are usually related to the three fundamental deformation modes of fracture where mode I is the opening mode, mode II is the shearing mode, and mode III is the tearing mode [25, 32]. Opening displacement v in the vicinity of the crack tip, as depicted in Fig. 8, is given by eq. (5): rffiffiffiffiffi  KI r θ 3θ K II v¼ ð2κ þ 1Þ sin  sin þ 2 2 4G 2π 4G rffiffiffiffiffi  r θ 3θ  ð2κ  3Þ cos þ cos 2π 2 2

(5)

where KI and KII are the stress intensity factors associated with modes I and II. G is the shear modulus, υ is Poisson’s ratio, v is the displacement in loading direction, and

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Fig. 7 Welding residual stresses [MPa] in the stiffened panel specimen

Fig. 8 Crack opening profile for a half crack model

r and Θ are local polar coordinates. The conversion factor κ for plain stress conditions is given by eq. (6): κ ¼ ð3  νÞ=ð1 þ νÞ

(6)

In eq. (5) the higher-order terms are neglected, and the equation is therefore only valid in the vicinity of the crack tip. It should be noted that stress distribution is singular for r = 0. When eq. (5) is applied to the half crack configuration illustrated in Fig. 8, the displacements v, across the faces of the crack, are given by eq. (7):

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Multiscale Fatigue Crack Growth Modeling for Welded Stiffened Panels

KI v¼ 2G

rffiffiffiffiffi r ð1 þ κ Þ 2π

1205

(7)

In the next step is described how K is obtained by using finite element results and the theoretical equations given above. Attention is restricted to calculating the KI factor for mode I displacements. The displacement v on the crack face of the half crack configuration depicted in Fig. 8 can be approximated by pffiffi v= r ¼ A þ Br

(8)

where A and B are constants determined from a linear curve fit of nodal displacements. Once A and B are determined, the limit r ! 0 is taken:  pffiffi lim v= r ¼ A

r!0

(9)

By combining eqs. (5) and (9) for the half crack model, SIF is obtained as pffiffiffiffiffi 2G 2πA KI ¼ ð 1 þ κÞ

(10)

Based on the above described technique, in the general post-processing procedure, the KCALC command was used to calculate SIFs. The mode I SIF values, KI, are determined for a stiffened panel specimen for a loading stress range Δσ = 80 MPa, assuming the presence of residual stresses as described above. SIF values with respect to half crack length a are given in Fig. 9. Kappl represents the SIF values due to the applied stress range only, without residual stresses. Ktot represents the SIF values for the case when the residual stresses are taken into account along with the external loading stress range. It can be seen that residual stresses significantly increase Ktot values for shorter crack lengths, where tensile residual stresses prevail. Between the stiffeners residual stresses reduce the Ktot values. A well-known method for predicting fatigue crack propagation under constant stress range is the power law (11) introduced by Paris and Erdogan [26]. In eq. (11) da/dN and ΔK represent the crack growth rate and the stress intensity factor range, respectively. C and m are material constants which are determined experimentally: da ¼ CðΔK Þm dN

(11)

Elber [30] and Donahue et al. [31] further developed Paris’ law assuming an effective stress intensity factor range, ΔKeff, as the crack growth driving force parameter. Donahue et al. [31] defined the effective stress intensity factor range as ΔKeff = Kmax  Kth, where Kmax is the maximum SIF value in a loading cycle and Kth is a SIF threshold value below which no crack propagation occurs. Božić et al. [48, 49] used this model to analyze fatigue crack propagation in plates damaged with

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K (MPa m1/2)

Fig. 9 Ktot and Kappl values

50

Ktot

45

Kappl

40 35 30 25 20 15 10

0

0.02

0.04

0.06

0.08

a (m)

a single and multiple cracks, respectively. Assuming the stress ratio R = 0 as applied in the experiment, the threshold SIF value for the used mild steel was taken as Kth = 6.8 MPa [50]. The Donahue model predicted well fatigue lifetime and crack growth rate for centrally cracked unstiffened plate specimens without welds and with a constant stress ratio R = 0 [48, 49]. However, this model does not take into account variable stress ratio R, which occurs in welded specimens due to residual stresses and can significantly influence the fatigue crack growth rate. Elber [30] observed that crack closure decreases the fatigue crack growth rate by reducing the effective SIF range. He proposed an equation for the effective stress intensity factor range which takes the load ratio R into account: ΔK eff ¼ ð0:5 þ 0:4RÞΔK appl

(12)

In the present study, the nominal ratio R in eq. (12) was replaced by the effective SIF ratio Reff, in order to take into account the influence of welding residual stresses on fatigue crack propagation rate. The effective SIF ratio Reff is defined as Reff ¼

K tot, min K appl, min þ K res ¼ K tot, max K appl, max þ K res

(13)

This superposition method based on the principle of LEFM was originally proposed by Glinka [51] and was recently implemented in the finite element modeling of fatigue crack growth rate in welded butt joints by Servetti and Zhang [52]. As the crack propagates under cyclic loading, the effective SIF ratio Reff changes due to the presence of residual stresses. The number of constant amplitude loading cycles due to which a crack propagates from its initial crack length, a0, to a final crack length, afin, can be determined by the integration of eq. (11), which becomes

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Table 3 Material’s constants* Model Donahue Elber

C 6.50  1011 1.67  1010

m 2.75 2.75

*The units for ΔK and Δa/ΔN are [MPa  m1/2] and [m], respectively

a

0.08

b

Simulation Experiment

0.07 0.06

a (m)

0.04 0.03

0.05 0.04 0.03

0.02

0.02

0.01

0.01

0

Simulation Experiment

0.06

0.05

a (m)

0.08 0.07

0

1

2

N (cycles)

3

4

5 x 105

0

0

1

2

N (cycles)

3

4

5 x 105

Fig. 10 Fatigue crack growth life for the applied stress range Δσ o = 80 MPa: (a) without residual stresses and (b) including residual stresses

N¼

ð afin a0

da C½ΔK eff m

(14)

The integration of eq. (14) can be performed numerically. Fatigue life was simulated for the specimen by integrating eq. (14), where the material constants C and m were as given in Table 3. The exponent m for the considered material was determined in a previous study by means of crack growth rate diagrams based on a-N data obtained for centrally cracked plate specimens [45]. The C constants from Table 3 were estimated in a way to provide a good agreement of simulated fatigue lifetime with experimentally obtained a-N curve. The Donahue model was implemented to the welded stiffened panel specimen, considering the effective SIF range values due to applied load only, ΔKeff = Kappl  Kth, without taking into account residual stresses. The Elber model takes into account the influence of residual stresses on fatigue crack growth rate by using the effective SIF range defined by eq. (12) and the effective SIF ratio Reff given by eq. (13). For the two cases considered, the fatigue crack propagation life was obtained as shown in Fig. 10a and b. The presented measured crack lengths a are to be considered as averaged half crack lengths. These values are obtained by averaging measured crack lengths of the two propagating crack tips with respect to applied number of cycles N. The averaged half crack lengths are used in order to be comparable with the simulation results for semicrack lengths obtained by the FE model where only one quarter of the specimen was modeled.

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The FE analysis for the stiffened panel specimens showed that high tensile residual stresses in the vicinity of a stiffener significantly increase Kres and Ktot, as shown in Fig. 9. Correspondingly, the simulated crack growth rate was higher in this region, which is in good agreement with experimental results, as can be seen in Fig. 10b. Compressive welding residual stresses decreased the total SIF value Ktot. The Donahue model, which does not take account of welding residual stresses, could not simulate high crack growth rates in the vicinity of the stiffener, as can be seen in Fig. 10a. Fatigue crack growth simulation based on the Elber model, which takes into account the welding residual stresses, provides thus better agreement with experimental results in terms of crack growth rate and total number of cycles. In conclusion, residual stresses in welded stiffened panels should be taken into account for a proper evaluation of SIFs and fatigue crack growth rates. In further work one should do microstructural analyses in experiment and simulation (micro and nano) in order to foster the multiscale procedure and use the method to predict materials with improved fatigue properties in the future.

5

Conclusions

Simulation of cyclic loading to model dislocation nucleation as an initial step in fatigue initiation is possible in MD. Already after the very few cycles, essential changes in the system behavior were observed under respective loading conditions. Contrary to the first cycles, where reversible changes were dominant, not dissolving restructuring occurs in the sense of dislocations and remaining lattice defects or in other words, plasticity. Numerical simulations of the fatigue crack initiation and growth of martensitic steel, based on a modified Tanaka-Mura model, were presented. A simulation model related to the micro-crack nucleation along slip bands was presented. Results obtained by using the proposed simulation model were compared to high cyclic fatigue tests and showed reasonably good agreement. Crack propagation simulation based on numerical integration of a power law equation, taking account of welding residual stresses, was implemented to welded stiffened panel specimens. The FE analysis of the stiffened panel specimens showed that high tensile residual stresses in the vicinity of a stiffener significantly increase Kres and Ktot. The simulated crack growth rate was higher in this region, which is in good agreement with experimental results. Compressive welding residual stresses decreased the total SIF value Ktot and the crack growth rate between the two stiffeners. Residual stresses should thus be taken into account for a proper evaluation of SIFs and fatigue crack growth rates in welded stiffened panels. Acknowledgments This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. Schm 746/132-1 and as part of the Collaborative Research Centre SFB 716 at the University of Stuttgart and by the Croatian Science Foundation Grant No. 120-0362321-2198. The support is gratefully acknowledged.

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Nomenclature

a a0 afin C CRSS d da/dN E
 ! F r ,t

half crack length initial crack length final crack length material constant of the Paris equation critical resolved shear stress slip band length crack growth rate Young’s modulus interatomic force

Fmax Fmin G K Kappl Kres Kth Ktot m m N Nf Ng

maximum applied force minimum applied force shear modulus stress intensity factor (SIF) stress intensity factor due to the applied load stress intensity factor due to welding residual stresses stress intensity factor threshold total stress intensity factor atomic mass material constant of the Paris eq. number of stress cycles for the fatigue crack propagation number of stress cycles for fatigue failure number of stress cycles required for crack nucleation in a single grain number of stress cycles needed for the initiation of a small crack stress ratio effective stress intensity factor ratio interatomic embedded atom method (EAM) pair potential

Nini R Reff


!



U r ,t Wc ΔF ΔK ΔKeff Δσ Δτ σ0

specific fracture energy per unit area applied force range stress intensity factor range effective stress intensity factor range average applied stress range average shear stress range on the slip band yield stress

References 1. Luo C, Chattopadhyay A. Prediction of fatigue crack initial stage based on a multiscale damage criterion. Int J Fatigue. 2011;33:403–13. 2. Curtin WA, Deshpande VS, Needleman A, Van der Giessen E, Wallin M. Hybrid discrete dislocation models for fatigue crack growth. Int J Fatigue. 2010;32:1511–20.

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3. White P. Molecular dynamic modelling of fatigue crack growth in aluminium using LEFM boundary conditions. Int J Fatigue. 2012;44:141–50. 4. Horstemeyer MF, Farkas D, Kim S, Tang T, Potirniche G. Nanostructurally small cracks (NSC): a review on atomistic modeling of fatigue. Int J Fatigue. 2010;32:1473–502. 5. Božić Ž, Schmauder S, Mlikota M, Hummel M. Fatigue Crack Growth Modelling in Welded Stiffened Panels under Cyclic Tension, 13th International Conference on Fracture, Beijing, China. 2013. 6. Stadler J, Mikulla R, Trebin HR. IMD: a software package for molecular dynamics studies on parallel computers. Int J Mod Phys. 1997;8:1131. 7. Bonny G, Pasianot RC, Castin N, Malerba L. Ternary Fe-cu-Ni many-body potential to model reactor pressure vessel steels: first validation by simulated thermal annealing. Phil Mag. 2009;89:3531–46. 8. Grottel S, Reina G, Dachsbacher C, Ertl T. Coherent culling and shading for large molecular dynamics visualization. Computer Graphics Forum Proc of EUROVIS 2010. 2010;29(3):953–62. 9. Stukowski A, Bulatov VV, Arsenlis A. Automated identification and indexing of dislocations in crystal interfaces. Modelling Simul Mater Sci Eng. 2012;20:085007. 10. Stukowski A. DXA user manual Version 1.3.4; 2010. http://dxa.ovito.org/README.txt. 11. Glodez S, Jezernik N, Kramberger J, Lassen T. Numerical modelling of fatigue crack initiation of martensitic steel. Adv Eng Softw. 2010;41(5):823–9. 12. Wood WA. Fatigue in aircraft structures. New York: Academic Press; 1956. 13. Fine ME, Ritchie RO. Fatigue-crack initiation and near-threshold crack growth. In: Meshii M, editor. Fatigue and microstructure. Materials Park: ASM; 1978. p. 245–78. 14. Laird C. Mechanisms and theories of fatigue. In: Meshii M, editor. Fatigue and microstructure. Materials Park: ASM; 1978. p. 149–203. 15. Klesnil M, Lukas P. Fatigue of metallic materials. New York: Elsevier; 1980. p. 57–80. 16. Mughrabi H. Rev Phys Appl. 1988;23:367–79. 17. Mughrabi H. In: Chan KS, Liaw PK, Bellows RS, Zogas T, Soboyejo WO, editors. Fatigue: David L. Davidson symposium. Warrendale: TMS; 2002. p. 3–15. 18. Davidson DL, Chan KS. Crystallography of fatigue crack initiation in astrology at ambient temperature. Acta Metall. 1989;37(4):1089–97. 19. Wang QY, Bathias C, Kawagoishi N, Chen Q. Effect of inclusion on subsurface crack initiation and gigacycle fatigue strength. Int J Fatigue. 2002;24(12):1269–74. 20. Murakami Y, Nomoto T, Ueda T. On the mechanism of fatigue failure in the superlong life regime (N>107 cycles). Part 1: influence of hydrogen trapped by inclusions. Fatigue Fract Engng Mater Struct. 2000;23(11):893–902. 21. Tanaka K, Mura T. A dislocation model for fatigue crack initiation. J Appl Mech. 1981;48:97–103. 22. Tanaka K, Mura T. A theory of fatigue crack initiation at inclusions. Metall Trans A. 1982;13(1):117–23. 23. Brückner-Foit A, Huang X. Numerical simulation of micro-crack initiation of martensitic steel under fatigue loading. Int J Fatigue. 2006;28(9):963–71. 24. Jezernik N, Kramberger J, Lassen T, Glodez S. Numerical modelling of fatigue crack initiation and growth of martensitic steels. Fatigue & Fracture of Engineering Materials & Structures. 2010;33:714–23. 25. Broek D. The practical use of fracture mechanics. Dordrecht: Kluwer Academic Publishers; 1989. 26. Paris P, Erdogan F. A critical analysis of crack propagation laws. J Basic Eng. 1963;85:528–34. 27. Dexter RJ, Pilarski PJ, Mahmoud HN. Analysis of crack propagation in welded stiffened panels. Int J Fatigue. 2003;25:1169–74. 28. Mahmoud HN, Dexter RJ. Propagation rate of large cracks in stiffened panels under tension loading. Mar Struct. 2005;18:265–88.

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29. Sumi Y, Božić Ž, Iyama H, Kawamura Y. Multiple fatigue cracks propagating in a stiffened panel. Journal of the Society of Naval Architects of Japan. 1996;179:407–12. 30. Elber W. The significance of fatigue crack closure, Damage tolerance in aircraft structures. ASTM STP 486. American Society for Testing & Materials; 1971. p. 230–242. 31. Donahue RJ, Clark HM, Atanmo P, Kumble R, McEvily AJ. Crack opening displacement and the rate of fatigue crack growth. Int J Fract Mech. 1972;8:209–19. 32. Swanson Analysis System (2009). Inc. ANSYS User’s Manual Revision 11.0. 33. Han T, Luo Y, Wang C. Effects of temperature and strain rate on the mechanical properties of hexagonal boron nitride nanosheets. J Phys D Appl Phys. 2014;47:025303. 34. Tapasa K, Bacon DJ, Osetsky YN. Simulation of dislocation glide in dilute Fe-cu alloys. Materials Science & Engineering A. 2005;400-401:109–13. 35. Kohler C, Kizler P, Schmauder S. Atomistic simulation of precipitation hardening in α-iron: influence of precipitate shape and chemical composition. Model Simul Mater Sci Eng. 2005;13:35–45. 36. Molnar D, et al.. Unpublished research. 2014. 37. Naveen Kumar N, Durgaprasad PV, Dutta BK, Dey GK. Modeling of radiation hardening in ferritic/martensitic steel using multi-scale approach. Comput Mater Sci. 2012;53:258–67. 38. Latapie A, Farkas D. Molecular dynamics simulations of stress-induced phase transformations and grain nucleation at crack tips in Fe. Modelling Simul. Mater. Sci. Eng. 2003;11:745–53. 39. Nakai Y. Evaluation of fatigue damage and fatigue crack initiation process by means of atomicforce microscopy. Mater Sci Res Int. 2001;7(2):1–9. 40. Zabett A, Plumtree A. Microstructural effects on the small fatigue crack behaviour of an aluminum alloy plate. Fatigue & Fracture of Engineering Materials & Structures. 1995;18(7–8):801–9. 41. Taylor D, Knott JF. Fatigue crack propagation behaviour of short cracks; the effect of microstructure. Fatigue & Fracture of Engineering Materials & Structures. 1981;4(2):147–55. 42. Miller KJ. The behaviour of short fatigue cracks and their initiation part II-A general summary. Fatigue & Fracture of Engineering Materials & Structures. 1987;10(2):93–113. 43. Bao R, Zhang X, Yahaya NA. Evaluating stress intensity factors due to weld residual stresses by the weight function and finite element methods. Eng Fract Mech. 2010;77:2550–66. 44. Croatian Register of Shipping. Rules for the Classification of Ships, Part 25 – Metallic Materials. 2012. 45. Faulkner D. A review of effective plating for use in the analysis of stiffened plating in bending and compression. J Ship Res. 1975;19:1–17. 46. Barsoum RS. On the use of Isoparametric finite elements in linear fracture mechanics. Int J Numer Methods Eng. 1976;10:25–37. 47. Henshell RD, Shaw KG. Crack tip finite elements are unnecessary. Int J Numer Methods Eng. 1975;9:495–507. 48. Božić Ž, Mlikota M, Schmauder S. Application of the ΔK, ΔJ and ΔCTOD parameters in fatigue crack growth modelling, Technical. Gazette. 2011;18(3):459–66. 49. Božić Ž, Schmauder S, Mlikota M. Fatigue growth models for multiple long cracks in plates under cyclic tension based on ΔKI, ΔJ-integral and ΔCTOD parameter. Key Eng Mater. 2012;488-489:525–8. 50. Liu Y, Mahadevan S. Threshold stress intensity factor and crack growth rate prediction under mixed-mode loading. Eng Fract Mech. 2007;74:332–45. 51. Glinka G. Effect of residual stresses on fatigue crack growth in steel weldments under constant and variable amplitude load, Fracture mechanics, ASTM STP 677, American Society for Testing and Materials; 1979. p. 198–214. 52. Servetti G, Zhang X. Predicting fatigue crack growth rate in a welded butt joint: the role of effective R ratio in accounting for residual stress effect. Engng Fract Mech. 2009;76:1589–602.

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element Analysis

38

Tetsuya Ohashi

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Plastic Shear Strain as a Result of Dislocation Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Dislocation Accumulation Due to Plastic Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Statistically Stored Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Geometrically Necessary Dislocations [2, 4, 16, 17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Models for the Critical Resolved Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Example of Finite Element Crystal Plasticity Analysis [24] . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Dislocations play a major role in plastic deformation and fracture of metallic materials. A number of metallographic aspects such as grain boundaries, precipitates, and others contribute to the dislocations’ behavior, and therefore, we have to consider their effects too when we intend to understand the mechanical behavior of metals with microstructure. Finite element method is a powerful tool to express the shape and arrangement of metal microstructures and analyze the deformation under a prescribed boundary and loading conditions. We tried to develop models for the movement, interaction, and accumulation of dislocations during plastic slip deformation in metal microstructure and implemented them to the framework of finite element method. Models originate from physics of discrete dislocations and are brought to dislocation density-based numerical models. In this chapter, the physical pictures and expressions of dislocation density-based models are shown. Some examples of analyses are also shown.

T. Ohashi (*) Kitami Institute of Technology, Kitami, Hokkaido, Japan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_74

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T. Ohashi

Introduction

Dislocations play a major role in plastic deformation and fracture of metallic materials. A number of metallographic aspects such as grain boundaries, precipitates, and others contribute to the dislocations’ behavior, and therefore, we have to consider their effects too when we intend to understand the mechanical behavior of metals with microstructure. Finite element method is a powerful tool to express the shape and arrangement of metal microstructures and analyze the deformation under a prescribed boundary and loading conditions. We tried to develop models for the movement, interaction, and accumulation of dislocations during plastic slip deformation in metal microstructure and implemented them to the framework of finite element method. Models originate from physics of discrete dislocations and are brought to dislocation density-based numerical models. In this chapter, the physical pictures and expressions of dislocation density-based models are shown. Some examples of analyses are also shown.

2

Plastic Shear Strain as a Result of Dislocation Movement

It is well known that dislocations are line defects in crystals. While grain boundaries are lattice defects expanding as interfaces in the microstructure, dislocations are the defects extending as lines. Dislocation lines can change their position under applied stress, and this is called the movement of dislocations, as if the dislocations are some kind of substances. There is a compilation of experimental results on the dislocation velocity v and applied shear stress τ for face-centered cubic and hexagonal close-packed metal crystals, and the following expression is considered to be a good approximation [1]: v ¼ Aτm , m  1:

(1)

Above equation holds when the applied stress exceeds a threshold value and the velocity rapidly decreases when the stress is smaller than the value. This lower limit of the shear stress is needed to obtain the dislocation velocity of approximately 1 m/ s, by which macroscopic slip deformation is realized. Physical background for that the exponent m is nearly equal to unity stands on the fact that emission of elastodynamic wave dissipates free energy of the system. Because the exponent is nearly equal to unity, the movement of dislocations is considered to be viscous. Let us consider the case when a number of linear dislocations parallel to each other move in the direction perpendicular to the sense of dislocation line. Spatial density of dislocations, ρ, is defined by the total length of dislocation lines in unit volume, and this is equal to the number of threading dislocations in unit area perpendicular to the dislocation lines. When these dislocations move, resultant plastic shear strain γ is given by ~ γ ¼ ρbx,

(2)

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Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

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where b~ and x denote the magnitude of the Burgers vector and the average of travel distance of dislocations, respectively. Equation (2) is called the Orowan equation, and when the average velocity of the dislocations is v, the strain rate is ~ γ_ ¼ ρbv:

(3)

After a free movement of dislocations under applied stress, all dislocations may go out from the specimen surface and no dislocation is left. In this case, no further plastic deformation by the movement of dislocations can take place. However in reality, this does not happen. In well-annealed crystals, dislocation lines are divided into segments and construct three-dimensional network. Only a part of dislocation segments lie on crystallographic planes where they can move. Under the application of shear stress, dislocation segments on the favorable crystal planes, which is called slip plane, move as shown in Fig. 1a. The dislocation segment in Fig. 1a is pinned at points A and B and expands on the slip plane as shown in Fig. 1c. If the applied stress is sufficiently large, the pinned segment will emit dislocation loops repeatedly through the process shown in Fig. 1b–f. It should be noted that the parts of curved dislocation Fig. 1 Schematic illustration of the movement of dislocation curve from a Frank-Read source. Reproduced from a textbook by Hull and Bacon [1]. (a) A dislocation segment AB lie on slip plane and other part of the dislocation line does not. The segment AB bows out on the slip plane under a shear stress, while points A and B are pinned and cannot move. (c–f) Dislocation line sweeps the shadowed area. When the applied stress is large enough, a closed loop is emitted and a dislocation segment is formed again between A and B

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line labeled m and n in Fig. 1e have opposite sign to each other and annihilate when they impinge. Shape of the dislocation lines after this annihilation is shown in Fig. 1f where a dislocation loop is emitted and a pinned segment is formed again. This mechanism is called Frank-Read mechanism, and the pinned segment is named the Frank-Read source. The minimum stress needed for this dislocation multiplication is called Orowan stress and approximately given by τ1 ¼

μb~ , λ

(4)

where μ denotes elastic shear modulus. Equation (4) means that the larger the λ is, the smaller the Orowan stress is. Therefore, the onset of plastic slip deformation in polycrystals will be brought by Frank-Read source with a large λ. However, grain boundaries and other metallographic factors contribute to the emission of dislocation loops when the microstructure length scale, such as the grain size or spacing of plastically less favorable phases embedded in crystal grains, is comparable to λ. When a dislocation line bows out from the Frank-Read source but is obstructed its free movement by the microstructure, some part of the dislocation line is forced to bend. Dislocation is a defect structure and accompanies strain energy. This results in an apparent tension in dislocation lines and a larger stress is needed to bend. Due to this reason, the most suitable sizes of Frank-Read sources to operate inside a crystal grain depend on the grain size. Similarly when one or more harder phase(s) disperse in crystal grains, the most suitable size of the Frank-Read source will be a function of the geometry and arrangement of these phases. Let us assume a cube-shaped single crystal specimen shown in Fig. 2a [2]. Six surfaces of the specimen correspond to grain boundaries, and movement of dislocations is confined to the interior of the specimen. The dimension of the slip plane that is trimmed by four grain boundary planes is d  d, and a dislocation segment of length λ is placed on it as shown in Fig. 2b. Both ends of the dislocation segment are pinned and cannot move. Movement and growth of the dislocation line were numerically simulated by a discrete dislocation dynamics simulation code [3]. We applied a constant shear stress of τ to the specimen. Figure 2c–e shows snapshots of dislocation movement for the case when λ = 0.2 μm, d = 1 μm, and τ = 1.027 τ1. Some parts of the dislocation line hit grain boundary planes (Fig. 2d), but the dislocation continues to expand and makes up a closed loop (Fig. 2e). Figure 3 shows the results when λ = 0.5 μm, d = 1 μm, and τ = 1.9 τ1. In contrast to the results shown in Fig. 2, growth of the dislocation arc stops after it hits grain boundaries and makes the shape shown in Fig. 3. Larger stress is needed to form a closed dislocation loop. Let us assume a crystal grain with a certain grain size, and there can be FR sources with any size smaller than the grain. Applied stress gradually increases from zero. On one hand, the FR source with the largest size will become activated before the smaller ones, because τ1 is inversely proportional to the size λ of the FR source. On the other hand, a large FR source with a size as large as the grain diameter cannot fully operate to form a closed loop at an initial stage of yield, since some parts of the

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Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

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Fig. 2 Simulation of dislocation loop emission by a discrete dislocation dynamics simulation. (a) Cuboidal-shaped crystal grain employed for the simulation. A dislocation segment aa’ lies on the slip plane and operates as a Frank-Read source. (b) Geometry of the slip plane, Frank-Read source and Burgers vector. (c–e) Growth of the dislocation line when d = 1 and λ = 0.2 μm and τ = 1.027τ1 is applied Fig. 3 Expansion of a dislocation arc when d/λ = 2 and τ = 1.9τ1

dislocation arc will hit the grain boundaries at an early stage of its growth and it cannot fully grow until shear stress much larger than τ1 is applied and, before this happen, smaller FR sources will be activated. The stress needed to form a closed loop is also dependent on the position of the FR source. Figure 4 shows the result when sizes for the FR source and the grain and the applied stress are the same to the ones used in Fig. 2. When the FR source is positioned close to the grain boundary, the minimum stress to emit a closed loop is larger.

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Fig. 4 Growth of dislocation arc that is generated from an FR source near the grain boundary. d = 1 and λ = 0.2 μm and τ = 1.02τ1 is applied

This discussion indicates that there is an optimum size of FR source between zero and the grain size, from which a fully developed dislocation loop can be formed at the minimum stress level. Such a FR source should be placed somewhere near the center of the grain. We simulate dislocation emission from a FR source that is placed at the center of the grain. Various combinations of sizes for FR source and grain are chosen. The minimum shear stress τthresh at which the FR source emits one closed loop is searched by dislocation dynamics simulation for various conditions of grain and FR source sizes (Fig. 5). Details of the simulations are described in [2]. The minimum shear stress needed to emit a dislocation loop is then given as a function of grain size d by τloff ¼ β

μb~ μb~ ¼ 3β , ðd=3Þ d

(5)

when there are no obstacles to the expansion of a dislocation arc other than grain boundaries. Coefficient β, which is defined by the ratio of the level-off stress and τ1, is dependent on the FR source length λ and other factors, but its dependence on λ is weak and can be approximated to be unity. Equation (5) gives the Orowan stress for dislocation loop emission where microstructure length scale contributes. In general, density of dislocations ranges from 1012 to 1015 m2 in structural metals. When the density is 1012 m2, for example, the total length of dislocations included in a cube-shaped crystal grain with lateral dimension of 10 μm is 1000 μm. In such case, movement and accumulation of dislocations as well as the strain hardening characteristics of slip systems should be constructed with physical pictures on the behavior of individual dislocation lines and also their collective behavior. On the other hand, when the grain diameter is 1 μm and the dislocation density is 1012 m2, for example, there is only one dislocation line in each grain. In such case, movement, multiplication, and accumulation of a single dislocation and the effect of grain boundaries are more important.

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

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Fig. 5 Threshold stress for the emission of a dislocation loop plotted as against the ratio of the grain diameter to the FR source length. When the ratio is smaller than a value, which is close to 3, the threshold stress increases rapidly

3

Dislocation Accumulation Due to Plastic Slip

Let us consider dislocation accumulation due to plastic slip deformation assuming that the grain diameter is relatively large. We consider two kinds of dislocation accumulations. One is “statistically stored” and the other is “geometrically necessary” dislocations [4].

3.1

Statistically Stored Dislocations

Let us consider a cube-shaped crystal grain with lateral dimension W, and n dislocation loops of rectangular shape are emitted from a dislocation source as shown in Fig. 6. If their movement sweeps an area of ΔA, the increment of plastic shear strain Δγ is given by Δγ ¼

b~ ΔA : W W2

(6)

If we denote the dimensions of the rectangular loop by d and αd, ΔA ¼ n  αd2 :

(7)

The mean travel distance of dislocation lines is
 d αd dð1 þ αÞ þ : L¼ =2 ¼ 2 2 4

(8)

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a

b

dislocation loops

W

+

ad

d W W Fig. 6 (a) Emission of rectangular-shaped dislocation loops in a cube shaped crystal with lateral dimension W. (b) Shape of the dislocation loop with aspect ratio α

Increment of the dislocation density after this slip process is given by Δρ ¼ ð2αd þ 2dÞ  n=W 3 :

(9)

Substitution of Eqs. (6, 7, and 8) into Eq. (9) yields Δρ ¼

ð1 þ αÞ2 1 Δγ: ~ 2α bL

(10)

Equation (10) gives the relation between increments of the dislocation density and plastic shear strain on a slip system. The dislocation density evaluated in this way is called the statistically stored dislocations or SS dislocations. The mean travel distance of dislocations, which is often called as the dislocation mean free path, will be discussed later. The aspect ratio α of dislocation loops is introduced to express the difference of velocities of edge and screw dislocations [1]. The minimum value 2 of the shape function (1 þ α)2/2α is obtained when α = 1, and the function is approximately given by a linear function of 1 þ α/2 when the aspect ratio is much larger than unity. The shape function is shown in Fig. 7. Let us generalize the Eq. (10) as follows: ðnÞ

dρS ¼ c

1 dγ ðnÞ : ~ bLðnÞ

(11)

Here, γ (n), ρs(n), and L(n) are plastic shear strain, density of the SS dislocations, and dislocation mean free path on the slip system n, respectively. The statistically stored dislocations consist of equal amount of plus and minus dislocations, and

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Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

Shape function, (1+a)2/2a

Fig. 7 Variation of the shape function in Eq. (10) with the aspect ratio of dislocation loops

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14 12 10 8 6 4 2 0

5

10

15

20

Aspect ratio of dislocation loops, a

the net Burgers vector of the SS dislocations is zero. Associated elastic field of dislocation lines with plus and minus signs cancels each other, and no long-range stress field is formed, but only short-range field remains. There is a possibility that positive and negative dislocations annihilate each other after a thermal activation process especially when the SS dislocation density is high, and this recovery effect is introduced in the model as follows [5, 6]: ðnÞ dρS


¼

 c D ðnÞ  ρ dγ ðnÞ : ~ ðnÞ b~ S bL

(12)

Parameter D defines the frequency of the dislocation annihilation and has the unit of length. Right-hand terms of Eq. (12) consist of the accumulation of dislocations after the movement of distance L, which is considered to have geometrical or statistical nature and thus athermal, and their annihilation process influenced by thermal activation [6]. Annihilation can take place not only by pair annihilation of plus and minus dislocations but also by other atomistic process such as absorption of dislocations by grain boundaries. Therefore, the parameter D is a function of temperature, strain rate, stress state, and other factors:   ^ ... , D ¼ D γ, _ T, σ,d,

(13)

where d^ denotes a microstructural length scale. Details of the function D are discussed recently [7], however, not yet fully incorporated into schemes of numerical analyses. The mean free path L(n) of moving dislocations depends on the character and density of accumulated dislocations in the material. Let us imagine a simple case when all the accumulated dislocations are parallel to the moving dislocations. In this case the moving dislocation does not intersect with the accumulated ones, and the mean free path will be quite large because the interaction between dislocations is delivered only through elastic field. Accumulated dislocations after their movement also interact with other moving dislocations during the subsequent slip process;

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however, the interaction will remain to be weak. This picture guides us to a model that the mean free path does not evolve with slip deformation but kept constant. If this is true, integration of Eq. (11) gives us that the accumulated dislocation density is proportional to the shear strain. When the geometrical arrangement of the accumulated dislocations is that they intersect with the moving ones, interaction of the accumulated and moving dislocations will be stronger and the mean free path will be shorter, accordingly. This discussion can be more precise if we consider the reaction of moving and accumulated dislocations. Details of the dislocation interaction will be discussed in later section, but the following model is one possibility in this direction [8–10]: c LðiÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    X  ðjÞ  ðjÞ wðijÞ ρS þ ρG 

(14)

j

Here, the mean free path of dislocations on slip system i is determined by the density of dislocations   not only on the slip system i but also by the densities  ðnÞ  on other systems. ρG  denotes the density norm of the geometrically necessary dislocations, on which we will discuss later. w(ij) gives the weight of contribution from accumulated dislocations on j-th slip system on the mean free path of dislocations on the i-th system. Larger density of accumulated dislocations results in shorter mean free path, but the contribution from accumulated dislocation on individual slip system is controlled by the weight w(ij), and this weight could be determined in relation to the dislocation reaction. c* is a numerical constant which could range pffiffiffi from 10 to 100 [11], and 1= ρ gives the average spacing of accumulated dislocations with density ρ. Therefore, Eq. (14) corresponds to a physical picture that the mean free path is given by the effective average spacing of accumulated dislocations multiplied by the factor c*. The effectiveness of the accumulated dislocations on different slip systems is controlled by the weight w(ij). Let us consider a simple situation where only a slip system i is active and the weight is given by w(ij) = 1  δij. δij denotes the Kronecker delta and its value is 1 when i = j and 0 otherwise. Dislocation densities on all the slip system, except the i-th system, are kept constant during deformation because slip takes place only on the i-th system. The mean free path for the i-th system is kept constant since w(ii) = 0, and this will result in a linear increase of the dislocation density. If we further apply this result to the Taylor hardening rule [12, 13], we will obtain a parabolic relation between plastic shear strain and the plastic flow stress. When a multiple slip starts, the mean free path on the i-th system will decrease due to the evolution of the dislocation density on j(6¼i)-th system and the strain hardening rate will be larger. When the microstructure length scale, such as the grain diameter, is at the same order to the mean spacing of accumulated dislocations, the mean free path will depend also on the length scale. We introduced the following model [2, 14]:

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Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

2

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3

6 7 6 7 c 7 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LðiÞ ¼ Min6 , d    6 X 7,  ðjÞ  ðjÞ 4 5 ðijÞ w ρ þ ρ  S

(15)

G

j

where d* is the representative length scale of the microstructure and the function Min[. . .] selects the minimum value from the members listed in the parenthesis. When d* is smaller than the effective average spacing of accumulated dislocations, the mean free path is equal to d*. As an example, let us imagine a case when c* = 10 and the weighted sum of the accumulated dislocation density is 1014 m2. The first term in the right-hand side of Eq. (15) is 1 μm in this situation. Therefore, the accumulated dislocations determine the mean free path when the microstructure length scale d* is larger than 1 μm, while the microstructure determines the mean free path when the length scale is smaller than 1 μm. The following model is somewhat similar to Eq. (15): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    X  ðjÞ  ð jÞ wðijÞ ρS þ ρG  1 ¼ LðiÞ

j

c

þ

1 d

(16)

Figure 8a shows the mean free path of dislocations plotted against the weighted sum of dislocation densities when c* = 10 and d* = 10 μm. Difference between the results obtained by Eqs. (15) and (16) is small, and the decrease of the mean free path is moderate after the dislocation density is larger than 1x1014 m2. Figure 8b shows the evolution of the mean free path when d* = 1 μm. In contrast to the results shown in Fig. 8a, the mean free path obtained by Eqs. (15) and (16) differs largely. This means that selection of mean free path model is sensitive when the microstructure length scale d* is small. Various crystallographic factors will contribute to d*. For single-phase polycrystals with equiaxial grains, it could be natural to use a fraction of the grain diameter for d*; however, it is not straightforward to determine when the grain shape deviates from the equiaxial one to pancake, needle, or more complicated shape [15]. For materials with fine particles dispersed in crystal grains, it is somewhat simpler; we took the average spacing of dispersed particles for d*, and this enabled us to reproduce macroscopic strain hardening ratio by numerical simulations [14].

3.2

Geometrically Necessary Dislocations [2, 4, 16, 17]

Let us consider a rectangular shaped single crystal specimen and divide into four segments. Figure 9 schematically illustrates movement of dislocations and distribution of resultant plastic shear strain. Dislocations are supposed to move from left to right and pass the segments 1 and 2 and positioned inside the segment 3. Plastic

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a

5 Dislocation mean free path, mm

Fig. 8 Dislocation mean free path plotted against the weighted sum of the accumulated dislocation density when the microstructure length scale d* =10 μm (a) and 1 μm (b)

T. Ohashi

c*=10, d*=10 m

4

Eq. (15) Eq. (16)

3 2 1 0 0

Dislocation mean free path, mm

b

2 4 6 Weighted sum of accumulated dislocation density, m–2

8x 1014

1.2 c*=10, d*=1 m

1

Eq. (15)

0.8

Eq. (16)

0.6 0.4 0.2 0 0

2 4 6 Weighted sum of accumulated dislocation density, m–2

8 x 1014

shear strain in the segments 1 and 2 is a constant because the dislocations have already passed through those. Dislocations are not yet entering into the segment 4 and the plastic shear strain is zero. In the segment 3, the plastic shear strain varies from a finite value to zero, and dislocations are there. That is, dislocations exist where the plastic shear strain changes spatially. These dislocations are called geometrically necessary dislocations, or GNDs. Let us next consider in a three-dimensional space. In Fig. 10a, ξ axis is parallel to the slip direction and the slip takes on ξ-ζ plane. If we denote the Burgers vector and the slip plane normal vector as b and n, respectively, ξ and ζ axes are parallel to b and bxn, respectively.When a dislocation loop a-b-c-d is emitted, plastic shear strain is a constant inside this loop and zero outside. Line profile of the plastic shear strain along AA’ and BB’ will be something like those shown in Fig. 10b and c. Dislocation segments a–b and d–c lie perpendicular to the slip direction, as shown in Fig. 10a, and therefore, these segments are edge dislocations of opposite sign. Similarly, segments a–d and c–b are parallel to the slip direction, and these are screw dislocations of opposite sign. This consideration shows that edge and screw components of the geometrically necessary dislocations are accompanied by the spatial gradient of plastic shear strain in the direction ξ and ζ, respectively. Density components of the GNDs are obtained after some quantitative analysis as follows:

38

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

1225

a

1

2

3

4

Plastic shear strain

b

Distance

Fig. 9 Schematic illustrations of geometrically necessary dislocations and distribution of plastic shear strain. (a) Dislocations pass through the segments 1 and 2 of a crystal and exist in the segment 3. (b) Distribution of the plastic shear strain after the movement of dislocations. The strain is a constant in the segments 1 and 2 and there is a spatial gradient in the segment 3. Dislocations exist where the plastic shear strain vary

b A' S

S

A Positive screw segment

Negative screw segment

a

c B' a

B' Positive edge segment

b b bxn

A

A'

Negative edge segment

c

d B

B

Fig. 10 (a) Schematic illustration of a rectangular-shaped dislocation loop a-b-c-d emitted on a slip plane parallel to ξ-ζ plane. Slip direction is assumed to be parallel to ξ axis. (b, c) Distribution profiles of plastic shear strain along AA’ and BB’, respectively. Dislocations with positive and negative signs exist where there are strain gradients.

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T. Ohashi

ρg, edge ¼ 

1 @γ ðnÞ , ðnÞ be @ξ

(17a)

ρðgn,Þscrew ¼

1 @γ ðnÞ : b~ @ζ ðnÞ

(17b)

ðnÞ

Here, ξ(n) and ζ(n) denote the direction parallel to b and bxn of the slip system n, respectively. These components   can take positive and negative values. Norm of the  ðnÞ  GND density denoted as ρG  is from Eq. (17):   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  ðnÞ  ð nÞ ðnÞ ρG, screw þ ρG, edge : ρG  ¼

(18)

Relative magnitude of the edge to screw components, or vice versa, reflects the character of the GNDs and the characteristic angle between the tangent of the dislocation line, and the Burgers vector φ is ðnÞ

ρG, screw  cos φðnÞ ¼   ðnÞ  , ρG 

ðnÞ

ρG, edge  sin φðnÞ ¼   ðnÞ  ,  ρG 

(19)

or ðnÞ

φðnÞ ¼ arctan

ρG, screw ðnÞ

ρG, edge

:

(20)

pffiffiffi When the screw component is 3 times larger than the edge component, for example, the characteristic angle is π/3, and the GNDs have 60 mixed character.

4

Models for the Critical Resolved Shear Stress

When only one straight dislocation exists in a crystal and lies on a slip plane, the Peierls-Nabarro stress is needed to move the dislocation, and detailed studies on this have been made by first-principles calculations. If there is a Frank-Read dislocation source, shear stress needed to operate the source and emit a dislocation loop is evaluated by Eqs. (4) or (5), and this value is usually much larger than the PeierlsNabarro stress especially for metals with face-centered cubic crystal structure. If there are a lot of dislocations accumulated in the crystal as well as point-wise obstacles for the movement of dislocations, the critical resolved shear stress for the onset of plastic slip deformation is approximately given by the following model: pffiffiffiffiffi θðnÞ ¼ θ0 þ aμb~ ρa

(21)

38

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

1227

where θ(n) is the critical resolved shear stress or CRSS for slip system n. θ0 on the right-hand side of Eq. (21) represents the resistance to the movement of dislocations arising from solute atoms, other point-wise obstacles, and the PeierlsNabarro stress. Movement of dislocations against these obstacles includes thermal processes, and therefore, θ0 is assumed to be a function of temperature and strain rate. ρa is the total density of dislocations accumulated in the crystal and evolves with the plastic slip deformation. The second term is the Taylor hardening term and a is a numerical factor of order 0.1 [1]. The model given in Eq. (21) is sometimes called Bailey-Hirsch model. For Taylor hardening term, it is possible to separate the contributions of dislocations accumulated on different slip systems. One model is [8, 18] θðnÞ ¼ θ0 ðnÞ ðT Þ þ

N X

qffiffiffiffiffiffiffiffi ðmÞ ΩðnmÞ aμb~ ρS :

(22)

m¼1

where ρS(m) stands for the SS dislocations accumulated on the slip system m and Ω(nm) is a component of interaction matrix between slip systems n and m. Interaction of dislocations on slip systems will be different depending on the combination of slip systems. We call Eq. (22) the modified Bailey-Hirsch model. Interaction of dislocations lying on the same slip plane could be weak compared to the one between intersecting dislocations, for example. Classification of the dislocation interaction is effectively done if we consider reactions of dislocations. Let us consider the case for face-centered cubic crystals assuming that slip deformation takes place on {111} slip plane and in direction. Table 1 shows 12  combinations of slip plane and directions and their notations. Here, the 111 [101] slip system is assumed to be the primary one, and its Schmid-Boas notation is B4. Slip systems B2 and B5 are called coplanar system because their slip plane is equal to the plane for the primary system. There are critical, cross-slip, conjugate, and coplanar systems. Interactions of dislocations on two slip systems are summarized in Fig. 11 where R1, R2, R3, R'3, and R4 denote the interactions of dislocations given in Table 2. If we assign some relevant values to R1, R2, R3, R'3, and R4 and define the interaction matrix of Ω(nm) as shown in Fig. 11, we have the interaction intensity between dislocations on different slip systems as relative magnitude to the interaction between the same slip systems. Experimental studies on the latent hardening of slip systems help to decide the values [19]; however, there are a lot of points left for further study [20]. Contribution to the slip resistance from microstructure length scale, for example, the grain diameter, is widely known under the name of Hall-Petch effect [21]. Such effects are not yet included in the modified Bailey-Hirsch model of Eq. (22). Dislocation loop emission is largely influenced by the presence of grain boundaries as shown in Sect. 2, and when there is no obstacle to the movement of dislocations except grain boundaries, the shear stress needed to emit a dislocation loop is approximately given by Eq. (5). This effect is added to the modified Bailey-Hirsch model as [2]

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T. Ohashi

Table 1 {110} slip systems for face centered cubic crystals Slip system name Critical system

Slip system number 1

Notation by Schmid and Boas A2

Slip plane (111)

Critical system

2

A6

(111)

Critical system

3

A3

(111)

Cross-slip system Cross-slip system

4 5

D1 D6

(111)  (111)

Cross-slip system Coplanar system

6 7

D4 B2

Coplanar system

8

B5

Primary system

9

B4

10

C1

(111)   111   111   111    111    111    111

Conjugate system Conjugate system

11

C5

Conjugate system

12

C3

Fig. 11 The interaction matrix Ω for slip systems {111}, in facecentered cubic crystals.

Slip direction

  11 0

 01 1

 10 1



[110]

 01 1



[101]

  11 0 [011] [101] [110] [011]

 10 1

A2 A6 A3 D1 D6 D4 B2 B5 B4 C1 C5 C3 A2 A6 A3

1 R1 1 R1 R1 1

D1 D6 D4

R2 R3 R3 1 R3 R4 R3 R1 1 R3 R3 R2 R1 R1 1

B2 B5 B4

R4 R3 R3 R2 R3 R3 1 R3 R2 R3 R3 R2 R3 R1 1 R3 R3 R2 R3 R3 R4 R1 R1 1

C1 C5 C3

R2 R3 R3 R4 R3 R3 R2 R3 R3 1 R3 R2 R3 R3 R2 R3 R3 R4 R3 R1 1 R3 R3 R4 R3 R3 R2 R3 R3 R2 R1 R1 1

θ ð n Þ ¼ θ 0 ð n Þ ðT Þ þ

qffiffiffiffiffiffiffiffi μb~ ðmÞ ΩðnmÞ aμb~ ρS þ 3β , d m¼1

N X

symmetric

(23)

where the third term on the right-hand side adds the resistance arising from the presence of grain boundaries and could be called the Orowan stress term. The first and second terms are the lattice friction stress and Taylor’s strain hardening terms, respectively. The quantity d in the third term is introduced by basing the phenomenon that the movement of dislocations is prevented or largely hindered by grain

38

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

1229

Table 2 Dislocation interactions and components of the interaction matrix Ω(nm) 1 R1 R2 R3 R3’ R4

Dislocation interaction Primary as against primary Primary as against coplanar Hirth lock formation Glissile jog Lomer-Cottrell sessile dislocations Kink formation

Elastic interaction Junction formation

boundaries. Microstructural elements other than grain boundary will also prevent or hinder the movement. In such circumstance, one parameter modeling could be modified further to include information from three-dimensional arrangements of grain boundaries, precipitates, softer or harder phases, and other factors [22]. Also, it is interesting to discuss the relation of the length scales d in Eq. (23) and d* in Eq. (15) [14]. Constitutive equation by Hill [23] is used and implemented to a crystal plasticity finite element software code: " σ_ ij ¼

Seijkl

o1 X Xn ðnÞ ðmÞ þ hðnmÞ Pij Pkl n

#1 e_kl :

(24)

m

Infinitesimal strain approximation is used and crystal rotation is not included, though this effect is important in some cases. S denotes the elastic compliance tensor. P denotes the Schmid tensor and is defined as follows: ðnÞ

Pij ¼

f

g

1 ðnÞ ðnÞ ðnÞ ðnÞ b n þ bj ni , 2 i j

(25)

where b and n are unit vectors parallel to slip and slip plane normal directions, respectively. Strain hardening parameter h(nm) is obtained from the dislocation density-based models as follows: h

ðmmÞ



1 ðnmÞ 1 c ðmÞ ¼ Ω aμ qffiffiffiffiffiffiffiffi ðmÞ  DρS : 2 ðmÞ ρS L

(26)

The strain hardening parameter becomes zero when the density of accumulated ðmÞ dislocations reaches the equilibrium value ρS, eq : ðmÞ

ρS, eq ¼

c , DLðmÞ

(27)

and at this situation, the slip deformation will be inherently instable. Figure 12 shows some curves for the equilibrium density as a function of the annihilation distance D.

1230

T. Ohashi 2.5 × 1015

Equilibrium density of SS dislocations, m–2

L=0.1 µm L=1 µm

2.0 × 1015

L=10 µm 1.5 ×

1015

1.0 × 1015 5.0 × 1014 0 0

5 × 10–9

1 × 10–8

1.5 × 10–8

2 × 10–8

Annihilation distance, m Fig. 12 Equilibrium density of SS dislocations, ρS,eq plotted as against the annihilation distance D. c = 1 is used and curves are shown for cases when the dislocation mean free path L is 0.1, 1, and 10 μm. When the SS dislocation density reaches the equilibrium density, the strain hardening in Eq. (26) becomes zero and the slip deformation will be inherently instable

5

Numerical Example of Finite Element Crystal Plasticity Analysis [24]

Let us consider a model of rolled sheet of alpha-titanium alloy shown in Fig. 13a. RD, ND, and TD denote rolling, normal, and transverse directions, respectively. Dimension of the model is 100  100  100 μm3. The model is divided into 250 crystal grains by Voronoi tessellation, and we cut out a thin sheet of 2 μm thickness for the deformation analysis as shown in Fig. 13b. The number of crystal grains in the sheet is 54, and the average grain diameter is approximately 14 μm. Alpha-titanium has a hexagonal closepacked (HCP) crystal structure, and its rolled sheet has one or more textures. Typical texture is the basal type where the direction of c-axis (i.e., (0001) direction) is parallel to the ND. According to Inoue [25], there are five types of texture including the basal type. In this example, we examine slip deformation when the crystal orientation has TD-split texture or T-textures. Figure 14a shows the pole figure for the TD-split texture  [25]. Direction of c-axis deviates    approximately 40 from ND toward TD or the opposite direction of TD. 101 0 direction is nearly parallel to the RD. This is schematically illustrated in Fig. 14b where the deviation angle φ is nearly equal to 40 or – 40 . In the numerical model, we assign crystal orientation of each grain so that  the direction the exact angle of φ = 40 and,  ofc-axis fluctuates 15 from  similarly, 101 0 direction fluctuates 15 from RD. Resultant pole figure is shown in Fig. 14c. If we start from the situation that φ = 90 and give similar fluctuations, we obtain a texture model for the T-type. We use experimental values of CRSS [26], which are 420, 370, and 490 MPa for the basal, prismatic, and first pyramidal systems, respectively. We do not consider

38

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

a

1231

y ND

b x RD

TD z Fig. 13 (a) Voronoi tessellation of a cube shaped area and (b) thin plate specimen cut out from the cubic region. Dimension of the cut specimen is 100  100  2 μm and the mean grain diameter is approximately 14 μm

twin and second pyramidal slip deformations. Because the grain diameter is relatively large, the Orowan stress term in Eq. (23) is not significant and is approximately 3 MPa. If we assume the density of the initial dislocations as 1011 m2 on each slip systems, the Taylor hardening term in Eq. (23) is close to 9 MPa. Therefore the largest contribution to the CRSS comes from the lattice friction term. We use 408, 370, and 478 MPa for the lattice friction stress term for basal, prismatic, and first pyramidal systems, respectively. Interaction of slip systems is assumed to be isotropic and Ω(nm)  1. Weight for the dislocation mean free path is defined as w(ij) = 1- δij, where δij is the Kronecker delta. We apply uniform tensile displacement in the RD direction on the edges perpendicular to the RD axis. Remaining surfaces are traction-free. Figure 15 shows the numerically obtained nominal stress-nominal strain curves for the TD-split texture and T-texture models. Figure 16 shows histograms of the normal stress component in the RD direction when the nominal tensile strain is 0.5%. Deformation is purely elastic at this deformation stage. However, the stress is not uniform in the microstructure because of the elastic anisotropy. Both data for TD-split texture and T-texture show distribution of Gauss type. Distribution is wider in the TD-split texture model, and its width is approximately 10% of the average stress. Initial slip deformation takes place where the stress is high and crystal orientation is favorable. Sites near grain boundary plane, grain boundary triple junction, or quadruple point are candidates for such activity. The first slip deformation takes place when the nominal tensile strain is about 0.67%. As shown in Fig. 15, apparent deviation from the linear elastic range takes place when the nominal strain is approximately 0.7%. Figure 17 shows slip activity when

1232

T. Ohashi

RD

a

RD

(1010)

(0001)

RD

ND

b

0001 1120 TD

0001

c

1010

2110 1010

RD

ND 0

TD 90 45

Fig. 14 (a) Pole figure of an α titanium alloy with TD-split texture [24]. (b) Schematic illustration of the crystal orientation. Angle φ between the ND and c-axis of the hcp crystal axis is approximately 40 . (c) Numerically generated crystal orientation with the TD-split texture. Red and blue    colored dots indicate (0001) and 101 0 orientations, respectively

the nominal strain is 0.7%. Two prismatic systems operate near grain boundary triple junctions in TD-split texture model, and slip activity expands more extensively inside some grains in the T-texture model. However, it is noted that the magnitude of shear strain is at the order of 104 and the tensile deformation is mostly delivered by elastic strain components. Figure 18 shows developments of slip deformation when the nominal strains are 0.8 and 1%. The average elastic strain is approximately 0.75–0.8% when the nominal strain is 1% (Fig. 15). Therefore, the average plastic strain is 0.2–0.25%, whereas the plastic shear strain is larger than 5% in some areas and the plastic deformation is highly nonuniform. In the TD-split texture model, slip deformation appears to be significant in regions extending in two slanting directions, while in the T-texture model, slip

38

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

1233

900

Fig. 15 Numerical results of the nominal strain-nominal stress curves for TD-split and T texture models

TD-split texture 800

T texture

Nominal Stress [MPa]

700 600 500 400 300 200 100 0

0

0.005

0.01

Nominal Strain

7000 TD-split texture

Frequency [elements]

6000

T texture

5000 4000 3000 2000 1000 0

450

500 550 RD Normal Stress [MPa]

Fig. 16 Histograms of the normal stress component in RD direction when the nominal strain is 0.5% and the deformation is purely elastic throughout the specimen

1234

T. Ohashi

0.02

a

[%]

b

(1100) [1120] c

0.01

(1100) [1120]

(0110) [2110] d

x z

0 y

(0110) [2110]

Fig. 17 Distribution of plastic shear strain on two prismatic systems in TD-split (a, b) and T texture (c, d) models. The nominal tensile strain is 0.7%

deformation develops in horizontal and wider spread regions. It is also noted that these regions of larger slip strain do not have close connection with the local slip initiation observed in Fig. 17. This comparison shows that the slip deformation initiates at sites with elastic stress concentration, but subsequent slip activity shown in Fig. 18 is more influenced by anisotropy of slip deformation. Figure 19 schematically shows crystal orientation of some grains indicated in Fig. 18b. Thick arrows indicate slip directions and prismatic planes where slip deformation takes place. It is interesting to note that the directions of the regions with larger slip strain extend approximately perpendicular to the slip direction, except the grain G536. It is known that slip deformation sometimes develops in a strip-shaped region extending parallel or perpendicular to the slip direction. Definition of the terms shear and kink bands is not clear enough; however, it is commonly understood that they extend parallel and perpendicular to the slip direction, respectively. Therefore, it could be possible to understand the stripshaped regions formed in grains G102, G215, G316, G446, G608, and others in Fig. 18b correspond to the kink-type deformation bands. In the T-type texture, slip direction is nearly parallel to the x-axis (i.e., RD), and again the widespread regions extending in z-direction (i.e., TD) could be called the deformation bands of kink type.

38

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

a

1235

b G102

0.02

G215 G449 G536 G316

0.01

G608

c

d

0 x y

z

        Fig. 18 Distribution of plastic shear strain on prismatic slip systems 11 00 112 0 and 011 0

  2 110 . Only one of these slip systems operates in most grains. (a, b): Results obtained for the TD-split texture model when the nominal tensile strain is 0.8 and 1%, respectively. (c, d): Results obtained for the T texture model when the nominal tensile strain is 0.8 and 1%, respectively

Figure 20 shows density distributions of SS and GN dislocations when the nominal tensile strain is 1%. Distribution of the SS dislocation has a strong correspondence to the distribution of slip strain shown 18b. This is because   in  Fig.       the plastic slips mostly take on one slip system of 11 00 112 0 or 011 0 2 110 in each grain, and multiplication of these slips takes place only in a small portion in some grains. Figure 20(b–d) shows distribution of the density norm of GN dislocations and their edge and screw components, respectively. Comparison of Fig. 20a and b shows that the density of GNDs is higher than that of SSDs. Density of SSDs is relatively low because the slip takes only on one system in most regions and therefore the dislocation mean free path is kept at large values in such regions. If we compare the distribution of the density norm of GNDs in G215 with the distribution of slip strain shown in Fig. 18b, it is shown that two strip regions of high GND density are formed along the deformation band. These dislocations are mostly edge type of opposite sign as shown in Fig. 20c with their Burgers vector directing approximately perpendicular to the direction of the deformation band (cf. Figs. 18b and 19). Screw component is relatively small compared to the edge

1236

T. Ohashi

G102

X

G449

X

Z

G215

X

Z

G536

X

Z

G316

X

Z

G608

Z

X

Z

Fig. 19 Crystal orientations of six grains denoted in Fig. 18(b). Thick arrows and prismatic planes having the arrows indicate the slip directions and slip planes of the activated systems

component. This result again indicates that the deformation bands observed in G215 and others are kink type. Density distribution of the screw component (Fig. 20d) shows complicated pattern but mostly localized near grain boundaries and their triple junctions.

Dislocation Density-Based Modeling of Crystal Plasticity Finite Element. . .

38

1012 [m-2] a 8

SS dislocations

b

1237

GN dislocations

4

0 1012

[m-2] c 8

GND edge component

d GND screw component

0

x y

z

-8

Fig. 20 Density distributions of dislocations on the prismatic systems in the TD-split model when the nominal tensile strain is 1%. (a): SS dislocations, (b): density norm of GN dislocations, (c, d): edge and screw components of the GN dislocations, respectively

Acknowledgments This work was partly supported by the Council for Science, Technology and Innovation (CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), “Fundamentals for fatigue and fracture in structural metals” (funding agency: JST).

References 1. Hull D, Bacon DJ. Introduction to dislocations. 4th ed. Oxford: Butterworth Heinemann; 2001. 2. Ohashi T, Kawamukai M, Zbib H. A multiscale approach for modeling scale-dependent yield stress in polycrystalline metals. Int J Plast. 2007;23:897–914. 3. Zbib HM, Diaz T, Rubia D. A multiscale model of plasticity. Int J Plast. 2002;18:1133–63. 4. Ashby MF. The deformation of plastically non-homogeneous materials. Philos Mag. 1970;21:399–424. 5. Kocks UF. Laws for work-hardening and low-temperature creep. Trans ASME J Eng Mat Tech. 1976;98:76–85. 6. Mecking H, Kocks UF. Kinetics of flow and strain-hardening. Acta Metall. 1981;29:1865–75. 7. Yasnikov IS, Vinogradov A, Estrin Y. Revisiting the Considere criterion from the viewpoint of dislocation theory fundamentals. Scr Mater. 2014;76:37–40. 8. Ohashi T. Numerical modelling of plastic multislip in metal crystals of f . c . c . type. Philos Mag A. 1994;70:793–803.

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9. Ohashi T. A new model of scale dependent crystal plasticity analysis. In: Kitagawa H, Shibutani Y, editors. IUTAM symposium on mesoscopic dynamics of fracture process and materials strength. Osaka: Kluwer; 2003;97–106. 10. Ohashi T. Crystal plasticity analysis of dislocation emission from micro voids. Int J Plast. 2005;21:2071–88. 11. Kuhlmann-Wilsdorf D. Theory of plastic deformation: - properties of low energy dislocation structures. Mater Sci Eng A. 1989;113:1–41. 12. Taylor GI. The mechanism of plastic deformation of crystals. part I. theoretical. Proc R Soc A Math Phys Eng Sci. 1934;145:362–87. 13. Mott NF. A theory of work-hardening of metal crystals. Phil Mag Ser. 7. 1952;43:1151–78. 14. Okuyama Y, Ohashi T. Numerical modeling for strain hardening of two-phase alloys with dispersion of hard fine spherical particles. Tetsu-to-Hagane. 2016;102:396–404 15. Yasuda Y, Ohashi T. Crystal plasticity analysis considering dislocations’ behavior in ferrite/ cementite lamellar structure. ISIJ Int. 2017;57:573–580. 16. Fleck N, Muller G, Ashby M, Hutchinson J. Strain gradient plasticity: Theory and experiment. Acta Metall Mater. 1994;42:475–87. 17. Gurtin ME, Ohno N. A gradient theory of small-deformation, single-crystal plasticity that accounts for GNDinduced interactions between slip systems. J Mech Phys Solids. 2011;59:320–43. 18. Ohashi T. Computer simulation of non-uniform multiple slip in face centered cubic bicrystals. Trans JIM. 1987;28:906–15. 19. Jackson PJ, Basinski ZS. Latent hardening and the flow stress in copper single crystals. Can J Phys. 1967;45:707–35. 20. Hiura F, Niewczas M. Latent hardening effect under self- and coplanar dislocation interactions in Mg single crystals. Scr Mater. 2015;106:8–11. 21. Hansen N. Hall–Petch relation and boundary strengthening. Scr Mater. 2004;51:801–6. 22. Yasuda Y, Ohashi T. Crystal plasticity analyses of scale dependent mechanical properties of ferrite/cementite lamellar structure model in pearlite steel wire with bagaryatsky or pitsch-petch orientation relationship. ISIJ Int. 2016;56:2320–6. 23. Hill R. J Mech Phys Solids. Generalized constitutive relations for incremental deformation of metal crystals by multislip. J Mech Phys Solids. 1966;14:95–102. 24. Ohashi T, Amagai R, Okuyama Y, Kawano Y, Mayama T. Crystal plasticity finite element analysis of slip deformation in the polycrystal models of textured Ti alloys. In: JSME M&M 2016 conference. Kobe, Japan: Japan Society of Mechanical Engineers; 2016:8–10. 25. Inoue H. Texture of Ti and Ti alloys. Mater Sci Technol “Kinzoku” (in Japanese). 1999;69:30–38. 26. Bridier F, McDowell DL, Villechaise P, Mendez J. Crystal plasticity modeling of slip activity in Ti-6Al-4V under high cycle fatigue loading. Int J Plast. 2009;25:1066–82.

Competing Grain Boundary and Interior Deformation Mechanisms with Varying Sizes

39

Wei Zhang, Yanfei Gao, and Tai-Gang Nieh

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Deformation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Low-Temperature Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 High-Temperature Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Competition Between Grain Interior and Grain Boundary Processes . . . . . . . . . . . . . . . . . . . . . 3.1 Rate Equations Versus Explicit Finite Element Simulations . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Breakdown of the Hall-Petch Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Competing Deformation Mechanism with Varying Grain Sizes . . . . . . . . . . . . . . . . . . . . 4 Summary and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In typical coarse-grained alloys, the dominant plastic deformations are dislocation gliding or climbing, and material strengths can be tuned by dislocation interactions with grain boundaries, precipitates, solid solutions, and other defects. With the reduction of grain size, the increase of material strengths follows the classic Hall-Petch relationship up to nano-grained materials. Even at room temperatures, nano-grained materials exhibit strength softening, or called the inverse Hall-Petch effect, as grain boundary processes take over as the dominant deformation mechanisms. On the other hand, at elevated temperatures, grain boundary processes compete with grain interior deformation mechanisms over a wide range of the applied stress and grain sizes. This book chapter reviews and compares the W. Zhang · T.-G. Nieh Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA Y. Gao (*) Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_75

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rate equation model and the microstructure-based finite element simulations. The latter explicitly accounts for the grain boundary sliding, grain boundary diffusion and migration, as well as the grain interior dislocation creep. Therefore the explicit finite element method has clear advantages in problems where microstructural heterogeneities play a critical role, such as in the gradient microstructure in shot peening or weldment. Furthermore, combined with the Hall-Petch effect and its breakdown, the above competing processes help construct deformation mechanism maps by extending from the classic Frost-Ashby type to the ones with the dependence of grain size. Keywords

Deformation mechanism map · Grain boundary deformation processes · HallPetch effect and breakdown · Microstructure-based finite element method

1

Introduction

At elevated temperatures, creep deformation as the rate-dependent plastic behavior depends critically on the micro- and nano-structures. In polycrystalline metals, the creep deformation results from a synergy of thermally activated processes, including dislocation movements, diffusional flow, and grain boundary sliding. Each of these mechanisms may occur independently, but from the geometric point of view, grain boundary sliding requires kinematic accommodation from either dislocation creep or diffusional flow. The competition between the grain interior and grain boundary mechanisms has been extensively studied for both coarse-grained and fine-grained materials. In coarse-grained metals, the plastic flow is mediated by the Coble creep through grain boundary diffusion [1] or the Nabarro-Herring creep through lattice diffusion [2, 3] under low applied stress, by grain boundary sliding at intermediate applied stress, and by dislocation creep under high applied stress. For nano-grained materials, because of the increased volume fraction of grain boundary materials with decreasing grain size, the deformation of nanocrystalline materials can be dominated by grain boundary diffusion (Coble creep) and other possible grain boundary-related mechanisms, such as grain boundary sliding, migration and grain rotation [4–7], and dislocation emissions and annihilations at the grain boundaries [8–12]. The most common low-temperature plastic deformation behavior in coarsegrained materials is through the continuous dislocation nucleation from FrankRead sources and crystalline slips on well-defined slip systems. These materials are usually strengthened by the interaction with obstacles, including forest dislocations, precipitates or solid solution, grain boundaries, or the lattice resistance. Following the classic Hall-Petch effect [13, 14], the plastic resistance increases with decreasing grain size as a result of interactions between dislocations and grain boundaries. When the grain size reduces to ~20 nm, however, the nanocrystalline materials show strength softening, or called the inverse Hall-Petch effect (see, e.g., [15–19]). In the past several decades, extensive studies have been conducted to

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interpret the inverse Hall-Petch effect and to determine the underlying deformation mechanisms. It is now commonly accepted that the intergranular grain boundary processes overtake the grain interior dislocation activities as the dominant deformation mechanisms, because the fraction of grain boundary atoms increases significantly when grain size changes from micrometer to nanometer. The deformation mechanism maps have been extensively investigated in Frost and Ashby [20], which considers the competition mainly between diffusional flow and dislocation plasticity. The dependence on the grain size, as shown by the HallPetch effect in coarse-grained metals and the Hall-Petch breakdown in nanograined metals, will certainly add another dimension in these maps. In this book chapter, Sect. 2 presents the fundamental deformation mechanisms of crystalline solids, including dislocation glide at low temperatures, diffusional flow, grain boundary sliding, and dislocation creep at high temperatures. As these deformation mechanism maps are constructed by rate equations, they clearly are unable to address issues such as microstructural heterogeneities. Along these lines, the microstructure-based finite element method has been developed [7, 21, 22], which simulates the above deformation processes in an explicit microstructural model. Section 3 will reveal the competing grain boundary and grain interior deformation mechanisms with varying grain sizes and extend the Frost-Ashby maps by considering the Hall-Petch effect and its breakdown in nanocrystalline metals. Finally, the concluding remarks and perspectives will be given in Sect. 4, with a particular emphasis on the size dependence of plastic deformation mechanism of polycrystalline solids.

2

Deformation Mechanisms

This section describes the contribution from intragranular and intergranular deformation mechanisms with varying grain sizes. Plastic flow is determined by the kinetic process that occurs on the atomic scale, including the glide motion of dislocation lines, coupled dislocation gliding and climbing, diffusional flow of individual atoms, and grain boundary sliding (involving diffusion and defect motion), among many others [20, 21]. These deformation mechanisms are given in two groups: low-temperature plasticity and high-temperature plasticity. In the low-temperature range, plastic deformation is controlled by the propagation of dislocation lines on closely packed planes, and plastic flow is always hindered by obstacles, such as other dislocations, precipitates or solute atoms, and grain boundaries, which leads to increased strength. In high-temperature range, the following deformation mechanisms are dominant over different regimes of stress, temperature, and strain rate: diffusional flows controlled by lattice diffusion (Nabarro-Herring creep) and grain boundary diffusion (Coble creep) dominating at low-stress and high-temperature regime, dislocation creep limited by lattice or dislocation core diffusion dominating at high stress level, and grain boundary sliding dominating at intermediate stress regime especially in materials with fine grain sizes.

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Low-Temperature Plasticity

Plastic deformation in coarse-grained materials is caused by dislocation motion on a number of independent slip systems. The crystal structure itself determines the number and type of slip systems with unique Burgers vectors, and it also determines the lattice friction stress which controls the intrinsic strength. The stacking-fault energy determines the dislocation dissociation, which affects the cross-slip and the subsequent strain hardening rate. Also solute atoms and fine particles are important additions that can effectively impede the movements of dislocations and give rise to improved strength in polycrystalline materials. Frost and Ashby [20] constructed deformation maps based on rate equations, which mainly describe the mechanism of interaction between dislocation motion and obstacles. Based on their examination, the determination of strain rate falls into two groups: discrete obstacle-controlled plasticity, such as dislocation interaction with solid solutions and other dislocations, and lattice resistance-limited plasticity, in which dislocations overcome lattice friction collectively. For the discrete obstaclecontrolled plasticity, the velocity of dislocations is determined by the strength and density of the discrete obstacles, and the rate equation is given by γ1 ¼



 α τ 2 ΔGðτÞ βbνexp  b G kT
 τ ΔGðτÞ ¼ ΔF 1 

(1)

(2)

^τ

where γ_1 is the shear strain rate, α and β are dimensionless constants, and ν is a frequency. ΔF is the activation energy required to overcome the obstacle without external stress. ΔG(τ) represents the Gibbs free energy of activation for cutting or bypassing of an obstacle, and the material property ^τ is the stress which reduces ΔG to zero, that is, the dislocation pass through the obstacle without assistance from the thermal activation process. The velocity of dislocation in a polycrystalline material is also influenced by the atomic structure itself, that is, lattice resistance-controlled plasticity. The rate equation was obtained by Frost and Ashby [20] as γ_2 ¼ γ_p


2
 τ ΔGðτÞ exp  G kT

ΔGðτÞ ¼ ΔFp


3=4 !4=3 τ 1 ^τ p

(3)

(4)

where γ_p is the reference strain rate, ΔFp is the Helmholtz free energy, and ^τ p is the flow stress at 0K, and it depends on the crystal structure and slip systems which are activated when the polycrystalline solids deform.

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High-Temperature Plasticity

As the mobility of atoms increases rapidly with temperature, the diffusion-controlled process is often associated with creep as the dominant plastic deformation mechanism at high temperatures, including the diffusional creep, dislocation creep, and grain boundary sliding. The characteristic creep strain-time curve contains three stages: the primary creep during which the creep rate decreases with time, the steadystate creep with a nearly constant creep rate which results from a balance of competing processes of strain hardening and recovery, and the tertiary creep in which void formation and necking begin to develop and lead to the final failure. Studies in the past were mainly focused on the development of deformation mechanisms for steady-state creep, which can be generally described by the following equation [23]: e_ ¼



 ADGb b p σ n kT d G

(5)

where e_ is the tensile creep rate; σ is the flow stress; T is the absolute temperature; G is the shear modulus; d is the grain size; p is the inverse grain size exponent; n is the stress exponent, which is equal to 1/m (where m is the strain rate sensitivity exponent); A is a dimensionless constant; and D is the diffusion coefficient, given by D ¼ D0 expðQ=kT Þ

(6)

where Q is the activation energy for the diffusion flow. Each diffusional creep mechanism can be described by Eq. (5) with specific values of D, p and n. The diffusional creep mechanisms and their corresponding rate equations are summarized as follows.

2.2.1 Diffusional Creep The diffusional creep at high temperature is a result of stress-driven migration of vacancies (or atoms). When vacancies move through the lattice, the corresponding Nabarro-Herring creep [2, 3] is given by e_NH ¼ ANH

Dl σΩ d2 kT

(7)

where Ω is the atomic volume, Dl is lattice self-diffusion coefficient, and ANHis a dimensionless constant which depends on the shape of grain. The Coble creep [1] involves vacancy flow along grain boundaries, and the rate equation is described by e_C ¼ AC

Dgb δσΩ d3 kT

(8)

where AC is a dimensionless constant, Dgb is the grain boundary diffusion coefficient, and δ is the grain boundary thickness. Note that Nabarro-Herring creep and

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Coble creep are of the same stress exponent n = 1, but Coble creep ( p = 3) shows a stronger grain size dependence than the Nabarro-Herring creep ( p = 2). The activation energy of grain boundary diffusion is much lower than that of lattice diffusion, and therefore the required temperature at which Coble creep dominates is lower than that of Nabarro-Herring creep. In nano-grained materials, the diffusional creep rate increases monotonically with decreasing grain size, and grain boundary diffusion-governed creep becomes one of the dominant deformation mechanisms due to the significantly increased volume fraction of grain boundaries materials [10, 11, 15, 16].

2.2.2 Dislocation Creep Compared to the diffusional flow, dislocation glide involves dislocations moving along slip planes in a more collective manner, e.g., all atoms above the slip plane move by the same distance with respect to the bottom half crystal. Dislocation creep, at the atomic level, is the result of the diffusion of single vacancies to or from the dislocation climb, which is controlled by lattice or dislocation core diffusion. The rate equation for dislocation creep can be described in the generalized power-law equation [20]: e_D ¼


 AD DGb σ n kT G

(9)

where AD is a dimensionless constant and D is the lattice or dislocation core diffusivity. For the majority of alloys, at high temperatures (T > 0.5 TM), climbing is required when the gliding dislocation is held by physical obstacles [24], in which transport of matter is through lattice diffusion Dl, and the stress exponent n varies from 4 to 5. It is usually seen in the dispersion-strengthened alloys, the stress exponent is very high n > 8, and it results from the existence of threshold stress [25]. At lower temperature or higher stresses, the dislocation core diffusion (with diffusivity Dc) contributes significantly to the overall diffusive transport of matter and becomes the dominant deformation mechanism. The dislocation creep is the favorable deformation mechanism at high-stress regime in coarse-grained materials.

2.2.3 Grain Boundary Sliding Grain boundary sliding is characterized by a stress exponent of n = 2: e_gbs ¼



 Agbs DG b p σ 2 d G kT

(10)

and the diffusion coefficient is derived either from lattice diffusion or from grain boundary diffusion, with grain size dependence p = 2 or 3, respectively. It is important to note that grain boundary sliding cannot occur in absence of other deformation mechanisms. It requires accommodation processes from diffusional flow, dislocation slip, and/or grain boundary migration. Many quantitative models have been developed to describe the grain boundary sliding with accommodated

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plastic flow (e.g., [26, 27]). Grain boundary sliding contributes to the plastic deformation in solids of all grain sizes, and it becomes the dominant mechanism especially in materials of small grain sizes [28–30].

3

Competition Between Grain Interior and Grain Boundary Processes

As the grain size varies from micrometer to nanometer, the dominant mechanism changes from dislocation-mediated deformation to grain boundary-associated processes [15, 31–34]. Especially, in the nanometer range, these grain boundary processes include grain boundary diffusion, grain boundary sliding, and rotation motion. In this section, the competition between grain boundary and grain interior deformation processes is discussed over different ranges of grain sizes.

3.1

Rate Equations Versus Explicit Finite Element Simulations

Frost and Ashby [20] constructed the deformation maps for various materials based on superposition of the above rate equations in Sect. 2, revealing the dominant deformation mechanisms in different ranges of stresses and temperatures. The contribution from the grain boundary sliding was, however, not treated explicitly in their work. For a typical polycrystalline material with micrometer grain size, the low-temperature plasticity is controlled by discrete obstacle-controlled glide through the rate equations Eqs. (1, 2) and lattice resistance-controlled glide with Eqs. (3, 4). At high temperature, power-law creep, diffusional flow, and grain boundary sliding, as three independent flow mechanisms, compete in different ranges of stresses and temperatures. The total strain rate can be obtained by summing the rate equations in Sect. 2:   e_ ¼ e_diffu þ greatest of e_plas , e_D þ e_gbs

(11)

It should be noted in each deformation mechanism the strain rate is governed by the faster one, that is, the contribution of dislocation motion is either from dislocation glide or from power-law creep. At low-temperature regime, dislocation glide is important, otherwise power-law creep contributes more to the total plastic deformation. e_diffu involves contributions from both Nabarro-Herring creep and Coble creep equations. Based on the material properties of pure nickel (reproduced based on Frost and Ashby [20]), contour of constant strain rate is plotted with the axes of log10(σ s/G) and T/TM as shown in Fig. 1. The deformation map for pure nickel is divided into three principle fields: low-temperature plasticity, power-law creep, and diffusional flow field. The power-law creep appears at high stress level, and this field is subdivided into two regions, one is core diffusion-controlled creep at low temperature and the other is lattice diffusion-controlled dislocation creep at high temperature. Diffusional flow is important at high-temperature and low-stress regime, in

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Fig. 1 Deformation mechanism map for a pure nickel with grain size of 20 μm, as constructed from Frost and Ashby [20]

which grain boundary diffusion-controlled Coble creep dominates at lower temperatures and lattice diffusion-controlled Nabarro-Herring creep dominates at higher temperatures. The deformation mechanism map provides a simple and direct way to distinguish which deformation mechanism is dominant under given combination of stress and temperature. In contrast to the rate equation model, Bower and Wininger [21] developed a finite element method to explicitly model the grain boundary sliding, grain boundary diffusion, grain boundary migration, and dislocation creep of polycrystalline materials at high temperatures. In their work, crystal plasticity is used to model the thermally activated dislocation motion within grain interiors. Grain boundary sliding, assumed to be a thermally activated process, is described by a linear viscous constitutive law. The flux of materials from one grain to another in both normal and tangential direction is driven by the chemical potential gradient, which is the basic idea for modeling grain boundary diffusion and migration. As shown in Fig. 2, jn represents the volumetric flux of materials transferred in the normal direction of grain (+) and grain (-), and jt denotes the flux of material tangent to each interface. In response to the chemical potential gradients, atoms near the grain boundary may diffuse along the boundary, detach, and reattach to one of the adjacent grains through grain boundary diffusion and migration. The normalized flow stress versus strain rate curve for a model polycrystalline material with grain size d = 1 μm is replotted as the solid lines in Fig. 3. In addition, an approximation based on the superposition of rate equations of grain boundary diffusion-mediated Coble creep, grain boundary sliding, and power-law creep is shown in the dash-dot lines for comparison. The representative material parameters used in the explicit modeling and rate equation

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Fig. 2 The explicit simulation of grain boundary diffusion and migration, plotted based on Bower and Wininger [21]

Grain (+) S

n t

Grain (–) Fig. 3 The relationship between the normalized applied stress and the strain rate for a polycrystal with grain size of 1 μm. Finite element simulation results were taken from Bower and Wininger [21]. Rate equation approximation is based on Eq. (11). See Table 1 for constitutive parameters used in these models

approximation are listed in Table 1. The results of the explicit modeling and superposition of rate equations agree quite well. It is observed that diffusional flow with strain rate sensitivity m  1 (m = 1/n) is important in the low stress range, the grain boundary sliding with m  0.33~0.5 dominates in the intermediate stress range, and dislocation creep with low strain rate sensitivity m  0.2 contributes most to the total plastic strain when the stress is high. Figure 4 shows the contribution of dislocation creep, grain boundary sliding, and grain boundary diffusion for grain size d = 1 μm and d = 5 μm. The trend in Fig. 4 is similar to that observed in Fig. 3, showing grain boundary diffusion and sliding are the favorable deformation mechanisms at low stress level and the contribution of dislocation creep becomes greater as stress increases. Comparing Fig. 4a, b, it is found the constitutive response of material is strongly dependent on its grain size. For example, Fig. 4b shows the fraction of strain rate for a material with grain size d = 5 μm, in which grain boundary diffusion does not do significant contribution to the overall strain rate, grain boundary sliding makes about 20% contribution, and dislocation creep is the dominant deformation mechanism over the whole stress range. These explicit simulation results reflect that there exists a transition from

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Table 1 Constitutive parameters used in the explicit microstructure-based finite element method [21] and the rate equation approximation in Fig. 3 Parameter Atomic volume Ω Burgers vector b Melting temperature TM Young’s modulus E Poisson’s ratio ν Shear modulus μ Critical resolved shear stress τ0 Grain boundary diffusion pre-exponential δGBDGBt Grain boundary sliding pre-exponential η0 Lattice diffusion pre-exponential D0l Grain boundary diffusion activation energy QGBt Grain boundary sliding activation energy QGBs Lattice diffusion activation energy Ql Stress exponent n Temperature dependence of modulus TEM0

dE dT

Value 1.66  1029 m3 2.48  1010 m 933 K 70 GPa 0.3 26.9 GPa 100 MPa 2.86  1015 m3/s 40 m/s 1.7  104 m2/s 84 KJ/mole 84 KJ/mole 142 KJ/mole 5 1.09

dislocation creep controlled plasticity to grain boundary sliding and diffusiongoverned deformation mechanism. It is also shown that grain boundary diffusion and sliding processes show strong size dependence, the smaller the grain size, the higher the strain rate, and therefore refining grain size will considerably increase the creep rate. Later, Cipoletti et al. [22] applied the same simulation framework to study the deformation in magnesium alloy at 450  C, and their work shows the size dependence of strain rate that causes transition of deformation mechanism from dislocation creep to grain boundary sliding. There are several advantages of the explicit model over the rate equations. It is capable of modeling polycrystalline materials with complex microstructures, such as those with gradient of grain sizes (e.g., shot peening or weldment), and materials in the presence of second phase and voids, among many others [7, 35]. As the explicit model accounts for grain boundary diffusion as well as surface diffusion, it can also be used to simulate diffusive or plastic void growth in the crystalline solid. Compared with molecular dynamics simulation, the explicit model requires reduced computational costs as it deals with much fewer degrees of freedom. Furthermore, the explicit model can be easily adapted to many other chemomechanical processes, such as lithiation processes in Li-ion batteries, microstructural evolution in heteroepitaxial thin films, and others.

3.2

Breakdown of the Hall-Petch Effect

The strength of conventional polycrystalline materials with grain size in micrometer scale follows the Hall-Petch relationship:

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Fig. 4 The contribution of grain boundary sliding, grain boundary diffusion, and grain interior dislocation creep to the total strain rate for a polycrystal with grain size of (a) 5 μm and (b) 1 μm [7, 21, 22]

σ ¼ σ 0 þ kd 1=2

(12)

where σ 0 is the lattice friction stress, k is the material constant, and d is the grain size. The Hall-Petch relationship predicts grain refinement and leads to higher strength. However, experimental results exhibit the inverse Hall-Petch behavior (strength softening) with further reduction of grain sizes. Figure 5 shows the Hall-Petch plot for Cu and Ni taken from Kumar et al. [36] and Masumura et al. [16]. An inverse trend (strength softening) is observed when the grain size decreases to about 25 nm, indicating a transition of underlying deformation mechanisms of nanocrystalline materials. Great efforts have been done to understand the breakdown of Hall-Petch effect. In the early theoretical model [37], nanocrystalline materials were assumed as two-phase composites consisting of grain

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Fig. 5 Normalized flow stress as a function of the grain size d, with the Ni and Cu data compiled from Kumar et al. [36] and Masumura et al. [16]

interior and grain boundaries, and the yield was calculated by the weighted sum of the yield stresses characterizing the grain interior and grain boundary phase. But such a rule of mixture approach is too approximate and sheds less insight on the actual mechanisms. Schiøtz et al. [31, 32] used atomistic simulations to study nanocrystalline copper and found the softening was caused by the sliding of atomic planes at grain boundaries through a large number of uncorrelated events and little dislocation activity was seen in the grain interiors. They concluded that the softening at small grain sizes resulted from a significant fraction 30–50% of atoms at grain boundaries. The study of nanocrystalline aluminum [33] revealed the onset of plastic flow by partial dislocations nucleating from grain boundaries. However, when grain size reduces to about 20 nm, the nucleation of complete dislocation was no longer possible, and grain boundary-associated processes became the favorable deformation mechanism. Later, Yamakov et al. [34] observed deformation twinning in nanocrystalline aluminum, which was formed by the successive emission of partial dislocations from the same grain boundary. And they pointed out mechanical twinning might play an important role in the deformation of nanocrystalline aluminum. Similar observations were also made in nanocrystalline copper by Yang et al. [38]. Trelewicz and Schuh [19] used the nanoindentation techniques to investigate the deformation behavior of nanocrystalline Ni-W. They found the material not only obeyed the Hall-Petch relationship but also exhibited softening (Hall-Petch breakdown) when the grain size was less than 20 nm. In their work, the breakdown is attributed to a mechanistic transition to inhomogeneous glass-like flow (shear band) based on the observation of shear offsets around the impression sites and discrete discontinuities in the indentation responses, as the finest grain sizes approach the amorphous limit. The above studies all suggest that dislocation activities cannot account for the major deformation mechanism due to the limited lattice geometries which suppress dislocation motion and multiplication, which indicates a transition in the underlying deformation mechanism of crystalline solids.

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Competing Deformation Mechanism with Varying Grain Sizes

It is useful to construct deformation mechanism maps to show the transition of dominant deformation mechanisms when the grain size varies from micrometer to nanometer. Figure 6 plots such maps for pure nickel based on the superposition of rate equations as discussed in Sect. 2 and Fig. 1. Similar maps can be found in Zhu and Li [12], which however did not show the inverse Hall-Petch effect and competition among the I–IV regimes as in Fig. 6. Figure 6a shows a schematic plot of the Hall-Petch effect and breakdown, with a critical transition at grain size of about 20 nm. At grain size d1 = 20 μm, the deformation map clearly shows three principal fields, including low-temperature plasticity, power-law creep, and diffusional flow governed regions. The power-law creep consists of core diffusion-controlled creep at low temperature and the lattice diffusion-controlled dislocation creep at high temperature. Diffusional flow is important at high-temperature and low-stress regime, involving grain boundary diffusion-controlled Coble creep and lattice diffusioncontrolled Nabarro-Herring creep at extremely high temperature (~0.8TM). When grain size decreases to 0.5 μm, the diffusional flow expands at the expense of powerlaw creep, and grain boundary diffusion becomes dominant; as a result, the lattice diffusion-controlled dislocation creep shrinks, and Nabarro-Herring creep disappears from the deformation map. As the grain size decreases further to 10 nm, grain boundary diffusion (Coble creep) contributes most to the total deformation, and dislocation activities are severely confined due to the small grain size. It is clearly seen that, in the high-temperature regime, grain boundary diffusion gradually becomes the dominant deformation mechanism when grain sizes decreases from 20 μm to 10 nm. In the low-temperature plasticity field, the strength increases as the grain size decreases from 20 μm to 0.5 μm following the grain boundary strengthening Hall-Petch relationship. But as the grain size decreases further to 10 nm, the strength shows softening behavior due to the lack of strengthening mechanism in absence of dislocation multiplication. As discussed in the Hall-Petch breakdown section, it is still questionable which deformation mechanism dominates in the low-temperature plasticity when the grain size is less than the critical grain size for the Hall-Petch breakdown.

4

Summary and Perspectives

This book chapter reviews the competing deformation mechanism of polycrystalline materials with varying grain sizes. Deformation maps and fully explicit model are used to identify the dominant mechanism under different ranges of stresses, temperatures, and strain rates for crystalline materials with grain size varying from micrometer to nanometer scale. Both fully explicit simulation results and rate equationbased deformation maps capture the transition from dislocation creep-dominated flow in coarse-grained materials to grain boundary sliding and diffusion-dominated flow in materials with fine grain sizes. Most of the literature studies attribute the critical transition to the limited grain interior dislocation activity and increased

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Fig. 6 (a) Schematic illustration of the Hall-Petch effect and its breakdown at nanoscales. (b) Deformation mechanism maps for pure nickel with grain sizes of 20 μm, 0.5 μm, and 10 nm, where region I and region II represent dislocation creep dominated by dislocation core diffusion and lattice diffusion, respectively, and region III is diffusional flow by grain boundary diffusion. The dependence of the flow stress at room temperature on the grain size follows the schematic in (a)

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fraction of grain boundary atoms which ease the grain boundary diffusion and sliding in nanocrystalline materials. This mechanistic transition has been demonstrated to correlate with the inverse Hall-Petch effect. In materials with grain size of about 20 nm, dislocation emitted from grain boundaries has less chance of cross-slip and multiplication, as the dislocation travels through the grain and annihilate at the opposite grain boundary, with no significant work hardening. In this regime, the grain boundary processes dominate the plastic deformation. As the grain size approaches the amorphous limit, the material gradually changes from nanocrystalline to an amorphous state, in which the deformation is dominated by microscopic shear bands. So far, most findings of deformation mechanisms of nanocrystalline material have been done by using molecular dynamics simulations; more experimental evidence is in urgent need to support the computation findings and to reveal the underlying deformation mechanisms. The microstructure-based finite element simulations, on the other hand, are capable of large-scale simulations and can investigate microstructural heterogeneities much more efficiently than the atomistic simulations. Acknowledgments This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.

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14. Petch NJ. The cleavage strength of polycrystals. J Iron Steel Inst Jpn. 1953;174:25–8. 15. Chokshi AH, Rosen A, Karch J, Gleiter H. On the validity of the Hall-Petch relationship in nanocrystalline materials. Scr Mater. 1989;23:1679–84. 16. Masumura RA, Hazzledine PM, Pande CS. Yield strength of fine grained materials. Acta Mater. 1998;46:4527–34. 17. Nieh TG, Wadsworth J. Hall-Petch relation in nanocrystalline solids. Scr Metall Mater. 1991;25:955–8. 18. Pande CS, Cooper KP. Nanomechanics of Hall-Petch relationship in nanocrystalline materials. Prog Mater Sci. 2009;54:689–706. 19. Trelewicz JR, Schuh CA. The Hall-Petch breakdown in nanocrystalline metals: a crossover to glass-like deformation. Acta Mater. 2007;55:5948–58. 20. Frost HJ, Ashby MF. Deformation-mechanism maps. Oxford: Pergamon Press; 1982. 21. Bower AF, Wininger EA. Two-dimensional finite element method for simulating the constitutive response and microstructure of polycrystals during high temperature plastic deformation. J Mech Phys Solids. 2004;52:1289–317. 22. Cipoletti DE, Bower AF, Krajewski PEA. Microstructure-based model of the deformation mechanisms and flow stress during elevated-temperature straining of a magnesium alloy. Scr Mater. 2011;64:931–4. 23. Mukherjee AK, Bird JE, Dorn JE. Experimental correlations for high temperature creep. Trans ASM. 1969;62:155–79. 24. Weertman J. Dislocation climb theory of steady-state creep. Trans ASM. 1968;61:681–94. 25. Lund RW, Nix WD. High temperature creep of Ni-20Cr-2ThO2 single crystal. Acta Metall. 1976;24:469–81. 26. Ball A, Hutchison MM. Superplasticity in aluminum-zinc eutectoid. Met Sci J. 1969;3:1–6. 27. Langdon TG. Grain boundary sliding as a deformation mechanism during creep. Philos Mag. 1970;178:689–700. 28. Conard H, Narayan J. Mechanics for grain hardening and softening in Zn. Acta Mater. 2002;50:5067–78. 29. Conard H. Grain-size dependence of the flow stress of Cu from millimeters to nanometers. Metall Mater Trans A. 2004;35:2681–95. 30. Van Swygenhoven H, Derlet PM. Grain boundary sliding in nanocrystalline fcc metals. Phys Rev B. 2001;64:224105. 31. Schiøtz J, Di Tolla FD, Jacobsen KW. Softening of nanocrystalline metals at very small grain sizes. Nature. 1998;391:561–3. 32. Schiøtz J, Vegge T, Di Tolla FD, Jacobsen KW. Atomic-scale simulations of the mechanical deformation of nanocrystalline metals. Phys Rev B. 1999;60:11971. 33. Yamakov V, Wolf D, Salazar M, Phillpot SR, Gleiter H. Length-scale effects in the nucleation of extended dislocations in nanocrystalline Al by molecular-dynamics simulation. Acta Mater. 2001;49:2713–22. 34. Yamakov V, Wolf D, Phillpot SR, Mukherjee AK, Gleiter H. Dislocation process in the deformation of nanocrystalline aluminum by molecular-dynamics simulation. Nat Mater. 2002;1:1–4. 35. Yu CH, Bird MW, Huang CW, Chen CS, Gao YF, White KW, Hsueh CH. Micromechanics modeling of creep fracture of zirconium diboride-silicon carbide composites at 1400–1700  C. J Eur Ceram Soc. 2014;34:4145–55. 36. Kumar KS, Van Swygenhoven H, Suresh S. Mechanical behavior of nanocrystalline metals and alloys. Acta Mater. 2003;51:5743–74. 37. Carsley JE, Ning J, Milligan WW, Hackney SA, Aifantis A. Simple mixtures-based model for the grain size dependence of strength in nanophase metals. Nanostruct Mater. 1995;4:441–7. 38. Yang XS, Wang YJ, Zhai HR, Wang GY, YJ S, Dai LH, Ogata S, Zhang TY. Time-, stress-, and temperature-dependent deformation in nanostructured copper: creep tests and simulations. J Mech Phys Solids. 2016;94:191–206.

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40

Victor E. Panin, Valerii E. Egorushkin, Tamara F. Elsukova, Natalya S. Surikova, Yurii I. Pochivalov, and Alexey V. Panin

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Role of the Crystal Structure in the Formation of Micropores and Crack Development Under Fatigue Fracture of Commercially Pure Titanium . . . . . . . . . . . . . . . . . . 2.1 Gauge Theory of Defects in Solids Taking into Account a Crystal Lattice Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Cyclic Loading of the Untreated Ti Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Influence of Hydrogenation of the Ti Surface Layers on Its Structure and Fatigue Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Influence of Ultrasonic Impact Treatment and Carbon Alloying of Ti Samples’ Surface Layers on Its Structure and Fatigue Life . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Influence of Crystal Lattice Curvature on Rotation Mode of Plastic Flow . . . . 2.6 Conclusion 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Multiscale Translation-Rotation Sliding and Shear Banding in Al Polycrystals Under Inhomogeneous Plastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Multiscale Interaction of Grain Boundary Sliding and Intragranular Sliding Under Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1256 1258 1260 1265 1266 1269 1270 1274 1275 1276

V. E. Panin (*) Laboratory of Physical Mesomechanics and Non-destructive testing, Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, Russia National Research Tomsk Polytechnic University, Tomsk, Russia National Research Tomsk State University, Tomsk, Russia e-mail: [email protected] V. E. Egorushkin · T. F. Elsukova · N. S. Surikova · Y. I. Pochivalov Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia e-mail: [email protected]; [email protected]; [email protected] A. V. Panin Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia National Research Tomsk Polytechnic University, Tomsk, Russia e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_77

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3.2 Dislocation Substructure, Multiscale Fragmentation, and Shear Banding in Al Foils Under Alternate Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Multiscale Model of Strain-Induced Defects Generation in a Polycrystal Under Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 General Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The problem of plastic shear propagation in conditions of high crystal lattice curvature has been researched theoretically and experimentally on the basis of the gauge theory. It has been shown that in conditions of high crystal lattice curvature, the flows of deformation defects reveal plastic distortion, vorticity of plastic shears, and the possibility of their non-crystallographic propagation by a shear banding. Experimental research on the mechanical behavior of the commercially pure Ti samples under alternating bending demonstrated the strong influence of their structural state on the development of high crystal lattice curvature and shear banding. The high crystal lattice curvature under a cyclic loading appears in the hydrogenated surface layers, where shear banding and microporosity develop and fatigue life is greatly reduced. High crystal lattice curvature in surface layer of Ti samples produced by ultrasonic processing determines the fourfold increase of Ti fatigue life. The translation-rotation deformation against developed grain boundary sliding in highpurity A999 Al polycrystals under creep and in A999 Al foils glued on commercial A7 Al plates under alternate bending was studied. The stage of steady-state creep provides intragranular sliding in the material mainly by dislocation mechanisms. The stage of tertiary creep causes multiscale fragmentation, non-crystallographic sliding, and fracture. Under alternate bending, the Al foils are involved in rotations by dislocation mechanisms only up to a strain of ~50%, and as they become highly corrugated, shear bands propagate in them. The observed shear banding provides the generation of elastoplastic rotations in zones of high lattice curvature. Keywords

Crystal lattice curvature · Ti and Al polycrystals · Fatigue fracture · Creep · Shear banding

1

Introduction

At the conventional physics of plastic deformation of solids lies the deformation defects theory [1–7]. All the deformation defects types have traditionally been examined in translationally invariant crystals, in which dislocations and disclinations play the main role. The role of point defects, even in conditions of creeping [8–10], has been considered to be low. According to the multiscale approach of physical mesomechanics [11–13], the primary plastic shears in a deformable solid develop in its planar subsystem

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(surface layers and all the internal boundaries). They produce the rotary deformation modes, which should be compensated with the rotary modes of an opposite sign under the deformation defects motion in a 3D crystal (the law of conservation of angular momentum). It causes crystal lattice curvature (CLC) development in the deformable solid, in which the translational invariance decreases during a plastic deformation. In severe plastic deformation conditions, the CLC growth makes the translational motion of the dislocations impossible, and shear banding develops. The mechanism of shear band propagation is widely discussed in the literature but within the framework of translationally invariant crystal. According to [11–13] the crystal lattice curvature, which radically changes the mechanisms of the deformation defects flows, should be considered in the mechanics of shear banding. In [11–13] a concept is put forward, according to which in the CLC zones there appear interstitial structural states, which determine the plastic distortion effect in the deformable solid. It means that there appears a new degree of freedom and growth of the role of point defects in the CLC zones, which was not considered in [8–10]. The important role of the crystal lattice curvature and the nonequilibrium point defects related to plastic distortion in CLC zones are considered in this chapter for commercially pure titanium (hereafter referred to as CP-Ti) subjected to alternate bending. Translation and rotation sliding processes in inhomogeneous plastic deformation are well-known in literature [2–4, 14–25]. In hcp metal polycrystals under ambient temperature creep, such sliding produces rotations of up to 5 deg. within nearboundary zones [14]. In metal polycrystals under creep load, grain boundary sliding (GBS) develops such that the plastic flow in their grains experiences rotations of opposite sign according to the angular momentum conservation law. Polycrystals under cyclic load generate a very specific internal dislocation structure with a low degree of strain hardening [17]. The features of dislocation substructures arising in cyclically loaded solids are now of much research interest [17, 21, 22]. It has been found that under cyclic load resulting in persistent slip bands, both the strengthening and the scaling coefficients are more than two times lower than their values under monotonic load. The low strengthening coefficients at low cyclic stresses, when a dislocation microstructure evolves, are explained by higher rates of increase in dislocation density during cycling, higher density of collinear dislocation segments, and partial reversibility of cyclic strains [17]. By now, there is no consensus on the rotation mechanisms of deformation. As is generally accepted, the rotations evolve through developing cellular dislocation substructures in which the cell misorientations increase progressively with strain. However, severe plastic deformation can trigger other rotation modes such as crystal lattice curvature, refinement, fragmentation, and microporosity, the understanding of which is still poor. One more well-known rotation phenomenon in deformed solids is surface corrugation, but the phenomenon still needs a multiscale description for clarification. These problems are studied in detail in this chapter for high-purity polycrystalline Al under creep and alternate bending.

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The Role of the Crystal Structure in the Formation of Micropores and Crack Development Under Fatigue Fracture of Commercially Pure Titanium

In this study the task is to examine the mesomechanics of shear banding development in deformable solids under the conditions of high crystal lattice curvature. The accounting of the crystal lattice curvature is produced on the basis of the gauge theory of the strain-induced defects. At the same time in the gauge theory, the mathematical condition is formulated that in conditions of high crystal lattice curvature, dislocations cannot appear. It changes the constitutive and quasi-elastic equilibrium equations of the gauge theory and allows the introduction of summands, which describe the crystal lattice curvature, plastic distortion, and vorticity of non-crystallographic atomic flows. The cyclic loading of the CP-Ti samples, whose surface layer is ultrasonic processed accompanied by alloying with carbon or hydrogenated, allows us to experimentally reveal the CLC zones where interstitial structural states occur. The hydrogen and titanium produce the weak interatomic bonds. It is supposed to activate the basal sliding, the appearance of the CLC zones, and the plastic distortion of Ti crystal lattice into the interstitial structural states in the CLC zones and to determine the microporosity development in the surface layers of the deformable samples. The influence of the carbon should be inverse. Interatomic Ti-C bonds should block the basal sliding and initiate the fragmentation of deformation twins. Consequently, the hydrogenation of the surface layers of the CP-Ti samples should decrease their fatigue life. The carbon alloying of the surface layers of the CP-Ti samples subjected by ultrasonic impact treatment should increase their fatigue life. The results of the experimental research should correlate with the mesomechanic predictions of shear banding. The material under experimental study was CP-Ti whose plastic deformation and fracture are well understood [26–29]. This material has an hcp structure in which plastic shear develops in basal slip planes accompanied by twinning. Accommodation basal slip activated by sliding mode cracking is bound to induce strong torque moments. Such torques can produce high local crystal lattice curvature, but its development can be controlled by external actions. Two types of surface layer processing of the CP-Ti samples were used: electrolytic hydrogenation and ultrasonic impact treatment with carbon alloying of the Ti surface layers. The hydrogenation of the surface layer of the Ti samples should promote the development of basal sliding accommodated by rotary deformation modes, which are necessary for the disclosure of the main fatigue crack under the cyclic loading. This should decrease the fatigue life of the Ti samples with hydrogenated surface layers. Electrolytic hydrogenation was used, conducted at room temperature in 1М of sulfuric acid solution at a current density of 1А/sm2 within a period of 60 min. The hydrogen concentration in the surface layer was 400 ppm. The ultrasonic processing and carbon alloying of the Ti samples’ surface layers should block the basic accommodative slipping during the disclosure of the main fatigue crack. That is why the increase of the fatigue life of the CP-Ti samples, the

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surface layers of which were ultrasonically treated and alloyed with carbon, was expected. Ultrasonic impact treatment was used, where during processing the indenter is attached to a pivotal waveguide and creates vibrations with the magnetostriction transducer amplitude. This amplitude is equal to 15 μm. The deforming tool with a 5 mm diameter is pressed to the surface of the processed plate with a 100 N load. The ultrasonic impact treatment allows the creating of folded undulating corrugation with protrusions of up to 1.5 μm (Fig. 1a) in the surface layer of the CP-Ti plate. The research of the folded structure with the aid of atomic force microscopy revealed smaller folds in the structure of every large fold (Fig. 1b). During the displacement of the small folds under the influence of the maximum shear stress τmax, refinement of their grain structure to 200 mm occurs (Fig. 1c), and high crystal lattice curvature appears in the surface layers of the CP-Ti samples. In such surface layers, the carbon is actively dissolved, and titanium nanocarbides can be formed, which in high CLC conditions should effectively delay the disclosure of the main fatigue crack. There should be no plastic distortion of Ti atoms and microporosity emergence in the CLC zones in such surface layers. The connection between the examined effects and emergence of the interstitial structural states in the CLC zones will be examined in more detail below. CP-Ti was spark cut as flat dumbbell-shaped specimens with a gauge section of 37
8
1 mm3. The specimens were preliminary annealed in a vacuum at 800 C for 1 h to remove surface defects and internal stresses. The grain size in the annealed

Fig. 1 Morphology of the surface (а, b) and the corresponding profilogram (c) of CP-Ti samples subjected to ultrasonic impact treatment; optical profilometry (a) and atomic force microscopy (b)

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specimens was 50 μm. The specimens were subjected to high-cycle fatigue by alternate bending; that is, the bending frequency (8 Hz) and amplitude (4.5 mm) were chosen so that the total number of cycles to fracture would be no less than 105. This parameter is used as a quantitative characteristic of fatigue life (given the cycling frequency, the parameter can be converted to time units). The specimen surface for structural examination was smoothened by electropolishing. On the polished specimen surface, a coordinate grid was superimposed, making it possible to analyze shear and rotational strain components and strain distributions at the polycrystal surface. Structural analysis of a specimen surface layer, threedimensional images of the mesoscopic substructure, and crystal lattice curvature formed on the specimen surface layers were taken using an Axiovert 25CA light microscope with a DIC device, Quanta 200 3D scanning electron microscopy, a New View 6200 interference profilometer, and JEOL2100 transmission electron microscopy.

2.1

Gauge Theory of Defects in Solids Taking into Account a Crystal Lattice Curvature

The absence of translation invariance in lattice curvature zones and the formation of interstitial structural states in these zones provide a possibility for deformation processes to change from dislocation-induced plastic flow to non-crystallographic shear banding. The new deformation mechanism is attributed to the cluster-type atomic structure of lattice curvature zones which lack translation invariance. Similar mechanisms, as applied to the structure of amorphous materials and deformed thin films, were considered in detail elsewhere [30–32]. The mesomechanics of structural transformations involved in shear banding should take into account an additional term in its constitutive and quasi-elastic equilibrium equations, namely, the term Pβν Cμν αβ =E descriptive of plastic distortion in the regions of interstitial structural states. Let us show that this is the case. In gauge theories of plastic deformation, the initial microstructure of a translationinvariant medium remains unchanged (6). According to the gauge theories, the equations of motion for mechanical fields have the vector form
 div Sa ¼ f abc Bb  Sc  gij λai , κ ηακ ηαj ; a μ

ðrot R Þ ¼ f

abc




(1a)

    b  1 @Sa μ c μ B
R þ 2 þ gij λai , κ ηακ @ ν δlj  Bbν λbj , l ηβl c @t

Cμν αβ E

;

(1b)
 divRa ¼ f abc Bb  Rc ;

(1c)

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Multiscale Translation-Rotation Plastic Flow in Polycrystals

a μ

ðrot S Þ ¼ f

abc



B
S b

 c μ

1261

 a μ @R  , @t

(1d)

where Sa ¼  1c @B @t is the rate of variation in plastic distortion (gradient velocity); Baμ  Bjμ, i is the gauge field (connectivity in geometric terms); Raα ¼ 12 eαμν σ aμν is the dimensionless disclination density; σ aμν is a 54-component stress tensor of the mechanical field σ aμν ¼ @ ν Baμ  @ μ Baν  f abc Bbν Bcμ (curvature tensor) characterizing ð a

Rαν dsν ¼ Θα is the Franck

the disclination density in the material (6) such that s

vector; а = 1, . . ., 9 is the index of the group GL(3, R); ημi is a local lattice reference of quasi-elastic microdistortion corresponding to structural elements of the medium; i and j are internal indices
and  α, β, μ, ν = 1, 2, 3 are external indices (global coordinate system); λa  λij lκ ¼ δil δjκ are generators of the group GL(3, R) in the Weyl basis [33]; f abc  f ij, sm, κr = δsiδmκδjr  δiκδsrδjm are structural constants of μν the group GL(3, R); gij = (ηi  ηj) is the internal metric tensor; gij ¼ g1 ij and Cαβ are the elastic moduli of the medium with structural elements; E is the Young’s modulus μ0 of structural elements; Cμν is for all other zero indices; ρ is the material αβ ¼ Eδαβ δ density in structural elements; and c is the velocity of sound in structural elements. The sources of gauge fields ημl ðx, tÞ related to elastic displacements can be determined using the equations [34]:

 giκ ηαj Dμ ηγi Dν ην jβ Cμν γβ þ

(

 ) gij Baμ λai , κ Dν ηβj þ gκj @ μ Dν ηβj κm γj

mj κγ

ðg η þ g η


  where Dμ ηγi ¼ @ μ δγi  Baμ λai , l ηγl .

Þ@ μ ηγm Dν ηβj

Cμν αβ ¼ 0,

(2)

The system of equations (1) and (2) represents globally: quasi-elastic equilibrium (2) and plastic equilibrium (1b), continuity conditions for Sa (1a) and Ra (1c), and deformation compatibility condition (1d). Thus, this system of equations is no different in its form from conventional dynamic equations of a continuous medium. Surface integration of (1d) gives the law of conservation of the Franck vector. þ

    α dΘα ¼ dxa f abc Bb
Sc
ðr  r0 Þ  Saα : dt L

In the integrated equation, Θα= constant where the rate of translational plastic deformation compensates the rate of rotational plastic deformation (deformation compatibility). The quantity f abc[Bb
Sc]
(r  r0) should be considered as the dislocation flux density. In the absence of nonlinearity (summation over a), the vector Θiκtransforms to the Burgers vector bα. By discarding the external sources related to ημi in (1) and by changing the group to S0(3)
T(3) and hence the generators a, b, and c, we obtain the Eqs. (6) and (7) in which the nonlinear parts f abc(Bb  Sc), f abc(Bb  Rc) of (1) are the dislocation and disclination charges and

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f abc[Bb
Sc], f abc[Bb
Rc] are the dislocation and disclination currents due to the distribution of displacement vector jumps δu(x, t) over the material surface S. Along with these sources of plastic deformation, the equations for solids with crystal lattice curvature contain nonuniform spatial and temporal microstructure distributions, in particular fluxes and nonuniform distribution of point defects in microscopic crystal regions. In the above statement, the stresses specified by the energy-momentum tensor Tλτ have the form   ij μ α T λτ ¼ δλτ Cμν Dν ηβj αβ g Dμ η i   h i ij ασ þCμν Dν ηβj δσγλτ @ σγλτ μ þ δσμ, λτ @ γ  δμγ , λτ @ σ Dν ηβi ηγj αβ g g 1 (3) Saλ Saτ  Raλ Raτ þ δλτ ðSa Sa þ Ra Ra Þ, 2
 where δαβ, λτ ¼ 12 δαλ δβτ þ δατ δβλ . In the general case, the mechanical field Sa, Ra is a unified system in which all the components are interrelated, and one or another form of defect motion can be defined only in a series of special physical conditions. Equations for a medium with local lattice curvature which lacks translation invariance can be derived from Eqs. (1) and (2) as follows. Let us constrain the group GL(3, R) with generators (λij)lκ such that ij (λ )lκ ! (λii)lκ = δilδiκ = δlκ. Then, the new transformations are orthogonal and are specified by three rather than by nine generators. It is easily seen that the structural constants f abc, in this case, vanish indicating that the generators commute with each other. Thus, we exclude the charges of dislocations and disclinations and their currents from Eqs. (1) and (2). By denoting the indices (kk) = 11, 22, 33 in terms of α = 1, 2, 3, we have Raμ ! ααμ , Saμ ! jαμ with both indices belonging to the same space. Equations (1) and (2), in this case, have the form @lnψ μ ðx, tÞ @ α ; j ¼ @xα μ @t eμxδ

eμxδ

(4a)

@ααμ @jαδ ¼ ; @x @t

(4b)

@ααμ ¼ 0; @xα

(4c)

μν α Cμν @ααδ 1 @jμ @lnψ β ðx, tÞ Cαβ αβ þ  Pβν ; ¼ 2 @xν @xx c @t E E

μν μν 2 1 @ 2 lnψ α ðx, tÞ @ lnψ β ðx, tÞ Cαβ @Pβν Cαβ ¼  :  c2 @t2 @xμ dxν E @xμ E

(4d)

(4e)

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If for simplicity we assume that the quasi-elastic variations ψ(x, t) are scalar functions (volume changes of structural elements), summation of (1) over the group index gives div J ¼ 

@lnψ ðx, tÞ ; @t

(5a)

@α ; @t

(5b)

rotJ ¼ 

div α ¼ 0, ðrotαÞμ ¼

μν X Cμν 1 @J μ @lnψ ðx, tÞ X Cαβ αβ β þ  P ; ν c2 @t @xν E E α αβ

μν μν 1 @ 2 lnψ ðx, tÞ @ 2 lnψ ðx, tÞ X Cαβ @Pβν X Cαβ ¼ :  c2 @t2 @xμ @xν αβ E @xμ α E

(5c) (5d)

(5e)

Unlike the expressions derived by convolution of (1)-type Eqs. (6) and (7), expressions (4) and (5) contain no currents and charges caused by nonlinearity but contain quasi-elastic sources of plastic deformation associated with interstitial structural states. Each of systems (4) and (5) constitutes a complete system of dynamic equations for an elastoplastic medium with moving defects.
 In Eq. (5), Pβν is the plastic componentof distortion, α αxβ ¼ exμν @ μ Pβν is the tensor of interstitial structural states, and J jαμ is the tensor of the current of point

defects associated with these states. Relations (4a) and (5a) are the continuity equations of the medium. Relations (4b) and (5b) are the deformation compatibility condition. Relations (4c) and (5c) are the continuity equations for interstitial structural states. Relations (4d), (5d), and (4e), (5e) are the constitutive equations of plastic deformation and quasi-elastic equilibrium, respectively. Equations (4e) and (5e) can be written as 1 @υα @σ αμ ¼ , c2 @t @xμ @lnψ

Cμν

(6)

Cμν

where σ αμ ¼ @xν β Eαβ  Pβν Eαβ is the quasi-elastic stress tensor; υα = @ ln ψ α/@t; on the other hand, (5e) is the continuity equation for ψ(x, t). Similarly for (5d), we have 1 @V μ X ¼ σ αμ  ðrot αÞμ : c2 @t α

(7)

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Thus, the time variation in the rate of plastic distortion is determined by the difference in σ αμ  eμxγ @ x ααγ, and the larger the quasi-elastic stresses Cμν Pβν Eαβ

μν @lnψ β Cαβ @xν E

and the

smaller the moment for clusters of interstitial structural states as well as the anisotropy of their distribution (rot α), the higher the quantity V. Equations (4d) and (5d) with continuity conditions (4c) and (5c) represent a system of equations for the density of interstitial structural states at given ψ = (x, t) (ψ α), Jðx, tÞ jαμ . From (4d) and (5d), we can find the force F acting
on the atomic clusters in the ~ ij : ~j α region of interstitial structural states with a density α ( ) μν Cμν 1 @V μ 1 X @lnψ β Cαβ β αβ ~ μκ :  Pν  2 α Fκ ¼  2 α, μ6¼κ c @t @xν E E

(8)

The latter equation is a generalization of the well-known Peach-Koehler equation [35] to fit the dynamic behavior of defects. From (5a)–(5d) and (4a)–(4d), it is easy to obtain the equations for plastic strain waves: ! X Cμν X Cμν 1 @2Jμ @2J @ @ @ αβ β αβ   2¼ lnψ ðx, tÞ  lnψ ðx, tÞ Pν , c2 @t2 @xμ @t @xμ @xν E E α αβ ! μν μν 1 @ 2 αμ @ 2 α @ 2 lnψ ðx, tÞ X Cαβ X @Pβν Cαβ   2 ¼ eμxσ , c2 @t2 @xμ @xx @xν E @xx E α αβ

(9)

(10)

and the compatibility condition, instead of (5b), for the sources: @Lμ @I m þ eμlm ¼ 0, @t @xl

(11)

where I is the right-hand side of (9), and L is the right-hand side of (10). The last term in the right-hand side of (9) represents the flow due to transformations of atomic clusters in the region of interstitial structural states, and the other two represent the vacancy flow due to dynamic variation in the volume of structural elements. The source of interstitial structural states, i.e., the right-hand side of (10), is represented by local lattice curvature and vorticity of atomic flows in the region of interstitial structural states. Thus, non-crystallographic propagation of shear bands is initiated in lattice curvature zones which lack translation invariance, assume cluster distributions of atoms, and form interstitial structural states. The atomic flows propagating in inhomogeneous internal stress fields through interstitial structural states are qualified as shear bands. The mechanism of their propagation is described by nonlinear wave Eqs. (9) and (10) and is radically different from crystallographic dislocation motion in a translation-invariant crystal.

40

2.2

Multiscale Translation-Rotation Plastic Flow in Polycrystals

1265

Cyclic Loading of the Untreated Ti Samples

In the maximum amplitude zone of alternating bending, bands of localized plastic deformation develop. Their distribution along the conjugate directions of the maximum shear stress provides alternation of the samples’ plastic stress-strain shifts under alternating bending. The fatigue main crack develops in a zigzag manner along the macrobands of localized deformation. The disclosure of the fatigue crack MN under the Ti cyclic loading causes the development of the accommodative rotary modes in the adjacent material, Fig. 2a. On the surface one can see three grains A, B, and C, rolled on each other. All the three grains experienced accommodative rotary modes in the form of complex corrugation (Fig. 2b, c). At the same time, the separate zones of the B grain (ab type, in Fig. 2b) preserved their initial flat character and experienced clockwise rotation. Conjugate zones of strong curvature underwent fragmentation and experienced shifts with material anticlockwise rotation. The surface of B grain obtained a zigzag profile, as shown in Fig. 2b. In Fig. 3 the pattern of the fatigue failure of the initial Ti sample is shown. It is seen that during the failure the main crack of sliding mode, accompanied by local material delamination in the separate zones (indicated by arrows), is propagated.

Fig. 2 Accommodation buckling of grain B in the initial CP-Ti surface layer on disclosure of main fatigue crack MN: optical image,
510 (a); buckling profile along KL; 3D–image of buckled grain B, optical profilometry (c)

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Fig. 3 SEM image of the fracture surface of the initial CP-Ti with a propagating main crack

The results of the research on translation-rotation deformation modes during the alternating bending of the initial Ti with the help of ТЕМ are represented in Fig. 4. It is seen that the formation of cellular dislocation substructure and the intensive development of deformation twins can entirely execute the translation-rotation modes of plastic flow in a crystal characterized by the translation invariance.

2.3

The Influence of Hydrogenation of the Ti Surface Layers on Its Structure and Fatigue Life

During the fatigue failure of the CP-Ti samples with the hydrogenated surface layers, the main crack of the sliding mode is also propagated. However, the effects of the material delamination in the transverse shear zones are expressed very strongly, as in Fig. 5. The delamination of the material lamellae is accompanied by their fragmentation. As shown in Fig. 5, AB is a fracture band of the D layer of the material. The crack propagation occurred along the C surface of the lower sample half. After that the D layer experienced a fracture along the AB band, and the crack continued to propagate along the D surface. The next fracture occurred along the EF band and the crack movement continued along the G surface. The surfaces of all the breaks had a microporous structure. The main crack on its top creates a curvature, where the nonequilibrium interstitial structural states appear [13]. In the AB band, the concentration of the nonequilibrium interstitial structural states is so high that microporosity forms. The fracture of the material occurs in this zone and the crack movement continues along the D surface. Abovementioned result convincingly illustrates the role of the nonequilibrium interstitial structural states, which appear on the top of the crack, in causing microporosity formation and the development of structural phase decomposition of the material condensed state as a mechanism of its failure [36]. The accommodative rotary modes in the hydrogenated surface layer, determined by the disclosure of the main fatigue crack KL, appear as a strong corrugation and ab, cd, ef, and other shear

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Fig. 4 TEM image and corresponding microdiffraction pattern of dislocation substructure and extended twins in CP-Ti sample under cyclic bending, N = 1.9
106

Fig. 5 SEM image of the fracture surface of the surfacehydrogenated CP-Ti sample

banding development (Fig. 6). There are already no flat surface zones of initial sample grains at the border zone with the fatigue crack. A very important issue is connected with the intensive corrugation mechanism of the hydrogenated surface layer. The translation-invariant crystal is unable to create macrocurvature, which is shown in Fig. 6. Grain C (Fig. 6) undergoes a strong anticlockwise rotation due to the disclosure of the main fatigue crack KL running perpendicular to the specimen axis. Subsurface grains D and E severely constrain the rotation of surface grain C, resulting in its multiscale corrugation and layer-wise delamination with the formation of broad shear-rotation bands M, N, O, P, etc. As grain C rotates, each of the bands acquires a lamellar structure in which individual lamellae are displaced relative to each other. At band boundaries ab, cd, ef, etc., macroscale shear bands develop. The material within the macroscale shear bands reveals high crystal lattice curvature (Figs. 7 and 8). Early in the bending, shear band LMN (Fig. 7) assumes highly pronounced microporosity, and at a larger number of cycles, similar porosity develops in the

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Fig. 6 Three-dimensional image of the hydrogenated surface layer of grain C in the zone of main fatigue crack KL; optical profilometry [36]

Fig. 7 Development of the crack LM in 3D shear band LMN; a hydrogenated CP-Ti surface layers, cyclic bending, N = 1.8
105 loading cycles, T = 293 K; optical profilometry

3D shear bands and results in cracking. This actually means that the crystal with high lattice curvature undergoes structural decomposition [37]. From TEM measurements made on discrete misorientations of shear bands (Fig. 8), it is seen that the 3D shear band in the bright-field image in Fig. 8a is a central meso-band propagating against the background of a multipath of crossing planar shear bands which had developed in conjugate directions of maximum tangential stresses τmax. It means that the 3D shear band develops as a multiscale process. At the first stage of surface layer corrugation emergence, bands of defect substructure, the fragment size of which is 40–60 nm, with continuous misorientations appear (Fig. 8). Next, in the bands of the nanofragmented structure, macro shear bands propagate, and there are signs of nanofragmented defect substructure on the lateral boundaries of the macro shear bands. In the dark-field images in Fig. 8bd, it is seen that within the central meso-band, a microstructure had developed with large discrete misorientations having undergone sub-micrometer fragmentation (Fig. 8c). Lateral sub-bands b and d were measured to have goniometer tilt angles of 15.15 and 31.07 , respectively. This suggests that the central meso-band has produced a curvature of ~16 /μm. Such an elastoplastic curvature of

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Fig. 8 TEM image of 3D shear band (a) consisting of left subband (b), central subband (c), and right subband (d); hydrogenated surface layers of CP-Ti, alternating bending; N = 1.8
105 cycles

surface grain corrugation determines the decrease of the fatigue life of the CP-Ti samples with a hydrogenated surface layer by 3.7 times (Table 1). The influence mechanism on the fatigue life of CP-Ti is considered in the discussion.

2.4

The Influence of Ultrasonic Impact Treatment and Carbon Alloying of Ti Samples’ Surface Layers on Its Structure and Fatigue Life

In Fig. 9 the surface of a CP-Ti sample corrugated by being ultrasonically treated and alloyed with carbon after its fatigue destruction is shown. There are no any signs of plastic deformation on this surface, though the sample experienced N = 1.92106 cycles of alternating bending. Also the mechanism of fatigue destruction changed

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Table 1 The fatigue life of CP-Ti samples at different structural states of the surface layer Surface layer state Number of cycles before the destruction

Initial 4.4
105

Hydrogenated 1.2
105

Ultrasonic impact treatment + carbon alloying 19
105

Fig. 9 A fragment of the nanostructured surface layer of a flat CP-Ti sample, which experienced fatigue fracture; optical profilometry

radically: the destruction occurred during the distribution of the opening mode crack, and the surface of destruction represents a typical dimple rupture (Fig. 10). Local delamination of the material is virtually absent at the fracture pattern. This is in good agreement with nanohardness measurements H and elasticity modulus E of the surface layer, which is ultrasonically treated and carbon-alloyed: the H value increases by 1.4 times and is equal to 4.56 GPa; the E value increases by 1.2 times and is equal to 150 GPa. Such a surface layer does not change the folded profile of its curvature created by ultrasonic treatment, which is necessary for the nucleation and disclosure of the main crack of fatigue destruction. It means that the accommodative rotary processes in the surface layer in Fig. 9 are elastic and reversible during the disclosure of the KL main fatigue crack. As a result, a fourfold increase was observed in the fatigue life of the Ti samples with ultrasonically treated and carbon-alloyed surface layers (Table 1). Let us consider the mechanism of the different influence of hydrogenation and ultrasonic impact treatment of Ti samples’ surface layers on their fatigue life.

2.5

The Influence of Crystal Lattice Curvature on Rotation Mode of Plastic Flow

This paper discusses, theoretically and experimentally, a conceptual state, according to which an important role in the durability and plasticity of solids is played by zones of crystal lattice curvature, where interstitial structural states (ISS) appear. On the basis of the gauge defect theory, it has been demonstrated that allowing for the appearance of ISS causes plastic distortion effects, vorticity of deformation defects flows, and shear banding development, which concludes with the destruction of the material. This conception has been experimentally verified using the example of

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Fig. 10 SEM image of the fracture surface of the surfacenanostructured CP-Ti sample

alternating bending of polycrystalline α  Ti samples. The increase of ISS concentration in the α  Tisamples during alternating bending should cause in it the development of short-range order of displacements of the HCP-BCC transition. The ISS in the zones of crystal lattice curvature are nonequilibrium, so they reveal themselves only locally. A very important experimental result is Ti nanocarbide formation in the surface layers of the samples, which are ultrasonically treated and carbon-alloyed. TiC is isomorphic to NaCl [38]. Therefore the short-range order of displacements of the HCP-BCC should promote nanocarbides formation. Let us consider a possible mechanism for the appearance of short-range order of displacements of the HCP-BCC in Ti on the basis of model [39]. Model [39] is presented in Fig.11a, b. Fig. 11a shows a Ti13 nanocluster which corresponds to the 12 nearest neighbors of atom 7 in the basal plane of the hcp lattice. The electronic structure of the two types of short-range order of displacement was calculated in [39]: where the Ti13 nanocluster is free and where it is placed in the potential field of 122 atoms of a Ti135 nanofragment. The probability distribution functions of the electronic charge in both cases were found to be very similar. This allows one to describe the local short-range order of displacement hcp!bcc transitions by a simple model, i.e., with a free Ti13 nanocluster. In the basal plane of the hcp α-Ti13 cluster, four nearest neighbors 2, 3, 5, and 6 of the central atom 7 correspond to those in the plane (110) of the quadruple bcc cell of short-range order. Atoms 1 and 4 can correlatively move in opposite directions in the basal plane (001) and become second nearest neighbors in the plane (110) of the virtual bcc cell of short-range order. In the former positions of atoms 1 and 4, there are two virtual vacancies α and β (Fig. 11b). This process proceeds in local curvature zones of the deformed hcp titanium. Atoms 1 and 4 approach atoms in the immediate surrounding of the adjacent nanocluster in the hcp crystal lattice curvature zone and locally increase the positive charge which is to be screened by electron gas from the regions α and β between atoms 3–5 and 2–6. The interatomic bonds of the latter get weaker, and an excess atomic volume arises in regions α and β in which new allowed

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Fig. 11 Cluster models for an hcp in titanium (a) and an hcp cell transformation into a bcc one with appearance of the nonequilibrium interstitial structural states α and β (b) under cyclic bending [39]

11

12 13

2

3

1

7 6

4 5

8

9 10

11

12 13

2 1

3 7

a 6

b

4

5

8

9 10

structural states are formed as described above. If the new allowed interstitial structural states are occupied by hydrogen or carbon atoms, their s  d interaction with titanium atoms 8, 11, and 9, 12 forms a bcc cell (likely distorted) of atoms 2, 8, 6, 11, 12, 3, 9, and 5 near central atom 7. This type of local short-range order structural transformations may take place in the deformed α-titanium. Eventually, the local crystal lattice curvature of the hydrogenated Ti surface layer results in propagation of shear bands by the mechanism of short-range order transformation. The distribution process of shear bands in hydrogenated CP-Ti surface layers causes the curved crystal structure fragmentation and the microporosity generation. But carbide bonds in Ti-C are very high. Therefore the Ti nanocarbide formation on the basic slip planes will block the accommodative shifts, which are

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Multiscale Translation-Rotation Plastic Flow in Polycrystals

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Fig. 12 TEM bright-field (a) and dark-field images (b) of microstructure of outmost surface layer of CP-Ti specimen subjected to ultrasonic impact treatment and carbon alloyed

necessary for disclosure of the main fatigue crack. Consequently, the fracture pattern in Fig. 10 shows opening mode crack disclosure during the fatigue destruction of CP-Ti samples, the surface layers of which are ultrasonically treated and carbon-alloyed. The formation of Ti nanocarbides, which appear in the thin surface layers of samples during the ultrasonic impact treatment and carbon alloying, is experimentally verified (Fig. 12). As Fig. 12 shows, nanograins of α  Ti with a size of 100 nm appear in the surface layers of CP-Ti samples. Ultrasonic impact treatment creates high crystal lattice curvature and short-range order of displacements with β  Ti structure in such grains (Fig. 11). Filling of the interstitial structural states of α and β type in the zones of short-range order of displacements with carbon atoms causes Ti2C nanocarbides formation. The nanosizes of such carbides are stabilized by shortrange order of displacements of β  Ti type. It determines the increase of all the mechanical characteristics of the surface layer: nanohardness by 1.4 times, elasticity modulus by 1.2 times, and fatigue life by 4 times. If a strongly nonequilibrium surface layer of α  Ti samples, ultrasonically treated and carbon-alloyed, is exposed to an alternating bending cyclic loading, then in the developed system of deformation twins of the surface layer, material fragmentation with nanostructure formation develops (Fig. 13). Strong crystal lattice curvature and high concentration of the interstitial structural states in the conditions of alternating bending form nanofragmentation with an average fragment size of less than 100 nm in the deformation twins. The carbon presence in the surface layer creates short-range order of displacements of β  Ti type. It determines the reversible inelasticity effect in the nanostructured deformation twins and multiplexly increases the fatigue life of CP-Ti. It is very important to note that the structure of the thin layer of CP-Ti samples, which are ultrasonically treated and carbon-alloyed, proved to be very nonequilibrium. Besides nanocarbides, it contains martensite particles, ω-phases, and

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Fig. 13 ТЕМ image of nanostructured deformation twin in surface layer of CP-Ti sample subjected to ultrasonic impact treatment followed by cyclic loading, N = 1.9
106 cycles

enhanced concentration of carbon in a solid solution. Their connection with crystal lattice curvature and interstitial structural states will be described in a separate paper.

2.6

Conclusion 1

On the basis of the gauge theory of defects, wave equations of plastic shears in the deformable solid taking into account crystal lattice curvature were obtained. It was shown that in the CLC conditions, the flows of deformation defects reveal plastic distortion effects, vorticity of plastic shears, and the non-crystallographic character of their propagation. Such flows of defects are classified as shear bands and observed in severe plastic deformation conditions. Experimental research was conducted on non-crystallographic (shear banding) mechanisms of plastic deformation of polycrystalline samples of CP-Ti during multicyclic alternating bending, when high CLC forms in the surface layers of the samples. Three structural states of surface layers were researched: initial, hydrogenated, and ultrasonically treated and carbon-alloyed. The maximum CLC during the cyclic loading appears in the hydrogenated surface layers, where the shear banding and microporosity develop and fatigue life greatly decreases. Crystal lattice curvature, which is created in the surface layers of CP-Ti samples by preliminary ultrasonic processing with simultaneous carbon alloying, causes their strong hardening (Ti nanocarbide appearance, the emergence of the martensitic phase particles, an increase of nanohardness, and modulus of elasticity). This excludes the development of shear banding and microporosity and determines the fourfold increase of CP-Ti fatigue life. The experimental results are in qualitatively good agreement with the developed theory of non-crystallographic plastic flow in the conditions of crystal lattice curvature, where the principally important role should be played by nonequilibrium interstitial structural states.

40

3

Multiscale Translation-Rotation Plastic Flow in Polycrystals

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Multiscale Translation-Rotation Sliding and Shear Banding in Al Polycrystals Under Inhomogeneous Plastic Deformation

The present paragraph reports on a multiscale study of translation-rotation deformation in high-purity A999 Al polycrystals under creep and in thin A999 foils glued on commercial A7 Al plates (two-layer A999/A7 composites) under alternate bending. Under creep, highly pronounced grain boundary sliding develops in the material, activating rotation modes in grain structures [18, 20]. Under alternate bending, the elastically loaded A7 plate prevents the thin A999 foil from fracture up to very high degrees of inhomogeneous plastic strain. Thus, it is possible to uncover a wide range of rotation modes in highly corrugated Al foils. It is significant that when corrugated, the crystal structure of Al foils losses its translation invariance and assumes two types of local states under tensile and compressive normal stresses [40]. These structural changes greatly complicate the translation-rotation mechanisms, giving rise to elastoplastic lattice rotation, microporosity, microcracks, and fracture. Our aim here is to clarify what mechanisms create elastoplastic rotation in a crystal when it loses its translation invariance under periodically distributed tensile and compressive normal stresses, how the crystal structure transforms when it gains high elastoplastic rotation, and how the curvature in the Al polycrystal structure increases under creep and alternate bending. Another point is to investigate the distinction between the rotation modes in high-purity A999 Al under creep and in corrugated A999 Al foils of A999/A7 composites under alternate bending, to ascertain the nature of this distinction, and to analyze the role of crystal lattice curvature in the fracture of the materials. The test material was high-purity polycrystalline aluminum of grade А999, featuring high shear stability of its crystal lattice due to its high shear modulus G (270 GPa) and high stacking fault energy γ (250 mJ/m2). The material was sheet rolled and pressed to obtain flat dumbbell-shaped specimens with a gauge section sized to 37
8
1.2 mm3. For removal of internal stresses, the specimens were annealed at 240  С for 30 min. Their surface was electropolished and covered with a square grid, allowing one to investigate the strain distribution in a specimen as whole and in individual grains, to analyze the shear and rotation strains, and to quantitatively estimate different strain components. The grid method is an efficient tool for detecting the zones in which grain boundary sliding develops lengthwise the surface plane (GBSII). The zones of grain boundary sliding normal to the specimen surface (GBS⊥) were investigated using a New View 6200 interference profilometer. The kinetics of self-consistent grain interaction over a large specimen area was studied using composite photography. The specimens were subjected to creep at a tensile stress of 9–13 MPa and T = 343K. The elongation of the specimens was measured with an indicating gauge to an accuracy of 1 μm.

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Multiscale Interaction of Grain Boundary Sliding and Intragranular Sliding Under Creep

3.1.1

Grain Boundary Sliding and Crystallographic Intragranular Sliding in Steady-State Creep Although there is no consensus on the decisive role of grain boundary sliding in creep [14, 18–20, 23–25], abundant experimental evidence is available to prove the primacy of this process in loaded polycrystals [20, 23, 24], which is particularly evident in high-purity metal polycrystals subjected to creep. The creep curves of polycrystalline А999 aluminum at P = 7–10 kg and T = 343K are standard, comprising successive stages of primary, steady-state, and tertiary creep. Under load with σ = 13 MPa and Δt = 22.2
103 h, the material is fractured at ε = 32.4%. The surface of all specimens reveals highly developed grain boundary sliding the structure of which is detected by interference profilometry and atomic force microscopy. Figure 14 shows the boundary of grain B displaced normal to the specimen surface, producing a system of individual vertical folds of width ~2 μm. The vertical folds are due to quasiperiodic atomic clusters formed at the grain boundary [13]. The Al2O3 film does not allow detection of slip bands at the specimen surface in Fig. 14, but they are clearly seen in Fig. 15 showing the surface of grain B displaced by GBS⊥ to ~8 μm. The grain boundary sliding developing in creep also produces strong surface corrugation (Fig. 16), which is not produced in Al polycrystals under low-temperature uniaxial tension. Such strong surface

Fig. 14 GBS along boundary MN of grains A and B in creep a – surface image, b – section normal to MN; optical profilometry

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Fig. 15 Self-consistent GBS and crystallographic slip lines in grain B in creep; optical profilometry

Fig. 16 Corrugated Al surface layer in creep: a – surface image, b – section along KL; optical profilometry

corrugation under creep arises to accommodate the intense grain boundary sliding in the polycrystals by intragranular sliding. Figure 17 presents typical dislocation structures formed in Al grains at the stage of steady-state creep. The degree of GBS differs from grain to grain and so does the degree of its accommodation by intragranular sliding. In the grains deformed as

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Fig. 17 TEM bright-field (a, b) and dark-field images (c) of dislocation substructure (a, b) and the individual shear band (c) in grains of Al polycrystals generated in the second stage of creep

shown in Figs. 14 and 15, the rotations are associated with the formation of a cellular dislocation substructure (Fig. 17a, b) in which the cell misorientation increases with plastic strains. However, in individual Al polycrystal grains even at the stage of steady-state creep, the lattice curvature becomes so high that band structures start developing against the dislocation substructure (Fig. 17c). In pure fcc metals (Cu, Al, Ni, etc.) under uniaxial tension, no band structures are observed. Thus, creep and attendant severe grain boundary sliding greatly increases the role of rotations in high-purity aluminum, producing curvature in its crystal lattice. Such curvature at the macroscale can readily be seen in Fig. 16. Evidently, the lattice curvature and the loss of translation invariance decrease the thermodynamic stability of crystals: the crystal lattice changes its metric, thus changing the mechanisms of plastic deformation, and the stage of tertiary creep greatly enhances these effects.

3.1.2

Grain Boundary Sliding and Non-Crystallographic Intragranular Sliding in Tertiary Creep Of particular interest is the interplay between grain boundary sliding and accommodating intragranular sliding in tertiary creep, which culminates in fracture. At this stage of creep, macroscale shearing develops inside grains in the directions of τmax (Fig. 18). As can be seen from the data of atomic force microscopy, the lamellae between shear bands α, β, γ are displaced relative to each other by translation modes of plastic flow such that the Al2O3 film is destroyed in the slip traces but is kept intact

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Multiscale Translation-Rotation Plastic Flow in Polycrystals

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Fig. 18 AFM image (a) and surface profile (b) of Al sample subjected to 2 years of creep, ε = 32.4%; optical profilometry

Fig. 19 Mesosubstructure of individual Al grains in tertiary creep at T = 50  C, ε = 30%: а, b – grain with subgrains А, В, С (dark-field TEM imaging in reflection 202 at different tilt angles α, diffraction zone [141]; c – submicron fragmentation; d – dislocation loop at the stage of dynamic recrystallization (bright-field imaging, zone [001])

at the tip of the lamellae. After 2-year creep at T = 343K, the fine transverse cracks readily detectable in the oxide film in Fig. 16 transform into weak inclined bands. Transmission electron microscopy shows that the main rotation mechanism in grains at the stage of tertiary creep is associated with their multiscale fragmentation (Fig. 19). Figure 6a and b presents dark-field images of subgrains A, B, and C formed inside a coarse grain at e = 30% (reflection 202). Their misorientation changes the image contrast at different goniometer tilt angles. For subgrains B and C, the

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misorientation is ~1.6 . Figure 19c shows the pattern of submicrometer fragmentation of subgrain B. The generation of dislocation loops in a recrystallized grain is illustrated in Fig. 19d. The formation of a recrystallized grain in a subgrain is exemplified in Fig. 20. Figure 21 shows the morphology of sliding at the Al polycrystal fracture surface: at boundary KL of grains А and В and at boundary MN of grains C and D. What is fundamentally important is that the GBS along boundary KL is accommodated by non-crystallographic sliding in near-boundary zones of grains А and В (Fig. 21a). The near-boundary sliding, which accommodates the GBS along boundary MN of grains C and D, is accompanied by shear splitting (Fig. 21b) and multiscale fragmentation of the crystal structure in the grain volume. This self-consistent evolution of GBS, accommodating non-crystallographic sliding in near-boundary

Fig. 20 Recrystallized grain (marked by an arrow): а – dark-field TEM image in reflection 111 in zone [112]; b – microdiffraction pattern for zones [112] and [554]

Fig. 21 SEM images of GBS along boundary KL of grains А and В (а) and along boundary MN of grains С and D (b) at the fracture surface of Al polycrystals in creep; accommodation by non-crystallographic sliding and structural splitting in near-boundary zones of grains А and D

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Multiscale Translation-Rotation Plastic Flow in Polycrystals

1281

zones of polycrystal grains, multiscale intragranular fragmentation, and local dynamic recrystallization provides anomalously high plastic strain rates in the polycrystal at the stage of tertiary creep. Note that the rate of steady-state creep, in which the GBS is accommodated by crystallographic intragranular sliding (Fig. 14–17), is orders of magnitude lower than the rate of tertiary creep. This means that the transition to tertiary creep in the polycrystal is associated with a steep decrease in its thermodynamic stability, particularly near the grain boundaries involved in GBS. Thus, the polycrystal assumes a new structural state that allows it to accommodate intense GBS by non-crystallographic intragranular sliding. Such transformations owe to the rise of high lattice curvature that creates new allowed states in the interstitial space of grain boundary zones and provides plastic distortion there [13]. The effect of high lattice curvature near grain boundaries can readily be observed in hcp metal polycrystals under ambient temperature creep [14]. In plastically deformed fcc materials, this effect is less pronounced but can be detected by analyzing the microporosity in grain boundary regions. Below we present respective data on the effect of lattice curvature, interstitial states and vacant lattice sites formed in lattice curvature zones, and associated plastic distortion in the polycrystalline A999 Al foils of А999/Ti composites deformed by alternate bending.

3.2

Dislocation Substructure, Multiscale Fragmentation, and Shear Banding in Al Foils Under Alternate Bending

Transmission electron microscopy (TEM) of cyclically bended А999/А7 composites shows that at a small number of cycles (N ~ 102–104), different structural states arise in individual A999 foil grains: dislocations with Burgers vectors b = 1/2a h110i in slip planes {111} generate several microstructures of dislocation networks (Fig. 22a), chaotically distributed dislocation pileups (Fig. 22b), and strongly fragmented structures with an average fragment size of 200 nm and misorientation of ~2 –5 (Fig. 22c, d). Such inhomogeneity of plastic deformation under cyclic bending is much contributed by rotation modes which cause multiscale fragmentation and internal stress relaxation. In some grains, the dislocation density reaches ρ ~ 2
1010 mm2. The strain lies in the range ε ~ 50–55% as estimated by the Taylor formula ε = ρLb [41], where L  103 mm is the dislocation mean pffiffiffi path and b = a 2/2 = 2.86
107 mm. When corrugated, the A999 foil of the A999/A7 composite radically changes its internal structure. Extended fiber-like planar shear bands with a width of 200–300 nm develop in its surface layer (Fig. 23a). Inside the shear bands, the material is fragmented to a fragment size of 30–50 nm (see insert image in Fig. 23b). Such nanoscale fragmentation inside the shear bands provides efficient accommodation of their high mesoscale curvature, which corresponds to the momentum conservation law.

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Fig. 22 TEM dark-field images of dislocation network (а), chaotically distributed dislocation pileups (b), and multiscale fragmentation (c, d) in Al at N~2000 cycles: а, b – h001i zone, c – h112i zone, d – h110i zone

When the foil becomes highly corrugated and the local curvature of its lattice increases greatly, bulk or 3D shear bands come into play. The structure and the type of rotations in these bands differ much from those in planar shear bands, as can be seen from Figs. 24 and 25. Figure 24 shows two shear bands of width ~500 nm covering a large area of the material involved in mesoscale fragmentation. The first band comprises regions A, B, C, and C0 (Fig 24a) misoriented relative to each other by ~0.2 –0.3 . In region A, there is an extinction contour of the tilted foil, suggesting that the planes (002) perpendicular to the foil plane and parallel to the incident electron beam are curved. This region experiences internal elastic stresses. Regions C and C0 are separated by a tilt boundary (white arrow in Fig. 24a–d), which is likely due to a disclination. A similar boundary is observed in second band D (two white arrows in Fig. 24d). The material in band D rotates as a whole with respect to the first band, gradually reaching a rotation angle of about 1 per micrometer (Fig. 24). In the grain zones containing tens of microscale shear bands, the total rotation of the material adjacent to the band can measure tens of degrees. At N = 104–106 cycles, shear bands with crystallographic planes continuously misoriented through a low angle develop in individual Al foil areas (Fig. 25). Figure 25b–d shows dark-field images of shear bands A and B and their surroundings in the Al foil at different goniometer tilt angles.

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Multiscale Translation-Rotation Plastic Flow in Polycrystals

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Fig. 23 TEM images of 2D (a) and 3D shear banding (b) in the Al foil of the A999/A7 composite under cyclic bending at N = 4
105; foil thickness h = 170 μm

Let us consider the section of band B perpendicular to the foil plane along line СD (Fig. 25b). At a foil tilt angle α = 3 , the planes (020) in the region of bright contour 1 are oriented parallel to the electron beam, and on either side of the contour, the crystallographic regions are out of the reflection position (2, 3), i.e., tilted at a small angle to the beam for which the reflection position is held (4, 5). Thus, there are two large regions 1 and 6 in which the diffraction conditions differ. At α = 2 (Fig. 25c), contour 1 is broken into a series of alternate small contours being all extinguished at α = 1.5 (Fig. 25d). These facts suggest that the crystallographic planes (020) in band B have an orientation gradient not only along the band but also in depth of the foil. Normal to the foil plane, there are at least three distinguishable layers which can be described in terms of diffraction conditions in Figs. 25b–d. Assuming that the thickness of each layer is ~0.03 μm (foil thickness h  0.1 μm), β  90 , and Δα ~ 1 , we can estimate the curvature component χ31 in crystallographic planes perpendicular to the foil plane and the local internal stress σ within the band by formulae (12) and (13) [42]. The expression for χ31 has the form χ 31 

Δα sin β  Δγ , h

(12)

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Fig. 24 TEM images of 3D shear bands in the corrugated Al foil under cyclic bending at N = 4
105 cycles

where Δα is the interval of specimen tilt angles which provide an extinction contour, β is the angle between the effective reflection and the tilt axis projection onto the foil plane, and Δγ is the angular size of diffraction (its peak) normally taken equal to 0.5 . The internal stress is estimated as σ ¼ G χ 31 η,

(13)

where G is the shear modulus, χij is the curvature component of crystallographic planes, and η  0.05 μm is the width of a dislocation affected zone. Estimates by (12) and (13) give that the curvature of crystallographic planes in each shear band layer is χ31  17 μm1, suggesting a rather high local stress

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Fig. 25 TEM images of 3D shear banding with crystallographic planes continuously misoriented through a low angle (a); section of band B along line CD (b)

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σloc ~ G/100 in the band. From this it follows that the generation of the shear band is associated with elastoplastic rotations within the zone of local lattice curvature.

3.3

Multiscale Model of Strain-Induced Defects Generation in a Polycrystal Under Loading

Let us consider the stress distribution in Al polycrystals under uniaxial tension using a model of four adjacent grains [43]. It is seen in Fig. 26 that the distribution of normal stresses σ n, shown by a dotted curve, is highly nonuniform. The stresses σ n peak at triple junctions А and В. Along boundary АВ of grains I and II, oscillating tensile and compressive normal stresses alternate. The stress concentrators arising at triple junctions A and B initiate structural transformations at grain boundary АВ. Early in the process, the stress concentrators at triple junctions А and В are efficiently relaxed by grain boundary sliding, and this provides high rates of primary creep. At the stage of steady-state creep, the stress intensity at triple junctions А and В decreases, allowing self-consistent development of GBS and intragranular sliding. The GBS produces clusters of positive ions in the zones of tensile normal stresses. The excess positive charge of these clusters is screened by electron gas from nearboundary regions of grain I. Thus, the interionic spacings in the grain increase and interstitial states arise there [13]. Due to Coulomb repulsion in the clusters of positive ions, ions from the grain boundary are injected into the interstitial space of the grain, forming a dislocation core. The above multiscale model of dislocation generation agrees well with experimental data on the generation of plane dislocation pileups at the grain boundaries of a deformed polycrystal [2, 3]. At the same time, the model provides a good qualitative description of non-crystallographic sliding in near-boundary zones as an accommodating mechanism of intense grain boundary sliding. As the concentration of interstitial states in lattice curvature zones becomes high, plastic distortion starts developing in these zones such that atoms can freely displace from lattice sites to interstices and provide accommodating non-crystallographic sliding. The coalescence of vacant sites in lattice curvature zones creates microvoids. Such microvoids were really arised in Fig. 26 Formation of clusters of positive ions in zones of tensile normal stresses at grain boundary AB under tension with grain boundary flows of structural transformations generated by stress concentrators at triple junctions A and B

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our experiment under creep of A999 Al polycrystals within grain boundary and localized slip band, Fig. 27. At present, similar phenomena are widely discussed in literature as a precursor to ductile fracture [44]. Because the vacant sites in lattice curvature zones are athermal, their concentration has no relation to temperature and is determined by the degree of lattice curvature. This fact, in our opinion, explains cold dynamic recrystallization in Al in tertiary creep at 343 K (Figs. 19d and 20). Such effects are widely discussed by researchers but there is still no consensus on the issue [45–47]. To additionally verify the role of lattice curvature in the formation of microporosity, a special experiment was performed in which a composite consisting of a thin A999 Al foil and a Ti substrate was subjected to alternate bending at room temperature. When loaded, the foil experienced severe plastic deformation, while the Ti substrate experienced elastic deformation. Under alternate bending at N = 2.5
104 cycles, micropores arose in the Al foil, being most evident in the region of grain boundaries (Fig. 28). Note that at T = 343 K, the formation of micropores cannot be associated with thermal vacancies. The grain boundaries in polycrystals lack translation invariance and feature periodically distributed normal and tangential stresses [13, 40]. Their cluster structure shows up as vertical folds on the denuded grain boundary in Fig. 14. The structural transformations involved in GBS are described by a theory similar to the Belousov–Zhabotinsky theory [48]. The generation of dislocations in a 3D crystal grain is similar to the production of an inhibitor in the Belousov–Zhabotinsky reaction. The GBS as a process responsible for clusters of positive ions at grain boundaries and for lattice curvature in near-boundary zones of the grains constitute the basis not only for multiscale generation of dislocations but also for non-crystallographic sliding in near-boundary zones with high lattice curvature. The fracture of a polycrystal in creep and alternate bending is associated with its structural phase decomposition [37]. The zones of lattice curvature are involved in plastic distortion, dynamic recrystallization, and multiscale fragmentation such that their thermodynamic state becomes highly nonequilibrium. The structural phase Fig. 27 Generation of the microvoids in the zone of grain boundary triple junction along grain boundaries and in the localized slip band; optical microscopy

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Fig. 28 Cold dissolution of Al foil at grain boundaries and in near-boundary zones for two-layer A999/α  Ti composites under alternate bending at N = 2.5
104 cycles; T = 293 K; optical profilometry

decay of the crystalline state in the lattice curvature zones is accompanied by cracking and fracture. This issue was discussed in detail at the 19th European Conference on Fracture Mechanics (2012) and at the 11th All-Russian Congress on Theoretical and Applied Mechanics (2015). It was concluded that the fracture of a solid should be qualified as a structural phase transition. The results of the present study agree well with the above conclusion. However, the new approach to fracture mechanics requires theoretical substantiation of the role of lattice curvature in the structural state of a deformed solid. Related research is now underway at the Russian Academy of Sciences [49–52].

3.4

Conclusion 2

Thus, we have investigated the translation-rotation modes of plastic deformation in high-purity A999 Al polycrystals under creep and in A999 Al foils glued on commercial A7 Al (A999/A7 composites) under alternate bending. Under low temperature creep, the basic process is GBS accommodated by intragranular sliding. At the stage of steady-state creep, the deformation develops through dislocation motion with the formation of misoriented cellular dislocation substructures and individual shear bands. At the stage of tertiary creep, the polycrystal grains are involved in non-crystallographic plastic deformation: multiscale grain fragmentation, non-crystallographic sliding near the grain boundaries, local dynamic recrystallization, and formation of microporosity. All these accommodating mechanisms along with intense GBS are responsible for fracture of the material under creep. The regular alternation of tensile and compressive normal stresses at the interface of the А999/А7 composite under alternate bending highly disperses the rotation modes of plastic flow in the Al foil. Under alternate bending up to e~50%, the material forms a dislocation substructure and undergoes homogeneous submicron fragmentation. At e>50%, the corrugated Al foil is involved in pronounced shear banding with the formation of high local elastoplastic lattice rotation. These facts

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allow us to conclude that shear banding is a rotation mechanism that transforms elastic lattice curvature to elastoplastic rotations. Multiscale fragmentation is also a rotation mechanism of deformation. The atomic mechanisms of shear banding and multiscale fragmentation require detailed structural studies.

4

General Summary

The methodology of mechanical behavior of polycrystals as multiscale hierarchically organized systems is described. It is very important to consider the mechanical behavior of 3D (crystal subsystem) and 2D (planar one) (surface layers and all interfaces particular grain boundaries). The primary plastic sliding is related to 2D – planar subsystem which is characterized by the lack of crystal lattice invariance. The 2D–planar subsystem generates couple stresses which act on 3D–crystal subsystem and create curvature of its lattice. According to the angular momentum conservation law, the generation of strain-induced defects in the 3D–crystal subsystem should provide the condition N X

rotJ i ¼ 0

(14)

i¼1

for any fluxes Ji of structural transformations in the multiscale system. In the 3D–crystal subsystem, rotational deformation modes develop, and their scale defines the type of generated defects and the stages of plastic deformation. Plastic deformation by dislocation motion represents the first mesoscale at which rotation modes evolve through misorientation of cellular dislocation substructures. As the scale of lattice curvature increases, the 2D–planar subsystem generates shear bands into the 3D–crystal subsystem. These shear bands provide translation-rotation modes which represent the second mesoscale. Finally, macrobands of localized plastic deformation arise and trigger macroscale rotational modes. Thus, the description of plastic deformation should allow for self-organization of the entire scale hierarchy of rotational deformation modes. Because in the general case the hierarchical self-consistency of rotational modes is incomplete, the lattice curvature in a solid increases progressively such that the solid is eventually switched to a blow-up mode which culminates in local structural phase decomposition of its crystalline state. It is these processes that are the essence of fracture. The basic for the concept is evolution of lattice curvature and rotational deformation modes as a major driving force in plasticity and strength of solids. The scientific substantiation of bifurcation interstitial structural states within zones of crystal lattice curvature is a very important discovery. It is the new kind of the strain-induced defect which allows to take part the plastic distortion of atoms within a localized plastic flow. First of all, it allows to understand the possible mechanisms of a localized band propagation in crystal lattice. But the most important is related to existence of the non-crystallographic plastic flow and arising of

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athermal vacancies in the zones of crystal lattice curvature. The athermal vacancies experience the coalescence with arising of voids which culminate in plastic collapse and fracture of solids. The new methodology predictions of mechanical behavior of multiscale hierarchically organized polycrystals were specifically supported in this chapter NX for titanium and high-purity aluminum. The role of crystal lattice curvature in shear banding and fatigue fracture was convincingly shown in these polycrystals subjected to alternate bending. The interaction between grain boundary sliding (which is the primary process) and the intragranular plastic flow specifies the different stages of low-temperature creep of high-purity Al polycrystals. At the steady-state creep, the intragranular sliding is related to the crystallographic mechanisms of plastic flow. The intensive tertiary creep culminated by Al polycrystal fracture is related to development of the non-crystallographic mechanisms of plastic flow: non-crystallographic sliding, multiscale fragmentation, dynamical recrystallization, microvoid appearance. The general conclusion of this chapter is multiscale mechanical behavior of polycrystalline materials as hierarchically organized systems must be described on the basis of nonlinear mechanics.

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Micromechanics of Hierarchical Materials: Modeling and Perspectives

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Leon Mishnaevsky Jr.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Load Sharing and Multiscale Computation Techniques: Idealized Models of Hierarchical Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Biological Hierarchical Materials and Their Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hierarchical Nanoengineered Composites and Their Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Multiple and Added Functionalities: Sensing, Biocomposites, Wear Resistance, etc. . . . 6 Localization of Nanoreinforcements in Hierarchical Materials: Nanostructured Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Hierarchical materials represent a new, promising direction of the materials development, inspired by biological materials and allowing the creation of multiscale materials design and multiple functionalities and achieving extraordinary material properties. In this article, a short overview of possible applications and perspectives on hierarchical materials is given. Several examples of the modeling of strength and damage in hierarchical materials are summarized. The main areas of research in micromechanics of hierarchical materials are identified, among them, the investigations of the effects of load redistribution between reinforcing elements at different scale levels, possibilities to control different material properties and to ensure synergy of strengthening effects at different scale levels and using the nanoreinforcement effects.

L. Mishnaevsky Jr. (*) Department of Wind Energy, Technical University of Denmark, Roskilde, Denmark e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_78

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Introduction

Since ancient times, there have been efforts to improve properties of materials by changing their structures, starting from Damascus swords or straw-based buildings in Mali. These forts led to the development of composites, materials consisting of two or more dissimilar constituents with different properties [1, 2]. At first, the perspectives of composite materials seemed infinite; however, later, it became clear that the potential for improvement of composite properties by adding or rearranging reinforcements is still limited. While some properties (e.g., stiffness) are improved by increasing the volume content of hard reinforcement in composites, other properties (fracture toughness) degrade [1]. Further, it became clear that the composite constituents have their own, sometimes complex structures which can in turn be modified. This motivated modifications of structure and properties of composite constituents, and control at several scale levels was suggested [3]. In his classical paper, Lakes [4] summarized the main ideas of hierarchical material structure as a “basis for synthesizing new microstructures which give rise to enhanced or useful physical properties.” Further, the extraordinary properties of biomaterials, like wood or bones, attracted the interest of the scientific community [5]. It was observed that one of the main sources of such extraordinary properties of biocomposites is their complex hierarchical structure [5, 6]. That gave the impetus to the development of new, bioinspired materials, based on biomimicking principles. With the development of nanotechnology, new possibilities emerged to actually manufacture hierarchical materials with nanomodified constituents. The potential advantages of hierarchical composites over common composites and materials include, among others: • Synergy between structural elements at several scale levels. For example, fibers control the tensile strength and stiffness in hierarchical composites, while secondary nanoparticles in the matrix control the shear and compression strength [7]. • Possibility to improve competing properties of materials. For example, hierarchical ceramics demonstrate both higher strength and higher toughness [3]. • Potential to improve the interface-controlled properties, like compression and fatigue strength, by placing nanoparticles on interfaces or fiber sizing. • Combination of advantages of nanomaterials and composites. These materials show great perspectives, as extra-durable composites, multifunctional materials, conducting protective coatings, highly wear resistant machining tools [8]. The extraordinary performance of the new materials is determined by their structures at micro- and nanoscale. Optimizing these structures, one can control the properties of new materials. This can be done using the methods of computational micromechanics and numerical experiments [1, 9]. In this chapter, concepts and methods of the micromechanics of hierarchical materials are reviewed.

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Micromechanics of Hierarchical Materials: Modeling and Perspectives

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Load Sharing and Multiscale Computation Techniques: Idealized Models of Hierarchical Materials

A number of mathematical models of hierarchical materials have been developed in the last decades. Apart from a large group of models reflecting the structures of real biological or man-made hierarchical materials (and considered below), there are a number of works which seek to answer general questions – what is the effect of hierarchical structure on the material response (in general case)? And how to simulate such structures (also, in general case)? Mechanisms of the peculiar properties of hierarchical composites are related to the interaction between structural elements at several scale levels. So, under mechanical loading, the load transfer between the levels takes place, especially when the microstructure changes (i.e., due to damage or deformation). In order to study the effect of hierarchical structures of composites on their properties, taking into account the peculiarities of load transfer directly, the hierarchical load sharing (HLS) models can be used [10]. According to this model, the load is transferred from the upper elements of the hierarchical “tree” (“roots”) to the lower (“branches”) and down to the lowest elements of the material (fibers, in the case of long fiber-reinforced composites). The load is shared equally among all the sub-elements of a given branch (as long as they are intact) or among remaining intact sub-elements, after some of them fail. In simplest case, this load rule can be directly introduced into the analytical fiber bundle models of composites [11, 12]. Figure 1 shows the schema of five level hierarchical fiber bundle model. In the computational simulations of hierarchical fiber bundle model [12], the effect of hierarchization and structure of the self-similar materials on their damage resistance were investigated. It was observed that the damage resistance of the multiscale self-similar fiber-reinforced composites increases with increasing the amount of hierarchy levels in the material (as differed from the case of “hierarchical tree” considered in [10, 12], where increasing the amount of hierarchical levels means reduced damage resistance). Further, the hierarchical fiber bundle model was generalized to include the case of particulate reinforcement, using the “embedded equivalent fiber” model of particles [12]. Fig. 1 Schema: multiscale (fives level) fiber bundle model (Reprinted from [12] with kind permission from Elsevier)

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In the “fractal bone” model [13], self-similar composite structure is simulated. The model as applied to defects in bones, “with slender hard platelets aligned in a parallel staggered pattern in a soft matrix.” One of the conclusions of this model was that a hierarchical material “can be designed to tolerate crack-like flaws.” The analytical and probabilistic models, including fiber bundle, hierarchical load sharing, and self-similar models, allow general evaluations of the effect of hierarchical structures on the material behavior.

3

Biological Hierarchical Materials and Their Modeling

Natural biological materials demonstrate often extraordinary strength, damage resistance, and hardness. In many studies, sources of such unusual extraordinary properties of biological materials have been analyzed [14, 15]. Among the peculiar structures of biological materials, several features were established: staggered brick-and-mortar structure and interlocked platelets of nacre [16–18] layered structure of the spicules, with randomly distributed layer thicknesses, functionally graded structure (graded distributions of reinforcement) of bamboo [19] and tooth, fiber/fibrils with varied distribution and density, cellular multilayered structure of timber, and so on. In many cases, the biological materials represent hierarchical materials, with varied, but still complex (lamellar, cellular, staggered, porous) structures at all the scale levels [1]. A prominent example is nacreous mollusk shells, which consist of 95% CaCO3 (by volume), have double the strength, and exhibit a work of fracture that is 3,000 times higher than that of the monolithic CaCO3 [20]. Nacre has been traditionally most apparent and well-studied example of biomaterial, whose extraordinary strength is in clear contrast to the brittleness of its components. Among the microstructural peculiarities of nacre, responsible for its unusual properties, the brick-andmortar structure, interlocking of platelets, layered configuration, thin organic layers, etc. can be mentioned [21, 22]. A number of micromechanical models of nacre have been developed, e.g., 3D micromechanical model of nacre with hexagonal aragonite platelets each surrounded by organic layers [23], the discrete lattice model based on continuous damage random threshold fuse network [24], 3D finite element (FE) unit cell model of nacre (two layers with 150 tablets in each layer) with actual tablet contours obtained from micrographs [25], homogenization model for anisotropic and biomodular mechanical behavior of nacre, based on the BM unit cell in [26], etc. The microstructures of bones are much more complex than those of nacre. Bone is a porous, cellular material consisting of multilayered lamellas, built in turn of fibrous layers with different orientations and thicknesses and with various microgeometries for different types of bones (cortical and cancellous). At the nanolevel, the bone is seen as the collagen fibers, surrounded by mineral [27].Among widely used approaches to the modeling of the microstructure-properties relationships of bones, one can mention the multiscale homogenization-based models which include typical microstructures at different scale levels. An example of this approach is the multiscale micromechanical model of bones from [28, 29], which is based on random homogenization theory and includes bone microstructures via six-step homogenization

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procedure (at the scales from 10 nm to 1 mm). Further, several research groups south to carry out the direct reproduction of complex real microstructures in FE models using high-resolution finite element programs. This approach allows to analyze the bone properties at the mesolevel, including the effect of porosity, other microstructural parameters on the mechanical properties, and strength of bones. Example of this approach is the voxel-based micro finite element (μFE) model from [30], or works by Niebur et al. [31], in which 3D models of bones were rendered from real images. Wood, similarly to bones, represents multiscale, graded, cellular materials with different types of reinforcements at different levels (see Fig. 2). Hofstetter et al. [33, 34] developed a micromechanical model based on a three- [34] and four-step [33] homogenization schemes for wood. Their approach includes two continuum homogenization steps (random homogenization) and one step based on the unit cell method (periodic homogenization). The elastic properties are determined using the self-consistent scheme and the Mori-Tanaka scheme for the two continuum homogenization steps, respectively. The unit cell method is applied to analyze the assembly of tube-like cellular into the softwood structure. This approach takes into account the microstructure of wood, covering several orders of magnitude, from the cell wall structure, to the structure of fibers, to the macroscopic defects. However, the homogenization method has some limitations when applied to the analysis of damage and nonlinear deformation of wood. Astley and colleagues [35, 36] developed multiscale models and carried out three-dimensional finite element simulations of representative sections of the softwood cell structure. Considering cell walls as seven-layered material, each layer consisting of concentric orthotropic lamellae, they analyzed interrelationships between the macroscopic elastic properties of softwood and the local microstructural characteristics of cells. Further micromechanical models of elastic properties of wood were developed by Bergander and Salmén [37, 38] (using the classical lamination theory and semiempirical Halpin-Tsai equations), Perré [39] (microstructure-based FE

S3

z 2 3

Secondary S2 wall

1

S1

Middle lamella

Primary wall y

x

a

b

Fig. 2 Multiscale finite element model of wood (Reprinted from [32] with kind permission from Elsevier)

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meshes, homogenization), and others (see detailed review elsewhere [40]). As different from the homogenization-based models, the discrete multiscale continuum mechanical models make it possible to model damage and strongly nonlinear and time-dependent behavior of the elements of the wood microstructures. So, the computational model of wood developed in [41–44] takes into account the different structural features of the wood at different scale levels. The model includes four levels of the heterogeneous microstructure of wood: (a) macro-level, annual rings are modeled as multilayers, using the improved 3D rule-of-mixture; and (b) mesolevel, the layered honeycomb-like microstructure of cells is modeled as a 3D unit cell with layered walls. The properties of the layers were taken from the microlevel model, (c) submicrolevel and microlevel. Each of the layers forming the cell walls was considered as unidirectional, fibril-reinforced composite. Taking into account the experimentally determined microfibril angles and content of cellulose, hemicellulose, and lignin in each layer, the elastic properties of the layers were determined with the use of Halpin-Tsai equations. Figure 2 gives an example of the FE unit model of the wood. Using the developed model, the effect of microstructural parameters of wood on its deformation behavior was studied. In particular, the influence of microfibril angle (MFA) and wood density on the deformation behavior was considered. From the computational studies, it was concluded that the variation of microfibril angles represents a rather efficient mechanism of the natural control of stiffness of the main shear load-bearing layer of the cell wall. By increasing the MFAs, the drastic increase of shear stiffness in 1–2 directions is achieved, without any sizable losses of the transverse Young modulus and shear modulus in the 23 planes. The 3D hierarchical model was also extended to include the damage process and combined with phenomenological approach toward the fatigue modeling [32]. With such multiscale models, the effects of different microstructural parameters and the synergy between microstructures at different scales can be studied in “virtual experiments.” The connection between the extraordinary performance of biological materials and their hierarchical structure inspired research toward new man-made materials using the same principles [45], including tough and strong alumina ceramic-based materials with nacre-like brick-and-mortar structure [46], polyurethane-based composites with nanoclay and inorganic particles distributed in polyurethane matrix [47] for stretchable electronics, and materials with hierarchical porous structures (like sponges or bones) showing high permeability and surface area, to be used in catalytic and gas adsorption applications, bone implants, and materials for impact protections.

4

Hierarchical Nanoengineered Composites and Their Modeling

Composite materials are used more and more widely in different areas of industry and everyday life, from furniture to airplanes. The improvement of composite performance can improve the reliability and lifetime of structures, and, thus, is an

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important subject for the industry. So, there are a number of works, seeking to link multiscale hierarchical structure of composites to their properties [48–52]. One group of hierarchical composites are recursively reinforced materials, in which reinforcing elements are recursively composed of matrix and smaller, lower level reinforcements (which are therefore itself composites at a finer scale). Carpinteri and Paggi [53] used top down approach, the rule of mixture, and the generalized Hall-Petch relationship (for hardness) and demonstrated that “a hierarchical material is tougher than its conventional counterpart” and that the material hardness increases with increasing the amount of hierarchy levels. Pugno and Carpinteri [54] employed “quantized fracture mechanics” (an energy-based theory, in which differentials in Griffith’s criterion are substituted with finite differences) to analyze a self-similar particle-reinforced composite with fractal-like structure and derived formulas for fracture energy and failure stress depending on size scales. Joshi and Ramesh [55] used the multiscale secant Mori-Tanaka method (with subscale terms, in particular, grain size, particle size, and dispersoid strengthening) to analyze the “trimodal” composites and computed overall response of the materials. Habibi et al. [56] calculated the overall yield strength of hierarchical Mg matrix composites, as a square root of a sum of squared strength contributions from the GND, Hall-Petch, and Orowan strengthening. An important group of hierarchical composites is the multiple reinforced materials, with reinforcing inclusions of qualitatively different properties and sizes [57]. Ceramic composites with ductile nanoparticles enhancing the composite toughness also belong to this group [58]. In the framework of the Japanese “Synergy Ceramics Projects” [3], Kanzaki et al. [3] presented an example of an improved material which has both high strength and toughness achieved by combination of aligned anisotropic grains (at microlevel) with the intragranular dispersion of nanoparticles (at nanolevel). Many models of ceramic nanocomposites are based on the fracture mechanics, stress field analysis around the crack/nanoparticles or models of dislocation evolution [59–62]. Awaji et al. [61] analyzed the nanoparticle toughening and the residual thermal stresses in intra-type nanocomposites, using a spherical particle inside concentric matrix sphere model, and demonstrated that the thermal expansion coefficients mismatch has a strong effect on the toughening of ceramic nanocomposites. In fiber-reinforced composites with secondary nanoreinforcements, additions of small amount of nanoparticles (e.g., nanoclay, carbon nanotubes, or graphene) ensure the strong improvement in the matrix-dominated composite properties (like compressive or fatigue strength), additionally to the high stiffness and axial strength provided by strong fibers [63]. Most often, the nanoreinforcements are distributed in the matrix or grown on the fiber surfaces (or in fiber sizing). While the methods of modeling of mechanical behavior and strength of fiber-reinforced composites are well known [64], the main challenges in the modeling of such hierarchical composites lie in the modeling nanoreinforcement clusters and in taking into account the atomistic properties of nanoreinforcements and their interaction with the matrix. To take into account atomic structure of nanoreinforcement, interphase, and polymer matrix in the micromechanical models of nanocomposites, atomistic, molecular

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mechanics or molecular dynamics based representative equivalent element models and material laws are used [65–67]. Multiphase (e.g., three-phase) models, including the matrix, interfacial region, and fillers, or matrix, the exfoliated clay nanolayers, and the nanolayer clusters [66, 68], as well as the effective particle idealization and the dilute dispersion of cluster models [69], can be employed to take into account both the nanoreinforcement clustering and the interphase effects. In [70–73] a computational multiscale model was developed, which includes the fiber/matrix interaction at the higher scale level (microlevel) and nanoparticle/epoxy matrix interaction on nanolevel. The macro (upper level) unit cells contain three phases: the matrix, fibers, and “third phase” interface layers (which characterizes the interface roughness, interphases). Both matrix and the interface layer might contain nanoreinforcements. The microscale (lower level) unit cell includes the nanoreinforcement in matrix and/or interfaces, polymer matrix as well as the “effective interface layers” [74, 75]. The macro-model analysis saves the time-dependent values of variables (i.e., stress and displacement field) of the “seed nodes” which are located on the boundary between macro-model and micro-model after the macrolevel analysis. The micro-model inherits these data as the initial boundary condition and together with other added boundary condition (if needed) to go on the analysis. Figure 3 shows an example of a unit cell model. In these works, it was demonstrated that the secondary nanoparticles in composites enhance the fatigue performance (maximum stress and lifetime) in all considered composites. For the very low cycle loading, the CNT reinforcement leads to 25%. . .43% increase in the stress, while for the millions of cycles, the CNT effect increases the stress by 64. . .120%.

5

Multiple and Added Functionalities: Sensing, Biocomposites, Wear Resistance, etc.

Multiple functionalities: Secondary nanocarbon particles, distributed in the polymer composite matrix, enhance not only enhanced strength, toughness, and fatigue lifetime but also create electrically conductive percolating networks [76–78], making it possible to detect damage accumulation by electric measurements, thus adding the sensing functionality and enabling the remote structural health monitoring [79]. Furthermore, the availability of conducting electric networks from nanoscale particles can add protective properties, like de-icing properties [80] and lightning strike protection [81] if the composites are used for wind turbines or aircraft structures. The modeling of the multiphysics effects in hierarchical nanocarbon particlereinforced composites should include the electrical tunneling effect and can be carried out considering nanotube matrix resistors and electrical circuit model [82–84].As a result, both mechanical behavior, strength and electrical resistance of nanotube networks are evaluated [85]. Bio-based biodegradable composites. There have been intensive efforts to replace the petroleum-based composites (and also metals) in structural applications by environmentally neutral, biodegradable, and bio-based materials. The efforts range from the car body produced from soya bean-based plastics by Henry Ford in

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a

GFRC

EI

HC

fibers CNT

CNT EI

fiber with CNT reinforcements

CFRC

GFRC: glass fiber reinforced composite HC: hybrid composite CFRC: carbon fiber reinforced composite EI: effective interface

b

carbon nanotube

fiber bridging

crack surface

fiber breaking fiber pull-out

Fig. 3 Example of unit cell model of CNT reinforced hybrid composite (a) and Crack interaction with CNTs (b) observed in simulations (Reprinted from [23] with kind permission of Elsevier)

1938, wood-aluminum chassis by Morgan Motor [86], flax/PE bio-based panels in Mercedes C and A [87], wood/polymer composites to multi-material mix [88], and so on. Still, the ultimate goal (not just replacement of some constituents or layer in a structural part by bio-based/degradable sources, but making the full part/structures biodegradable/bio-based) is still not fully achieved. Currently available material solutions for structural/automotive/aircraft applications, developed with view on biodegradability/sustainability, are typically combinations of (often non-load-bearing) biocomponents with load-bearing traditional materials [89]. The natural reinforcement is an important tool to enhance the performances of polymer materials. Additionally to the rather well-studied natural fibers, nanocellulose reinforcement attracted a growing interest of research community and industry since several years [90].Bio-based resins based on polylactic acid (PLA), thermoplastic starch,

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polyhydroxyalkanoate (PHA), and cellulose show only moderate mechanical and service properties, if compared with traditional petroleum-based plastics. The natural fiber composites have stiffness comparable to glass fiber composites, but much lower strength (especially lower compressive, tensile, and impact strength) [91], and just here the secondary nanoparticle reinforcement (e.g., on fiber surfaces) can drastically improve the properties. Okubo and colleagues produced PLA/bamboo fiber/microfibrillated cellulose (MFCs) composites, with additions of just 1–2 wt% of MFC [92]. Pommet et al. [93] produced hierarchical biocomposites by grafting bacterial cellulose fibrils which were in situ onto the surfaces of natural fibers. The computational models of nanocellulose-reinforced polymers are based typically on common micromechanics of composite approaches (theory of laminates, rule of mixture, Halpin-Tsai models [94, 95], Cox-Krenchel model for randomly oriented short fiber composites [96]), but also on fiber network models (developed earlier in the framework of paper studies), like Poisson fiber network [97] and percolation models [98]. Several promising directions are connected with combining the network models and micromechanics of composite methods, like the version of rule of mixture when representing the NC/polymer composite as polymer/“NC paper” composites [95], finite element model of fiber network [99], or the network model based on fiber-fiber bond model combined with rule of mixture approach [94]. Josefsson et al. [100] developed a multiscale analytical model of cellulose nanofibril-reinforced composites, using the laminate theory, self-consistent Mori-Tanaka model, and micromechanical models. They considered the composites as a matrix reinforced fibril aggregates shaped as oblate spheroids. Figure 4 shows a unit cell model of nanocellulose/polymer composites taking into account the snakelike shapes of nanocellulose and the availability of hydrogen bonds between the “snakes” (from [101]). Materials for machining tools: The productivity and wear resistance of machining tools determine the productivity and costs of many industrial technologies,

Fig. 4 3D unit cell model of polymer reinforced by 30 “snake-shaped” nanocellulose particles (left) with hydrogen bonds between closely located segments of the particles (right) (Reprinted from [101])

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including drilling, cutting, grinding, etc. In [8], it was proposed to use nanoengineered drilling tool materials to increase the productivity and wear resistance of machining technologies. Drilling bits with diamond grains and metallic binder reinforced by nanoparticles of different nature (carbon nanotubes, nano-sized diamonds, nanoparticles of disk-shaped hexagonal boron nitride hBN) demonstrated that the cutting speed of a tool containing the nanotube-modified binder is 50–70% higher than that of a tool made of material containing no nanotubes [102, 103]. In order to simulate the effect of nanoparticles on the drilling rate, multiscale finite element micromechanical model was developed [103]. It was demonstrated that the nanoreinforced binder ensures improved mechanical properties and performances of the metallic binders.

6

Localization of Nanoreinforcements in Hierarchical Materials: Nanostructured Interfaces

As demonstrated above, hierarchical structures of materials have a great potential to improve the material properties, especially if combined with nanoengineering of structural elements of materials. However, the idea that it is enough to “throw nanoparticles into composite” to receive a material with improved properties is wrong. The structures of hierarchical materials should be optimized and tailored in order to achieve the property improvement and not degradation. In the numerical experiments in [70, 71], as well as in a number of experimental works, it has been demonstrated that nanoparticles (nanoclay or carbon nanotubes) in fiber/matrix interfaces (fiber sizing) of unidirectional composites, aligned and oriented, allow to increase the fatigue lifetime of the composites drastically, while nanoparticles added into the polymer matrix can also lead to property degradation (due to the clustering effect). It is of interest to look again at some typical structures of hierarchical materials. So, Song and colleagues [104, 105] showed that the availability of nanostructures in the nacre interfaces (i.e., mineral bridges between aragonite platelets located in the biopolymer layers) is one of the reasons for the high toughness of nacre. The mineral bridges reinforce the weak interface and control the crack propagation in the interfaces (biopolymer layers). Layers of constrained disturbed polymers surrounding nanoparticles in polymer ensure the drastic enhancement of nanocomposite properties (like 200% increase in stiffness or strength achieved at 0.5% nanoparticle content) [74]. One of reasons of extraordinary strength and toughness of nanocrystalline ultrafine grained metals is the high content of grain boundary phases in these materials. Their properties can be further enhanced if the grain boundaries are nonequilibrium, with high density of dislocations and especially with foreign atoms/ precipitates [106, 107]. The availability of precipitates strongly delays the damage growth and ensures 83% increase in the critical strains due to the precipitates, and around 300% increase due to the precipitates located in grain boundaries. Figure 5 summarizes these examples.

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a

b

Precipitates randomly distributed in GB

Aragonite platelets Biopolymer layer

Precipitates located on GB/GI interface Mineral bridges in biopolymer layer

c

Fig. 5 Examples of materials with nanostructure interfaces and grain boundaries: (a) nacre structure (brick mortar structure with polymer layers bridged by mineral nanoparticles), (b) ultrafine grained metal (titanium) with atomistic size particles of grain boundaries, and (c) fiber-reinforced composites with secondary nanoreinforcement (Reprinted from [108, 109]with kind permission of Elsevier)

On the basis of generalization of these observations, it was demonstrated in [110] that the purposeful nanostructuring of interfaces and grain boundaries represents an important reserve of the improvement of the materials properties. Since the material deformation is often localized in and around defects (interfaces and grain boundaries), the structuring of these regions (adding specially arranged and oriented nanoreinforcements, or adding nanoscale defects, changing the local properties) allows to control the deformation and fracture behavior of these weak areas, thus, determining the degradation process in the whole material. Quite often, the interfaces with low stiffness lead to the localization of deformation, while the internal structures of the interfaces (like mineral bridges in nacre or nanoplatelets in sizing of fiber-reinforced composites) allow to control the deformation, damage initiation, and fracture processes locally.

7

Concluding Remarks

Hierarchical design of materials is a new, promising direction of the materials development. The initial impetus to the development of hierarchical materials was given by the observations of natural, biological materials, and by the requirement to

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develop multifunctional materials, with properties exceeding the properties of usual composites. In this article, a short overview of possible applications and perspectives on hierarchical materials is given. Several examples of the modeling of strength and damage in hierarchical materials are summarized. The main areas of research in micromechanics of hierarchical materials are identified, among them, the investigations of the effects of load redistribution between reinforcing elements at different scale levels, possibilities to control different material properties and to ensure synergy of strengthening effects at different scale levels and using the nanoreinforcement effects. Further, the extraordinary properties of biomaterials, like wood or bones, attracted the interest of the scientific community [5]. It was observed that one of the main sources of such extraordinary properties of biocomposites is their complex hierarchical structure [5, 6]. That gave the impetus to the development of new, bioinspired materials, based on biomimicking principles. With the development of nanotechnology, new possibilities emerged to actually manufacture hierarchical materials with nanomodified constituents. The potential advantages of hierarchical composites over common composites and materials include, among others: • Synergy between structural elements at several scale levels. For example, fibers control the tensile strength and stiffness in hierarchical composites, while secondary nanoparticles in the matrix control the shear and compression strength [7]. • Possibility to improve competing properties of materials. For example, hierarchical ceramics demonstrate both higher strength and higher toughness [3]. • Potential to improve the interface-controlled properties, like compression and fatigue strength, by placing nanoparticles on interfaces or fiber sizing. • Combination of advantages of nanomaterials and composites. These materials show great perspectives, as extra-durable composites, multifunctional materials, conducting protective coatings, and highly wear-resistant machining tools [8]. The extraordinary performance of the new materials is determined by their structures at micro- and nanoscale. Optimizing these structures, one can control the properties of new materials. This can be done using the methods of computational micromechanics and numerical experiments [1, 9]. In this chapter, concepts and methods of the micromechanics of hierarchical materials are reviewed.

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77. Kang I, et al. A carbon nanotube strain sensor for structural health monitoring. Smart Mater Struct. 2006;15:737–48. 78. Wood JR, et al. Carbon nanotubes: from molecular to macroscopic sensors. Phys Rev B. 2000;62:7571. 79. Fiedler B, Gojny FH, Wichmann MHG, Bauhofer W, Schulte K. Can carbon nanotubes be used to sense damage in composites. Annal de Chim-Sci des Materiaux 2004;29(6):81–94. 80. Smart anti-icing system for rotor blades. Research News / 1.12.2014. 81. U.S. Patent Application 20100134946; Carbon nanotubes used to form improved lightning strike protection for wind turbines, 36.2010, U.S. Patent Application 20100134946. 82. Li C, Thostenson ET, Chou TW. Sensors and actuators based on carbon nanotubes and their composites: a review. Compos Sci Technol. 2008;68(6):1227–49. 83. Hu N, Karube Y, Yan C, Masuda Z, Fukunaga H. Tunneling effect in a polymer/carbon nanotube nanocomposite strain sensor. Acta Mater. 2008;56(13):2929–36. 84. Grabowski K, et al. Electrical conductivity of nanocomposites based on carbon nanotubes: a 3D multiscale modeling approach. In: Proceedings SPIE 9803, Sensors and smart structures technologies for civil, mechanical, and aerospace systems 2016, 98030O (April 20, 2016); https://doi.org/10.1117/12.2219275 85. Li C, Chou TW. Modeling of damage sensing in fiber composites using carbon nanotube networks. Compos Sci Technol. 2008;68(15–16):3373–9. 86. Franzen M, et al. Naturfaserverstärkte Spritzgießmaterialien für den praktischen Einsatz in der Automobilindustrie. Vortrag: 4a Technologietag 2014. Schladming, 25 Feb 2014. 87. Akampumuza O, et al. Review of the applications of biocomposites in the automotive industry. Polym Compos. 2016. https://doi.org/10.1002/pc.23847. 88. Löwer C. Die neue Holzklasse. PM. 2012; 12: 92–8. see also http://www.projekt-hammer.de/. 89. Roy SB, et al. A review on biocomposites. Int J Innov Res Sci Eng Technol. 2014; 3(10):27–37. 90. Siró I, Plackett D. Microfibrillated cellulose and new nanocomposite materials: a review. Cellulose. 2010;17(3):459–94. 91. Berglund LA, Pejs T. MRS Bull. 2010;35:201. 92. Okubo K, Fujii T, Yamashita N. Improvement of interfacial adhesion in bamboo polymer composite enhanced with micro-fibrillated cellulose. JSME Int J Ser A. 2005;48(4):199–204. 93. Pommet M, Juntaro J, Heng JYY, Mantalaris A, Lee AF, Wilson K, et al. Surface modification of natural fibers using bacteria: depositing bacterial cellulose onto natural fibers to create hierarchical fiber reinforced nanocomposites. Biomacromolecules. 2008;9:1643. J. Juntaro et al, Nanocellulose enhanced interfaces in truly green unidirectional fibre reinforced composites, Compos. Interfaces 14, 753 (2007). 94. Aitomaki Y, Oksman K. Reinforcing efficiency of nanocellulose in polymers. React Funct Polym. 2014;85(SI):151–6. 95. Lee K-Y, Aitomäki Y, Berglund LA, Oksman K, Bismarck A. On the use of nanocellulose as reinforcement in polymer matrix composites. Compos Sci Technol. 2014;105:15–27. 96. Jogi BF, et al. Dispersion and performance properties of carbon nanotubes (CNTs) based polymer composites: a review. J Encapsulation Adsorpt Sci. 2012;2:69–78. 97. Aström. Elasticity of Poissonian fiber networks. Physical Review E. The American physical society. 2000;61(5):5550. 98. Favier V, Canova GR, Shrivastava SC, Cavaille JY. Mechanical percolation in cellulose whisker nanocomposites. Polym Eng Sci. 1997;37(10):1732–9. 99. Denoyelle Th. Mechanical properties of materials made of nano-cellulose. KTH Degree; Stockholm, 2011. 100. Josefsson G, et al. Stiffness contribution of cellulose nanofibrils to composite materials. Int J Solids Struct. 2014;51:945. 101. Mishnaevsky L Jr, Madsen B, Aitomäki Y. Computational simulations of nanocellulose reinforced polymers. EU FoF.NMP.2013-10 project INCOM. Report Deliverable D6.5; 2016. 19 pp.

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102. Sidorenko D, et al. Carbon nanotube reinforced metal binders for diamond cutting tools. Mater Des. 2015;83:536–44. 103. Loginov P, Mishnaevsky Jr L, Levashov E, Petrzhik M. Diamond and CBN hybrid and nanomodified cutting tools with enhanced performances: development, testing and modelling. Mater Des. 2015;88:310–9. 104. Song F, Bai YL. Effects of nanostructures on the fracture strength of the interfaces in nacre. J Mater Res. 2003;18(8):1741–4. 105. Song F, Soh AK, Bai YL. Structural and mechanical properties of the organic matrix layers of nacre. Biomaterials. 2003;24:3623–31. 106. Mishnaevsky L Jr, et al. Nanostructured titanium based materials for medical implants: modeling and development. Mater Sci Eng R Rep. 2014;81:1–19. 107. Liu HS, Mishnaevsky L Jr. Gradient ultrafine-grained titanium: computational study of mechanical and damage behavior. Acta Mater. 2014;71:220–33. 108. Mishnaevsky L Jr, Dai GM. Hybrid carbon/glass fiber composites: micromechanical analysis of structure-damage resistance relationship. Comput Mater Sci. 2014;81:630–40. 109. Blaker J, Lee KY, Bismarck A. Hierarchical composites made entirely from renewable resources. J Biobased Mater Bioenergy. 2011;5:1–16. 110. Mishnaevsky L Jr. Nanostructured interfaces for enhancing mechanical properties of materials: computational micromechanical studies. Compos Part B. 2015;68:75–84.

Modelling the Behavior of Complex Media by Jointly Using Discrete and Continuum Approaches

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Sergey G. Psakhie, Alexey Yu. Smolin, Evgeny V. Shilko, and Andrey V. Dimaki

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Joining Movable Cellular Automaton Method and Finite-Difference Method for Modeling Fluid-Saturated Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 MCA-Based Model of the Mechanical Response of Fluid-Saturated Solid . . . . . . . . 2.2 Equations of State and Filtration Transfer of Fluid in “Micropores” . . . . . . . . . . . . . . . 2.3 Simulation of Mass Transfer Between Micro- and Macropores . . . . . . . . . . . . . . . . . . . . 2.4 Verification of the Model of Fluid-Saturated Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Study of Strength of Fluid-Saturated Porous Elastic-Brittle Material Under Uniaxial Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dependence of Uniaxial Compressive Strength on Material Permeability and Sample Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Generalized Expression for the Strength of a Sample of Fluid-Saturated Elastic-Brittle Material under Uniaxial Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Sergey G. Psakhie: deceased. S. G. Psakhie (*) Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia Tomsk Polytechnic University, Tomsk, Russia Skolkovo Institute of Science and Technology, Skolkovo, Russia e-mail: [email protected] A. Y. Smolin · E. V. Shilko Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia Tomsk State University, Tomsk, Russia e-mail: [email protected]; [email protected] A. V. Dimaki Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_79

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4 Study of Strength of Fluid-Saturated Samples Porous Elastic-Plastic Material Under Shear Accompanied by Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337 5 Future Directions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1341 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342

Abstract

Usually, computer simulation of the behavior of materials and complex media is based on the continuum approach, which uses highly developed mathematical apparatus of continuous functions. The capabilities of this approach are extremely wide, and the results obtained are well known. However, for of a number of very important processes, such as severe plastic deformation, mass mixing, damages initiation and development, material fragmentation, and so on, continuum methods of solid mechanics face certain hard difficulties. As a result, a great interest for the approach based on a discrete description of materials and media has been growing up in recent years. Because both continuum and discrete approaches have their own advantages and disadvantages and a great number of engineering software has been created based on continuum mechanics, the main line of discrete approach development seems to be not a substitute but a supplement to continuum methods in solving complex specific problems based on a joint using of the continuum and discrete approaches. This chapter shows an example of joining discrete element method and grid method in an effort to model mechanical behavior of complex fluid-saturated poroelastic medium. The presented model adequately accounts for the deformation, fracture, and multiscale internal structure of a porous solid skeleton. The multiscale porous structure is taken into account implicitly by assigning the porosity and permeability values for the enclosing skeleton, which determine the rate of filtration of a fluid. Macroscopic pores and voids are taken into account explicitly by specifying the computational domain geometry. The relationship between the stress-strain state of the solid skeleton and pore fluid pressure is described in the approximations of a simply deformable discrete element and Biot’s model of poroelasticity. The capabilities of the presented approach were demonstrated in the case study of the shear loading of fluid-saturated samples of brittle material. Based on simulation results, a generalized logistic dependence of uniaxial compressive strength on loading rate, mechanical properties of the fluid, and enclosing skeleton and on sample dimensions was constructed. The logistic form of the generalized dependence of the strength of fluid-saturated elasticbrittle porous materials is due to the competition of two interrelated processes, such as pore fluid pressure increase under solid skeleton compression and fluid outflow from the enclosing skeleton to the environment. Another application of the presented approach is the study of the shear strength of a water-filled sample under constrained conditions. An elastic-plastic interface was situated between purely elastic permeable blocks that were loaded in the lateral direction with a constant velocity; periodic boundary conditions were applied in the lateral direction. In order to create an initial hydrostatic compression in a volume, a

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pre-loading was performed before shearing. The results of simulation show that shear strength of an elastic-plastic interface depends nonlinearly on the values of permeability and loading parameters. An analytical relation that approximates the obtained results of numerical simulation was proposed. Keywords

Computational solid mechanics · Discrete and continuum approaches · Coupling of modeling methods · Fluid-saturated media · Poroelasticity · Fracture

1

Introduction

Many natural and man-made materials, such as permeable rocks (including coal and oil bearing strata) [1–4], bone tissue [5], filtering materials [6], endoprostheses [7], and other, are fluid-saturated porous or cracked porous media. The mechanical response of fluid-saturated permeable materials exhibits some features that differentiate them from solid composite materials where phases also have different elastic and rheological characteristics. These features are associated with the ability of the liquid phase to redistribute in cracks and pores of the enclosing material (the macroscopic shear modulus of fluid is conventionally assumed to be zero). As a result of the redistribution, mean stress can either level off in different regions of the material or strongly oscillate due to dilatancy and filling of new discontinuities or due to fluid exchange with the surrounding medium (for permeable boundaries). A usual theoretical approach to the study of fluid-saturated permeable materials is to use coupled models that account for three very important aspects of functioning of such systems: (1) relation of solid skeleton deformation with volume and porosity variation in the pore and crack volume, (2) relation between pore pressure and stresses in the enclosing volume of a solid, and (3) fluid redistribution in the pore and crack volume. The most famous representatives of this approach are analytical macroscopic models of poroelasticity whose theoretical basis was first discussed by Biot [8–10]. They were further developed by taking into account a range of scales in real materials, damage accumulation, dilatancy, and their influence on the skeleton elastic properties and pore fluid pressure [11–16]. The possibilities of the mentioned and analogous macroscopic coupled models are as a rule limited to the estimation of the stress-strain state of permeable materials (first of all, geological media) in stationary conditions. A broad range of problems remains unaddressed which concern nonstationary and transient processes occurring under dynamic loading, dynamic variation of boundary conditions, or during fracture. The main tool for the theoretical solution of these problems is a numerical simulation with the use of coupled models based on the concept of poroelasticity. Today numerical simulation of fluid-saturated permeable materials is mostly performed using continuum mechanics methods (finite element and finite-difference methods) [17–30], which allow the construction of various rheological (coupled and multiscale) models of solids to study the mechanical response of materials at

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different scales. Although the continuum approach has obvious and well-known advantages, it entails difficulties in studying the processes in fluid-saturated materials, which are accompanied by multiple fracture of a solid skeleton and contact interaction of formed surfaces. The computational methods based on a discrete representation of the medium have much fewer disadvantages in fracture modeling [31–38]. A well-known representative of this family of models is the discrete element method (DEM) which represents the modeled material as an ensemble of interacting finite-size particles [32–34, 38–44]. Its major advantages result from the ability of discrete elements to change their surroundings, which is crucial for the modeling of complex phenomena such as contact interaction, cracking, and fragmentation of solids, flow of granulated media, and others. In fact, the term DEM implies a large group of modeling techniques that treat a solid as an ensemble of deformable or rigid bodies of arbitrary shape. The main differences between various representatives of the DEM group are in the approximations to the description of element shape and deformability and in the principles of formulating motion equations (explicit or implicit) [41]. In this chapter, an explicit DEM formulation is used. Various explicit DEMs differ in the approximations used to describe strain distribution within the discrete element volume and the influence of element shape/geometry on its kinematics and interaction with the surroundings [41]. A common approach to describing the form of an element is its interpretation as an equivalent circular disc (in 2D) or sphere [32, 40, 41]. Such simplification of a discrete element is associated with a well-known distinct element method proposed by Cundall [32, 43]. Based on the ideas of Cundall here will be considered equiaxed or nearly equiaxed elements in an approximation of equivalent circular discs and the term “discrete element” will be used to mention the given simplified representation of element shape. The discrete element method has a simple mathematical formulation and apparent advantages in modeling deformation and fracture. However, its application to the numerical solution of problems in the mechanics of fluid-saturated media is usually limited by the microscopic scale where pores, cracks, and channels are taken into account explicitly. In this case, coupled techniques [45, 46] are widely used. In particular, the DEM is applied to simulate the mechanical response of solid skeleton, and the lattice Boltzmann method or grid method is applied to simulate the flow of weakly compressible fluids [47–51]. The use of DEMs to simulate the response of permeable materials on higher scales is highly restricted because the mathematical formalism of this method is insufficiently developed for an adequate description of multiscale systems with complex rheological properties. For example, the most widespread approximation in describing the deformation of discrete elements is to represent them as nondeformable rigid bodies. In this approximation, the interaction between discrete elements is described using pair forces/potentials. This complicates the modeling of solids with complex rheological properties as well as elastic-plastic materials. In [52–54] another approach is proposed to describing the mechanical response of materials with complex rheological properties (including elastic-plastic and

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viscoelastic ones) in the DEM framework in which a many-body approximation is used to derive relations for element-element interaction forces. In this approach, the deformation of a discrete element is described in the approximation of homogeneous strain distribution within its volume (it is conventionally termed an approximation of simply deformable element [48, 55]). This implementation of DEM was called the movable cellular automaton method (MCA). It was shown that the derived relations enable the implementation of various models and criteria of elasticity, plasticity, and strength in the MCA method (including macroscopic rock plasticity models). Further development of the proposed formalism expanded the DEM application to a wider range of spatial and structural scales [53, 54, 56–58] and yielded a coupled numerical method for studying permeable media at the meso- and macroscale levels. In this coupled method, the mechanical response of the enclosing solid is described by the MCA method; fluid filtration and diffusion in cracks and pores of the solids (taken into account implicitly) are described by the finite-difference method. A similar coupled model of gas-bearing materials was developed in [59, 60]. Below it is shown how to apply the proposed coupled method for simulation of fluid-saturated materials at the meso- and microscale. The model accounts for the mutual influence of solid skeleton deformation and fluid redistribution in the pore space. The possibilities of this numerical model are demonstrated by a parametric study of the dependences of the strength of fluid-saturated elastic-brittle materials on pore size, sample geometry, and strain rate.

2

Joining Movable Cellular Automaton Method and Finite-Difference Method for Modeling Fluid-Saturated Media

The joint using of two different methods is based on dividing the complex problem into two subproblems: (1) to describe the mechanical behavior of the enclosing solid (skeleton) and (2) to describe fluid transfer in the filtration volume of the solid represented by a system of connected channels, pores, cracks, and other discontinuities. The mechanical response of the enclosing solid is described using the movable cellular automaton (MCA) method which is an implementation of the method of simply deformable elements [53, 54]. Depending on the structural features of the considered permeable material and on the simulated scale, the dimensions of discrete elements can be much larger than the linear dimensions of discontinuities in the solid; they also can be comparable to them or smaller. The influence of “micropores” in the solid skeleton (i.e., pores, channels, and other discontinuities whose characteristic size is smaller than the discrete element size) on the mechanical properties and response of a discrete element is accounted for explicitly. Additionally, the MCA method is used to solve the problem of filtration fluid transfer in the network of connected “micropores” of the enclosing solid. Fluid mass transfer between “micropores” and “macropores,” which are considered as the regions between spaced and

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Fig. 1 The layers formed by discrete elements (movable cellular automata) and finite-difference grid (a), grid cells on the solid skeleton boundary and macropores (b)

noninteracting discrete elements, is calculated using a finer grid embedded in a global coordinate system (Fig. 1a). The same grid is used to calculate “macropore” volumes (Fig. 1b).

2.1

MCA-Based Model of the Mechanical Response of Fluid-Saturated Solid

2.1.1 General Statements The MCA method used to simulate the deformation and fracture of solid skeleton considers the material as an ensemble of interacting particles (elements) of finite size. Interacting pairs of neighboring elements are considered as being in contact. To model a consolidated fragment of a solid, these contacts are assumed to be initially bonded. The contacts between damage/crack faces are considered as unbonded [53, 54 ]. Switching between these two contact states is based on the assigned force or displacement criteria. Within the used formalism, interaction forces between two elements are divided into two independent components: central (oriented along the line connecting the centers of mass of the elements) and tangential (in the plane transverse to the mentioned line). This approximation is appropriate for equiaxed or nearly equiaxed polygons or polyhedrons and is unconditionally correct for circular discs (in a 2D problem statement) or spheres (in 3D). Discrete elements are therefore interpreted as equivalent discs or spheres. Within this approximation, a pair of elements that have common edges/faces (i.e., two adjoining equivalent discs/spheres) is assumed to be an interacting pair. The interaction between these elements is realized through plane faces. The face area S of the interacting pair is set equal to the area of the common face of corresponding polyhedrons. A 2D

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Fig. 2 An example of regular close packing of discrete elements in a 2D problem statement. The circles denote equivalent discs

example shown in Fig. 2 demonstrates p that ffiffiffi the area of interaction between adjacent discrete elements i and j is Sij ¼ 2Rh= 3. The evolution of an ensemble of discrete elements is defined by a numerical solution of the system of Newton-Euler equations of motion: 8 ! ! Ni  > ! !  d2 r i dvi X > n τ > m ¼ ¼ m F þ F > i i ij ij > < dt dt2 j¼1 Ni > ! > d! ωi X > > ¼ J Mij > i : dt j¼1

,

(1)

! ! where r i , v i , and ! ω i are the radius vector, velocity vector, and angular velocity of discrete element i, mi is the element mass, Ji is the moment of inertia of an equivalent ! ! disc or sphere, F nij and F τij are the forces of central (normal) and tangential interaction !

!

between element i and its neighbor j, and Mij is the torque. The both constituents (F nij !

and F τij ) of the interaction force between discrete elements i and j in Eq. 1 include !

!

!

!

τp nv τv potential ( F np ij and F ij ) and dissipative/viscous ( F ij and F ij ) contributions [54]:

8 ! !np !

i > < Pi ¼ σ mean 3 s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi       2  2 2 2  2 1 > 2 > > σ ieq ¼ pffiffiffi σ ixx  σ iyy Þ þ σ iyy  σ izz þ σ izz  σ ixx þ 6 σ ixy þ σ iyz þ σ ixz : : 2

(8) According to Newton’s third law, the reaction forces of elements i and j must be equal in magnitude (σij=σji and τij=τji). This yields the following system of equations for the calculation of the central and tangential interaction forces in the pair i–j: (

 pair  σ ij ¼ σ pair ij  eiðjÞ   Ai Pi  σ ji ¼ σ ji ejðiÞ  Aj Pj , τij ¼ τpair γ iðjÞ  τji ¼ τpair γ jðiÞ ij ji

(9)

Equation 9 coupled with Eq. 6 can be used to calculate the contributions of elements i and j to the total values of pair overlap (εi( j ) and εj(i)), relative shear τp displacement (γi( j ) and γj(i)), and interaction forces Fnp ij and Fij . A particular form of pair, np pair, τp the pair-wise contributions Fij , Fij , and parameter Ai is defined by the   i i assigned constitutive law σ αβ eαβ .

2.1.2 MCA-Based Model of Brittle Solids It is known that depending on conditions all solids can exhibit both elastic-brittle and elastic-plastic behavior. Conventionally, the solids whose inelastic deformation mechanisms are associated with the formation and operation of a system of different scale damages are referred to brittle materials. The inelastic response of such materials is often called quasi-plastic or plastic. The mechanical behavior of these materials is characterized by a strong dependence of inelasticity and strength parameters on local pressure. In this chapter, a generalized   Hooke’s law for isotropic materials is taken as the constitutive relation σ iαβ eiαβ for the elastic response of simply deformable discrete element i:   σ iαβ ¼ 2Gi eiαβ þ 1  2Gi =Ki σ imean δαβ ,

(10)

where α,β = x,y,z; Gi and Ki are the shear and bulk elastic moduli of the material filling element i. The system of expressions that determine the values of specific

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reaction forces in the pair i–j and satisfy the generalized Hooke’s law for mean stresses and strains reads [53, 54]:

 8 2Gi pre pre > cur > σ ¼ σ ij þ Δσ ij ¼ σ ij þ 2Gi ΔeiðjÞ þ 1  Δσ imean ¼ > > < ij K i

 2Gj (11) pre pre cur > ¼ σ þ Δσ ¼ σ þ 2G Δe þ 1  ¼ σ Δσ jmean ji j jðiÞ ji ji > ji > Kj > : Δhij ¼ ΔeiðjÞ Ri þ ΔejðiÞ Rj , 8 pre pre cur > < τij ¼ τij þ Δτij ¼ τij þ 2Gi Δγ iðjÞ ¼ pre pre ¼ τcur (12) ji ¼ τji þ Δτji ¼ τji þ 2Gj Δγ jðiÞ : > : Δlij ¼ Δγ q þ Δγ q iðjÞ ij jðiÞ ji shear Here, the relations for calculating the central and tangential interaction forces are written in increments (in a hypoelastic form). The superscripts cur and pre denote the values of specific interaction forces at the current and previous time steps of integrating the motion equations. Eqs. 11 and 12 are first solved for strain increments Δei( j ), Δej(i), Δγ i( j ), and Δγ j(i). The found strain increments are then used to calculate τp the current values of the specific (σij and τij) and total (Fnp ij and Fij ) reaction forces. i Components of the average strain tensor eαβ are derived from Eq. 10 through the calculated average stress σ iαβ : 8 σ iαα 2Gi  K i i > i > > < Δeαα ¼ Δ 2G þ 2G K Δσ mean i i i i > σ > αβ > : : Δeiαβ ¼ Δ 2Gi

(13)

The calculated average strains eiαβ are used to update the values of the volume Ωi of simply deformable discrete element i and contact area Sij:  8   < Ωi ¼ Ω0i 1 þ eixx 1 þ eiyy 1 þ eizz    , : Sij ¼ S0 1 þ eij0 0 1 þ eij0 0 ij xx zz

(14)

where Ω0i is the “initial” volume of the element (in an undeformed state), and S0ij is the initial value of the area corresponding to the pair of undeformed elements i and j. The strain tensor eijα0 β0 is determined at the plane face of interaction between the elements in an instantaneous (local) coordinate system X0 Y0 Z0 of the considered pair (with the Y0 axis located at the center of mass of element i and oriented along the line connecting the centers of mass of elements i and j) [53]. It is calculated by linear interpolation of corresponding components of the average strain tensors (eiαβ and ejαβ)   to the central point of this area eijαβ and subsequent transformation from the global   coordinate system to the coordinate system X0 Y0 Z0 eijαβ ! eijα0 β0 .

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The behavior of brittle solids beyond the yield point is simulated using a rock plasticity model with a nonassociated flow rule and Mises-Schleicher yield criterion (Nikolaevsky’s model) [61, 62]: Φi ¼

βi i J þ 3 1

qffiffiffiffiffi σ ieq i J 2  Y i ¼ βi σ imean þ pffiffiffi  Y i ¼ 0, 3

(15)

where Yi is the shear yield point of the material of element i, βi is the internal i i friction coefficient, and J 1 and J 2 are the first principal invariant of the average stress tensor in the volume of discrete element i and the second principal invariant of the deviatoric part of the average stress tensor. A feature of Nikolaevsky’s model is that it postulates a linear dependence between the rates of the bulk and shear plastic strain components with the proportionality factor Λ, termed the dilatancy factor. This model was chosen because it adequately describes the response of a large group of brittle materials (geomaterials and ceramics) at different scales with regard to the contribution of “lower” structural scales. Nikolaevsky’s model is adjusted to the MCA formalism by using the radial return algorithm of Wilkins [63]. According to the algorithm, a numerical solution of an elastic-plastic problem at each time step is reduced to an incremental solution of an elastic problem and subsequent reduction of the average strain tensor σ αβ in the i volume of discrete element i to the limiting surface if the inequality Φi > Yi holds true. The reduction is carried out by adjusting the reduced values of the central and tangential interaction forces for element i and its neighbors j [53]: n

  σ 0ij ¼ σ ij  σ imean Mi þ σ imean  N i Þτ0ij ¼ τij Mi ,

(16)

  σ 0iðjÞ , τ0iðjÞ are the reduced values of the specific reaction forces of an pffiffiffi  3=σ ieq ð3Gi ðΦi  Y i Þ=ðK i Λi βi þ 3Gi ÞÞ is the reduction automaton, Mi ¼ 1 

where

factor of stress deviators, Ni = KiΛi(Φi  Yi)/(KiΛiβi + Gi) is the correction of the mean stress value calculated in an elastic approximation, and Λi is the dilatancy factor of the material of automaton i. Relations for the correction of interaction forces strictly meet the necessary requirements of Nikolaevsky’s model imposed on the values of local pressure and stress deviator [62]. The material rheological properties of a discrete element are assignedpby ffiffiffi the dependences Y(ems), β(ems), and Λ(ems). Here, ems ¼ emean Kβ=ð3GÞ þ eeq = 3 is a combination of mean strain and equivalent strain (this parameter can be conventionally called the Mises-Schleicher strain parameter). The developed approach to the modeling of many-body interaction allows multiparameter failure criteria (Mohr-Coulomb, Drucker-Prager, and other) to be used as a criterion of bond breaking (failure) in pairs of interacting discrete elements. Below the Drucker-Prager failure criterion is used which is a form of notation of the MisesSchleicher yield criterion:

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  σ ijDP ¼ 0:5 λij þ 1 σ ijeq þ 1:5 λij  1 σ ijmean ¼ σ ijc ,

(17)

where λij ¼ σ ijc =σ ijt is the bond strength ratio for the  pair  of discrete elements i and j  ij ij under uniaxial compression σ c and tension σ t , and σ ijmean and σ ijeq are the corresponding invariants of the stress tensor σ ijα0 β0 . The tensor σ ijα0 β0 is a local stress tensor at the area of interaction (contact area) of the considered pair i–j. It is determined in the same way as the strain tensor eijαβ [53]. A detailed procedure of calculating the strength criterion for a pair of bonded discrete elements is described elsewhere [53, 54].

2.1.3 MCA-Based Model of Fluid-Saturated Porous Solids The stress state of a permeable solid with a system of connected pores, channels, and cracks is complex. It is defined both by the specific volume of discontinuities and by their geometry and spatial distribution [64]. In case a pore volume has no particular geometry and orientation (texture), an adequate approximation is the one that takes into account the contribution of pore fluid pressure only to hydrostatic pressure in the skeleton (hydrostatic tension) on the meso- and macroscales. In this approximation, the influence of fluid inside “micropores” (i.e., fine discontinuities within the discrete element volume) on the mechanical response of a discrete element can be accounted for through the following modification of Eq. 11 for the central interaction forces [76]:



 ΔPifluid 2Gi pre pre σ cur ¼ σ þ Δσ ¼ σ þ 2G Δe  þ 1  Δσ imean ij i iðjÞ ij ij ij Ki Ki



 2Gj ΔPifluid pre pre cur ¼ σ ji ¼ σ ji þ Δσ ji ¼ σ ji þ 2Gj ΔejðiÞ  þ 1 Δσ jmean , Kj Kj (18) where Pifluid and Pjfluid are the contributions of pore fluid pressure inside “micropores” to the mean stress within the volume of discrete elements i and j, respectively. Notice that the substitution of Eq. 18 into the definition of components of average stress tensor (3) leads to the following form of Hooke’s law for average stresses and strains:

σ iαβ

¼ 2Gi

eiαβ



 Pifluid 2G i i  δαβ eαβ þ 1  =Ki σ imean δαβ : Ki

(19)

Expression (19) is a form of notation of Hooke’s law in the Biot linear model of poroelasticity [10]. Particularly, this expression yields a modified expression for diagonal average strain tensor components. In the hypoelastic form it reads: Δeiαα ¼

Δσ iαα 2Gi  K i i Pi þ Δσ mean þ fluid : 2Gi 2Gi K i Ki

(20)

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The quantity Pifluid is linearly related to the average pore fluid pressure Pipore inside “micropores” of discrete element i: Pifluid ¼ ai Pipore ,

(21)

where ai = 1  Ki/Ks, i. Here, Ks,i is the bulk modulus of the nonporous material (or, in other words, solid walls of the skeleton) of discrete element i. Taking into account the pore fluid pressure Pipore , the Mises-Schleicher yield criterion (15) takes the following modified form:   σi eq Φi ¼ βi σ imean þ bi Pipore þ pffiffiffi  Y i ¼ 0, 3

(22)

where bi is the dimensionless coefficient. The stresses applied to the solid skeleton change the volume of the enclosing solid and, correspondingly, the volume of the system of connected discontinuities (also called the open porosity or filtration volume). The specific volume of the system of connected “micropores” within the volume of discrete element i is characterized by “microporosity” ϕi which is formally determined as: φi ¼

Ωipore Ωi

,

(23)

where Ωipore is the filtration volume. The initial value of Ωipore is equal to the filtration volume of undeformed discrete element i (the corresponding microporosity value will be designated as ϕ0i ). During element deformation, the filtration volume Ωipore can undergo both reversible (due to elastic skeleton deformation) and irreversible (due to material dilatancy) changes. Here it is assumed that the discontinuities formed/evolving during inelastic deformation are integrated into the filtration volume. Then the parameter Ωipore can be divided into two components: the volume of the initial system of connected pores, cracks, and channels Ωipore, el and the volume of new pores Ωipore, pl formed due to “quasi-plastic” deformation of the material (i.e., due to an irreversible opening of microscopic discontinuities): Ωipore ¼ Ωipore, el þ Ωipore, pl :

(24)

The quantity Ωipore, el undergoes reversible changes during elastic deformation of the porous material. The value of these changes in the Biot linear model of poroelasticity [10] is determined by initial “microporosity” ϕ0i , mean stress in the skeleton σ imean, pore pressure Piporeand by the ratio of the elastic moduli of porous solid skeleton Ki and nonporous solid grains Ks,i that form the skeleton [76]:

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" Ωipore,el

¼

Ωi φ0i

þ

3σ imean

1 1  K i K s,i



þ

3Pipore

1 1 þ φ0i  Ki K s,i

# :

(25)

The irreversible change of pore volume Ωipore, pl due to material dilatancy can be expressed through the plastic component of the bulk strain of discrete element i: Ωipore,pl ¼ Ωi θiplast ,

(26)

where θiplast is the plastic component of the bulk strain of discrete element i. The quantity θiplast is formally determined as the difference of the total bulk strain of the element θi and its elastic component θielast :   ( θiplast ¼ θi  θielast ¼ eixx þ eiyy þ eizz  θielast (27)  θielast ¼ 3 σ imean þ Pifluid Þ=K i Equations 18, 19, 20, 21, 22, 23, 24, 25, 26, and 27 comprise the mathematical formulation for the model of interaction between solid skeleton and fluid in the “micropore” space, which is implemented within the movable cellular automata method. An important point in using this model is the determination of dimensionless coefficient bi for the studied material. Its value is determined based on a priori assumptions about the geometrical shape of pores and their spatial distribution in the solid skeleton [65, 66]. For materials whose pore space permeability and configuration allows uniform hydrostatic pressure distribution in the solid skeleton, the coefficient bi is conventionally assumed to be equal to 1 [65, 66] (the condition of chemically neutral fluid is also valid in this case [67, 68]). It is also assumed that new damages (whose formation defines macroscopic quasi-plasticity of brittle materials) are integrated into the system of connected existing micropores. Features of the pore space and microscopic structure of the solid skeleton may lead to a more complex relationship between pore pressure and damage accumulation for which the value of bi is less than 1 and is a function of porosity parameters and the ratio of applied stress to pore pressure [66, 68–72]. The lower boundary of parameter bi can be the initial “microporosity” φ0i of undeformed material [65]. Modified failure criteria that take into account local pore fluid pressure can be used as criteria of bond failure in pairs of interacting fluid-saturated discrete elements. For example, the numerical simulation results given below were obtained using a modified variant of the Drucker-Prager failure criterion (17):     σ ijDP ¼ 0:5 λij þ 1 σ ijeq þ 1:5 λij  1 σ ijmean þ bij Pijpore ¼ σ ijc , (28) where bij is the dimensionless coefficient that has the same meaning as the corresponding coefficient in Eq. 22, and Pijpore is the pore pressure in the contact

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point of the pair i–j obtained by a linear interpolation of the corresponding values in elements i and j: Pijpore ¼

Pipore qji þ Pjpore qij r ij

:

(29)

Notice that although yield criterion (22) and failure criterion (28) have the same structural form, the meaning of the coefficients at the term σ mean in them is different. The value of the coefficient β in Eq. 22 defines the degree of pressure influence on the nucleation and evolution of a network of microscopic damages and cracks within the material volume. The coefficient 1.5(λ  1) in Eq. 28 defines the influence of pressure on the condition of localized coalescence of individual microdamages into a single meso- or macrofracture. Since the role of local pressure in damage evolution usually increases with the growing spatial scale of damage, the values of coefficients β and 1.5(λ  1) are not necessarily consistent. This also holds true for the coefficients bi and bij at the term Ppore in Eqs. 22 and 28.

2.2

Equations of State and Filtration Transfer of Fluid in “Micropores”

The discussed model of liquid phase transfer is based on the following approximations to describing the fluid state and filtration volume of the solid: 1. Fluid is assumed to be compressible. 2. Fluid can occupy the total pore volume or its part. 3. Fluid adsorption to pore walls, capillary effects, and the effect of strength reduction through adsorption in a permeable solid are not taken into account. 4. Size distribution of “micropores” (i.e., connected discontinuities within the discrete element volume) is not taken into account. Within the last approximation, the pore and crack volumes of a discrete element are described by two parameters: open “microporosity” ϕ and characteristic diameter of filtration channel dch. It should be noted that in the general case the quantity dch is not equal to the average transverse size of discontinuities but to the size of the narrowest channels in the pore space which determine the rate of fluid filtration through the permeable volume of the discrete element. If the parameter dch is determined correctly, the proposed simplification for describing the pore and crack volume is adequate for a wide range of permeable solids, including geomaterials. Compressible fluid in the developed model is described by the equation of state [73]:  ρðPÞ ¼ ρ0 1 þ ðP  P0 Þ=K fl

(30)

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where ρ and P are the current fluid density and pressure, ρ0 and P0 are the equilibrium values of fluid density and pressure under atmospheric conditions, and Kfl is the bulk modulus of fluid. An essential fluid property is its ability to occupy a part of the available pore volume (while gas occupies the total available volume). When fluid fills a part of the pore volume, its pressure on the solid skeleton can be assumed equal to atmospheric. Strictly speaking, in this case, fluid pressure on pore walls is distributed nonuniformly and defined by the filled pore volume and fluid mass; these effects are neglected in the considered model. Correspondingly, if the ratio of fluid mass in a pore to the occupied pore volume is less than ρ0, fluid pressure is assumed to be equal to P0. The quantity P0 is defined by external conditions. For example, for a sample immersed in a gaseous or liquid medium, P0 is numerically equal to the pressure of this medium and compensates the mechanical effect of the external medium on the sample (compression). In the numerical simulation results given below, the compressive effect of the external medium on model samples was not taken into account, and hence P0 was assumed to be zero. Expression (30) defines the relationship between fluid density and pressure both in “macropores” (explicitly taken into account as regions between surfaces of unbonded and not contacting discrete elements) and in “micropores” within the volume of discrete elements. In the latter case, the permeability and configuration of the “micropore space” in a discrete element are assumed to provide uniform hydrostatic pressure distribution in its solid skeleton. This allows volume averaged density ρi and pore pressure Pipore of retained fluid to be used for each discrete element i (this approximation corresponds to the formalism of the simply deformable discrete element). In the developed model, Eq. 30 is used to calculate fluid pressure Pipore in the micropore space of discrete element i through local fluid density ρi. The current value of density ρi is determined as the ratio of fluid mass in the micropore space of discrete element i to micropore volume Ωipore . In filtration modeling, the fluid pressure gradient is assumed to be a driving force of this process. Gravitational effects being neglected, the equation of filtration (fluid mass transfer) in the “micropore” space can be written as follows [73]: ϕ

  @ρ k ¼ K fl ∇ ∇ ρ @t η

(31)

where η is the fluid viscosity, and k is the permeability coefficient of the solid skeleton which can be found as [72]: k ¼ ϕd 2ch :

(32)

Equation 31 is numerically solved using the Euler method (or a higher-order numerical method) on a grid formed by an ensemble of interacting discrete elements. Here grid nodes are the centers of mass of discrete elements. With regard to the used approximations, there is no mass transfer between the grid nodes in which ρ  ρ0.

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The approximations of the proposed model imply that fluid mass transfer between discrete elements can be neglected due to capillary forces. This is valid for the majority of rocks in the case of “micropores” with the characteristic size dch not exceeding several micrometers. At large dch values, the “capillary” contribution to filtration can be accounted for by an additional relation.

2.3

Simulation of Mass Transfer Between Micro- and Macropores

The description of the solid skeleton as an ensemble of particles raises additional problems such as determining the volume of macroscopic pores and voids in the solid skeleton and calculating the length of interfaces between the microporecontaining solid skeleton and macropores. As has been said, a macropore (void) in the discussed model is a pore whose size is larger than the size of a movable cellular automaton (discrete element of the solid skeleton). To define the location and size of macropores, a system of discrete elements is projected onto a special finite-difference grid (Fig. 1b) embedded into the global coordinate system. The grid cell size is assumed to be much smaller than the discrete element size (in the calculations below, their ratio equals to 0.1). To calculate fluid transfer through the “solid skeleton–macropore” interface, the interface grid nodes have to be defined as the nodes belonging to discrete elements and the boundary cells, which have nodes belonging to macropores (Fig. 1b). The number of such nodes for each discrete element defines the length of its contact boundary with a macropore (or several macropores). The calculation of fluid transfer between macropores and bordering discrete elements that contain micropores is carried out on the basis of Eq. 31. Fluid redistribution in the volume of one macropore or between connected macropores is described using an approximation of equal pressure. This approximation assumes that fluid density and pressure are distributed uniformly in a closed macrovolume at each integration step. The simplification is adequate at low rates of the processes and in the case of partially filled macrovolumes. In a more general case, fluid mass transfer in a system of connected macropores can be modeled using, e.g., the lattice Boltzmann method [50, 51].

2.4

Verification of the Model of Fluid-Saturated Material

The presented fully coupled model was verified in a series of numerical tests [76], such as: 1. Fluid filtration through a thin layer of permeable material 2. Fluid discharge from a sample under rapid uniaxial compression (without failure) Fluid filtration through a thin layer was modeled for a sample of porous elasticbrittle material of length L = 0.94 m and width Ws = 0.0035 m using the following initial

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Fig. 3 A schematic of the permeable sample for fluid filtration modeling

Pðt ¼ 0Þ ¼ 0 MPa and boundary conditions in the layer: Pðx ¼ 0Þ ¼ 0:43 MPa Pðx ¼ LÞ ¼ 0 MPa The filtered fluid was water. The initial microporosity of the sample was ϕ0 = 0.1. At the initial time point, the micropores in the sample were assumed to be completely filled with water (pore fluid pressure was assumed to be zero). The schematic of the modeled sample is given in Fig. 3. The simulation results were compared with the estimates obtained with the use of the following semi-analytical model. If the fluid is assumed to be weakly compressible and pore deformation under the action of fluid is neglected, pressure distribution in a narrow permeable sample may be approximated by the series [74]:

   pπ 2 kK liq Pðx, tÞ ¼ Pðx ¼ 0Þ  ð1  x=LÞ þ t ð1Þ exp  L ϕ0 μ p¼1 1 X



sin ðpπx=LÞ : pπ

p

(33)

By calculating a sufficient number of terms in the series (33), we can approximate pressure distribution in the sample to a required accuracy. In order to verify the model adequacy and to study how an explicit account of poroelasticity affects pressure distribution in the sample, fluid filtration was modeled in the following approximations: 1. Of constant (not depending on fluid pressure and stress state of the material) microporosity ϕ = ϕ0 = const 2. With regard to the microporosity-pressure dependence described by Eqs. 23, 24, 25, 26, and 27 In the both cases, the numerically simulated pore fluid pressure profiles differed from the semi-analytical estimate (33) by no more than 10% (Fig. 4). As can be seen from Fig. 4b, taking into account poroelasticity leads to a pore pressure decrease compared to the model in which the change of pore volume in a solid is not taken into account. This effect is observed even at relatively low pore pressure values. The obtained results demonstrate the necessity of taking into account the effect of

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Fig. 4 Pore pressure distribution along the length of a permeable sample at different points of time t: (a) approximation of constant porosity and (b) with regard to the dependence of porosity on pore fluid pressure. The initial fluid pressure in micropores was assumed to be zero

poroelasticity in calculating the mechanical state of fluid-saturated samples subjected to high pore fluid pressure or high mechanical loads. The mechanical response of a fluid-saturated porous sample is governed by the following factors: (1) elastic modulus of the solid skeleton, (2) presence of fluid in pores and its bulk modulus, and (3) fluid filtration from the solid skeleton to the environment. The third factor provides a decrease in pore fluid pressure and, consequently, internal stress reduction in the solid skeleton. A result of this effect is a decrease in the reaction force of samples under compression. The dynamics of stress state variation in fluid-saturated samples during fluid discharge to the environment was analyzed in the following numerical test. We modeled rapid uniaxial deformation of a permeable water-saturated elastic-brittle sample depicted in Fig. 3 (ϕ0 = 0.1) to an assigned strain with the subsequent fixing of punches (the sample remained in the deformed state, Fig. 5a). Rapid deformation here means the sample loading to an assigned axial strain for the time which is much shorter than the characteristic time of fluid filtration through the sample cross section. After termination of the sample deformation and fixing of punches, the

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Fig. 5 Time dependence of the axial strain applied to the samples (a) and time dependences of the sample pressure on the punches (b). The initial fluid pressure in pores of water-saturated samples was assumed to be zero

sample pressure on the punches gradually decreases (Fig. 5b) as a result of fluid outflow to the environment. The dependence of porosity on pore pressure being neglected, it can be assumed that the decrease of sample pressure on punches occurs exponentially. In this case, the initial pressure corresponds to the pressure induced by the deformed water-saturated sample in the absence of filtration, and the final pressure corresponds to the pressure acting on punches from a “dry” sample. The above assumptions were also used to model uniaxial compression of a watersaturated sample with the effective diameter of filtration channels dch = 0 (which imitates the absence of filtration fluid transfer between micropores and its discharge to the environment) and a sample with no fluid in its pores. It is seen from Fig. 5b that the numerical simulation results fully correspond to the above-described form of the time dependence of the reaction force of the deformed sample. Detournay et al. showed in [10] that for a loaded sample of fluid-saturated poroelastic material the following condition holds true after the transient process governed by fluid discharge to the environment is terminated:

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Modelling the Behavior of Complex Media by Jointly Using Discrete and. . .

ζ ¼ aθ,

1331

(34)

where ζ is the change of fluid volume in a unit volume of the porous medium, θ is the bulk strain of a unit volume of the medium, and coefficient a is found from Eq. 21. In the above-considered case, the value of ζ is approximately determined as: ζ ¼ ϕ0

m1  m0 , m0

(35)

where m0 is the initial amount of fluid in the pore space of the sample and m1 is the amount of fluid in the sample after fluid discharge to the environment is terminated. The residual volume of fluid in the sample was ζ   4.1  104. The bulk strain of the sample after the transient process termination was θ   6.32  104. The ratio ζ/e = 0.648 corresponds with good accuracy to the assigned value of the coefficient a = 1  K/Ks  0.649. Thus, the results described substantiate the adequacy of the presented model of fluid-saturated porous medium and its applicability to studying the mechanical response and strength properties of fluid-saturated brittle materials.

3

Study of Strength of Fluid-Saturated Porous Elastic-Brittle Material Under Uniaxial Compression

3.1

Dependence of Uniaxial Compressive Strength on Material Permeability and Sample Dimensions

The presented model was applied to model uniaxial compression of porous elasticbrittle samples with water-filled filtration volume [76]. The samples were fixed between the fixed matrix (at the bottom) and the punch (at the top) that moved downwards and compressed the sample with constant velocity Vy. The compression direction coincided with the vertical sample axis (Fig. 6). The problem was solved in 2D statement in the plane stress approximation. The sample structure was assumed to be homogeneous, without pores and inclusions. The sample height was H = 0.1 m and width was W = 0.05 m. Numerical experiments were performed using the following parameters of the model material: K = 37.5 GPa, Ks = 107 GPa, G = 5.77 GPa, ρ = 2000 kg/m3, λ = 7, σ c = 70 MPa, and Kfl = 2.2 GPa. Since these tests were carried out for an elastic-brittle material, the yield criterion (22) was not used. Different values of parameter b in strength criterion (28) were considered ranging from b = ϕ0 = 0.1 to b = 1. The simulation results revealed that pore fluid pressure exerts a strong influence on the mechanical response of the brittle porous samples. Another factors that significantly influence the sample strength are the loading rate, characteristic filtration channel diameter (this quantity, along with open porosity, determines material permeability), and geometrical dimensions of the sample [76].

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Fig. 6 A schematic of uniaxial compression of the modeled sample

Fluid pressure in pores of a solid under uniaxial compression is governed by two competing processes: 1. Solid skeleton deformation accompanied by pore volume reduction according to Eq. 25 and by pore pressure increase (and, consequently, increasing the influence of fluid on the stress state of the skeleton) 2. Fluid discharge to the environment through the lateral faces of the sample, due to which pore pressure decreases and the influence of fluid on the stress state of the skeleton also decreases A balance between these two processes is governed, particularly, by the value of material permeability coefficient associated with the characteristic filtration channel diameter dch. Fig. 7 illustrates a typical numerically simulated dch dependence of the strength of the model material samples. The dependence has three portions. On portion I (at low sample permeability), fluid outflow from the sample does not compensate fluid density increase during pore deformation. As a result, fluid pressure in the filtration volume constantly increases in the course of deformation, due to which the effective sample strength decreases (see Eq. 28). At large values of dch (portion III), the rate of fluid outflow from the sample is sufficient for pore pressure to reduce to zero. The influence of fluid on the stress state of such samples is nearly absent, and their strength tends to the strength of a “dry” sample. On intermediate portion II, the rate of fluid density decrease caused by fluid outflow from the sample is comparable in magnitude to the rate of fluid density increase due to solid skeleton compression.

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Fig. 7 A typical dependence of the uniaxial compressive strength of samples of model elastic-brittle material on average filtration channel diameter at the loading rate Vy = 0.01 m/s and zero initial pore fluid pressure

It is obvious that the sample strength is also defined by fluid pressure distribution in the pore volume throughout the sample cross section. This distribution depends on the ratio of sample width W to its height H. The dch dependence of sample strength at different W/H ratios is clearly demonstrated by the results in Fig. 8a. One can see that the dependence of strength on the average filtration channel diameter flattens out with the increasing sample width. This effect can be explained as follows. The material volume which is adjacent to the lateral face and through which fluid escapes to the environment decreases with the increasing sample width. It is evident from the examples of pore pressure distribution in the horizontal section of samples of different width given in Fig. 8b. Correspondingly, the specific amount and pressure of the fluid retained in pores at the beginning of fracture increase with the growing W/H ratio. This leads to an unexpected conclusion: other factors being equal, the strength of a water-saturated sample of larger width appears to be lower than the strength of a “narrower” sample.

3.2

Generalized Expression for the Strength of a Sample of FluidSaturated Elastic-Brittle Material under Uniaxial Compression

As has been said above, the strength of fluid-saturated permeable samples is governed by the competition of mechanical sample deformation under applied external load and fluid discharge from the pore space to the environment. To reveal general mechanisms of this competition, the effect of loading rate on the uniaxial compressive strength of fluid-saturated samples was studied. Uniaxial compression of fluid-saturated samples with different dch values was modeled for different loading rates Vy. The initial pore fluid pressure Pinit was assumed to be zero (pore space of samples was completely filled with fluid, and initial fluid density was ρ0). Two hypotheses of micropore distribution in the solid skeleton were considered with regard to the influence of pore pressure on sample strength.

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Fig. 8 Dependence of uniaxial compressive strength of samples of model elasticbrittle material on average filtration channel diameter dch at different values of sample width W (a) and pore pressure distribution in the horizontal section of samples of different width at the beginning of fracture (b). Distributions in Fig. (b) are given for the same horizontal section of samples. In all cases, Vy = 0.1 m/s, Pinit = P0

1. Micropores are distributed homogeneously. Their size is much smaller than the characteristic size of damages and cracks formed during fracture. In this case, it can be assumed for relatively low porosity values that damage formation is not necessarily related to the location of micropores and to a large extent defined by the presence of other larger-scale defects (compared to micropores) in the solid skeleton. It can be assumed within this approximation that the effect of pore fluid on skeleton strength is governed by the porosity value and is taken into account by a subsequent determination of coefficient b in failure criterion (28): b = ϕ. 2. Micropores are distributed inhomogeneously, and damages in the material are formed through coalescence of several micropores. In this case, the effect of pore fluid on skeleton strength is directly defined by the pore pressure value: b = 1. Figure 9 gives the dependences of sample strength on the characteristic filtration channel diameter at different strain rates. The value of parameter b that determines the pore pressure contribution to failure criterion (28) is seen to strongly affect the strength of fluid-saturated samples. For example, within the approximation of

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Fig. 9 Generalized dependence of water-saturated sample strength on filtration channel diameter at different loading rates: (a) approximation of homogeneous micropore distribution (b = ϕ = 0.1), (b) approximation of relatively large micropores whose evolution leads to macrocrack formation (b = 1). The initial fluid pressure in solid skeleton pores is Pinit = P0. In all calculations, the aspect ratio of samples was W/H = 0.5

homogeneous micropore distribution in the solid skeleton (b = ϕ = 0.1, Fig. 9a) the pore pressure contribution is rather low (maximum decrease in sample strength does not exceed 25%). Within the second approximation (b = 1, Fig. 9b), the strength of water-saturated samples at low permeability values (at low characteristic filtration channel diameters dch) can decrease severalfold. The dch strength dependence flattens out in this case due to a stronger influence of residual fluid in the pore space on sample strength. Analysis of the obtained dependences of sample strength on the effective filtration channel diameter dch at different loading rates and different W/H ratios revealed that they can be reduced to a single dependence of strength on the reduced effective filtration channel diameter: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  d ch = ðW=H Þ e_yy =e_ 0yy ,

(36)

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where e_yy ¼ V y =H is the strain rate, and e_0yy is the scale multiplier that has the dimension of strain rate. The curves shown in Fig. 9 are plotted in these variables (e_0yy was assumed to be equal to 1 s1). It is known that the processes whose occurrence is governed by the competition of several factors or phenomena (e.g., biological population growth, etc.) are often described by a logistic function [75]. Based on the above assumption about the decisive role of the competition between pore pressure increase and fluid outflow from the sample, the following logistic function was used to approximate the dependence of the strength of uniaxially compressed water-saturated samples on the reduced effective filtration channel diameter [76]:  σ c dch , e_yy ¼ σ min þ c

σ 0c  σ min c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip 1 þ dch = d0 ðW=HÞ e_yy =e_ 0yy 



(37)

where σ 0c is the sample strength under uniaxial compression in the absence of fluid in the pore space, σ min c is the water-saturated sample strength in the absence of the fluid mass transfer, d0 is the parameter of the approximating function having the dimension of distance, ε_ yy is the axial strain rate of the sample, and p is the exponent. The parameters of Eq. 37 are defined by the elastic moduli of the solid skeleton and fluid, fluid viscosity, porosity value, and so on. As one can see, logistic function (37) allowed us to approximate with good accuracy the numerical calculation data given in Fig. 9 (at d0 = 1.62 μm, p = 3.7 for the curve in Fig. 9a and at d0 = 3.07 μm, p = 4 for the curve in Fig. 9b). Notice that the quantity σ min c can be analytically estimated based on Eqs. 25 and 28:  σ min ¼ σ 0c c

1 þ 1:5ðλ  1Þ

 bK f l 1 1  : ɸ0 K K s

(38)

As follows from Eq. 38, the strength of fluid-saturated samples is affected not only by the strength properties of the solid skeleton but also by the properties of fluid in the pore space, as well as by porosity structure and specific contribution of pore pressure to the skeleton stress state. For the used material parameters of the solid skeleton and fluid, the estimate of σ min is c min σ min ¼ 52:23 MPa for b = 0.1 and σ ¼ 15:83 MPa for b = 1, which agrees c c well with the simulation results at dch !0 (Fig. 9). Slight differences between the numerical simulation results and estimate (38) are due to the fact that this estimate does not account for the influence of stressed material regions adjacent to the punch and the matrix. In the general case, it is logical to take reduced material permeability k for the parameter that defines fluid filtration rate (the above dependences were plotted with the use of the effective filtration channel diameter related with permeability by Eq. 32):

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Modelling the Behavior of Complex Media by Jointly Using Discrete and. . .

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 e_yy W 2 k H2 e_yy ! dch : e_yy W 2 φ0 e_0yy H 2

1337

(39)

With this formulation, the parameter (36) and approximating function (37) take a more general meaning and can be applied to permeable materials with the different structure of filtration space. In sum, complex relations between the parameters that characterize the mechanical response of the solid skeleton, physical and mechanical properties of fluid, and its filtration redistribution dynamics in the system of pores define the nonlinear dependence of sample strength on a combination of these parameters and necessitates the application of numerical methods to study the mechanical response of fluidsaturated porous materials.

4

Study of Strength of Fluid-Saturated Samples Porous Elastic-Plastic Material Under Shear Accompanied by Compression

The liquid in pore volume influences on the strength of porous permeable media, both brittle and elastic-plastic [5, 60]. Despite the dilation of the latter, a pore pressure of a liquid can remain non-zero under constrained shear loading. As a result, a liquid pressure contributes into stress distribution, and, correspondingly, into strength. On the other hand, liquid pressure depends on transfer properties of the medium namely on porosity and permeability, the latter, in turn, is determined by a characteristic diameter of filtration channels in the solid skeleton. A superposition and interplay of filtration of the liquid and its compression in pores originate a dynamical distribution of pore pressure in bulk of the material [77]. In this section, the described coupled approach is applied to study the dependence of the shear strength of a water-filled sample under constrained conditions. An elastic-plastic interface was situated between purely elastic permeable blocks that were loaded in the lateral direction with a constant velocity. There were periodic boundary conditions in the lateral direction. In order to create an initial hydrostatic compression in a volume, a pre-loading was performed before shearing. The elastic blocks and the sample had the same values of porosity, permeability, elastic moduli, etc. Let’s consider a shear loading of an infinitely long sample under constrained conditions (see Fig. 10a). Pore volumes of permeable elastic blocks and elasticplastic interface have been saturated with water under initial atmospheric pressure. The diagrams of uniaxial loading of the materials of the elastic blocks and the elasticplastic interface are given in Fig. 10b. The considered sample is mounted between thin impermeable layers, to which an external loading is applied. There are periodic boundary conditions in the lateral direction.

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Fig. 10 (a) Scheme of loading; (b) diagrams of uniaxial loading of the materials of the blocks and the interface Table 1 The physical-mechanical parameters of the solid skeleton Parameter name Open porosity of a skeleton ϕ Bulk modulus of a porous skeleton K Bulk modulus of monolithic grains KS Density of a porous skeleton Poisson ratio of a porous skeleton

Value 0.1 37.5 MPa 107.5 MPa 2000 kg/m3 0.3

Parameter name Compression strength σc Tensile strength σt Dilation coefficient Λ Internal friction coefficient β Parameter b

Value 70 MPa 23.3 MPa 0.36 0.57 0.1

The values of physical-mechanical parameters of the material are given in Table 1. The values of compressible and tensile strengths are given for the elasticplastic interface; the elastic blocks are considered as indestructible. The total height of the considered sample was L = 0.3 m, and the height of elasticplastic interface L0 was varied from 0.0087 to 0.0312 m. The loading was performed in two stages. At the first stage, an initial pre-loading with compression normal force FN was performed. After that, the loading was fixed until fading of elastic waves in the sample. At the second stage, a shear loading in the lateral direction with the constant velocity Vx was applied till fracture of the sample. At that, the top and bottom layers were fixed in the vertical direction. It was found that under relatively small values of the normal pre-loading the fracture of the elastic-plastic interface occurs before the plastic deformation in the interface begins. At the certain value of the normal pre-loading, the fracture of the interface occurs after the plastic deformation begins and takes place at relatively high values of plastic strain (see Fig. 11). The latter demonstrates a “brittle-to-ductile” transition that takes place in real materials, in particular, in geological media. The results presented below have been obtained in the “ductile” zone of fracture, in other words, when the fracture occurs significantly after reaching the yield point. At that, the dependence of shear strength on the normal confining pressure can be approximated with the following equation:

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Fig. 11 A typical dependence of shear strength on normal load when dch = 0

qffiffiffiffiffiffiffiffiffiffiffiffi  τc ¼ σ0 1 þ σN =σy

(40)

where σ0 is the scale factor having the strength dimension and σy is the yield strength. Under the conditions described above, the dependencies of the shear strength on the permeability of the material demonstrate an exponential decrease and further growth with an increase of the value of the permeability (see Fig. 12a). The dependencies of shear strength on a permeability (that depends on a square of characteristic diameter of the filtration channel in solid skeleton) obtained for different values of shear rate can be reduced to a single dependence of shear strength on an effective filtration channel diameter dch pffiffiffiffiffiffiffiffiffiffiffiffiffiffi V x =V x0

(41)

where Vx0 is the scale factor having the velocity dimension. The abovementioned growth of the shear strength τc(dch) with an increase of the permeability manifests itself the most strongly for the samples with the relation of L0/L = 0.026 (see Fig. 12b), in other words, for the samples with the least thickness of the elastic-plastic interface. The above mentioned nonmonotonicity of the dependencies of the shear strength on the permeability has the following explanation. At relatively small values of permeability, the liquid pressure in elastic-plastic interface rapidly decreases down to zero due to the increase of the pore volume under dilation of elastic-plastic material. At that, the effective stiffness of the elastic-plastic interface remains constant and depends only on the elastic moduli of the solid skeleton. In turn, due to filtration of the liquid from the elastic blocks to the elastic-plastic interface, the liquid pressure in the elastic blocks decreases. As a result, the effective stiffness of the elastic blocks

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Fig. 12 The dependencies of shear strength on the characteristic diameter of filtration channel and deformation rate: (a) under different values of normal load σN and L0/L = 0.052 and (b) for different values of L0/L and σN = 41.7 MPa

decreases that leads to the reduction of the mean stress in the interface and, correspondingly, to the reduction of its strength in accordance with the DruckerPrager criterion (28). At relatively high values of permeability, the liquid pressure in the interface remains non-zero due to a rapid inflow of the liquid from the bulk of elastic blocks. At that, an effective stiffness of the blocks decreases while the effective stiffness of the interface increases. As a result, the mean pressure in the interface and, correspondingly, its strength grows up. A competition of these processes results in the occurrence of a minimal value of shear strength, where the rate of liquid filtration is still not enough to provide a non-zero liquid pressure in the whole cross-section of the elastic-plastic interface and, at the same time, it is enough to significantly decrease the liquid pressure in the bulk of elastic blocks. So, one can conclude that shear strength of the interface depends on a cooperation of the following processes: (1) increase of the mean stress in the medium under shear, (2) dilation of the elastic-

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plastic sample after reaching the yield point, (3) mass transfer of the fluid in the pore volume of the sample and elastic blocks, and (4) redistribution of the liquid pressure due to its mass transfer. The observed effects together with the results of numerical simulations allow one to suggest a binomial dependence of the shear strength of the elastic-plastic interface on permeability and shear rate for a given normal load σN:  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi τc ¼ σ0 þ σ1 exp c1 dch = V x =V x0 þ

σ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffip , 1 þ c1 dch = V x =V x0 

(42)

where the value (σ0 þ σ1) corresponds to the strength of the impermeable waterfilled sample (in so-called undrained conditions) and the value (σ0 þ σ2) represents the strength of a “dry” sample. Parameters c1 and c2 characterize the rate of change of exponential and sigmoidal branches of the dependence (42) with a change of permeability. Note that values σ0, σ1, and σ2 are not constants but depend on the thickness of the elastic-plastic interface, the physical-mechanical parameters of the material, and the boundary conditions. The first term of the dependence (42) describes the exponential decrease of strength of a sample due to the decrease of effective stiffness of the elastic blocks under outflow of the liquid into an excess pore volume of the sample. Note that the latter leads to the decrease in the degree of constraint. The second term characterizes the influence of filtration on the increase of effective stiffness of the interface due to the growth of pore pressure of the liquid. This, in turn, leads to the increase of the degree of constraint of the interface. The parameters of the given dependence represent the combinations of the values of loading, the interface width, and physical-mechanical properties of the sample, including permeability and physical-mechanical properties of the liquid.

5

Future Directions and Open Problems

This chapter demonstrates a wide potential of the described DEM-based coupled model of poroelasticity and shows the importance of numerical modeling in the study of mechanical properties (including strength) of dynamically loaded fluidsaturated materials. However, the presented model is valid only for brittle and quasi-plastic permeable materials and applicable mainly for special kind of geological materials like coals, fault zones, etc. To use this approach for modeling, for example, clay, one has to modify the used plasticity model and take into account pore collapse. Generally speaking, one of the most open problems in this field is an adoption of the described coupled formalism to the simulation of heterogeneous materials within a wide spatial scale range, from microscopic scale to macroscopic one. A solution of this problem needs taking into account additional interactions between the skeleton and the fluid. For example, a model of nanoporous materials has to account for adsorption and osmotic effects due to small pore sizes and curvature of pore walls.

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Another interesting problem concerns the study of orientation dependence of the mechanical response for anisotropic heterogeneous materials with anisotropy of the permeability and multiscale porous structure under dynamic constrained shear and torsion with compression/tension. The next direction of the development of the presented approach is taking into account the temperature effects including different temperatures of the skeleton and the fluid. This will enable a wide range of applications in engineering and geophysics. And the last problem that has to be mentioned here is the extending the presented approach for heterogeneous materials containing gas or a fluid/gas mixture within the solid skeleton.

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Spectral Solvers for Crystal Plasticity and Multi-physics Simulations Pratheek Shanthraj, Martin Diehl, Philip Eisenlohr, Franz Roters, and Dierk Raabe

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fundamentals and Thematic Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Finite Strain Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Constitutive Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Formulation/Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Thermo-mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Damage-Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Calculation of Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Reference Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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P. Shanthraj (*) Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany The School of Materials, The University of Manchester, Manchester, UK e-mail: [email protected] M. Diehl · F. Roters · D. Raabe Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany e-mail: [email protected]; [email protected]; [email protected] P. Eisenlohr Chemical Engineering and Materials Science, Michigan State University, East Lansing, MI, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_80

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6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Mechanics: Kirsch’s Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Thermo-mechanics: Plastic Dissipation and Thermal Expansion in a Polycrystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Damage-Mechanics: Polycrystalline Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The local and global behavior of materials with internal microstructure is often investigated on a (representative) volume element. Typically, periodic boundary conditions are applied on such “virtual specimens” to reflect the situation in the bulk of the material. Spectral methods based on Fast Fourier Transforms (FFT) have been established as a powerful numerical tool especially suited for this task. Starting from the pioneering work of Moulinec and Suquet, FFT-based solvers have been significantly improved with respect to performance and stability. Recent advancements of using the spectral approach to solve coupled field equations enable also the modeling of multiphysical phenomena such as fracture propagation, temperature evolution, chemical diffusion, and phase transformation in conjunction with the mechanical boundary value problem. The fundamentals of such a multi-physics framework, which is implemented in the Düsseldorf Advanced Materials Simulation Kit (DAMASK), are presented here together with implementation aspects. The capabilities of this approach are demonstrated on illustrative examples.

1

Introduction

The local and global behavior of materials with internal microstructures is often investigated on a (representative) volume element. Typically, periodic boundary conditions are applied on such “virtual specimens” to mimic the situation in the bulk of the material. While, in general, different types of boundary value solvers can be used to solve for mechanical equilibrium, spectral methods have been established as a powerful numerical tool especially suited for this task [for application examples see 1–7]. Starting from the pioneering work of Moulinec and Suquet [8], several improvements in performance and stability have been achieved for solving mechanical boundary value problems [9–13]. Recent advancements of using the spectral approach to solve coupled field equations enable the modeling of multi-physical phenomena such as fracture propagation, temperature evolution, chemical diffusion, and phase transformation in conjunction with the mechanical boundary value problem. The fundamentals of such a multi-physics framework, which is implemented

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in the Düsseldorf Advanced Materials Simulation Kit (DAMASK) [14, 15], are presented in the following together with implementation details and illustrative examples.

2

Fundamentals and Thematic Classification

Spectral methods are – like the finite element method (FEM) and the finite difference method (FDM) – algorithms to solve partial differential equations (PDEs) numerically [extensive introductions can be found in 16–18]. The main difference between FEM/FDM and spectral methods is the way in which the solution is approximated. The FEM takes its name from the discretization into elements on which local ansatz functions are defined. The ansatz functions of the FEM are usually polynomials of low degree (p < 5) with compact support, meaning they are non-zero only in their domain (i.e., in one element). The approximate solution on the whole domain can be computed from the assembly of the element matrices into a global matrix. The matrix links the discrete input values with the discrete output values on the sampling points. Thus, the FEM converts PDEs into algebraic equations. The resulting matrix is sparse because only a few ansatz functions are nonzero on each point. In the threedimensional case, the elements are typically tetrahedral or hexahedral with edges of arbitrary lengths and thus can conform to arbitrarily shaped bodies. Hence, the FEM is able to approximate the solution of partial differential equations on domains with a complex shape. The FEM has low accuracy (for a given number of sampling points N) because each ansatz function is a polynomial of low degree. To achieve greater accuracy, three different modifications can be used for the FEM [17]: • h-refinement: Subdivide each element to improve resolution over the whole domain. • r-refinement: Subdivide only in regions where high resolution is needed. • p-refinement: Increase the degree of the polynomials in each subdomain. In contrast, spectral methods take a global approach and approximate the whole domain by one (large) set of basis functions. Thus, spectral methods can be loosely described as an approach where p-refinement is applied, while the number of elements is limited to one. Spectral methods use global ansatz functions ϕn ðxÞ in the form of polynomials or trigonometric polynomials of high degree p. In contrast to the low-order shape functions used in the FEM, which are zero outside their respective element, ϕn ðxÞ are nonzero over the entire domain (except at their roots). The high order of the ansatz functions gives high accuracy for a given number of sampling points N. Since increasing N results in a combined h- and p-refinement, spectral methods show a superior convergence behavior, sometimes termed “exponential convergence”. The general procedure of approximating a function by a combination of global ansatz functions is given in Example 1. If the approximation is done using trigonometric polynomials, it is equivalent to using a Fourier series. The global approach of spectral methods has the disadvantage that nonsmooth solutions are difficult to describe.

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Example 1

Discretization of an unknown function by a linear combination of ansatz functions. Taken from [17]. A function uðxÞ is expressed as a linear combination of global ansatz functions ϕn ðxÞ. If the number of ansatz functions is limited to N + 1, the approximation reads as: N X an ϕn ð x Þ (1) uð x Þ  n¼0

This series is then used to find an approximate solution of an equation in the form: L uð x Þ ¼ f ð x Þ (2) where L is the operator of the differential or integral equation and uðxÞ the unknown function. Approximate and exact solution differ only by the “residual function” defined as: ! N X Rðx; a0 , a1 , . . . aN Þ ¼ L an ϕn ð x Þ  f ð x Þ (3) n¼0

The residual function Rðx; an Þ is identically equal to zero for the exact solution. The different spectral methods have different approaches to minimizing the residual. According to Boyd [17], spectral methods can be classified into “interpolating” and “noninterpolating” types. While the “noninterpolating” variants include the Galerkin method, which also is the basis for the FEM, the “interpolating” approaches approximate the solution exact at each sampling point (but only at, not in between them). Another name of the “interpolating” approach is “pseudospectral” [17].

3

Finite Strain Framework

For continuum modeling, a finite strain framework is introduced that allows to capture the shape changes of the investigated body during deformation. In that framework, B denotes a body that occupies the region B0 in the reference configuration and Bt in the current configuration. The location of the material points in the reference state is given by x  B0 and in a deformed configuration by y  Bt. A deformation map X ðxÞ : x  B0 ! y  Bt maps points x in the reference configuration to points y in the current configuration. The displacement u of a material point is the difference vector between these configurations: uðx,t Þ ¼ X ðx,t Þ  x

(4)

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Focusing on a point in time, i.e., a fixed deformation state, allows simplifying the notation to uðxÞ ¼ X ðxÞ  x. A line segment dx in an infinitesimal neighborhood of a material point x is pushed forward by:   @y y þ dy ¼ y þ  d x þ O dx2 : (5) @x Neglecting terms of higher order, dy can be expressed as: dy ¼ ¼

@y  dx @x @X ðxÞ  dx @x

(6)

¼ |{z} ∇X  d x, ≕FðxÞ

where F ðxÞ is the deformation gradient and ∇ is the “del” operator. The deformation gradient maps the vector dx at x in the reference configuration to the vector dy at y in the current configuration. For a moving body, the position of the material points varies with time. The material velocity field is defined as:1 d uð x Þ v¼ dt ¼ u_ ¼ Ẋ :

(7)

u_ ¼Ẋ holds because the points in the reference configuration do not change their position, i.e., dx/dt = 0. The spatial gradient of the velocity field is L¼

@v , @y

(8)

where L is called the velocity gradient. Using the chain rule, it can be expressed as _ 1 : L ¼ FF

3.1

(9)

Constitutive Modeling

The modeling approach followed here is based on the work of Shanthraj et al. [19]. The deformation gradient introduced in Eq. (6) is multiplicatively decomposed as F ¼ FeFiFp,

(10)

To simplify the notation, in the following, the argument x is dropped whenever it is possible, i.e., F(x) is denoted as F only

1

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where Fp is a lattice preserving deformation to the plastic configuration, Fi is a lattice distorting, but stress-free, inelastic deformation, e.g., thermal expansion, to the inelastic configuration, and Fe is an elastic deformation to the deformed configuration. The Green–Lagrange strain Ee in the lattice (plastic) configuration is obtained from Fe and Fi as   1 Ee ¼ F i T F e T F e  I F i 2

(11)

and is work conjugate with the second Piola–Kirchhoff stress tensor Sp in the lattice configuration. Thus, Sp depends on Fe and Fi. For small elastic strains, Hookean elasticity can be assumed, and, therefore, Sp depends on the elastic Green–Lagrange strain through the anisotropic elastic stiffness ℂ.   1 S p ¼ ℂ Ee ¼ ℂ F i T F e T F e  I F i : 2

(12)

The second Piola–Kirchhoff stress tensor in the unloaded (inelastic) configuration is related to Sp by S i ¼ F i S p F i T det F i 1

(13)

and Sp is related to the first Piola–Kirchoff stress tensor P through P ¼ F e F i S p F p T ¼ F F p 1 S p F p T :

(14)

The plastic deformation gradient is given in terms of the plastic velocity gradient Lp by the flow rule F_ p ¼ Lp F p ,

(15)

where Lp is work conjugate with the Mandel stress in the lattice configuration, Mp  FiT Fi Sp (assuming FeTFe  I), and thus depends on Fi and Sp. In addition, Lp depends on the underlying microstructure represented by a state variable vector ξ of the plasticity model and possibly other variables such as strain rate and/or temperature. For example, a conventional viscoplastic phenomenological crystal plasticity model for face-centered cubic lattices [20, 21] provides Lp based on the assumption that plastic slip γ occurs on a slip system α when the resolved shear stress τα exceeds a critical value ξα. The critical shear stress on each slip system is assumed to evolve from an initial value ξ0 to a saturation value ξ1 due toslip on all h011i α α β = 1, . . ., 12 according to the relation ξ_ ¼ h0 γ_β 1  ξβ =ξβ1  sgn {111} βsystems  1  ξ =ξβ1 hαβ with initial hardening h0, interaction coefficients hαβ, and a numerical parameter a. The shear rate on system α is then computed as γ_α ¼ γ_0 jτα =ξα jn sgnðτα =ξα Þ with the stress exponent (inverse shear rate sensitivity) n and reference shear rate γ_0. The sum of the shear rates on all systems finally determines the plastic velocity gradient.

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Similarly, the inelastic deformation gradient is given in terms of the inelastic velocity gradient Li by the flow rule F_ i ¼ Li F i ;

(16)

where Li is work conjugate with the Mandel stress in the unloaded configuration, Mi  Si = Fi Sp FiT det Fi1 (assuming F e T F e  I), and thus depends on Fi and Sp. Similar to Lp, internal state variables Ξ introduce a history-dependent behavior. The set of nonlinear Eqs. (10) to (16) is solved iteratively in DAMASK using a Newton–Raphson method. More details about the implementation are given by Shanthraj et al. [19]. The time integration of the state evolution is performed in a staggered step to the iterative solution of Eqs. (10) to (16), i.e., Eqs. (10) to (16) are solved at a fixed internal state, followed by a state update, the solution of Eqs. (10) to (16), and so forth until a converged solution is achieved within the tolerance limits specified.

4

Formulation/Solution Strategy

In material mechanics, a spectral method using fast Fourier transforms (FFT) was introduced by Moulinec and Suquet [8]. Since it uses trigonometric polynomials as ansatz functions, its application is limited to periodic boundary conditions. It is based on reformulating the PDE into an integral equation with the help of a “reference” stiffness and falls into the class of “pseudospectral” methods. Lebensohn [22] extended the method to the context of crystal viscoplasticity and showed the capabilities of this approach as well as several applications in a number of studies [23, 24]. Crystal plasticity-based constitutive laws have been successfully employed by, e.g., Lebensohn et al. [25], Suquet et al. [2], and Grennerat et al. [26]. Simulations of heterogeneous materials, however, are limited by the slow convergence of the original fixed-point iterative method when it is applied to materials with a large contrast in the local stiffness [27]. Several approaches have been proposed to overcome this limitation. Accelerated schemes have been introduced by Eyre and Milton [28] and Monchiet and Bonnet [9] for materials with large property contrasts. Michel et al. [27] suggested a method based on augmented Lagrangians that also works in the case of materials with infinite property contrast. Using the original formulation and substituting the fixed-point method by advanced solution methods is another option to improve convergence as shown by Zeman et al. [12] and Brisard and Dormieux [29]. A systematic comparison to FEM approaches is given in [13, 30, 31]. The application of this approach to possibly time-dependent and interacting field equations representing coupled multi-physics systems has been relatively unexplored (see [32] for applications other than mechanical equilibrium). To address this shortcoming, recent work [19, 33] has focused on extending the spectral solution scheme for the time-dependent, reaction-diffusion equation, which is representative

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of several field equations of interest, such as thermal conduction, nonlocal damage, and more general phase-field equations.

4.1

Mechanics

The spectral method formulation given here has been presented by Eisenlohr et al. [34] and was improved by Shanthraj et al. [35]. It is based on Lahellec et al. [36] finite-strain extension of the original small-strain variant. To start, the deformation map X ðxÞ is expressed as a sum of a homogeneous deformation,  and a superimposed deformation characterized by a constant deformation gradient F, ~ fluctuation field w,  þ w~ðxÞ, X ðxÞ ¼ Fx

(17)

for which periodicity conditions hold, i.e., w~ ¼ w~þ on corresponding surfaces @B and @Bþ on B. Equation (17) allows writing the deformation gradient F as the sum of a spatially homogeneous deformation part F and a locally fluctuating component F~: F ¼ F þ F~:

(18)

The material response given in Eq. (14) is formally written as a relation between F and P through a strain energy density functional, W: P ðxÞ ¼

δW ¼ f ðx,F,ξ,Ξ, . . .Þ: δF ðxÞ

(19)

The equilibrated deformation field is obtained by minimizing W over all deformation fields that fulfill Eq. (17) for an externally prescribed average deformation. Static equilibrium expressed in real and Fourier2space follows as: min W ) ∇  P ðxÞ ¼ F 1 ½P ðkÞ i k ¼ 0, X

(20)

which is equivalent to finding the root of the residual body force field F ½X ðkÞ≔P ðkÞ i k ¼ 0:

(21)

The differential Eq. (21) in Fourier space is numerically difficult to solve because of its high condition number. Introducing, in the spirit of Eshelby and Mura [37], a linear comparison material of stiffness  allows reformulation of Eq. (21) into an 2

Quantities in real space and Fourier space are distinguished by notation Q(x) and Q(k), respectively, with x the position in real space, k the frequency vector in Fourier space, and i2 =  1. F 1[] denotes the inverse Fourier transform.

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equivalent problem P ðxÞ ¼ F ðxÞ ¼ ∇ X with better numerical properties, i.e., a lower condition number. Equilibrium in this reference material is fulfilled if, for a given deformation map X , the residual body force field vanishes P ½X ðkÞ≔½X ðkÞ  i k i k ¼ AðkÞX ðkÞ ¼ 0:

(22)

The acoustic tensor AðkÞ is a shorthand notation for AðkÞaðkÞ≔½aðkÞ  i k i k for any given vector field aðkÞ . It corresponds to an operator on a deformation map producing the body forces resulting in the reference material. The inverse A1 therefore gives the deformation map that would result from a known body force field in the reference material. This deformation map vanishes iff the body force field vanishes, i.e., in static equilibrium for a positive-definite . Next, an operator that results in the deformation map causing the same body force field in the reference material as a given deformation map in the original material is defined. This corresponds to a preconditioning operation of P 1 on the non-linear operator F : P is straightforward to invert since it is local in k, with P 1 ¼ AðkÞ1 . The preconditioned system thus reads (8k 6¼ 0): P 1 F ½X ðkÞ ¼ AðkÞ1 P ðkÞ i k ¼ 0:

(23)

The deformation gradient field corresponding to this deformation map is obtained from the gradient in real space of Eq. (23) h i P 1 F ½X ðkÞ  i k ¼ AðkÞ1 P ðkÞi k  i k ¼ 0:

(24)

This is equivalent to Eq. (23) except for a constant residual field, i.e., at k = 0, where the prescribed average deformation gradient is known to hold. Expressed in terms of the deformation gradient field, Eq. (24), reads F mech ½F ðkÞ≔ΓðkÞP ðkÞ ¼ 0,

(25)

where the “Gamma operator” ΓðkÞ is defined as a shorthand notation such that ΓðkÞT (k) ≔ [AðkÞ1T ðkÞik]  ik for a tensor field T ðkÞ. Other solution approaches based on a mixed variational formulation with improved numerical stability have been presented. A general large-strain framework has been introduced by Shanthraj et al. [35] and improved small-strain formulations can be found in [9, 27, 28, 38].

4.2

Thermo-mechanics

Thermal equilibrium describes a state where zero net heat flux occurs within a material volume. However, during a material deformation process heat may be generated, which perturbs the system from its thermal equilibrium state. The initial value problem associated with non-steady-state heat conduction takes the form:

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ϱC p T_ ¼ f temp  Div H:

(26)

The mass density ϱ, heat capacity Cp, heat flux H, and heat generation rate ftemp are provided by the material point model. Constitutive modeling of the heat flux is based on Fourier’s law, where a thermal conductivity tensor relates the temperature gradient with the heat flux, i.e., H =  K∇ T. In addition, the main considerations for thermo-mechanical coupling are incorporating thermal expansion strains and heat generation due to plastic dissipation. The rate of thermal expansion is modeled here as an inelastic velocity gradient, Li ¼ αT_ I , where α is the coefficient of thermal expansion, and the heat generation rate due to plastic dissipation is modeled as f temp ¼ κM p L_ p based on a fraction of dissipated work, κ. The spectral formulation presented in the Sect. 4.1 can be also used to solve this initial value problem. A backward Euler time discretization is used to express the time dependent Eq. (26) in the following semi-discrete form:

ϱC p

T ðx,t Þ  T ðx, t 0 Þ ¼ f temp ðxÞ  Div H ðxÞ: Δt

(27)

The flux H, is additively split into a linear homogeneous term and a fluctuating field:  T ðx,tÞ  τ~temp ðxÞ, H ðxÞ ¼ K∇ (28) where K is a defined homogeneous reference conductivity tensor, similar to the reference stiffness tensor introduced in Sect. 4.1. Similarly, the coefficients ϱ CP, are split into spatially averaged and fluctuating parts, i.e., ϱ C P ¼ ϱ C P þ ϱg C P . With these definitions, Eq. (27) can be expressed in Fourier space as 

  T ðk,t Þ ¼ τtemp ðkÞ, ϱ C P þ Δtk  Kk

(29)

where the thermal polarization field τtemp ðxÞ, which implicitly depends on T ðx, t Þ, is given by   τtemp ðxÞ ¼ Δt f temp ðxÞ þ Divτ~temp ðxÞ þ ϱC P T ðx,t 0 Þ  ϱC P T ðx,tÞ: (30) The solution of the thermal initial value problem can therefore be expressed as the root of the following residual function F temp ½φðk,t Þ≔T ðk,t Þ  G ðkÞτtemp ðkÞ ¼ 0,  where G1 ðkÞ ¼ ϱ C p þ Δtk  Kk.

(31)

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Damage-Mechanics

The prediction of crack initiation and propagation in a material is of critical importance in many applications. The field of Continuum Damage Mechanics [39, 40] and the Phase-Field Method for Fracture [33, 41, 42] have emerged as powerful and versatile tools to model the nonlocal damage evolution process. A scalar nonlocal damage variable φ is introduced, which indicates a phase transition between undamaged, i.e., φ = 1, and fully damaged regions, i.e., φ = 0. The initial value problem associated with the evolution of Φ takes the general form: μφ_ ¼ f damage  Div Φ:

(32)

The damage viscosity μ, flux Φ, and driving force fdamage are provided by the material point model. Constitutively, the flux is related to the energy of a newly created crack surface [41], and is given by the classical form, Φ ¼ Gl c ∇ φ in terms of the surface tension G and a length scale parameter lc. Similarly, the driving force for fracture, fdamage, is a competition between surface tension and elastic energy release, and is modeled as f damage ¼ G=l c  φS p  Ee . Here, it is assumed that the dissipation due to the damage process occurs in the form of a stiffness degradation. Therefore, the elastic constitutive law, Eq. (12), is modified as Sp = φ2ℂEe. Similar to the thermo-mechanical case, a spectral method can be formulated to solve the damage initial value problem. A backward Euler time discretization is used to express the time-dependent Eq. (32) in the following semi-discrete form: μ

φðx,t Þ  φðx, t 0 Þ ¼ f damage ðxÞ  Div ΦðxÞ: Δt

(33)

The flux Φ is additively split into a linear homogeneous term and a fluctuating field:  φðx,t Þ  τ~damage ðxÞ, ΦðxÞ ¼ D∇ (34)  is a defined homogeneous reference diffusion tensor. Similarly, the damage where D viscosity μ is split into spatially averaged and fluctuating parts, i.e., μ ¼ μ þ μ. ~ With these definitions, Eq. (33) can be expressed in Fourier space as  Þφðk,t Þ ¼ τdamage ðkÞ, ðμ þ Δtk  Dk

(35)

where the damage polarization field τdamage ðxÞ, which implicitly depends on φðx, t Þ, is given by   τdamage ðxÞ ¼ Δt f damage ðxÞ þ Div τ~damage ðxÞ þ μ φðx, t 0 Þ  μ~φðx,t Þ:

(36)

The solution of the damage initial value problem can therefore be expressed as the root of the following residual function

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F damage ½φðk,t Þ≔φðk,t Þ  G ðkÞτdamage ðkÞ ¼ 0,

(37)

 where G1 ðkÞ ¼ μ þ Δtk  Dk.

5

Implementation

In the following, the implementation of the methods presented in the previous sections in DAMASK is concisely presented.

5.1

Fast Fourier Transform

The problems are discretized to employ a discrete Fourier transform (DFT). For that, the hexahedral domain B0 with side lengths dx, dy, dz is discretized into a regular grid of Nx  Ny  Nz = N points. The solution field is approximated in the discrete Fourier space associated with this real space grid. To achieve a reasonable performance, the FFT is used. The term FFT refers to a group of algorithms that compute the DFT in OðN log N Þ operations. A free FFT implementation that has shown an excellent performance is the Fastest Fourier Transform in the West (FFTW) developed by Frigo and Johnson [43]. To save memory and computation time, the complex conjugate symmetry resulting from the purely real-valued input data is exploited.

5.2

Calculation of Gradients

While the fields of deformation, temperature, and damage are necessarily continuous, their spatial gradients are allowed to be discontinuous, e.g., across phase boundaries. A numerical artifact associated with Fourier representations of such discontinuous fields is the development of spurious oscillations, and is referred to as the Gibbs phenomenon. This can deteriorate the performance of the solution schemes as well as the quality of the solution [44]. Various techniques exist to reduce its effects, such as a low-pass filter to effectively dampen high-frequency modes associated with the spurious oscillations [45]. As outlined by Willot [46] and Kaßbohm et al. [47], finite difference approximations of the gradient fields can be easily obtained in Fourier space that also reduce such spurious oscillations. As introduced in Sect. 4.1, the gradient ∇ f of a field f ðxÞ is expressed in Fourier ~ Working with continuous derivatives results in the straightforward space as f ðkÞ  k. ~ relation k ¼ ik: The forward-backward (FWBW) finite difference variant introduced by Willot [46] reads then as:    N x  ik 1 k~1 ¼ e  1 eik 2 þ 1 eik 3 þ 1 , 4d x

(38a)

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   N y  ik 1 e þ 1 eik 2  1 eik 3 þ 1 , k~2 ¼ 4d y

(38b)

   N z  ik 1 k~3 ¼ e þ 1 eik 2 þ 1 eik 3  1 : 4d z

(38c)

An exhaustive discussion on alternative finite difference schemes applicable for this purpose has been carried out by Schneider et al. [48].

5.3

Reference Stiffness

The choice of the reference stiffness has a strong influence on the stability and convergence rate as shown by Michel et al. [27]. In the absence of an analytic expression for the large strain formulation, similar to Michel et al. [27], the reference stiffness  for the mechanical problem is selected as      d P  d P  1    ¼ argmax ðxÞ þ argmin ð xÞ   : 2 dF d F F F

(39)

Similarly, the choices for the reference thermal conductivity and damage gradient tensors are made as  1 K ¼ argmaxkK ðxÞkF þ argminkK ðxÞkF 2 (40)    ¼ 1 argmaxkGðxÞl c I k þ argminkGðxÞl c I k : D F F 2 Note that, unlike in the small-strain mechanical case, such convenient choice of the reference tensors does not guarantee convergence.

5.4

Boundary and Initial Conditions

The applied boundary conditions on B are volume averages, i.e., for the mechanical boundary value problem F is set to the applied load FBC. To do this, the desired change of FBC is conveniently expressed in rate form, i.e., F BC ¼ F_ BC Δt for a given time increment of length Δt. In order to allow the (component-wise) prescription of stress boundary conditions PBC, an iterative adjustment3 of FBC needs to be done until the stress boundary conditions are fulfilled. Within Δt, the volume element is then subjected to a set of complementary boundary conditions in terms of deformation rate F_ BC and stress PBC, where stress boundary conditions must not allow for rigid body rotations. Components of both are mutually exclusive and, 3 The solution for the deformation gradient field, i.e., the actual spectral method procedure, is performed in parallel to these iterations.

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when not defined, set to zero in the following. These mixed boundary conditions are translated into pure deformation boundary conditions at iteration n + 1 by setting fF BC gnþ1 ¼ fF gn¼0 þ F_ BC Δt 

(

@F @P

)



 fPgn  P BC :

(41)

n

The last term in Eq. (41) corrects for deviations from the prescribed stress boundary conditions. The average compliance @F=@P is estimated from the local responses @P/@F [34]. An approach how to define boundary conditions rotated to the discretization of B has recently been presented by Kabel et al. [49]. In contrast to the mechanical boundary value problem, the initial value problems describing temperature and damage require specification of their respective field values T ðx, 0Þ and φðx, 0Þ.

5.5

Numerical Solution

Nonlinear solution methods are required to solve the resulting systems of discretized equations. While originally a fix-point scheme was employed for that [50], more recent approaches use more advanced solving techniques [12, 51]. DAMASK allows the use of solvers provided by the PETSc framework developed by Balay et al. [52]. Shanthraj et al. [35] compared three solution methods implemented in PETSc: the nonlinear Richardson method [53], the nonlinear GMRES method [54], and the inexact Newton-GMRES method [55]. In these methods, an existing solution {□}n at iteration n is iteratively improved until the prescribed convergence criteria are satisfied. Here, the notation {□} is used as a placeholder that represents {F ðxÞ} for the mechanical problem, {T ðxÞ} for the thermal problem, or {φ(x)} for the damage problem.

5.5.1 Nonlinear Richardson The nonlinear Richardson method iteratively improves a solution {□}n found at iteration n through the update f□gnþ1 ¼ f□gn  F f□gn :

(42)

5.5.2 Nonlinear GMRES In the nonlinear GMRES method an updated solution {□}n + 1 is found as the linear combination of the mprevious solutions spanning a Krylov subspace Kn ¼ span

f□gn ,  , f□gnmþ1 for which the norm of the linearized residual is minimal    @F 

  þ F □ □  □ f g f g f g nþ1 n n :  f□gnþ1  Kn @ f□gn 2 min

This residual is approximated by PETSc in a Jacobian-free way [54].

(43)

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5.5.3 Inexact Newton-GMRES In Newton methods, an existing solution {□}n is incrementally updated to {□}n + 1 = {□}n + {Δ□}n with each Newton step given by    @F  @F  Δ□ þ F □ fΔ□gn ¼ F f□gn ,  f g f g n n  ¼ 0:  @ f□gn @ f □g n 2

(44)

To increase the efficiency of such an algorithm, Eq. (44) is solved only in an inexact fashion:    @F        Δ□ þ F □ f g f g n n   η n F f □g n 2 , @ f□g n 2

(45)

where the tolerance for a vanishing norm of the residual is gradually tightened, i.e., ηn ! 0, with increasing iteration count n [56]. Within each Newton iteration, the typically large linear system Eq. (45) is solved for {Δ□}n by means of the GMRES method where the kth GMRES iterate of the Newton step is given by    @F  k  min  fΔ□gn þ F f□gn   k fΔ□gn  Kk @ f□gn 2 where Kk ¼ span

n

@F @ f□gn

fΔ□gk1 n ,  ,

@F @ f□gn

fΔ□gkm n

(46)

o

 and K0 ¼ span F f□gn . Since the discretization of the Jacobian is dense and numerically difficult to handle, a finite difference approximation @F fΔ□gkn  @ f□ gn

h i F f□gn þ ϵfΔ□gkn  F f□gn ϵ

(47)

is used to update the Krylov subspace Kk .

5.6

Algorithm

For a given time interval Δt = t  t0, the solution for individual fields can be found iteratively as described in Sect. 5.5. The multi-physics coupling approach followed in this work involves solving the coupled system Eqs. (25), (31), and (37) within a staggered iterative loop until a consistent solution, within specified tolerances, is achieved for the time interval. The procedure is detailed in Algorithm 1. The advantage of such a staggered approach is that the solution scheme of each field can be described independently.

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Algorithm 1 Staggered Algorithm for Multi-physics Coupling in [t0, t] Data: F ðx, t 0 Þ, T ðx, t 0 Þ, φðx, t 0 Þ Result: F ðx,t Þ, T ðx,t Þ, φðx,t Þ 1 Initialization: F 0 ðx,t Þ ¼ F ðx, t 0 Þ, T 0 ðx,t Þ ¼ T ðx, t 0 Þ, φ0 ðx,t Þ ¼ φðx, t 0 Þ j¼1 2 while  j   j  j1 T ðx,t Þ  T j1 ðx,t Þ ϵstag  ðx,t Þ2 ϵstag temp and φ ðx,t Þ  φ damage do 2  3  solve F mech ½F j ðx,t Þ ¼ 0 using T j1 ðx,t Þ and φ j1 ðx,t Þ; 4  solve F temp ½T j ðx,t Þ ¼ 0 using F j ðx,t Þ and φ j1 ðx,t Þ; 5  solve F damage ½φ j ðx,t Þ ¼ 0 using F j ðx,t Þ and T j ðx,t Þ; 6 j ¼ j þ 1 7 end 8 F ðx,t Þ ¼ F j ðx, t 0 Þ T ðx,t Þ ¼ T j ðx, t 0 Þ φðx,t Þ ¼ φ j ðx, t 0 Þ

5.7

Convergence Criteria

To ensure that the resulting stress field is in equilibrium, the Root Mean Square (RMS) value of the divergence of the stress field is reduced below a prescribed tolerance. The corresponding equilibrium criterion reads:     max eeq,rel kPkmax , eeq,abs =m RMS kDiv P ðxÞk2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u N    ¼ t P kj i kj 2 =N 2 ,

(48)

j¼1

where eeq,rel and eeq,abs are the relative and absolute equilibrium tolerances. The fulfillment of complementary macroscopic deformation gradient and stress boundary conditions is checked by   max eBC,rel kPkmax , eBC,abs kΔP BC kmax ,  Aijkl ðF BC  FÞkl where ΔPBCij ¼ ðPBC  PÞij

if F BCij prescribed if PBCij prescribed

(49)

and eBG,rel and eBC,abs are the relative and absolute boundary condition tolerances. The residual norm of Eqs. (31) and (37) is evaluated as a convergence test for the initial value problems.

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6

Examples

6.1

Mechanics: Kirsch’s Plate

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To compare the two approaches for calculating gradients in Fourier space presented in Sect. 5.2, the stress field in a plate with a circular hole and deformed in uniaxial tension is computed. The analytic solution for an infinitely large plate with linear isotropic behavior, known as Kirsch’s plate [57], serves as reference. For both numerical approaches and the analytic benchmark, the plate is discretized by 4096  4096 grid points. The diameter of the circular inclusion is 512 grid points. In case of the numerical approaches, elastic constants for the plate are selected as C11 = 110.9 GPa and C12 = 58.3 GPa and the hole is approximated by assigning low stiffness values to the respective material points: C11 = 100.0 Pa, C12 = 66.7 Pa. The boundary conditions reflect uniaxial tension along the (horizontal) 1-direction: 2

1:01 0:0

F ¼ 4 0:0 0:0 0:0

3 2

0:0  P ¼ 4

0:0 5 and Pa







3

0:0



5



(50)

0:0

which results in an average stress of approximately 710 MPa along the loading direction. This stress value is taken as the “far-field value” of the analytic solution. A detail of the solution field around the hole is shown in Fig. 1b, c, d. The reference solution given in Fig. 1b shows the expected behavior of a stress amplification by a factor of three at the top and the bottom of the hole and zero stresses in loading direction left and right to the hole. The numerical solutions given in Fig. 1c, d clearly match this behavior. However, a closer look at the stress field above the hole reveals some undesired features for both numerical solutions (insets in Fig. 1b, c, d and Fig. 1a): A stress intensification of more than three occurs and some regions of the matrix material have a much lower stress than expected. The advantage of the FWBW finite difference approximation lies in reducing the number of pixels affected by these features and in efficiently removing spurious oscillations seen in the solution of the continuous approach (Fig. 1c, d). Moreover, the FWBW finite difference approach is advantageous as the number of iterations required for higher stiffness ratios between the inclusion and the matrix phase can be drastically reduced as shown in Fig. 2.

6.2

Thermo-mechanics: Plastic Dissipation and Thermal Expansion in a Polycrystal

To demonstrate the capabilities of the approach for thermo-mechanical coupling, a polycrystalline volume element is generated by a Voronoi tessellation of 20 randomly distributed seed points within a cubic domain of edge length 100 μm discretized into 64  64  64 grid points. All grid points belonging to a Voronoi cell are grouped together and assigned the same random crystal orientation. The phenomenological

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σ/ MPa

2840

2130 MPa

2130

1420

710

analytic continuous FWBW

0

128

256

384

512

d/pixel

(a) Stress along a vertical line starting above the horizontal center at the top of the hole.

(c) Solution field, continuous.

(b) Solution field, analytic.

(d) Solution field, FWBW.

Fig. 1 Comparison of the analytic solution to the continuous and the forward-backward (FWBW) finite difference approximation of the derivative operator in terms of the normal stress along the horizontal loading direction. Grey color indicates the applied load of 710 MPa, the minimum (blue) depicts a stress free state (0 MPa) and the maximum (red) is set to the expected stress amplification of 3, i.e., 2130 MPa

crystal plasticity constitutive model presented in Sect. 3.1 is used to describe the mechanical behavior of the volume element (the initial microstructure of the volume element and the material parameters are given in Fig. 6 and Table 3). In addition, heat generation due to plastic dissipation, heat conduction, and thermal expansion effects are considered. The corresponding material parameters are listed in Table 1. The following boundary conditions are chosen to reflect uniaxial tension along the 2-direction 2

3 2 3 0:0

0:0 0:0  P F_ ¼4 5 ¼ 4 0:0 1:0 0:0 5 and (51) Pa 103 s1



0:0 0:0

0:0

Spectral Solvers for Crystal Plasticity and Multi-physics Simulations

Fig. 2 Iterations needed for a converged solution in dependence of the stiffness ratio between inclusion and matrix for the presented gradient calculation approaches

continuous

102

FWBW 101

100 -4 10 10-3 10-2 10-1 100 101 phase contrast

Table 1 Constitutive parameters for thermomechanics: mass density ρ, specific heat Cp, conductivity K, thermal expansion coefficient α, coefficient of cold work κ

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103

number of iterations

43

Parameter ρ Cp K/I α κ

Unit Kg m3 J kg1 K1 Wm1 K1 mK

102

103

104

Value 2700.0 910.0 237.0 2.31 0.95

and a homogeneous temperature field of 300 K is set as the initial condition for the thermal conduction problem. The resulting stress in loading direction, total plastic shear, and temperature distributions at F ¼ 1:1 (i.e., after 100 s) are shown in Fig. 3. It can be seen that the thermal hot spots correspond to regions of high plastic deformation, and result in stress relaxation due to thermal expansion. The global stress–strain curve is shown in Fig. 4 and compared to a case without thermal expansion.

6.3

Damage-Mechanics: Polycrystalline Fracture

The phase-field damage model is applied to study crack nucleation phenomena and failure of a plastically deforming polycrystalline material. The same volume element as considered in Sect. 6.2 is used, with the additional damage material parameters listed in Table 2. The boundary conditions chosen to reflect uniaxial tension along the 2-direction are

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σ22 50

γ

0.1

200 MPa

(a) von Mises stress.

T 0.3

300

(b) Plastic strain.

307 K

(c) Temperature.

Fig. 3 The resulting normal stress along loading direction, plastic strain, and temperature distributions at F ¼ 1:1. The outlined cube indicates the undeformed configuration

Fig. 4 Comparison of the global stress–strain curves between the model with (solid line) and without (dashed line) thermal expansion

120

α=0

σ 22 / MPa

100

α≠0

80 60 40 20

0.00

0.02

0.04

0.06

0.08

0.10

ε22

Table 2 Constitutive parameters for damage-mechanics: interface energy G, characteristic length lc, and damage mobility μ Parameter G

Unit J m2

Value 1.0

lc μφ

μm s1

2.0 0.01

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Spectral Solvers for Crystal Plasticity and Multi-physics Simulations

F_ 103 s1

2



¼ 4 0:0 0:0

0:0 1:0 0:0

3 2 0:0 0:0  P ¼4

0:0 5 and Pa



1367





3







5

0:0

(52)

The nucleation and evolution of crack surfaces and the corresponding von Mises stress through a section of the polycrystalline sample is shown in Fig. 5. The crack nucleation is observed at a higher-order grain boundary junction due to the incompatibility of the plastic deformation modes in the respective grains forming the junction, which results in large stresses. Crack nucleation and its subsequent evolution during deformation strongly depend on the texture of the material due to the anisotropy of the plastic deformation modes, and highlight the need for accurate constitutive models. The nucleation and growth of a crack is followed by an unloading of the material behind the crack tip and a general redistribution of the stress field. The propagation of a nucleated crack is determined by the energetics of the material surrounding the crack tip. The crack tip stress field, and hence the elastic energy release rate, is maximum in the direction of the crack, which results in a strong driving force for the propagation of the crack in this direction. However, the crack propagation path in engineering materials is rarely straight at the microstructural scale, and a significant amount of crack kinking is predicted by the model due to

σvM 0

100 MPa

(a) von Mises stress.

(b) Crack surface.

Fig. 5 Nucleation and evolution (left to right) of crack surfaces (bottom) and the corresponding von Mises stress through a section of the polycrystalline sample (top). Uniaxial tension in vertical direction is applied. The driving force for crack propagation is governed by the Griffith criterion

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Fig. 6 The polycrystalline volume element containing 20 randomly oriented grains, indicated by color, of size 100  100  100 μm discretized with 64  64  64 grid points used for the thermo-mechanics and damage-mechanics example

Table 3 Material parameters used to model the elastic and plastic behavior of the polycrystal shown in Fig. 6: elastic constants Cab, reference shear rate γ_0 , stress exponent n, initial and saturation resistance ξ0 and ξ1, hardening parameters a, h0, and hαβ. (a) Elasticity Parameter C11

Unit GPa

C12 C44

GPa GPa

Value 168.0 121.4 28.34

(b) Plasticity Parameter γ_0 n ξ0 ξ1 a h0 Coplanar hαβ Non-coplanar hαβ

Unit s1 MPa MPa MPa

Value 1  103 20 31 63 2.25 75 1 1.4

the heterogeneity of the stress field in the neighborhood of the crack tip. Crack kinking is observed towards regions of stress concentrations in neighboring grain boundary junctions.

7

Summary

The spectral method, complemented by the flexible material model implemented as DAMASK, is a powerful tool for the prediction of the micromechanical response in structural materials. This method has been extended to solve the time-dependent reaction-diffusion equation, which is representative of several field equations of interest such as thermal conduction, nonlocal damage, and more general phasefield equations, and provides the means for further investigation of coupled multiphysics systems. The solution scheme is demonstrated in the context of a polycrystalline material undergoing coupled thermo-damage-mechanical deformation using

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available crystal plasticity, damage, and thermal constitutive laws. Such a simulation framework opens the door for further investigation of important open issues related to the micromechanics of polycrystals, such as damage initiation, deformation localization, micro-texture development, or nucleation of phase transformations. The software used in this work is open-source and freely available at https:// damask.mpie.de/. Acknowledgments PS and FR acknowledge funding through SFB 761 Steel – ab initio by the Deutsche Forschungsgemeinschaft (DFG). MD acknowledges the funding of the TCMPrecipSteel project in the framework of the SPP 1713 Strong coupling of thermo-chemical and thermomechanical states in applied materials by the DFG.

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Interface Delamination Analysis of Dissimilar Materials: Application to Thermal Barrier Coatings Yutaka Kagawa, Makoto Tanaka, and Makoto Hasegawa

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Thermal Barrier Coatings and Delamination Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Needs for New Test for Mode II Delamination Toughness . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Barb Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Principle of Barb Pullout Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Steady State Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Buckling Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Phase Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Beam Theory Assesment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Practical Barb Test Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Stress Distribution: FEM Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Thermal Stress Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Stress Distribution Change upon Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Stress Distribution Along Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Finite Element Procedure II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Energy Release Rate: The Steady State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Phase Angle of Barb Test Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Y. Kagawa (*) Katayanagi Advanced Research Laboratories, Tokyo University of Technology, Tokyo, Japan e-mail: [email protected] M. Tanaka Materials Research and Development Laboratory, Japan Fine Ceramics Center, Nagoya, Japan e-mail: [email protected] M. Hasegawa Division of Systems Research, Faculty of Engineering, Yokohama National University, Yokohama, Japan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_83

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5 Application to EB-PVD TBC System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Effect of Interface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam Theory Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1402 1406 1409 1410 1410 1411

Abstract

Recent progress of gas turbine technologies requires protection of superalloy components from harsh use environments. To satisfy this requirement, surface protection coatings have been developed. The coatings allow protection of superalloy substrate from use environments, which includes temperature. Among the coatings, thermal barrier coatings (TBCs) have been developed to protect high temperature metal components. TBCs are usually composed of oxide ceramic topcoat layer, metal or intermetallic bond coat layer, and substrate. Currently, yttria stabilized zirconia (YSZ) is believed the best material for TBCs. YSZ-TBCs reduce temperature of a metal component because of its low thermal conductivity and cooling of metal substrate, and allows safety operation of superalloy components. List of Abbreviations

htbc hs htgo h 1, h 2 Etbc Es Etgo μ νtbc νs νtgo Pa σa σ tbc σs σ tgo σ Ttbc σ Ts σ Ttgo σ xx σ yy τxy x, y, z

Thickness of TBC layer Thickness of substrate Thickness of thermally grown oxide (TGO) layer Thickness of coating layer (1) and substrate (2) for analytical model Young’s modulus of TBC layer Young’s modulus of substrate Young’s modulus of TGO layer Shear modulus Poisson’s ratio of TBC layer Poisson’s ratio of substrate Poisson’s ratio of TGO layer Applied load Applied stress Stress of TBC layer Stress of substrate Stress of TGO layer Thermal expansion misfit stress of TBC layer Thermal expansion misfit stress of substrate Thermal expansion misfit stress of TGO layer Normal stress component Normal stress component Shear stress component Coordinates

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u ψ ψ ss ϵ etbc es etgo ea Ltbc Ls a abuckle Gss G G0 Γss Γc αtbc αs αtgo ΔT Γ Ω η λ(a/Ltbc)

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Displacement Phase angle at crack tip Steady state phase angle at crack tip Bi-material constant In plane strain of TBC layer In plane strain of substrate In plane strain of TGO layer Applied strain Length of TBC layer Length of substrate Interface crack length Crack length at buckling Steady stare strain energy release rate (calculated value) Strain energy release rate (calculated value) Strain energy release rate at a ffi 0 Steady stare strain energy release rate (measured value) Critical strain energy release rate (measured value) Thermal expansion coefficient of TBC layer Thermal expansion coefficient of substrate Thermal expansion coefficient of TGO layer Temperature difference Strain energy release rate (measured value) Young’s modulus ratio Ω = Etbc/Es Thickness ratio η = htbc/hs Nondimensional shape factor

1

Introduction

1.1

Thermal Barrier Coatings and Delamination Toughness

Recent progress of gas turbine technologies requires protection of superalloy components from harsh use environments. To satisfy this requirement, surface protection coatings have been developed. The coatings allow protection of superalloy substrate from use environments, which includes temperature. Among the coatings, thermal barrier coatings (TBCs) have been developed to protect high temperature metal components. TBCs are usually composed of oxide ceramic topcoat layer, metal or intermetallic bond coat layer, and substrate. Currently, yttria stabilized zirconia (YSZ) is believed the best material for TBCs. YSZ-TBCs reduce temperature of a metal component because of its low thermal conductivity and cooling of metal substrate, and allows safety operation of superalloy components. Thermal barrier coatings (TBCs) consisting of low thermal conductivity yttria YSZ are now employed in most gas turbine engines to protect superalloy components from high temperature severe environment. For high performance systems, the TBCs are manufactured by electron-beam physical vapor deposition (EB-PVD) or

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Fig. 1 Simple model for definition of interface bond strength

plasma spraying (PS) on superalloy components. Delamination of TBCs is one of the important undesirable events because durability of various coating structures relies on heavily the integrity of interfaces inherent in TBCs. However, once TBC delaminates from a substrate, surface of the substrate is exposed to high temperature environment and damage on the surface of substrate accelerates. Temperature increase of a superalloy component also increases; this increase leads creep deformation of a superalloy. Thus, delamination event of TBC layer from superalloy component should be avoided in order to keep performance of superalloys. Various levels of approaches have been done on delamination problem of TBC layer from substrate. Figure 1 shows examples of measuring methods for delamination resistance of TBC from superalloy substrate. For example, tensile or shear fracture stress of interface between TBC layer and superalloy substrate is simply calculated and used as engineering standards [1–4]. In this case, delamination resistance is given by σ or τ ¼

Pa S

(1)

where σ and τ, respectively, are tensile fracture stress and shear fracture stress, and Pa and S are the applied load and the cross-sectional area, respectively. The problem of using such a simple stress is no clear physical meaning of the measured fracture stress; especially specimen shape and test method dependence of the stress could only be used for comparison purpose of the same test results. Only standardized methods have been used for prediction of delamination resistance between coating and substrate. To avoid this complicated situation, stress distribution is incorporated in the strength analysis, but the effect of specimen shape and test method is still difficult to solve, however. Recently, fracture mechanics approach has been applied for evaluation of delamination toughness in TBCs. The method essentially applies interface fracture mechanics for onset of unstable delamination condition. The delamination toughness has a dimension of “strain energy release rate” and this is possible to use as a criterion for computer simulation, such as FEM, etc. In the analysis, delamination toughness is obtained as a steady state value of the strain energy release rate, Γss, at

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which the delamination takes place stably. The toughness is not a single parameter, but rather a function of the mode mixity, ψ, which is a measure of the relative amounts of mode II and mode I loading acting on the delamination tip [5–7]. For a two-dimensional interface crack problem, plane strain complex stress intensity factor K is given by K ¼ K 1 þ iK 2 ,

(2)

where K1 and K2 are tensile and shear stress intensity factors at the crack tip, respectively. ðK 1 þ iK 2 Þ ¼ jðK 1 þ iK 2 Þjeiψ :

(3)

Here, the phase angle, ψ, is defined as ψ ¼ tan

1


 K2 : K1

(4)

The plane strain energy release rate for a bi-material interface is given by 

  1  β2  2 Γ¼ K 1 þ K 22  E

(5)

where β is a Dundurs’ parameter related to mechanical properties of the constituent materials. E is given by
 1 1 1 1 ¼ þ , E 2 Etbc Es

(6)

and Ei ¼ 

Ei : 1  ν2i

(7)

In the present chapter, energy release rate Γ is used to predict delamination resistance of coating layer from substrate, and the delamination resistance is called as “delamination toughness.” Two kinds of delamination toughness are used as a criterion to predict onset of delamination; (1) critical strain energy release rate, Γc, and (2) steady state strain energy release rate Γss. Figure 2 shows schematic drawing of both the toughness as a function of delamination length of TBC layer from substrate. Here, it is assumed that the phase angle is constant. Practically, the toughness becomes a function of phase angle. The delamination toughness usually shows delamination length dependence, i.e., R-curve behavior. Assuming that only one delamination crack path exists, delamination toughness depends on both toughening term Γ(a, ψ) > 0 and phase angle ψ(a). The delamination toughness is given by

Fig. 2 Resistance curve of delamination toughness and definition of delamination toughness. L0 initial toughness, Γc critical toughness, Γss steady state toughness

Y. Kagawa et al.

Energy release rate,Γ(a)

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Γss

Γc

Γ0 y : const. Delamination length, a

Γða, ψ Þ ¼ ΔΓ ða, ψ Þ þ Γ0 ðψ Þ

(8)

where Γ0 is the intrinsic delamination toughness, which only depends on the phase angle ψ. Critical delamination toughness indicates onset of the unstable fracture and the meaning is similar to that used fracture toughness GIc (or KIc). In the present chapter, experimentally obtained energy release rate is expressed as “Γ” and energy release rate obtained from theory is expressed as “G.” It should be pointed out that energy release rate Γ can only be obtained from experiment and difficult to predict theoretically. The value of measured toughness depends on the test method and test condition; thus, standardization of the measurement procedure is needed for common discussion among toughness users. On the contrary, steady state toughness is independent of the delamination length and ideally identical to strain energy release rate of constitutes. However, prediction of the steady state toughness needs experiment. If strain energy released by coating layer is dominate factor and ΔΓ(a, ψ) = 0, the steady state energy release rate under plane stress condition is identical to the strain energy released by coating layer; ’ σ 2c =2Ec where σ c is the coating layer stress and Ec is the in plane Young’s modulus of coating layer. The effect of additional toughening, ΔΓ(a, ψ) originates increase of friction on the interface over friction between rough delaminated surfaces [8]. Effect of crack deflection and wedging, usually considered as toughening effect of monolithic ceramics [9, 10], are also important factors to change the additional toughness. Bending of delaminated coating layer also changes these effects [9]. In the simple case (Fig. 3), the condition for onset of unstable delamination of a coating from a substrate is given by Gða, ψ Þ ¼ Γc ða, ψ Þ

(9)

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Fig. 3 Analytical model for calculation of steady state delamination toughness

Here, G(a, ψ) is the strain energy release rate at a front of delaminated crack and the value is calculated from shape, dimension, and external loading conditions; Γc(a, ψ) is the critical strain energy release rate, usually refered as “delamination toughness” of dissimilar interface. Both the values depend on both crack length, a, and phase angle, ψ. Note that the strain energy release rate is calculated, but delamination toughness can not be obtained theoretically and needs experimental verification. To obtain steady state strain energy release rate, experimentally stable delamination behavior should be achieved. If the delamination occurs unstably, the toughness underestimates steady state delamination toughness.

1.2

Needs for New Test for Mode II Delamination Toughness

A key concern for the experimental technique is to accurately simulate the loading conditions that drive failure in TBCs, as crack growth resistance is highly dependent upon the mode-mixity, ψ, of loading condition. Additionally, the toughness of a TBC is also highly anisotropic, depending strongly on the coating process and the resulting microstructure. For failures caused by large compressive stresses that develop within the coating, delamination crack growth occurs under principally mode II loading – particularly for longer cracks with fully developed crack wake contact zones. Hence, the durability or lifetime is a direct function of mode II toughness of the coating, and measurement of mode II delamination toughness is very important. Figure 4 summarizes experimental technique and phase angle relation for a number of test approaches that have been developed for measuring key toughness parameters in thin film and coating systems [6, 9, 11–13, 20]. These test methodologies rely on inducing film delamination in a controlled manner, allowing for a quantitative measurement of adhesion or crack resistance. However, each approach has limitations, both experimentally and in application to a particular materials performance issue. Examples of test approaches that have been developed for ceramic coating systems include: (1) blister or bulge propagation methods [9] that rely on pressurization from the underside of the film or coating; (2) bend tests of

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90°

y ª 90°

Indentation test

Mode II condition y ª 60°

Phase angle, ψ

Barb test y ª 45°

Push-out test 45°

y ª 40 ~ 45°

4-point bending test y ª 0 ~ 30°

Tension test 0°

Mode I condition

Test method Fig. 4 Phase angle and test method for TBCs

notched multilayer beams [9, 12–14]; and (3) indentation techniques [10, 11, 15, 16] that depend upon developing plastic strains in a ductile substrate and transmitting those strains to the coating. Each of these test methodologies is appropriate over a particular range of mode-mixities. For example, blister growth methods typically result in opening mode crack growth, with phase angles in the range of 0∘ < ψ < 30∘. Bending methods are only appropriate for mixed-mode cracking measurements, where the applied loading is equally opening and shear (phase angles of 40∘ < ψ < 45∘). The impression methods, typified by conical and wedge indentation, result in phase angles that approach ψ = 90 as the delamination front moves away from the indenter [11]. However, other challenges and limitations exist in using these techniques for evaluation of TBC failure. For example, the blister tests are hampered by the need for special sample preparation approaches (back-thinning of wafers), whereas the indentation experiments, although much more appropriate for evaluation of shear mechanical properties, are highly dependent upon the plastic properties of the substrate [15] and are very sensitive to the anisotropic (porous) structure of the ceramic top coat [10]. In this work, we describe a test that obviates many of the issues with existing protocols for measuring the fracture resistance of TBCs and related multilayer systems. The methodology is based on a Barb geometry and originates from similar approaches applied in the study of fiber reinforced ceramic and/or metal matrix composites [17, 18]. The approach is particularly useful in capturing crack growth behavior under mode II loading conditions and is amenable to measurement of both short-crack and long-crack behavior. It allows direct measurement of the relative opening and shear displacements, and direct correlation of crack length, applied load, and microstructural aspects. The technique is illustrated by investigating the interfacial shear mechanical properties of an electron-beam

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physical vapor deposition (EB-PVD) TBC system. The barb test is demonstrated to have utility in investigating the delamination behavior of this coating.

2

The Barb Test

As discussed in the previous section, the interface delamination toughness Γ(ψ) depends on the interface crack phase angle, ψ; great attention has been paid to developing experimental tests capable of altering ψ. Figure 4 shows some typical test methods and phase angle range. A number of methods reported for thin film and coating systems [9, 10, 14–16] include, but limited to: (1) blister or bulge propagation method [9], (2) bend tests of notched multiplayer beams [9, 14], and (3) indentation tests, among others [10, 15, 16]. Each approach has limitations and merits, for example, blister growth methods typically result in opening mode crack grown, with phase angles in the range of ψ = 0–30 , while bending tests give a range of ψ = 40–45 . In an earlier work [19], an experiment named “barb test” was proposed for assessing the TBCs delamination toughness to obtain interface toughness close to mode II delamination tip coatings. Discussions on the mechanical conditions will be done in the following sections. A schematic illustration of the test fixture for barb pullout method is shown in Fig. 5. The fixture is designed to provide flat surfaces that transmit force to the cut edge of the TBC layer during relative translation of the metallic substrate. Given that the ceramic coating layers are generally very thin (~200 μm), ensuring proper contact with the test fixture is a primary concern of the test technique. In the barb test, sapphire plates that could be translated normal to the specimen surface were used as load transfer or support blocks. Prior to performing the barb test, the position of the sapphire supports were adjusted using the fine-thread adjustment pins in the loading fixtures. The sapphire was positioned to provide optimal load transfer to the cut edge of the coating, free of a frictional load transfer to the metallic substrate. The barb tests were performed in ambient air at room temperature using a screw-driven mechanical test system with a constant cross head displacement rate: typically 0.1 mm/min. The force-displacement response during the testing was digitized and continuously recorded using a digital memory scope. Number of specimen was usually selected based on the scattering of experimental results. After test completion, the specimens were examined using scanning electron microscopy (SEM) to investigate the crack path selection and general morphology of the fracture surfaces. Shape and dimensions of the barb test specimen are shown in Fig. 6. Test specimens with typical dimensions of 40 mm (length), 6 mm (width), and 3 mm (thickness) were cut from TBC specimens (EB-PVD or APS), using a conventional mechanical cutting procedure. The cut faces were ground through standard metallurgical procedures to (i) obtain specimens with the desired final thickness, (ii) introduce parallel surfaces, and (iii) eliminate mechanical flaws introduced during the cutting process. The TBC layers were then notched at a distance of 3 mm from the end of the specimens, to define the length over which crack growth occurs. The remaining TBC segments were carefully removed with a mechanical polishing tool.

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Fig. 5 Experimental arrangement of a “barb” test

a

b

Front View

Side View

3

Lc hs

hc

TBC

40

Fig. 6 Shape and dimensions of barb test specimen

Wc / Ws

Bond Coat

Substrate

6

6

(Unit: mm)

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Two specimen pieces prepared in this manner were then affixed back-to-back, using an alignment tool and epoxy adhesive, to form the final barb test specimens.

3

Principle of Barb Pullout Test

3.1

Steady State Toughness

It is assumed that delamination cracks extend symmetrically along both interfaces (Fig. 7) [20]. For most coating situations of practical interest, the ratio of the coating thickness to the thickness of the substrate is small: η = h1/h2  0.1. For the loads and moments shown in Fig. 7 (no shear force), the steady state energy release rate is [21, 22]:   1 P21 M21 1 P22 Gss ¼ þ 12 3 þ : (10a) 2E1 h1 2E2 h2 h1   For plane strain, Ei ¼ Ei = 1  ν2i (i = 1, 2) with Ei and νi being the Young’s modulus and Poisson ratio, respectively. In the absence of a bending moment (namely, uniform end load at the supports), Eq. 10a reduces to: Gss 

1 P21 1 P22 þ 2E1 h1 2E2 h2

(10b)

It will be shown below that the moment is sufficiently small (relative to P1h1) that Eq. 10b provides an accurate approximation.

Fig. 7 The “barb” test model showing the axial load, bending moment, and shear force The parameters and the coordinate system (x-y) with origin located at the mid-line of the delaminated coating. The deflection w(x) is defined as the vertical displacement from the original centerline. The subscripts 1 and 2 refer to the coating and substrate, respectively

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The load quantities have two contributions: one from the thermal expansion misfit and the other from the applied stress. Expansion misfit induces the stresses: ΔαΔT η=E2 þ 1=E1

(11a)

ΔαΔT 1=ηE1 þ 1=E2

(11b)

σ T1 ¼  σ T2 ¼

where ΔT represents the temperature change from processing to ambient, αi ¼ ð1 þ νi Þαi with αi (i = 1, 2) is the coefficient of thermal expansion and Δα ¼ α1  α2. The stresses associated with the loading are: σ a2 ¼ σ a ,

(12a)

σ a1 ¼ σ a =η

(12b)

Inserting the stresses into Eq. 10b gives the final solution for the energy release rate: Gss ¼

 2  2 h2 ΔαΔT h1 1 ΔαΔT σa þ þ  σa  η 1=E1 η þ 1=E2 η=E2 þ 1=E1 2E2 2E1

(13)

In the ensuing assessments, all energy release rates are normalized by this estimate of Gss.

3.2

Buckling Condition

When sufficiently large, the delamination becomes susceptible to buckling due to the end force, P1. By regarding the end of the delaminated region as a clamped plate and, moreover, that the supports are representative of another clamped end, the delamination length for the onset of buckling would be [23]: Lbuckle  2π h1

sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi ηE1 E1 h 1  2π 3σ a 3P1

(14a)

or kLbuckle  2π (14b) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k ¼ 3P1 =E1 h31 . The applicability of this clamped result will be assessed below using both beam theory and finite-element analysis.

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Phase Angle

The phase angle for the loading situation depicted in Fig. 7 can be approximated by the solution for a bilayer [21]: ψ  tan 1



λ sin ω  cos ðω þ γ Þ λ cos ω þ sin ðω þ γ Þ

 (15a)

where ω is an angle tabulated in Ref. [21] for different η. The parameter λ measures the loading combination such as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 1 þ ð4η þ 6η2 þ 3η3 Þ P1 h1 λ : 12 M1

(15b)

The quantity Σ = (1 þ D1)/(1  D1) is related to the first Dundurs’ parameter [24]: D1 ¼

μ 1 ð χ 2 þ 1Þ  μ 2 ð χ 1 þ 1Þ μ 1 ð χ 2 þ 1Þ þ μ 2 ð χ 1 þ 1Þ

(15c)

where μi is the shear modulus, with χ i = (3  4νi) for plane strain: for thin coatings, η < 0.1, such that γ  0. The symmetry boundary condition applicable to the barb test (Fig. 7) differs from the stress-free surface used to derive Eqs. 15a, 15b, and 15c. This difference results in an extra moment that causes the phase angle to deviate from this estimate, as discussed later. In order to estimate the phase angle using Eqs. 15a, 15b, and 15c, an analytic method capable of determining the moment, M1, is required. A beam theory approach having this objective is presented in the following section.

3.4

Beam Theory Assesment

The displacement of the coating can be assessed using a one-dimensional beam analogy. The transverse deflection at the center-line, w(x), is given by [25]: wðxÞ ¼ C1 sin kx þ C2 cos kx þ C3 x þ C4

(16)

The coefficients, Ci, are determined from the following boundary conditions. At the supports (x = 0): w(0) = 0 and w’(0) = 0, giving C3 = C4 = 0. At the end of delamination (x = L and y = h1/2), the rotation angle is [26]: θ ¼ w0 ðLÞ ¼ c0P

P1 M1 ðLÞ V 1 ðLÞ þ cθM þ cθV 2 E1 h 1 E1 h 1 E1 h 1

(17a)

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and the transverse deflection [23] wðLÞ ¼ cwP

P1 M1 ðLÞ V 1 ðLÞ þ cwM þ cwV E1 E 1 h1 E

(17b)

1

The coefficients ci must be ascertained numerically. Their assessment is presented in the APPENDIX and summarized in Fig. A1. Once determined, Eq. 16 can be solved for C1 and C2 upon making the substitutions MðLÞ ¼ E1 Iw00 ðLÞ and ðLÞ ¼ E1 Iw000 ðLÞ. The bending moment is reexpressed in terms of the loading parameter, kL, as MðLÞ=ðPh1 Þ ¼ C1 sin kL þ C2 cos kL

(18a)

V ðLÞ=P ¼ kh1 ðC1 cos kL  C2 sin kLÞ

(18b)

and the shear force as

Typical trends in the moment and shear force with load parameter are plotted in Fig. 8b for a case representative of a TBC on a metallic substrate fD1 ¼ 0:65, D2 ¼ 0:26 ðE1 =E2 ¼ 0:22Þ, h1 =h2 ¼ 0:05, Lss =h1 ¼ 20

cθp ¼ 1:75, cθM ¼ 4:91, cθV ¼ 1:38, cwp ¼ 0:55, cwM ¼ 1:1, and cwV ¼ 1:86 g: Here D2 is the second Dundurs’ parameter [24]. The results reveal a rapid change at kL ! 6.2, as the buckling condition is approached (kLbuckle = 2π), consistent with the foregoing clamped/clamped assertion for the onset of buckling. The shear force is essentially zero (except as kL ! kLbuckle). The moment is nonzero. It is positive at small kL (with M1(0)/P1h1  0.05), diminishes as kL increases, becomes negative, and then becomes large as kL ! kLbuckle. This moment is sufficiently small that it has minimal effect on the steady state energy release rate prior to buckling (Fig. 8a) because of the (quadratic) scaling: i  h Gss e P21 =E1 1 þ 12ðM1 =P1 h1 Þ2 . Yet, it is large enough to have an influence on the phase angle (Fig. 8c) because of the different (linear) scaling evident from Eqs. 15a, 15b, and 15c: λ~P1h1/M1. Note that this angle (ψ ss  65) deviates appreciably from mode II (ψ ss  90), revealing that this is not a shear test, contrary to an earlier assumption [19]. The fidelity of these beam theory results is assessed in the following section.

3.5

Finite Element Analysis

The commercial code ABAQUS (version 6.6) [27] was used to calculate the energy release rate and the phase angle using approaches elaborated elsewhere [28, 29]. A typical finite-element discretization is shown in Fig. 9. Second-order quadrilateral iso-parametric elements with eight nodes and nine integration points were used. The

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

Fig. 8 Results obtained using beam theory with selected numerical results superimposed. In all cases, the abscissa is the load parameter kL. (a) The normalized energy release rate G/Gss. (b) The normalized crack-tip moment and shear force. (c) The phase angle

1387

(kL)buckle

a Energy Release Rate, G/Gss

44

Numerical Beam Theory

6.0

4.0

2.0 2π 0

2

3

Moment Shear Force

b 0.2

Phase Angle, Ψ (degrees)

6

V1(L)/P1

0 –0.2

M1(L) P1h1

–0.4 –0.6 –0.8 –1.0

c

4 5 Load Parameter, kL

2

3

70 60 50 40 30 20 10 0 –10

4 5 Load Parameter, kL

6

Numberical

Beam Theory

2

3

5 4 Load Parameter, kL

6

mesh details at the interface crack tip and at the support are indicated in Fig. 9b and c, respectively. A uniform tensile stress, σ a, is applied to the substrate and symmetry is imposed. At the supports (x = 0), clamped boundary conditions are used: ux = 0, uy = 0. at h2/2  y  h2/2. Calculations have been conducted for kL kLbuckle using typical dimensions (L/h1= 2, 10, 20, 30 for η= 0.05 and L/h1= 1, 5, 10, 15 for η = 0.1) for a case representative of TBCs on superalloy substrates: coating

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Fig. 9 Finite element analysis model for barb test: (a) FEA mesh, boundary conditions and definition of coordinate system, (b) detail of A, (c) detail of crack tip and a polar coordinate system

modulus, E1= 44, 80, 120, 200 GPa and substrate modulus, E2= 200 GPa. When the buckling load is approached, geometric nonlinearity is activated to take into account the influence of the relatively large deformations on the stress and deformation fields at the crack tip. The results are plotted in Fig. 10.

4

Practical Barb Test Analysis

Analytical half model of the barb test is shown in Fig. 11. A thickness ratio, η, is defined for later use as η¼

htbc hs

(19)

where, htbc and hs are the thickness of the TBC layer and substrate, respectively. The thickness ratio, η, is varied as η = 1/30, 1/15, 1/10, 2/15, and 1/6 which corresponding to htbc = 100, 200, 300, 400, 500 μm for a fixed hs = 3 mm, respectively. The length of TBC layer, Ltbc, is set to 3 mm. This length was chosen according to a set of preliminary experiments [19]. It should be noted that no buckling condition and no compressive damage in the practical test are experimentally proved [19]. A delamination crack is likely to be initiated from the lower side edge close to the support; therefore, a lower-side delamination crack with length a is assumed and varied in the range of 0 < a/Ltbc < 0.7. The Young’s modulus of the substrate is fixed at Es= 200 GPa [9, 10, 14–16, 19]. In order to examine the effect of Young’s modulus ratio, a parameter, Ω, is defined as,

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

Fig. 10 Numerical results for (a) the normalized energy release rates G/Gss and (b) the phase angle as a function of crack length, L/h1, for small kL

a Energy Release Rate, G/Gss

44

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1.2 1.0 0.8 0.6 0.4 0.2 0

0

4

8

12

16

20

16

20

Phase Angle, Ψ (degrees)

b 100 E1/E2 = 0.2

80 60

E1/E2 = 1

40 20 b 0

0

Ω¼

4

Etbc Es

8 12 Crack Length, L/h1

(20)

In this simulation, Ω is changed from 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0. It was reported that in-plane Young’s modulus of EB-PVD TBC is 40–60 GPa (Ω = 0.2–0.3) and APS-TBC is 30–200 GPa (Ω = 0.15–1.0) [30, 31]; therefore, the range 0.1  Ω  1.0 covers these Young’s moduli of broad TBC systems. Poisson’s ratio of the TBC layer and substrate are assumed to be νtbc = 0.2 and νs = 0.3, respectively [9, 10, 14–16, 19]. The materials are taken to be isotropic and completely linear elastic undergoing plane strain deformation.

4.1

Stress Distribution: FEM Analysis

The commercial finite element analysis program MARCMENTAT [32] was used to analyze a half of the barb test model (Fig. 12), where the x–y coordinate system is also defined in the same figure. Definition of an internal and a free edge is indicated

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Fig. 11 Analytical model for “barb test”

in Fig. 12. The geometry of the model is Ltbc/htbc = 40 and Ltbc/hs = 1/15, to simulate the previous experimental dimensions [19]. The finite element model consists of arbitrary quadrilateral elements with four nodes and four integration points. Element sizes were selected by continuously refining the mesh until approximate convergence of the numerical solution. The minimum mesh size is 0.0005htbc, which corresponds to an actual dimension of  0.1 μm (Fig. 12). A finer mesh model with about 4 times more elements gave a maximum stress difference of less than 3% at x = 0 and y/htbc = 104. The current mesh size was believed to be fine enough to compute the stress gradient near at the TBC ends with sufficiently high accuracy. Here, Es = 200 GPa [5, 15, 19, 33, 34, 43], and the thermal expansion coefficients of a Ni-based superalloy substrate and a Y2O3– ZrO2 TBC layer are taken to be αs = 15 106 K1 and αtbc = 11 105 K1, respectively [30, 31]. The modulus ratio, Ω, is changed to 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0 to take into account various thermal barrier coatings, e.g., in-plane Young’s modulus of EB-PVD TBC is reported to be 40–60 GPa [30, 31] and APS-TBC is reported to be 30–200 GPa [31]. Poisson’s ratios of the TBC and substrate are assumed to be νtbc = 0.2 and νs = 0.3, respectively [5, 15, 19, 33, 34, 43]. In addition, the TBC layer and substrate are assumed to be completely linear elastic materials undergoing plane strain deformation. The computation consists of two steps: (i) calculation of thermally induced stress in the entire specimen, and (ii) calculation of stress distribution under an applied

44

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

a

b

Y

8

Substrate

Substrate

1391

TBC Crack tip

TBC

8 a X A 8

Interface Crack

c Substrate

TBC

8

8

8

σa

Reaction Force

Fig. 12 FEA model for barb test: (a) FEA mesh, boundary conditions and definition of coordinate system, (b) detail of crack tip and a polar coordinate system, (c) detail of A

stress σ a = 90 MPa. This process is the same as reported previously by one of the authors [35]. To determine stresses at the interface, the stresses of TBC layer nodes nearest the interface were used [32]; detailed procedure of the definition of interfacial shear stress was reported elsewhere [35]. In the first step, thermal stresses are calculated under a reference temperature difference ΔT = 1000  C. This ΔT value was used by assuming that a stress free sample at an elevated temperature was cooled down uniformly to ambient temperature. The temperature-dependent material properties such as thermal expansion coefficients are known to influence magnitude of the thermal stress; however, such temperature-dependent behavior is ignored. The symmetrical axis (x =  hs) of the model is fixed and all other nodes are allowed to move. This boundary condition means that the specimen could change its shape according to thermal mismatches. In the next step, the boundary conditions are changed to reproduce actual support conditions (Fig. 11). The TBC layer is set to U = 0, V = 0 at 0.005  x/htbc  1, and y = 0. Here, U and V are displacements along the x and y axes, respectively. This condition corresponds to a high frictional contact between the TBC layer and support, so that the contact area could not move. The other extreme situation is frictionless at the support, that is, V = 0 while U is free to move at 0.005  x/htbc  1 and y = 0. The difference between the two cases was examined under a typical material case of Ω = 0.2 with an arbitrary applied load. It was confirmed that the shear stress was nearly the same, while fairly large

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a

b TBC

1/9 2/27 1/27 0 –1/27 –2/27 –1/9

s xx /s Ts 1 2/3 1/3 0 –1/3 –2/3 –1

s yy /s cT

Substrate

c TBC

0 –1/6 –1/3 –1/2 –2/3 –5/6 –1

s yy /s sT 3 7/3 5/3 1 1/3 –1/3 –1

s yy /s sT

Substrate

TBC

txy /s sT 2 4/3 2/3 Substrate 0 –2/3 –4/3 –2

txy /s sT 1/6 1/9 1/18 0 –1/18 –1/9 –1/6

Fig. 13 Thermal stress distribution of entire specimen: (a) stress contour of σ xx of substrate and BC layer, (b) σ yy of substrate and TBC layer, (c) τxy of substrate and TBC layer Fig. 14 Definition of shear stress signs

Substrate

TBC

differences in the normal stresses occurred only within the range 0 < y/htbc < 0.02, e.g., the normal stress for the high friction case is about two times larger than that of frictionless case at y/htbc  0.01. Hereafter, only the high friction boundary condition is considered.

4.2

Thermal Stress Distribution

Figure 13 shows typical thermal stress distributions of the entire specimen with Young’s modulus ratio Ω = 0.2. This Ω value is very close to the case of EB-PVD TBCs. The stresses σ xx, σ yy, and τxy indicate normal and shear components of the defined coordinates, respectively. The sign of τxy is defined in Fig. 14. These stresses are normalized by closed-form thermal stresses σ Ts and σ Ttbc, in the substrate and TBC layer, respectively. The thermal stress component σ xx (Fig. 13a) in both TBC layer and substrate appears only in the coated region (y  0), and it does not exist in the uncoated region (x < 0, y < 0). This seems natural because there is no constraint in the range of x < 0 and y < 0. However, thermal stresses are not completely symmetric with respect to the free and internal edges. The distribution of the normal stress, σ xx, in the substrate is almost symmetrical with respect to y  Ltbc / 2. Near the

44

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

1393

internal edge, σ xx stress concentration occurs in the substrate, whereas only slight stress concentration in the σ xx component appears at the edge of TBC layer. The σ yy stress component (Fig. 13b) shows different distribution from that of σ xx component. In-plane compressive stress appears in the TBC layer and substrate vice versa. This seems quite reasonable because the relationship of the thermal expansion coefficient between TBC layer (αtbc) and substrate (αs) satisfies αtbc < αs. The stress component σ yy in the substrate increases when approaching the interface, and the value approaches to  σ Ts at a half of the width separated from the interface. On the other hand, the σ yy component shows a uniform distribution with a value of  σ Ttbc throughout the entire TBC layer. Figure 13c shows the shear thermal stress distribution, τxy, in the substrate and TBC layer. Near the internal edges (y = 0), τxy stress concentration appears both in the substrate and the TBC layer; however, the sign of the shear stress is opposite at the free and internal edges (y = 0 and y = Ltbc, respectively). With separation from both the edges, shear components, τxy, completely disappear, this result suggests that the thermal shear stress concentration is limited only near the edge regions of the specimen.

4.3

Stress Distribution Change upon Loading

The stress changes within the barb test specimen upon loading are presented in Fig. 15. The normal stress distribution σ xx is only slightly changed after loading except near both the internal and the free edges (Fig. 15a). On the contrary, the in-plane compressive stress in TBC layer, σ yy, at the internal edge increases because both initial thermal and in-plane stress form by applied stresses to TBC layer are in compression (Fig. 15b). Tensile compressive stress in σ yy component appears in the substrate near the internal edge of TBC layer. Another region of the substrate in y direction is in tension (σ yy > 0); this result seems quite reasonable because pulling

Fig. 15 Stress distribution after mechanical loading: (a) stress contour of σ xx of substrate and TBC layer, (b) σ yy of substrate and TBC layer, (c) τxy of substrate and TBC layer

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load is applied to the substrate. Upon loading, shear stress concentration becomes significant especially near internal edge (y ~ 0) and it shows a positive stress concentration at the internal edge (y = 0), which is opposite with thermal stress distribution (c.f. Fig. 15c). However, the shear component in TBC layer shows a negative stress concentration at the internal edge (y = 0), and except for a small region near the edge, the stress distribution shows only slight change compared to the thermally induced shear stress distribution. It should be noted that the direction of the thermal shear stress is opposite to that caused by the applied loading. This behavior should be considered for analysis of stress distribution in various coating systems, where thermal misfit stress is effective.

4.4

Stress Distribution Along Interface

Figure 16 show the interfacial normal (Fig. 16a) and shear (Fig. 16b) stress distributions along interface between TBC layer and substrate with Young’s modulus ratio of Ω = 0.2 for pure thermal stress and with additional applied load. The stresses are normalized by ησ a(η = htbc/hs). Prior to applying the mechanical loading, both normal and shear stresses show singular behavior near the two edges (y/Ltbc= 0 and 1), but quickly approach to zero at positions away from the two edges. Here, the singular behavior means that the stresses reach infinite mathematically as commonly defined. When the pullout loading σ a is imposed, stress fields at the free edge (y/ Ltbc = 1) remain unchanged. The total normal stress near the internal edge (y/Ltbc = 0) moves to larger compressive stress, but the shear stress changes signs and are singular with an opposite direction. It is well known that interfacial shear stress singularity appears at internal edge of dissimilar bonded materials by thermal mismatch strain [35, 36]. In the present case, the effect of the modulus ratio Ω on the singularity is evaluated in the area y/Ltbc < 0.2, where strong singularity appears in the specimen. Here, it is assumed that the singularity is expressed as [36, 37].

τTxy ðr Þ ¼ τTxy r λðΩÞ ,

(21)

where λ(Ω) is a nondimensional constant parameter which depends on the Young’s modulus ratio, Ω. Figure 17 is plot of normalized thermal stress components τxy(r) at interface versus distance from an internal edge, r (edge is taken as x = 0). The normalization is done using in-plane stress of TBC layer given by σ Ttbc ¼ 

ΔαΔT : η=Es þ 1=Etbc

(22)

It is evident that the shear stress component, τxy(r), has nearly linear-logarithmic relation dependent on the distance r: this result means that rλ(Ω) singularity clearly appears in the interfacial shear stress component. Assuming that the slope of the relation in Fig. 17 is given by a straight line, order of the singularity, λ, is given by

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

a

1395

0.1 Thermal stress 0 Total stress -0.1

s ixx /hsa

Fig. 16 Distributions of interfacial stress caused by thermal mismatch and mechanical loading for Young’s modulus ratio of Ω = 0.2: (a) normal stress component, σ xx and (b) shear stress component, τxy

Noralized interfacial normal stress,

44

-0.2 -0.3 -0.4 Ω =0.2 -0.5 0

0.2

0.4

0.6

0.8

1

Normalized distance from internal edge, y/Ltbc 3 Ω =0.2

2.5 2

t ixy/hsa

Normalized interfacial shear stress,

b

1.5 1 0.5

Total stress

0

Thermal stress -0.5

0

0.2

0.4

0.6

0.8

1

Normalized distance from internal edge, y/Ltbc

slope of the line. The order of singularity, λ(Ω), is obtained using least squares fitting of the curve. The singularity order λ(Ω) slightly increases with the increase of Young’s modulus ratio Ω. The curve fitting of Fig. 18 for 0.1  Ω  1.0 under a plane strain condition yields, λðΩÞ ¼ 0:25 þ 0:36Ω  0:17Ω2 þ OðΩÞ:

(23)

where O(Ω) is the higher order term and could be assumed where O(Ω) is the higher order term and could be assumed nearly zero (O(Ω) ffi 0). Numerical fitting curve is also shown in Fig. 18 by a solid line. This equation is useful to obtain the initial shear stress distribution near an internal edge of the barb test specimen. After mechanical loading, the interfacial normal stress near the push end changes from tension to compression. However, the changed region is only in the range of 0 < y/Ltbc < 0.2,

0.8 0.6 Log(t ixy /sTrbc)

Fig. 17 Stress singularity at interface for internal edge of specimen

Y. Kagawa et al.

Normalized interfacial shear stress,

1396

Ω=0.2

0.4 0.2

0.6

0

-0.2 -0.4 -4

-3.5

-3

-2.5

-2

-1.5

-1

Logarithmic distance from internal edge, Logr

Fig. 18 Relationship between singularity order, λ, and Young’s modulus ratio, Ω

0.5

Singularity order, λ

0.4

0.3

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1

Young’s modulus ratio, Ω

and the stress in the free edge region is not affected by the mechanical loading. The shear stress originates by both thermal stress and mechanical loading, which only exists near a free edge and the value is zero in the middle part of the TBC coated region (0.2 < y/Ltbc 0.8). This result seems natural became interface shear stress by applied load is concentrated only at the internal edge and the direction of shear stress at the free edge is opposite. The magnitude of stress concentration at the internal edge depends on the modulus ratio, Ω. Using superimpose principle of stress, the interfacial shear stress is expressed as τxy ðyÞ ¼ τTxy ðyÞ þ τaxy ðyÞ,

(24)

where, τxy( y) is the shear stress by thermal misfit and τaxy ðyÞ is the shear stress by the applied load.

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

44

4.5

1397

Finite Element Procedure II

A crack face displacement extrapolation method [38] was used for computing fracture mechanics parameters. The displacements at a distance r behind the delamination tip are given by Δux þ iΔuy ¼

pffiffiffiffiffiffiffiffiffiffi 2½ð1  νtbc Þ=μtbc þ ð1  νs Þ=μs r=2π r ie Lie K, ð1 þ 2ieÞcoshðπeÞ

(25)

where K = k1 + ik2 with k1 and k2 being the mode I and mode II stress intensity factors of an interface crack tip, respectively. Δux and Δuy are the crack surface displacement in the direction of x and y, respectively, given by Δux = ux(r, π)  ux(r  π), and Δuy = uy(r, π)  uy(r  π). L is a length quantity taken to be delamination length, a, in this work, and μ is the shear modulus, subscripts “tbc” and “s” refer to TBC layer and substrate, respectively. The bimaterial constant, e, for the plane strain condition is defined as [39] e¼

1 ð3  4νtbc Þ=μtbc þ 1=μs ln , 2π ð3  4νs Þ=μs þ 1=μtbc

(26)

The energy release rate, G, is related to the stress intensity factors as follows [38]:

G¼

½ð1  νs Þ=μs þ ð1  νtbc Þ=μtbc KK 4 cos h2 ðπeÞ

(27)

where K is the conjugation of the complex intensity factor K. In the ensuing numerical assessment, all the energy release rates are normalized by steady state energy release rate, Gss, which is given by 

2
 σ a þ σ Ts hs 1 1þ Gss ¼ : 2Es ηΩ

(28)

The phase angle, ψ, which gives mode mixity, is defined as ψ ¼ arctan


 k2 k1

(29)

ψ = 0 represents a pure opening mode I and ψ = 90 refers to a pure shear mode II crack. In calculating ψ, the choice of reference length, L, for Eq. 25 is somewhat arbitrary. The freedom in the choice of L in the definition of phase angle is a consequence of the simple transformation rule from one choice to another. Assuming that phase angle ψ 1 associated with L1, and ψ 2 with L2. From the definition of phase angle, the difference between the two phase angles can readily show [38].

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ψ 2 ¼ ψ 1 þ elnðL2 =L1 Þ,

(30)

where, e is the bimaterial constant as defined in Eq. 26. Thus, it is a simple matter to transform from one choice to another. Besides, the bimaterial constant, e, for a typical coating system is sufficiently small (e  0); the shift will generally be negligible even for changes of L of several orders of magnitude.

4.6

Energy Release Rate: The Steady State Analysis

FEM analysis is used to obtain stress distribution during bend tests. Assuming that the delamination crack is long enough and that all the strain energy stored in the TBC layer and the substrate is used to create new surfaces. The steady state energy release rate, Gss, is given by [20] 

2
 σ a þ σ Ts hs 1 Gss ¼ 1þ 2Es ηΩ

(31)

where σ Ts is the thermal expansion misfit stress in substrate and σ a the stress  applied  to the substrate with Ω ¼ Ω for the plane stress condition and Ω ¼ Ω 1  ν2s =   1  ν2tbc for the plane strain condition. σ Ts is assigned zero without loss of generality since the contribution of the thermal stress to the energy release rate is merged to the loading stress when energy release rates are normalized by Gss in the following the previous section (Sect. 3.1), prior to coating buckling, the moment at the support has negligible effect on the energy release rate relative to the axial loading. Moreover, it should be noted that both the lower and upper interface edge effects may cause energy release rates to deviate from the steady state value since the specimen geometry does not meet the steady state condition of a htbc and a Ltbc. The steady state phase angle can be approximated as [20] ψ ss  tan 1



 κ sin ω  cos ω , κ cos ω þ sin ω

(32)

where ω is an angle tabulated in Ref. [21] (for different η). The parameter κ measures the loading combinations of the moment M with regards to the mid-point of the support line ,and the applied loading, σ a, is given by [20] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Ωð4η þ 6η2 þ 3η3 Þ σ a h2tbc κ : 12 ηM

(33)

Figure 19a and b show the relationship between normalized energy release rate, G/Gss, and Young’s modulus ratio, Ω. The delamination lengths are in the range of 0 < a/Ltbc < 0.7. It shows that larger thickness ratio, η, always gives a larger G/Gss value, when the normalized delamination length, a/Ltbc, is fixed at 1/3 as shown in

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

Fig. 19 The relationship between the normalized energy release rate G/Gss and Young’s modulus ratio Ω when (a) the thickness ratio η and (b) the normalized crack length a/Ltbc is changed

1.6 Normalized Energy Release Rate, G /Gss

44

a

1.5

1399

h = 1/6

a/Ltbc =1/3

1.4

2/15

1.3

1/10

1.2

1/15

1.1

1/30

1.0 1.5

b

1.4

h = 1/15

a/Ltbc =2/3

1.3

7/12

1.2

1/2 5/12 1/4

1.1 1.0

0

0.8 1 0.2 0.4 0.6 Young’s Modulus Ratio, Etbc /Es

1.2

Fig. 19a. With a fixed η = 1/15, delamination lengths of a/Ltbc = 1/4, 5/12, 1/2, 7/12, 2/3 are chosen for comparison in Fig. 19b. The increasing tendency of G/Gss becomes remarkable with the increase of a/Ltbc. Both graphs show that the energy release rate is always larger than the steady state value, Gss. Moreover, the normalized strain energy release rate, G/Gss, increases almost linearly with the increase of Ω. The G/Gss could be expressed as G=Gss  λða=Ltbc ÞΩ þ G0 ða=Ltbc Þ=Gss ,

(34)

where λ is a nondimensional function, which depends on the crack length. Figure 20 shows plots of λ(a/Ltbc) versus a/Ltbc. The parameter, λ(a/Ltbc), increases with the increase of the delamination length, a/Ltbc, while G0(a/Ltbc)/Gss  1 is constant. Therefore, the relation is expressed as G=Gss  λða=Ltbc ÞΩ þ 1:ð0:1 < a=Ltbc < 0:7Þ

(35)

From Eq. 31, the steady state energy release rate, Gss, is given as a function of the applied stress, σ a; thus, Eq. 35 becomes G



 σ 2a hs 1 a 1þ Ω þ 1 :ð0:1 < a=Ltbc < 0:7Þ λ Ltbc 2Es ηΩ

(36)

By using the least square method to fit the results as shown in Fig. 20, the parameter, λ(a/Ltbc), can be expressed as a polynomial equation for each thickness ratio as,

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Fig. 20 The nondimensional shape factor of the barb test specimen

1.25

h =1/6

1

2/15

0.75

1/10

0.5 1/15 0.25 0 0

1/30 0.1

0.5 0.6 0.2 0.3 0.4 Normalized Crack Length, a/Ltbc

0.7

Table 1 The constant, ci, in Eq. 37 η 1/30 1/15 1/10 2/15 1/60.0

c0 0.0052 0.0134 0.0015 0.0368 0.0876

c1 0.2916 0.3462 1.1681 1.3582 1.4390

c2 0.7232 2.1003 2.8857 3.2254 3.0612

c3 0.9910 2.7334 4.2074 5.3140 5.8041

λða=Ltbc Þ  c0 þ c1 ða=Ltbc Þ þ c2 ða=Ltbc Þ2 þ c3 ða=Ltbc Þ3 ð0:1 < a=Ltbc < 0:7Þ

(37)

where, the constants ci (i = 0, 1, 2, 3) for each thickness ratios are given in Table 1.

4.7

Phase Angle of Barb Test Specimen

The effects of the modulus ratio on the phase angle are demonstrated through deformation maps (Fig. 21a and b). Upon loading, the delaminated TBC layer shows bending deformation, and the delaminated crack tip is subjected to both opening and shearing deformation. The deformation is obviously larger with a larger modulus ratio, Ω. Figure 22a shows the trends of the phase angle, ψ, with respect to the crack length for various thickness ratios. Here, the modulus ratio is fixed at Ω=0.2. The phase angle gradually decreases from 90 (pure mode II) to the mixed mode with increasing of the normalized crack length. For a small delamination length, the mode II state exists in all thickness ratios. The phase angles decrease with increasing of modulus ratios (Fig. 22b), and for smaller Ω, pure mode II state exists in a range of small delamination lengths. Note that the phase angle of Fig. 22

44

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

1401

Fig. 21 Deformation of the delaminated region for (a) Ω = 0.2 and (b) Ω = 0.6

90

85

Phase angle, ψ (deg.)

Fig. 22 The relationship between phase angle ψ and the normalized crack length a/ Ltbc when (a) the thickness ratio η and (b) Young’s modulus ratio Ω is changed

h =1/30 1/15 1/10 2/15 1/6

80

a Ω=0.2 75 90 85

Ω=0.1

80

0.2

75

0.4 0.6 0.8 1.0

70 65 60

b h =1/15 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized crack length, a/Lc

corresponds to an experimental geometric ratio Ltbc /hs = 1 [19]. Steady state delamination conditions (a Ltbc and a htbc) are not likely to be satisfied for this geometric ratio. It may be of interest to see how the phase angle in this case differs from that of the steady state situation. Selected calculation using Eq. 32 for η = 1/15 with various Young’s modulus ratios is presented in Fig. 23. The moment, M, needed for computing ψ ss by Eq. 32 was calculated from distributed reaction axial and shear forces at the support output directly from the finite element analysis.

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Fig. 23 The phase angle for η = 1/15 with various Young’s modulus ratios Ω

90 h=1/15

Phase angle, ψ (deg.)

85

Ω=0.1 0.2

80

0.4 0.6

75 70 0.8

65

1.0 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized crack length, a/Lc

In the delamination length range a /Ltbc > 0.2, the phase angles by the finite element solution (Fig. 22b) and Eq. 32 (Fig. 23) show similar trends associated with Ω, while much larger differences in phase angles between Fig. 22b and Fig. 23 are seen in the delamination range a/Ltbc > 0.2. Note that even at a/Ltbc  0.7, the finite element results do not agree with the steady state value ψ ss  65 [20]. This is attributed to the edge effects caused by the short geometric ratio Ltbc/hs = 1.

5

Application to EB-PVD TBC System

To demonstrate the utility of the barb test technique, an EB-PVD TBC system was investigated in this study. The EB-PVD TBC system was supplied by the JFCC (Japan) of ~200 μm. The substrate was a MA738LC nickel-based superalloy, overcoated with a NiCoCrAlY bond coat layer, and sectioned into pieces averaging 55 mm 20 mm 2.5 mm in size. The bond coat was deposited by plasma spray deposition to a thickness of 100–150 μm, and had an overall composition of 32 wt.% Ni, 21 wt.% Cr, 8 wt.% Al, 0.5 wt.% Y, with the balance Co. The bond coat surface was then polished before the TBC deposition. Figure 24a shows a typical columnar structure observed in the EB-PVD TBC layer. A thermally grown oxide (TGO) layer, 0.5 μm thick, is present between the TBC layer and the bond coat after deposition of TBC layer. High-magnification images revealed the undulated nature of the bond coat/TGO interface, as shown in Fig. 24b (indicated by arrows). A schematic of the barb test specimen is the same as Fig. 6. Test specimens with dimensions of 40 mm 6 mm 3 mm were cut from the EB-PVD-processed TBC specimens, using a conventional mechanical cutting procedure. The cut faces were ground through standard metallurgical procedures to (1) obtain specimens with the desired final thickness, (2) introduce parallel surfaces, and (3) eliminate mechanical flaws introduced during the cutting process. The TBC layers were then notched at a

44

Interface Delamination Analysis of Dissimilar Materials: Application to. . .

Fig. 24 A typical columnar structure of the EB-PVD TBC (a), and a high-magnification image (b) showing the undulated nature of the bond coat/TGO interface

1403

a

TBC

TGO

Bond Coat

50 mm

b

5 mm distance of 3 mm from the end of the specimens, to define the length over which crack growth occurs. The remaining TBC segments were carefully removed with a mechanical polishing tool. Two specimen pieces prepared in this manner were then affixed back-to-back, using an alignment tool and epoxy adhesive, to form the final barb test specimens. The barb tests were performed in ambient air at room temperature using a screw-driven mechanical test system. Test was performed with a constant cross head displacement rate of 0.1 mm/min. The force-displacement response during the testing was digitized and continuously recorded using a digital memory scope. Five specimens were tested. After test completion, the specimens were examined using scanning electron microscopy (SEM) to investigate the crack path selection and general morphology of the fracture surfaces. A typical plot of response force versus crosshead displacement (P–u) relation, obtained from barb testing of the EB-PVD TBC specimen, is shown in Fig. 25. The force increases nearly linearly up to a maximum force, with the exception of an initial nonlinear segment corresponding to compliance associated with the test

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Load, P (N)

400

Pmax

300

200

100

0 0

50

100 150 Displacement, u (μm)

200

Fig. 25 A typical force-displacement curve obtained during the barb test for the EB-PVD TBC system

fixture and specimen alignment in the self-aligning mechanisms. The maximum force, Pmax to the point at which the applied force is sufficient to initiate delamination of the TBC layer from the underlying substrate. Direct observations show no visible delamination crack growth prior to the peak force point. Three or four instantaneous force drops are typically observed after the maximum force. This behavior suggests that propagation of the interfacial delamination occurs incrementally. Direct observation shows that the coatings on both sides of the sample remain intact, adhered to the substrate over part of the contact area after the initial force drops. Complete decohesion coincides with the final drop to zero response force. The measured force drop corresponds to direct observation of complete delamination. SEM micrographs of the underside of the delaminated TBC layer in the EB-PVD TBC system: (a) a low-magnification image showing a surface of TBC with patches of attached TGO, and (b) a high-magnification image showing micropores in the attached TGO as well as columnar structure of the TBC layer. Based on these observations, the delamination behavior is schematically illustrated in Fig. 26. To quantify the effective coating/interface toughness, the delamination strain energy release rate should be determined. For the test configuration used here, the thickness of the substrate larger than the thickness of the TBC dictates that the failure occurs more or less instantaneously. Under such conditions, an estimate of the strain energy release rate (1) that all of the strain energy stored in the TBC layer and the substrate prior to initiating delamination is released in developing new surface area during crack propagation, and (2) that the delamination is uniform along the crack tip without buckling. Neglecting the anisotropic nature of the TBC elastic properties, the strain energy release rate is approximated as [22]:

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Interface Delamination Analysis of Dissimilar Materials: Application to. . .

Fig. 26 Schematic illustrations showing the delamination behavior of a TBC layer during the barb test for the EB-PVD TBC system: (a) as-coated and (b) posttested

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a As-Coated TBCs

TBC

Bond Coat TGO Substrate

b

Post-Tested TBCs

TBC

TBC

TGO

Bond Coat

Substrate

Gi 

hs σ 2s htbc σ 2tbc þ Es Etbc

(38)

where σ s and σ tbc are the uniaxial stress in the substrate and TBC layer, respectively, Es and Etbc are Young’s modulus of the substrate and the TBC layer, respectively. Young’s modulus is taken to be Es = 200 GPa for the substrate [40] and Etbc = 44 GPa for the EB-PVD TBC [15]. From force balance in the specimen, the in-plane stress in the coatings, due to the applied force, can be calculated as: σ tbc ¼

P 2htbc wtbc

(39)

Substituting Eq. 39 into Eq. 38 yield
 P2 1 1 Gi  þ 4 Es hs w2s Etbc htbc w2tbc

(40)

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Because the maximum applied load obtained in the P–u curve is believed to correspond to the onset of the TBC layer delamination from the substrate, the strain energy release rate for delamination cracking of the TBC layer from the substrate is determined using Eq. 40 with the maximum force, instantaneous force drop behavior was neglected in determining the energy release rate using Eq. 40 although typical three or four load drops were observed after the maximum load (Fig. 25). Thus, the energy release rate determined with the peak load for initial delamination is the critical energy delamination criterion, i.e., toughness of interface. For the EB-PVD TBC system used, it is found that the measured critical strain energy release rates ranged from Eq. 40, with an average value of ~70 J/m2 present experimental method; these values are roughly the mode II strain energy release rates (ψ = 90 ) for either the TBC/bond coat interface or the TBC layer-dependent upon the observed fracture path selection. The mode II toughness of EB-PVD TBC has been measured at ambient temperature by both indentation testing [15] and wedge impression testing [10], and found to be of the order of C mode II delaminations occurred predominantly at the TGO/ bond coat interface [10, 15]. The value obtained in the present measurements is higher than that mentioned above for the mode II toughness reported elsewhere [10, 15]. It is known that the critical strain energy release rate is linked to the properties of the bond coat, TGO, and TBC as well as their interface properties such as the surface roughness and composition of the bond coat, thickness of TGO, residual stress at interface, and physical properties of TBC. In particular, the residual stress at the interface and the Young’s moduli used in determining the critical strain energy release value from Eq. 40 directly affect the measured value. However, these properties of TBC are not well characterized, and they could vary from specimen to specimen. In the present study, a possible reason for the higher toughness value that was obtained is correlated with observations of significant wear tracks on the post-test specimens. The influence of interfacial friction on the mode II toughness is documented in the literature: the mode II toughness value of TGO/bond coat interface 56 J/m2 [10]) was lower than that of the oxide mode II toughness is attributable to the bending effect of the TGO/TBC bilayer as well as to shortening of the zone subject to friction over the full length of the delamination. In the present measurement, a larger frictional contribution relative to the indentation and wedge impression measurements was expected along the wake of the mode II delaminations because of significant frictional affects, as observed in delaminated surface of specimens. This large effect due to friction results from the longer contact zone subjected to friction as well as larger normal compressive stress acting on the TBC layer, as a result of fixing the TBC layer delamination edges during the barb testing and preventing bending of the TGO/TBC bilayer resulting from delamination.

6

Effect of Interface Roughness

The test method was applied to different types of TBC systems, i.e., (1) PL-TBC200: BC polished with 200 μm TBC thickness; (2) PL-TBC500, BC polished with 500 μm TBC thickness; and (3) AS-TBC500, BC “as-sprayed” with 500 μm TBC thickness. In Fig. 27, SEM micrographs of the interface morphology between the TBC and the BC

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Interface Delamination Analysis of Dissimilar Materials: Application to. . .

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for the studied TBC systems are presented. A thin thermally grown oxide (TGO) layer is shown in darker contrast between the TBC and the BC. The formation of the TGO layer is a result of the inward diffusion of oxygen and the outward diffusion of Al during the deposition. For PL-TBC200 and PL-TBC500, the low-magnification image reveals that the TGO layer is relatively flat (Fig. 27a), and its thickness is approximately 0.5 and 0.8 μm, respectively. Under high-magnification image (Fig. 27c), although the TGO layer is smooth at the TBC side, at the BC side certain “blocky” particles 0.2–0.7 μm in diameter are present (indicated by arrows in Fig. 27c). This indicated that the TGO protruded into the BC in some locations. Differing from the PL-TBC200 and PLTBC500, for the ASTBC500 the interface reveals a saw-toothed shape (Fig. 27b), showing the undulated nature of the BC/TGO interface. The TGO layer is not uniform between the TBC and the BC, and its thickness has a range of 0.5–2 μm. High magnification image reveals details of the TGO, TBC, and BC near the interface (Fig. 27d). Block particles 1–3 μm in diameter are formed at the TGO front adjacent to the BC. In isolated locations, the TGO appears to form within the BC beneath the interface (indicated by arrows in Fig. 27d). The barb test is applied for measurement of interface toughness between TBC layer and substrate. From experimentally obtained load-displacement relations, the toughness is determined using Eq. 40. It is found that the measured average interface toughness was approximately 84 J/m2 and 106 J/m2 for the PL-TBC200 and PLTBC500, respectively. In the case of AS-TBC500, the obtained interface fracture toughness was approximately 123 J/m2. With the present experimental method,

Fig. 27 Typical SEM images of interface morphology between the TBC and the BC in the EBPVD-TBC systems: (a and c) PL-TBC500, and (b and d) AS-TBC500, showing a rough interface for AS-TBC500 compared with PL-TBC500

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these values are roughly the mode II strain energy release rates (ψ 90 ) for either the TBC/BC interface or the TBC layer, dependent upon the observed fracture path selection. The interface residual compressive stress measured along the TBC/TGO interface is summarized in Table 2. The interface compressive residual stress is larger for PL-TBC500 than for PL-TBC200, showing that the thick TBC layer led to larger interface residual compressive stress. In addition, it is found that the interface residual compressive stress is larger for PL-TBC500 than for AS-TBC500. It is known that the interface residual compressive stress is caused by the growth strains in TGO and the thermal expansion misfit between the TGO and the TBC. Thus, it is believed that the major cause of low residual compressive stress for AS-TBC500 is the presence of defects (pores and cracks, Fig. 27) in the TBC layer adjacent to the interface because these defects result in a relaxation of the interface residual compressive stress. It is known that the interface toughness is linked to the properties of the BC, TGO, and TBC as well as to the interface roughness and thickness of TBC. In the studied TBC system, the interface roughness and thickness of the TBC affect the residual compressive stress at the interface and in the TBC layer, in turn affecting the delamination crack propagation path as well as the interface toughness. It is found that the interface toughness of AS-TBC500 is the largest among the three studied TBC systems although the residual compressive stress at the interface is lower for ASTBC500 than for PL-TBC500 (Table 2). This finding suggested that the rough interface led to larger interface toughness because of the presence of a saw-toothed shape interface for AS-TBC500 compared with a flat interface for PL-TBC500 (Fig. 27). The roughness of fracture surfaces after delamination for the studied TBC systems was measured under a laser optical microscopy; the observed images are shown in Fig. 28. This clearly shows that the interface roughness is larger for AS-TBC500 than for PL-TBC500. The measured average roughness amplitude of the interface for three studied TBC systems is shown in Table 2. It is seen that the average roughness amplitude is significantly larger for AS-TBC500 than for PL-TBC200 and PLTBC500. This suggests that a possible reason for the high interface toughness obtained in AS-TBC500 is correlated with interface frictional contribution between the TBC and the BC evidenced by the presence of significant wear tracks observed in the SEM views of the post-tested specimens. The wear tracks observed on the failure surfaces show that the crack faces remained in contact during crack advance because the presence of the wear tracks indicated a sliding along the spalling interface during the testing. The influence of interface friction on the interface toughness is documented in Table 2 Average interface compressive residual stress, average interface roughness amplitude, and average interface toughness Coating materials PLTBC200 PLTBC500 ASTBC500

Average interface residual stress, σ r (GPa) 0.934

Average interface roughness amplitude, Ar (μm) 15.15

Interface fracture toughness Gi (J/m2) 84

1.284

16.20

106

1.216

22.95

123

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Interface Delamination Analysis of Dissimilar Materials: Application to. . .

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Fig. 28 Examples of interface roughness measured on fracture surfaces for the EB-PVD-TBC systems: (a) PL-TBC500, and (b) ASTBC500

the literature. Mumm and Evans [10] showed that the mode II toughness value of the TGO/BC interface (ΓII  56 J/m2) was lower than that of the oxide (ΓII  100 J/m2) [14]. This low mode II toughness is attributable to the bending effect of the TGO/TBC bilayer as well as to shortening of the zone subject to friction over the full length of the delamination. In the present study, for AS-TBC500 the delamination is located within the TBC adjacent to the interface and the rough interface is observed, compared with PL-TBC200 and PL-TBC500 which showed that the delamination located at the TGO/ BC and/or TBC/TGO interfaces had a flat interface. This suggests that the bending effect of the TGO/TBC bilayer is diminished and that the zone subject to friction is large for AS-TBC500, compared with PL-TBC500. Thus, larger frictional contribution for AS-TBC500 than for PL-TBC500 was expected along the wake of the delaminations because the rough interface for the former resulted in larger frictional resistance compared with the latter.

7

Conclusion

Delamination behavior of high temperature use coatings plays very important role on the performance of high temperature components. We introduce recent progress of quantitative understanding of the interface toughness for delamination of coating

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Fig. A1 Trends in the beam theory c coefficients defined in Eqs. 17a and 17b

layer from substrate, including definition, experimental measurement, and some applications. The analytical and experimental approaches have been applied for thermal barrier coatings (TBCs) because thermal barrier coatings have been widely used for various high temperature metal components. The same concept is useful for environmental barrier coatings (EBCs) [41, 42] on CMC substrate, and we hope that the concept could be applied for quantitative measurement of interface toughness in various coatings.

Appendix Beam Theory Coefficients The ci coefficients defined in Eq. 17 have been computed using the finite-element method (Sect. 3.5). This has been achieved by applying separately the axial load, P; bending moment, M; and shear load, V, to the edge (x = 0) when fixed along the xaxis. The average rotation angle along the line x = L has been calculated as well as the y-axis deformation at the point (x = L, y = 0). The coefficients are plotted in Fig. 29. Based on these coefficients, C1 and C2 can be obtained from beam theory as [25]:

where

C1 ¼

B1 A22  B2 A12 det jAj

C2 ¼

B2 A11  B1 A21 det jAj

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Interface Delamination Analysis of Dissimilar Materials: Application to. . .

1411

 ðkLÞ3 w ðkLÞ4  θ w CV CM  CθM CwV  C sin kL 4 144l 12l3 V , kL θ ðkLÞ2  w θ C ð sin kL  kL cos kLÞ CM ð1  cos kLÞ þ CV ð cos kL þ kL sin kL  1Þ  12l M 12l2 2 det jAj ¼ kL sin kL þ 2 cos kL  2 þ

kLÞ kL A11 ¼  cos kL þ 1 þ CθM 12l sin kL þ CθV ð12l 2 cos kL, 2

kLÞ kL A12 ¼ sin kL þ CθM 12l cos kL  CθV ð12l 2 sin kL, 2

3

kLÞ w ðkLÞ A21 ¼ sin kL  kL þ CwM ð12l cos kL, 2 sin kL þ CV 12l3 2

3

kLÞ w ðkLÞ sin kL, A22 ¼ cos kL  1 þ CwM ð12l 2 cos kL  CV 12l3 θ kL B1 ¼ CP 12l h1 , 2

kLÞ B2 ¼ CwP ð12l 2 h1 ,

l ¼ L=h1

References 1. Qian G, Nakamura T, Berndt CC, Leigh SH. Tensile toughness test and high temperature fracture analysis of thermal barrier coatings, Acta Mater. 1997;45:1767–84. 2. ISO14916 – Determination of tensile adhesive strength. 3. EN 15340, European Standard Method – Determination of shear load resistance of thermally sprayed coatings. 4. Marot G, Lesage J, Demarecaux P, Hadad M, Siegmann S, Staia MH. Interfacial indentation and shear tests to determine the adhesion of thermal spray coatings, Surf Coat Technol. 2006;201:2080–5. 5. Evans AG, Mumm DR, Hutchinson JW, Meier GH, Pettit FS. Mechanisms controlling the durability of thermal barrier coatings, Prog Mater Sci. 2001;46:505–53. 6. Hutchinson RG, Hutchinson JW. Lifetime Assessment for Thermal Barrier Coatings: Tests for Measuring Mixed Mode Delamination Toughness, J Am Ceram Soc. 2011;94:S85–95. 7. Hsueh CH, Luttrel CR, Lee S, Wu TC, Lin HY. Interfacial Peeling Moments and Shear Forces at Free Edges of Multilayers Subjected to Thermal Stresses, J Am Ceram Soc. 2006;89:1632–8. 8. Balint DS, Hutchinson JW. Mode II Edge Delamination of Compressed Thin Films, J Appl Mech. 2001;68:725–30. 9. Volinsky AA, Moody NR, Gerberich WW. Interfacial toughness measurements for thin films on substrates, Acta Mater. 2002;50:441–66. 10. Mumm DR, Evans AG. On the role of imperfections in the failure of a thermal barrier coating made by electron beam deposition, Acta Mater. 2000;48:1815–27. 11. Begley MR, Mumm DR, Evans AG, Hutchinson JW. Analysis of a wedge impression test for measuring the interface toughness between films/coatings and ductile substrates, Acta Mater. 2000;48:3211–20. 12. Charalambides PG, Lund J, Evans AG, McMeeking RM. A Test Specimen for Determining the Fracture Resistance of Bimaterial Interfaces, J Appl Mech. 1989;56:77–82. 13. Hsueh CH, Tuan WH, Wei WCJ. Analyses of steady-state interface fracture of elastic multilayered beams under four-point bending, Scr Mater. 2009;60:721–4. 14. Clyne TW, Gill SC, Residual Stresses in Thermal Spray Coatings and Their Effect on Interfacial Adhesion: A Review of Recent Work, Therm J. Spray Technol. 1996;5:401–18. 15. Vasinonta A, Beuth JL. Measurement of interfacial toughness in thermal barrier coating systems by indentation, Eng Fract Mech. 2001;68:843–60.

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16. Drory MD, Hutchinson JW. Measurement of the adhesion of a brittle film on a ductile substrate by indentation, Proc R Soc Lond A. 1996;452:2319–41. 17. Honda K, Kagawa Y. Interfacial Stress Transfer Mechanism in SiC Fiber-reinforced Ti Matrix Composites, J Japan Inst Metals. 1994;58:91–7. 18. Honda K, Kagawa Y. Debonding criterion in the pushout process of fiber-reinforced ceramics, Acta Mater. 1996;44:3267–77. 19. Guo SQ, Mumm DR, Karlsson AM, Kagawa Y. Measurement of interfacial shear mechanical properties in thermal barrier coating systems by a barb pullout method,Scr Mater. 2005;53:1043–8. 20. Liu YF, Kagawa Y, Evans AG. Analysis of a “barb test” for measuring the mixed-mode delamination toughness of coatings, Acta Mater. 2008;56:43–9. 21. Suo Z, Hutchinson JW. Interface crack between two elastic layers, Int J Fract. 1990;43:1–18. 22. Hutchinson JW, Suo Z. Mixed mode cracking in layered materials, Adv Appl Mech. 1991;29:63–191. 23. Freund LB, Suresh S. Thin film materials. Cambridge: Cambridge University Press; 2004. 24. Dunders J. Edge- bonded dissimilar orthogonal elastic wedges, J Appl Mech. 1969;36:650–2. 25. Timosheko SP, Gere JM. Theory of elastic stability. 2nd ed. New York: McGraw-Hill; 1961. 26. Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials, J Mech Phys Solids. 2004;52:193–214. 27. ABAQUS. User’s manuals: ABAQUS Inc.; 2006. 28. Yuuki R, Cho SB. Efficient boundary element analysis of stress intensity factors for interface cracks in dissimilar materials, Eng Fract Mech. 1989;34:179–88. 29. Liu YF, Tanaka Y, Masuda C. Crack deflection and penetration at an interphase interface, Acta Mater. 1998;46:5237–47. 30. Padture NP, Gell M, Jordan EH. Thermal barrier coatings for gas-turbine engine applications, Science. 2002;296:280. 31. Zhao JC, Westbrook JH. Ultrahigh-temperature materials for jet engines, MRS Bull. 2003;28:622. 32. MSC Software Corporation, MSC. Marc, 2003 Volume A and B. 33. Schlichting KW, Padture NP, Jordan EH, Gell M. Failure modes in plasma-sprayed thermal barrier coatings, Mater Sci Eng A. 2003;342:120. 34. Kim SS, Liu YF, Kagawa Y. Evaluation of interfacial mechanical properties under shear loading in EB-PVD TBCs by the pushout method, Acta Mater. 2007;55:3771. 35. Honda K, Kagawa Y. Analysis of shear stress distribution in pushout process of fiber-reinforced ceramics, Acta Metall Mater. 1995;43:1477. 36. Hein VL, Erdogan F. Stress singularities in a two-material wedge, Int J Fract Mech. 1971;7:317. 37. Bogy DB. Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions, J Appl Mech. 1971;38:377. 38. Rice JR. Elastic fracture mechanics concepts for interfacial cracks, J Appl Mech. 1988;55:98. 39. Williams ML. The stresses around a fault or crack in dissimilar media, Bull Seismol Soc Am. 1959;49:199. 40. Evans AG, He MY, Hutchinson JW. Mechanics-based scaling laws for the durability of thermal barrier coatings, Prog Mater Sci. 2001;46:249. 41. Aoki Y, Inoue J, Kagawa Y, Igashira K. A simple method for measurement of shear delamination toughness in environmental barrier coatings, Surf Coat Technol. 2017;321:213-8. 42. Arai Y, Aoki Y, Kagawa Y. Effect of cristobalite formation on the delamination resistance of an oxide/Si/(SiC/SiC) environmental barrier coating system after cyclic high temperature thermal exposure, Scr Mater. 2017;139:58–62. 43. Guo SQ, Tanaka Y, Kagawa Y. Effect of interface roughness and coating thickness on interfacial shear mechanical properties of EB-PVD yttria-partially stabilized zirconia thermal barrier coating systems, J Eur Ceram Soc. 2007;27:3425.

Part III Macromechanics

Smoothed Particle Hydrodynamics for Ductile Solid Continua

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Peter Eberhard and Fabian Spreng

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Discretization Principle of Smoothed Particle Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Function Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Particle-Based Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Kernel Function Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Discretized Equations of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Plasticity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Damage and Fracture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Methodological Enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Artificial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Variable Smoothing Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Kernel Function Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Solution of the Discretized Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Neighborhood Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Numerical Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exemplary Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Friction Stir Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Orthogonal Metal Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1416 1417 1418 1418 1419 1420 1421 1422 1423 1425 1426 1428 1433 1435 1435 1437 1440 1442 1443 1444 1445 1447 1447 1454 1460 1462

P. Eberhard (*) Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany e-mail: [email protected] F. Spreng Materials Testing Institute University of Stuttgart (MPA Stuttgart, Otto-Graf-Institut (FMPA)), Stuttgart, Germany # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_28

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P. Eberhard and F. Spreng

Abstract

In this chapter, a numerical simulation model for ductile solid continua is presented. It is based on the Smoothed Particle Hydrodynamics (SPH) method, which serves to spatially discretize and, thus, solve the governing equations of continuum mechanics. Due to the meshless, Lagrangian character of the SPH spatial discretization technique, the introduced model is naturally well-suited for the simulation of continua featuring large deformations, major changes in topology, material failure including structure disintegration, and/or a large number of contacts with the environment occurring at the same time. For this reason, it has the potential to become a beneficial complement to the well-established numerical solid models, which mainly make use of mesh-based methods. To that end, however, the original SPH discretization scheme is to be variously extended and modified as discussed in detail in the course of this chapter. Besides, also its efficient implementation, i.e. the efficient numerical solution of the SPHdiscretized governing equations of continuum mechanics, is addressed. The quality of the developed SPH formulation for ductile solids including its versatility and accuracy is demonstrated on the basis of two exemplary applications, namely, the industrial processes of friction stir welding and orthogonal metal cutting. It is shown as part of this contribution that, in either case, the proposed SPH model for ductile solid continua is capable of reproducing both the mechanical and the thermal macroscopic behavior of the real processed material in the simulation.

1

Introduction

In continuum mechanics [1], matter is considered to be continuously distributed within the region of space it occupies, whereas its actual atomic structure is ignored. In so doing, the macroscopic behavior of a real material can be described by a set of partial differential equations (PDEs), the so-called governing equations of continuum mechanics, and studied by solving them. In the case of mechanical systems of engineering relevance, it is a very tedious or even impossible task to calculate the desired solution by hand. That is why numerical methods are usually used to approximately solve the governing equations of continuum mechanics with the aid of computers. To that end, however, they need to be spatially discretized. For this purpose, different discretization techniques are available. One of them is the particle-based Smoothed Particle Hydrodynamics (SPH) method [2, 3]. Due to the meshless, Lagrangian character of the SPH spatial discretization scheme, it is naturally well-suited for the analysis of continua featuring large deformations, major changes in topology, material failure including structure disintegration, and/or a large number of contacts with the environment occurring at the same time. Nonetheless, the original SPH discretization technique needs to be variously extended in order to be able to accurately reproduce the macroscopic behavior of real ductile solids in the simulation. Therefore, the aim of this contribution is to introduce a numerical simulation

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model for ductile solid continua based on an enhanced SPH method. Thus, the present chapter is structured as follows. First, the basic spatial discretization principle behind SPH is explained in Sect. 2. In this context, the nature of the introduced particles is elucidated as well. Next, the previously discussed SPH discretization scheme is applied to the governing equations of continuum mechanics. That way, the reader is provided with numerically treatable formulations of these differential equations in Sect. 3. To be able to simulate the macroscopic behavior of real ductile solids, the original SPH method has to be extended and modified in terms of its methodology. In this regard, simple but proven concepts to solve the problems of particle clumping due to tensile instability, numerically induced fracture caused by a loss of contact between neighboring interpolation points, and a reduced approximation quality close to the free surfaces of a model resulting from an only partial coverage of the smoothing areas of the near-surface particles are integrated into the developed formulation as part of Sect. 4. How the SPH-discretized governing equations of continuum mechanics can be solved and, thus, the temporal evolution of the behavior of the considered ductile solid can be simulated within the framework of meshless methods is in the focus of Sect. 5. Here, different strategies regarding the three main tasks to be performed, namely, the neighborhood search, the interactions, and the numerical time integration, can be pursued. The ones that have been implemented into the particle simulation package Pasimodo [4], which provides the basis for all numerical investigations presented in this chapter, are explained in detail. Both the versatility and the accuracy of the developed SPH formulation for ductile solid continua are demonstrated in Sect. 6 when simulating two challenging examples, the industrial applications of friction stir welding and orthogonal metal cutting. Finally, the key results of the present contribution are summarized in Sect. 7. Furthermore, the topics of planned future studies are outlined.

2

Discretization Principle of Smoothed Particle Hydrodynamics

The meshless, Lagrangian SPH method can be used as a spatial discretization technique in order to solve PDEs [5]. For this purpose, the original system of time- and space-dependent PDEs is transformed into a set of ordinary differential equations (ODEs) in time only. This SPH-based transformation enables to numerically integrate the obtained equations with conventional integration schemes. It consists of the three steps: 1. Function convolution using the Dirac delta 2. Particle-based spatial discretization 3. Kernel function approximation and is concisely described hereinafter. Further details regarding the basic SPH theory can be found, for instance, in [6, 7].

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Function Convolution

The value of a given function named A that is continuous and either of scalar or vector character at the specific position ra in the d-dimensional real space ℝd can be expressed in an exact manner using the function value at any arbitrary position r and the convolution

ð Aðra Þ ¼ AðrÞδðra  rÞdΩr ,

(1)

Ω

where dΩr represents the differential spatial domain that is associated with position r, Ω is the entire domain, and δ (ra  r) denotes the Dirac delta function defined as  δðra  rÞ :¼

2.2

1 0

if r  ra else:

(2)

Particle-Based Discretization

In order to make (1) solvable in a numerical way, it is necessary to convert the integral over the entire domain into a sum over discrete parts of it, in other words, to spatially discretize Ω. To that end, a collection of abstract, i.e. non-tangible particles is introduced in SPH. Because each of them represents a specific part b of the initial spatial domain, it carries the same invariable portion of mass mb during the whole simulation, moves along with the related material, and is labeled with index b as well. Since the particle b serves as a sampling point while interpolating the function value of A at position ra, also all the other properties of the associated segment of material that are necessary for this purpose, e.g. its center of position rb and its mass density ρb, are assigned to it. Having discretized the domain Ω based on the SPH method, the integral in (1) can be replaced with a sum over all existing particles. This yields the approximation A^ðra Þ :¼

Xm b Aðrb Þδðra  rb Þ  Aðra Þ: ρ b b

(3)

Here, the fraction mb/ρb ensures that the employed sum over the entire set of SPH particles corresponds to a domain integral as requested. As a consequence thereof, the function value of particle b influences the interpolated value at reference position ra only according to its mass-density ratio. This means that, although they may have the same distance from the interpolation center, i.e. an identical Euclidean norm rab := ||rab||2 of the distance vector rab := ra  rb, a small SPH particle, that is one with a low mass-density ratio, affects the approximated value to a smaller extent than a large one does.

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2.3

Smoothed Particle Hydrodynamics for Ductile Solid Continua

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Kernel Function Approximation

Due to the fact that the Dirac delta function, which has been employed so far, is neither continuous nor differentiable, (3) does not allow for an adequate representation of a continuous field variable. Thus, the SPH transformation procedure is not yet complete, but a second approximation, which focuses on the delta function itself, is required. To enable the SPH interpolation to also reproduce continuous spatial distributions, the Dirac delta δ (ra  rb) is replaced by the so-called kernel function Wab (h) := W (rab, h), which shows the requested mathematical properties of continuity and differentiability. This yields the final SPH approximation to the original function A at position ra as A~ðra Þ :¼

Xm b Aðrb ÞWab ðhÞ  A^ðra Þ  Aðra Þ: ρ b b

(4)

According to (4), the contribution of each particle b to the approximation at position ra depends on both the directional distance between the two points and the actual shape of the kernel function. Therefore, W determines the intensity of spatial smoothing that is seen for the interpolated function values and, hence, can be considered as weight function. To illustrate the aforementioned fact, a cluster of adjacent SPH particles in the two-dimensional space (x, y), as well as the spatial distribution of an exemplary kernel function, is depicted in Fig. 1. The function value at the center of the particle a to be determined is interpolated using the value of any neighboring interpolation point b that is located within its smoothing area, which also includes the reference particle a itself, and the corresponding value of the kernel function. Here, the value of W (intensity of blue color) decreases with increasing distance from the interpolation center ra. Since the amount of decrease for a certain distance rab is found to be the same regardless of the specific direction of the distance vector, W is referred to as Fig. 1 Reference particle a (red color), its neighbors  b (black color), and the distribution of the kernel function W (blue color) over its smoothing area (dashed line)

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symmetric kernel. In order to save numerical effort by not taking into account the relatively minor contributions from distant particles, the kernel function reaches a value of zero at distance h (dashed line). This makes the smoothing area a compact support. In the literature, there is some confusion regarding the definition of the smoothing length h [8]. In the following, the smoothing length is taken as the radius of the kernel function defining its non-zero area, i.e. the maximum distance up to which adjacent SPH particles interact. A comprehensive study of different kernel functions typically used in SPH is found in [8]. For the presented examples, the Wendland C2 kernel [9] is chosen since it offers a reasonable compromise between achieved results accuracy and required computational effort [8]. This is an important aspect in the context of SPH simulations featuring a large number of particles. As a piecewise polynomial function of low order, the Wendland C2 kernel is computationally less effortful than, for example, the common Gaußian-bell-shaped [2] one and higher-order Wendland functions, and it is easy to implement as well. Furthermore, the Wendland kernels seem to provide the best results among the available ones when being used in combination with kernel function renormalization techniques [10]. Such a correction technique aiming at restoring the zeroth-order consistency, i.e. completeness [11], of the SPH interpolation will be introduced in Sect. 4.3. With the normalized distance ζab := rab/h, the Wendland C2 kernel function reads 8 < αd ð1  ζ Þ4 ð1 þ 4ζ Þ if 0  ζ < 1 ab ab ab WWC2 ðζ ab , hÞ :¼ hd : 0 else,

(5)

where αd := 7/π or αd := 21/(2π) in the two- or three-dimensional case, respectively. Accordingly, the spatial gradient of the Wendland C2 kernel relative to the local coordinates of the reference position ra evolves as 8 α r   < d ab 20ζ ab ð1  ζ ab Þ3 dþ1 ∇a WWC2 ðζ ab , hÞ ¼ h rab : 0

if 0  ζ ab < 1

(6)

else:

Equation (6) enables to determine the SPH approximation to the spatial gradient (scalar case) or divergence (vector case) ∇a of any continuous field variable. This is an important point if intending to discretize the governing equations of continuum mechanics based on the SPH method as will be discussed in the following section.

3

Discretized Equations of Continuum Mechanics

Any material, regardless of whether it is fluid or solid, is composed of discrete molecules being separated by empty space. The interactions between these molecules and, with that, the behavior of the regarded domain of discontinuous matter on the microscopic scale can be simulated using, for instance, methods from Molecular

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Dynamics. If dealing with macroscopic problems, however, it is much too effortful to individually treat every single molecule. For this reason, the material to be modeled is assumed to be continuously distributed within the region of space it occupies, which is the main idea behind continuum mechanics [1]. Accordingly, the state variables characterizing the considered material are smeared over the model domain and turn into field variables. This means that each distinct point within the continuum holds its own set of physical properties, which includes, for example, velocity and temperature. By continually subdividing the previously homogenized matter into infinitesimal small parts and assessing the in- and outflows observed for the state variables of such an element, a system of local balance equations describing its behavior, i.e. physical state, can be derived. This set of first-order PDEs is referred to as the governing equations of continuum mechanics. It consists of: • The continuity equation • The momentum equation • The energy equation Besides these fundamental equations of continuum mechanics, an additional relation is required in order to complete this system of PDEs. It takes the specific properties of the considered material into account and is, therefore, typically called constitutive equation [12]. To be able to simulate the temporal evolution of the mechanical and thermal behavior of the modeled material, the afore-stated time- and space-dependent PDEs need to be transformed into ODEs, which are numerically treatable. This can be achieved by applying the previously introduced SPH spatial discretization technique to them, what is in the focus of this section.

3.1

Continuity Equation

The continuity equation says that the balance between exchanged and contained mass must be maintained for any part of the model domain if there are no mass sources or sinks. There are different ways to mathematically express this requirement, as well as all the other conservation laws of continuum mechanics, depending on the chosen point of view. In case the considered control volume moves with the material (Lagrangian discretization), but the continuity equation is to be written in absolute, i.e. spatially fixed coordinates (Eulerian description), the material derivative operator [13] defined as D @ :¼ þv∇, Dt @t

(7)

where t is the time and v denotes the flow velocity vector, is needed. The continuity equation for compressible media in differential form then reads [14]

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Dρ ¼ ρ∇  v, Dt

(8)

which means that the temporal change in local mass density at the position of reference equals the net flow of density that is exchanged with adjacent control volumes. The continuity equation (8) can be approximated in a direct way using the introduced SPH interpolation scheme. This yields the relation Xm Dρa b ¼ ρa vb  ∇a Wab ðhÞ: Dt ρ b b

(9)

However, this interpolation rule shows some unwanted properties, e.g., Dρa/Dt does not vanish if the velocity vector is constant over space [6]. For this reason, an improved formulation was developed, which reads Xm Dρa b ¼ ρa ðva  vb Þ  ∇a Wab ðhÞ: Dt ρ b b

3.2

(10)

Momentum Equation

Choosing the same perspective and making the same assumptions as in the case of the previously regarded continuity equation, the Cauchy momentum equation [15], which is derived from Newton’s second law, can be written as Dv 1 ¼ ∇  σ þ g: Dt ρ

(11)

Here, σ denotes the Cauchy stress tensor, and g is the vector per unit mass that takes the contributions from both acting body forces and additional surface forces due to boundary contact into account. The Cauchy stress tensor represents the surface forces resulting from the interaction with neighboring parts of material. Therefore, it includes direction-independent stresses, e.g. thermally induced ones, and direction-dependent stresses, such as viscosity (see Sect. 3.5). In general, (11) only guarantees the balance of linear momentum by interrelating its change with the quantities of acting body and surface forces. If the stress tensor is symmetric, i.e., σ  σ T, the angular momentum is locally balanced as well [16]. A major demand on the SPH approximation to the Cauchy momentum equation (11) is the conservation of both linear and angular momentum. Thus, an improved interpolation scheme that complies to this requirement is typically used in SPH. It reads [6]   Dva X σ σ ¼ mb 2a þ 2b : ∇a Wab ðhÞ þ ga Dt ρa ρb b

(12)

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1423

and guarantees the requested conservation of momentum even in case of differing stress tensors σ a and σ b if a symmetric kernel is employed. In the presence of extreme spatial gradients and/or an insufficient quality of the deployed gradient approximation, discontinuities may arise in the distributions of the SPH-interpolated field variables. These discontinuities can lead to numerical instabilities that avoid the solution of the given problem. However, this is not an SPH-specific issue but also well-known in other spatial discretization techniques, e.g. Finite Difference methods. Here, it is standard practice to introduce an artificial viscosity to the momentum equation, which includes the real viscous forces, too [17]. That way, undesirable oscillations are damped and, hence, a remedy for the problem of numerical instability is provided as well. An advanced form of artificial viscosity that incorporates the nature of the SPH method and is, therefore, superior to classical shear/bulk or von Neumann-Richtmyer approaches in this context was introduced in [18]. It has been successfully applied to SPH solid simulations and is given as

Π ab

8   < 1 αv cs,ab μab þ βv μ2ab :¼ ρab : 0

if ðva  vb Þ  ðra  rb Þ < 0

(13)

else,

where ρab represents the arithmetic mean of the particle densities ρa and ρb, cs,ab denotes, in a similar manner, the mean value of the speeds of sound, and μab is defined as μab :¼

hðva  vb Þ  ðra  rb Þ : rab þ eh2

(14)

In (14), the term eh2 was added to the denominator for preventing a singularity in case of a vanishing particle distance rab, where e = 0.01 is a typical choice [19]. Besides, values of αv = 1 and βv = 2 are suggested in [6] for best results. The extended Cauchy momentum equation reads   Dva X σ σ ¼ mb 2a þ 2b  Π ab I : ∇a Wab ðhÞ þ ga : Dt ρa ρb b

3.3

(15)

Energy Equation

Starting from the Cauchy momentum equation (11), the mechanical energy balance for solid matter including the kinetic and the gravitational potential energy can be deduced [20]. By subtracting it from the general energy equation describing the change in total energy of a control volume over time, the energy equation concentrating on the specific internal energy u, often referred to as free energy or internal energy density, is found. This third governing equation takes the form [14]

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P. Eberhard and F. Spreng

Du 1 ¼ ðσ  d  ∇  q þ S Þ, Dt ρ

(16)

where q denotes the heat flux vector, S represents the contributions from heat sources and sinks, and d is the symmetric part of the velocity gradient tensor [21] d :¼

 1 ∇v þ ð∇vÞT : 2

(17)

As can be seen from (16), the local balance law for the specific internal energy says that u changes according to the quantities of isochoric and non-isochoric stressrelated power (first part), the heat exchanged with neighboring control volumes (middle part), and additional thermal energy flows (last part). In case of solid continua without phase change, the dominant mechanism of heat transfer is conduction [22]. Then, Fourier’s law [12] allows to model the heat flux q seen at a specific point within the considered continuum based on the molecular energy transfer caused by differences in temperature ΔT. For a homogeneous, isotropic solid, it follows the relation q ¼ k∇T ,

(18)

where k denotes the thermal conductivity of the material and T is the absolute temperature. As is the case with all material-dependent parameters appearing in this chapter, k is assumed to be position-, direction- and time-independent, and, hence, it becomes a scalar. With respect to (18), the temporal evolution of the specific internal energy can be written as Du 1 ¼ ðσ  d þ kΔT þ S Þ: Dt ρ

(19)

Modifying the energy equation (19) to include the additional pressure resulting from the artificial viscosity and discretizing it using SPH, it follows   X Dua 1 X σa σb ¼ ρb 2 þ 2  d ab  ρb Π ab trðd ab Þ 2 b Dt ρa ρb b 4 X mb k a k b r 1 ðTa  Tb Þ ab  ∇a Wab ðhÞ þ S a : þ 2 ρa b ρb k a þ k b ρa rab

(20)

Here, an advanced scheme is employed to discretize the second-order diffusive PDE of heat conduction, what leads to the third term in (20). It is derived from the first law of thermodynamics and was improved using Finite Differences [23]. Taking this approach, the discretized energy equation holds true also in case of a time- and space-dependent thermal conductivity k. With the tensor product operator , the contribution from the pair of particles ab to the symmetric part of the velocity gradient tensor of particle a can be expressed in the form

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

d ab :¼

 T  1 mb  ðvb  va Þ  ∇a Wab ðhÞ þ ðvb  va Þ  ∇a Wab ðhÞ , 2 ρb

where

X b

3.4

1425

d ab 

 1 ∇a va þ ð∇a va ÞT ¼ d a : 2

(21)

(22)

Heat Equation

If intending to observe the temperature distribution over the model domain and its change over time, the amount of thermal energy, which is related to the temperature of the material and no longer available for mechanical purposes, has to be extracted from the total specific internal energy u. However, this is not straightforward when dealing with deforming solids as the thermal energy and the remaining portion of u, i.e. the deformation energy, are not independent of each other but interrelated in that case. This phenomenon is termed thermomechanical coupling [12]. Following the approach of linear thermoelasticity, the coupled heat equation reads [12] c

DT 3 1 ¼  Kαth T0 trðd Þ þ ðkΔT þ S Þ , Dt ρ ρ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ¼: wth ¼: wel

(23)

where c denotes the heat capacity per unit mass, K is the bulk modulus, αth represents the coefficient of thermal expansion, T0 denotes the absolute temperature in the reference state 0, and tr() is the mathematical operator that gives the trace of a square matrix. Here, it has been assumed that the strain rate tensor is D«/Dt  d. This is considered to be a reasonable assumption if the strains per time increment are small [14]. To achieve consistency with the SPH-based formulation of the energy equation (20), the discretized heat equation, which enables to calculate the temperature of particle a, is stated as ca

  X K a αth,a T0,a K b αth,b T0,b DT a 3X ¼ ρb þ ρb Π ab trðd ab Þ trðd ab Þ  2 2 2 b Dt ρa ρb b |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ wel,a ¼: wvi,a þ

4 X mb k a k b r 1 ðT  Tb Þ ab  ∇ W ð hÞ þ S a : ρ a b ρb k a þ k b a ρa r2ab a ab |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ wth,a

(24)

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P. Eberhard and F. Spreng

This approximation formula guarantees both the conservation of specific thermal energy and an entropy increase due to viscous dissipation even if heat is exchanged between neighboring SPH particles [23].

3.5

Constitutive Equation

Up to now, the three fundamental equations of continuum mechanics, i.e. (8), (11), and (19), have been presented. However, they include the four unknowns ρ, v, σ, and u. In order to complete this system of PDEs and, thereby, enable to determine all the unknowns, another equation relating these quantities to each other is required. A convenient relationship can be found if taking the type of the material to be modeled into account by expressing the stress tensor σ as a function of the other state variables. Due to the fact that this additional relation gives the yet material-independent continuum mechanical description the material-dependent character, it is also often referred to as material equation of state (material EoS) [24]. Based on the knowledge that the total stresses acting on a control volume consist of direction-dependent and direction-independent portions [13], also the constitutive equation can be split into two parts. One of them represents the direction-independent contributions and is known as hydrostatic stress or negative pressure p due to its isotropic nature. The second part, which includes all the stresses that are dependent on the direction, is called deviatoric stress τ. As these two components of the stress tensor are not interrelated to each other, they can be determined separately. To finally obtain the stress tensor σ, they are added up using the identity matrix I according to the relation σ ¼ pI þ τ:

(25)

3.5.1 Hydrostatic Stress As to the first part of (25), a constitutive relation involving the pressure is needed. One that provides accurate results for a wide range of different solid materials even in case of both a major change in the temperature as well as large plastic deformation is the first-order Mie-Grüneisen EoS [24]. It is linear in mass density and in specific internal energy, i.e., it also includes thermally induced stresses, and reads p ¼ p0 þ Γρ0 ðu  u0 Þ þ

ρ0 c2K,0 ηv  ð1  sH ηv Þ2

1Γ

ηv  , 2

(26)

where Γ and sH are dimensionless, material-dependent parameters; the Grüneisen coefficient and the Hugoniot slope coefficient, respectively. The bulk speed of sound ffi pffiffiffiffiffiffiffiffiffiffi in reference condition is defined as cK,0 :¼ K=ρ0 and the volumetric compression

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

1427

strain as ηv :¼ 1  ρ0 =ρ. Because the current density and internal energy of an SPH particle result from numerical time integration and the Mie-Grüneisen EoS shows no further dependencies on other field variables or gradients, (26) can directly be evaluated without additional interpolation.

3.5.2 Deviatoric Stress As can be seen from the description of the Cauchy stress tensor σ given in (25), also its traceless deviatoric part represented by τ is required to quantify the total stresses acting on the considered control volume. Within the framework of fully Lagrangian simulation techniques, like SPH, it is beneficial to perform a differential update on the deviatoric stress tensor. That way, the initial, i.e. undeformed configuration of the model may be discarded. Assuming a linear elastic response of the deformed matter as well as small strain increments, the deviatoric stress rate Dτ/Dt can be calculated based on Hooke’s law [25] following the relation Dτ ¼ C : d0, Dt

(27)

where C denotes the symmetric fourth-order stiffness tensor (here in matrix notation) specifying the material’s resistance against deformation, to be more precise, against its deviatoric deformation expressed by the rate tensor d 0 := d 1/3 tr(d) I. In the case of a purely elastic response, C depends solely on the elastic parameters of the material and is, therefore, typically called elasticity tensor. For isotropic media, two independent elastic constants are sufficient to describe the answer of the solid continuum to a given strain or strain rate [25]. Considering three-dimensional problems, it is advantageous, since numerically more efficient, to rewrite (27) using Lamé’s first parameter and the shear modulus G, also known as Lamé’s second parameter. Because d 0 is a traceless tensor, this yields the equation Dτ ¼ 2Gd 0 : Dt

(28)

Although the deviatoric stress tensor τ itself shows objectivity, the afore-stated formulation for its time derivative does not. This results in an unphysical dependence of the obtained stress rates on the chosen frame of reference. Objectivity for the deviatoric response of the material can be guaranteed if employing the Jaumann stress rate [21]. This leads to the revised identity τ_ :¼ 2Gd 0  Ω : τ þ τ : Ω,

(29)

where the antisymmetric part of the velocity gradient tensor Ω reads [21] Ω :¼

 1 ∇v  ð∇vÞT : 2

(30)

In SPH form, the evolutionary equation for the deviatoric stresses evolves as

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P. Eberhard and F. Spreng

τ_ a ¼

X

ð2Ga d 0ab  Ωab : τ a þ τ a : Ωab Þ,

(31)

b

where d 0ab :¼ d ab  1=3trðd ab ÞI denotes the deviatoric deformation of the portions of material being associated to particles a and b, and Ω can be expressed as Ωab :¼

 1 mb  ðvb  va Þ  ∇a Wab ðhÞ  ððvb  va Þ  ∇a Wab ðhÞÞT , 2 ρb

(32)

where X

Ωab 

b

3.6

 1 ∇a va  ð∇a va ÞT ¼ Ωa : 2

(33)

Plasticity Model

If investigating ductile solids undergoing large deformations, a non-reversible response of the material is seen once a certain stress level has been surpassed. This means that the current deformation does not vanish entirely and the solid does not return to its original shape after unloading. Then, the material is entitled as plastically deformed. In this case, the total strain is composed of a temporary, elastic part «el, which still follows Hooke’s law, and a permanent, plastic part «pl, which will remain even after the applied loading has been removed. To be able to model a plastically deforming solid continuum, the equations of linear elasticity presented in the previous section are no longer sufficient and to be complemented by a so-called plasticity model [26]. Regardless of its actual design, any plasticity model has to answer three fundamental questions concerning the plastic behavior of the material. Accordingly, it consists of the three basic components: • Yield criterion (when?) • Flow rule (in which direction?) • Hardening law (how much?)

3.6.1 Yield Criterion As mentioned above, a plastic character of a modeled solid resulting in a permanent deformation is not observed before a critical level of loading, i.e. material stress, is exceeded. In the case of uniaxial loading, the limit stress at which the material begins to deform plastically can be quantified using the scalar σ y, which is known as yield strength [27]. By comparing the yield strength with the current stress level, it can be decided whether the solid’s response is still elastic or has already become plastic. However, any three-dimensional combination of stresses is possible in reality. Consequently, the one-dimensional concept of yield strength needs to be generalized in order to be able to identify a plastic behavior also in such cases. To that end,

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

1429

so-called equivalent stresses, which are scalar measures converting the real, multidimensional state of stress into a fictitious, uniaxial one, are introduced. That way, they allow for a comparison with the yield strength σ y, which is typically found from uniaxial material tests. One equivalent stress commonly used for isotropic ductile solids is the von Mises or octahedral shearing stress [27] defined as σ vM :¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  

σ 2xx þ σ 2yy þ σ 2zz  σ xx σ yy  σ xx σ zz  σ yy σ zz þ 3 σ 2xy þ σ 2xz þ σ 2yz , (34)

or using the principal stresses σ 1, σ 2, and σ 3, for which any shear stress disappears, σ vM

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 :¼ ðσ 1  σ 2 Þ2 þ ðσ 2  σ 3 Þ2 þ ðσ 3  σ 1 Þ2 : 2

(35)

According to this definition and based on the previously presented evolutionary equation for the deviatoric stress tensor τ a , the particle-related von Mises equivalent stress can be expressed as σ vM,a

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 τ τ : ¼ 2 a a

(36)

It is, therefore, solely related to the deviatoric part of the Cauchy stress tensor σ and, thus, a measure for the amount of energy spent to distort the considered control volume. As a result, the von Mises yield criterion is a hypothesis that predicts a plastic behavior of the material if the distortion energy reaches a critical value but not in case of a purely hydrostatic loading, i.e. a shape-preserving deformation. Accordingly, the critical stress states for which the character of the response changes from elastic to plastic form a cylinder of infinite length around the hydrostatic axis in the three-dimensional space of the principal stresses, as depicted in Fig. 2. If setting pffiffiffiffiffiffiffiffithe yield strength σ y as limit, the radius of the cylinder corresponds to RvM :¼ 2=3σ y. In other words, if the set of principal stresses found for the current loading state is located within the von Mises cylinder, i.e., σ vM < σ y, it is related to an elastic behavior, while plasticity is detected when it lies on or, temporarily, outside the cylinder, i.e., σ vM σ y.

3.6.2 Flow Rule Another integral part of any plasticity model is the flow rule. It addresses the question in which direction plastic deformation occurs. In the case of uniaxial loading, the answer to it is obvious, but it is not for multidimensional stress states. Basically, there are two approaches to tackle this issue; associated and non-associated flow rules [26]. While for the latter ones, the direction of plastic deformation can be derived from any plastic potential having been specially designed for this purpose, associated flow rules are more restricted and additionally have to meet the so-called normality condition. It says that the directional increment

1430

P. Eberhard and F. Spreng

Fig. 2 Von Mises yield surface (blue color) around the hydrostatic axis (dashdotted line) in the threedimensional space of the principal stresses

of plastic deformation has always to be coaxial with the normal vector to the yield surface that points to the current combination of principal stresses. As a consequence thereof, the plastic strains and the deviatoric stresses have common principal directions. From experimental investigations, this assumption was found to be reasonable when modeling plastically deforming metals and is, therefore, used in the following. For soils, rocks, and concrete, on the contrary, non-associated flow rules are to be preferred [28].

3.6.3 Hardening Law Having determined the direction of the material’s plastic response from the flow rule, the last information needed to complete the plasticity formulation is the intensity, i.e. the amount of plastic deformation, that is seen for the current stress state. As a differential update on the deviatoric stress tensor τ is employed (see (29)), it is not known in advance whether the updated stresses will be outside the yield cylinder or not. For this reason, the approach of radial return [26] is taken to correct them if necessary. The basic idea behind the radial return method is the following. The increase in deviatoric stress resulting from a further deformation of a material is first assumed to be fully elastic and, thus, calculated using Hooke’s law. To avoid possible confusion regarding the several stress variables occurring in both the present and the following sections, the temporary, elastically calculated deviatoric stresses are called trial stresses from now on, and they are, as well as all related quantities, marked with an asterisk. Next, the updated stress state represented by the von Mises equivalent stress σ vM is tested for a violation of the specified yield criterion. If a violation is detected, then the part of the Cauchy stress tensor that is responsible for the plastification of the material, i.e. the deviatoric trial stress tensor τ , has to be scaled in order to reduce its von Mises stress to the maximum allowed value σ y, which means the principal stress state is to brought back on the von Mises yield surface. This can be achieved by using the proportionality factor f given as

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

f :¼

σy : σ vM

1431

(37)

The deviatoric trial stresses τ are then limited following the relation τ ¼ f τ ,

(38)

which allows to assemble the final stress tensor σ based on (25). One isotropic hardening law that proved to be sufficiently accurate, easy-tohandle, and numerically efficient if simulating the behavior of metals that are subjected to large strains, high strain rates, and/or high temperatures is the JohnsonCook flow stress model (JC flow stress model) [29]. It describes the phenomenon of work hardening, which is a strengthening of the material due to prior plastic deformation, on the macroscopic scale by dynamically adapting the yield strength of a plastically deforming solid. In doing so, the JC model takes the influences of the applied strain rate and the current temperature into account. Hence, the evolution of the yield strength is described according to the equation   σ y,JC epl,vM , Depl,vM =Dt, T 0   nJC ¼ AJC þ BJC epl;vM @1 þ C JC ln

!1   Depl,vM =Dt A 1  T mJC : hom Depl,vM,0 =Dt

(39)

Here, epl,vM denotes the accumulated von Mises equivalent plastic strain, that means the part of the von Mises equivalent total strain, etot,vM that is associated with the plastic deformation of the solid. Hence, it can be stated eel,vM þ epl,vM ¼ etot,vM

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 0 « «, :¼ 2

(40)

where 1 «0 :¼ «  trð«ÞI: 3

(41)

Moreover, Dεpl,vM/Dt is the corresponding strain rate, Dεpl,vM,0/Dt denotes the reference von Mises equivalent plastic strain rate, AJC, BJC, nJC, CJC and mJC are material-dependent constants, ln() is the mathematical operator determining the natural logarithm to the given argument, and the homologous temperature Thom is defined as Thom :¼

T  T0 Tm  T0

(42)

with the melting temperature Tm. By comparing the yield strength σ y,JC with the trial von Mises equivalent stress σ vM , a plastic response of the simulated material can be identified. However, the

1432

P. Eberhard and F. Spreng

JC-shifted yield strength cannot be determined using (39) in its present form as it contains three more unknowns, i.e. the equivalent plastic strain epl,vM, the equivalent plastic strain rate Dεpl,vM/Dt, and the temperature T. Since these quantities are not independent of each other, the number of unknowns can be reduced to two based on the heat equation (23) and the temporally incremental formulation of the von Mises equivalent plastic strain rate given as Depl,vM ðt þ Δt Þ epl,vM ðt þ Δt Þ  epl,vM ðt Þ ¼ : Dt Δt

(43)

Furthermore, the consistency condition     ! σ y,JC epl,vM , Depl,vM =Dt, T ¼ 3Geel,vM ¼ 3G etot,vM  epl,vM

(44)

relates the two variables σ y,JC and epl,vM to each other, and, thus, it reduces the number of unknowns to one. Equation (44) arises from the very definition of the yield strength as limit stress at which the material begins to deform plastically. This meaning holds true even in case of a prior plastic deformation, i.e. a shifted yield strength. So, no further plastic deformation occurs for equivalent stresses lower than or equal to σ y,JC, and the response of the material up to this level of stress is from then on entirely elastic. For this reason, the current yield strength can be determined based on either the current plastic strain epl,vM (left-hand part of (44)) or the current elastic strain eel,vM (middle part). Rewriting eel,vM in the form etot,vM  epl,vM makes it possible to determine the current equivalent plastic strain and, thus, yield strength based on the JC flow stress model. However, to actually solve the system consisting of (23), (43), and (44) is not straightforward. Detailed information on its numerical solution is provided, for example, in [30]. Although the energy equation (19) still holds true for plastically deforming solid continua, the heat equation (23) needs to be adapted because of the increase in dissipated energy that results from the material plastification. Since a hydrostatic loading will never lead to a plastic response when applying the von Mises yield criterion, the thermoelastic coupling term wel remains the same. Due to the effect of work hardening, the increment of deviatoric strain d 0 consists of an elastic and a plastic part. Hence, only a certain amount of the total portion of energy spent to distort the control volume causes an irreversible response of the modeled solid. For this reason, the thermoplastic coupling term denoted by wpl includes the proportionality factor βpl ðt þ Δt Þ :¼

Depl,vM ðt þ Δt Þ=Dt epl,vM ðt þ Δt Þ  epl,vM ðt Þ ¼  ½0, 1, Detot,vM ðt þ Δt Þ=Dt etot,vM ðt þ Δt Þ  etot,vM ðt Þ

(45)

which allows to extract the part of the deviatoric strain tensor that is related to the plastic deformation of the material. Based on this extension, the heat equation in linear thermoelastoplasticity [12] reads

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

c

DT 3 1 ¼  Kαth T0 trðd Þ þ ðkΔT þ S Þ þ Dt ρ ρ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ¼wth ¼ wel

1 β β σ  d0 ρ pl TQ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ¼:wpl

1433

(46)

Here, another multiplication factor, i.e. βTQ, has been integrated into the thermoplastic coupling term. This is because of the observation made in [31] that for metals not the entire plastic power is available for heating the material. Instead, a certain portion of it, commonly called cold work, is required for the physical processes leading to the changes in the crystal structure of the matter that result in the phenomenon of work hardening. If phase transforming is not considered, the factor βTQ, which is typically termed Taylor-Quinney coefficient (TQ coefficient), takes a value 0  βTQ  1. Since the amount of cold work was found to be invariant over a wide range of plastic deformation, βTQ is most often treated as a constant. If considering the TQ coefficient, the SPH-compliant formulation of the thermoelastoplastically coupled heat equation evolves as   X K a αth,a T0,a K b αth,b T0,b DT a 3X ca ¼ ρb þ ð Þ  ρb Π ab trðd ab Þ tr d ab 2 2 b Dt ρ2a ρb b |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ wel,a ¼wvi,a 4 X mb k a k b r 1 þ ðTa  Tb Þ ab  ∇a Wab ðhÞ þ S a 2 ρa b ρb k a þ k b ρa rab |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼wth,a   1X σ σ þ ρb βpl,a βTQ,a 2a þ 2b  d 0ab : (47) 2 b ρa ρb |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼wpl,a

3.7

Damage and Fracture Model

If intending to investigate ductile solid continua undergoing large deformations, an additional model that is dedicated to the description of the process of ductile damage and fracture needs to be incorporated into the presented SPH solid formulation. Since the degree of triaxiality of the current stress state, i.e. the ratio of the hydrostatic to the deviatoric stress, has been identified to primarily influence the ductility of a solid [32], a model that considers this ratio as the key factor in describing the damage and fracture behavior of a material is employed. Just as the JC flow stress model discussed before, the complementary ductile damage and fracture model takes further the effects of the applied strain rate and the current temperature on the solid’s response into account. The present damage state of the material is represented by the scalar quantity DJC. Here, a value of DJC = 0 indicates a pristine condition of the solid, whereas it ultimately fails for DJC = 1. The current value of the state variable DJC is determined based on the temporally, i.e. strainhistory-related evolutionary equation

1434

P. Eberhard and F. Spreng

DJC ðt þ Δt Þ :¼

tþΔt X ts

ΔtDepl,vM =Dt  , ef ,JC σ m , σ vM , Depl, vM =Dt, T

(48)

which allows the damage variable to grow from the beginning of the simulation denoted by ts until the current point of time t + Δt under both tension and compression. For this purpose, the ratio of the present increment of equivalent plastic strain Δt Depl,vM/Dt to the quantity of critical equivalent fracture strain under the current conditions ef,JC, which is given as   ef ,JC σ m , σ vM , Depl,vM =Dt, T

 ! σm ¼ D1,JC þ D2,JC exp D3,JC σ vM 0 !1   Depl,vM =Dt A 1 þ D5,JC Thom , @1 þ D4,JC ln Depl,vM,0 =Dt

(49)

is required. Here, D1,JC to D5,JC are material-dependent constants, and exp() is the mathematical operator determining the value of the exponential function to the given argument. Furthermore, σ m denotes the mean value of the three normal stresses, which is used to characterize the level of hydrostatic stress in relation to the extent of deviatoric one represented by the von Mises equivalent stress σ vM. The original purpose of the JC cumulative-damage fracture model as it was introduced in [32] is to predict when failure of a material occurs while not affecting its state up to that point. This means that the material abruptly fails when the critical value of DJC = 1 is reached, whereas the effect of stress reduction due to prior damage is not considered. However, an immediate complete failure without any previous indication of impending fracture does not correspond to the experimental observations made for ductile solids. Instead, a gradually decreasing loading capacity is found. Even though the JC model was originally designed as a pure fracture model, it allows to simulate the growth in material damage and its influence on the material behavior as well. To that end, the damaged stress tensor defined as [33] σ D :¼ ð1  DJC Þσ

(50)

is introduced. The undamaged tensor σ is then to be replaced by its damaged counterpart in all equations belonging to the presented SPH solid model except the ones that are related to the determination of the JC-shifted yield strength σ y,JC (see Sect. 3.6.3). According to the definition of the damaged stress tensor given in (50), the material, i.e. a particle in the context of SPH simulations, experiences a gradual reduction in stress as the value of the damage variable DJC increases until it, finally, reaches the state of complete damage and can no longer take any load in case DJC = 1.

45

4

Smoothed Particle Hydrodynamics for Ductile Solid Continua

1435

Methodological Enhancements

In the previous section, the SPH-discretized governing equations of continuum mechanics have been discussed. Furthermore, additional models describing the plastic, as well as the damage and fracture behavior of ductile solid continua have been introduced. Besides these physically motivated extensions, different improvements and modifications to the basic methodology of the original SPH discretization technique are required in order to be able to correctly simulate ductile solids. In this respect, simple but proven concepts to solve the problems of particle clumping due to tensile instability, numerically induced fracture caused by a loss of contact between neighboring interpolation points and a reduced approximation quality close to the free surfaces of a model resulting from an only partial coverage of the smoothing areas of the near-surface particles will be integrated into the developed formulation in the following.

4.1

Artificial Stress

If a solid is stretched, the material resists this deformation by establishing attractive forces between its components. If compressing a solid on the contrary, repulsive forces that try to return it to its original shape are evoked. This circumstance is also described by the fundamental equations of continuum mechanics. However, SPH-discretized solid continua as introduced in Sect. 3 sometimes fail to model this behavior in case of solids that are subjected to tensile loads. Then, a reduced amount of repulsive force acting between neighboring interpolation points is seen. This unphysical reduction in force makes the SPH particles to approach and, finally, to form clumps, what leads to a significant degradation in the quality of the SPH interpolation. The afore-described unwanted effect of particle clustering is termed tensile instability [34] and considered one of the main weaknesses of SPH solid simulations. To the best knowledge of the authors, the problem of tensile instability is not yet completely understood and needs a thorough consideration in the future. So far, it is known that the tensile instability results from inaccuracies in the SPH-interpolated momentum equation, which induce the already mentioned clustering of SPH particles under tension in the direction of maximum loading. In order to be able to accurately model the behavior of deforming solids using the SPH method, an effective remedy to the issue of tensile instability is required. As the reason for it is not entirely clear, different attempts to solve this problem have been made. A simple, but proven and widely applicable approach in this respect is to mimic the missing repulsive forces by introducing a short-range artificial pressure, i.e. negative stress, to the SPH momentum equation (15) [34]. It then reads   Dva X σa σb n ¼ mb 2 þ 2  Π ab I þ Rab f ab : ∇a Wab ðhÞ þ ga , Dt ρa ρb b

(51)

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P. Eberhard and F. Spreng

where the artificial stress consists of two parts. The first one, Rab := Ra + Rb, takes the states of stress of the two particles a and b into account while determining the distance-independent magnitude of the artificial stress. Because the issue of tensile instability solely arises in case of particles that are under tension, the artificial stress should only act then and vanish otherwise. Therefore, the character of a particle’s state of stress needs to be identified. This can be done by evaluating the signs of its principal stresses [35]. To obtain the principal stresses, the stress tensor σ is to be diagonalized by rotating its coordinate system until no more shear stresses are seen. In the general, i.e. three-dimensional case, the principal stresses can be found through an eigenvalue decomposition of the stress tensor. Next, each element on the diagonal of the principal stress tensor σ is analyzed in terms of its sign, and the corresponding entry Rxx=yy=zz of the rotated artificial stress tensor R is calculated following the rule Rxx=yy=zz ¼

8 < :

γ 0

σxx=yy=zz ρ2

if σxx=yy=zz > 0

(52)

else,

where γ is a scale factor that determines the magnitude of the artificial stress. The artificial stress tensor R in the particles, i.e. absolute coordinate system is, finally, obtained by rotating the coordinates R¯ back. If the damaged stress tensor σ D is used instead of its undamaged counterpart in (51), the artificial stress is to be determined on this basis. n The second part of the artificial stress, i.e. f ab , ensures that the additional pressure is restricted to the close vicinity of the reference particle a and rapidly decreases for larger distances. According to that, fab is related to the shape of the kernel function W and defined as f ab :¼

Wab ðhÞ , W ðΔL,hÞ

(53)

where ΔL denotes the average particle spacing, which is proportional to h based on a constant ratio. From linear perturbation analysis of the governing equations of continuum mechanics, n = 4 was found to be a reasonable choice as it ensures that just the closest neighbors of an SPH particle are affected by its artificial stress [35]. This is a desired property as only for them the unphysical decrease in repulsive force in case of the uncorrected momentum equation is observed. The specification of the parameter γ is a problem-dependent task, and it needs to be within a certain range to guarantee a high accuracy of the obtained results. This circumstance is the main weakness of the taken approach and requires a further consideration. Due to the fact that an additional pressure term is added to the momentum equation, also the balance equations for the specific internal energy, (20), and the temperature of a particle, (47), are to be extended accordingly, which yields the relations

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

  X X Dua 1 X σa σb n ¼ ρb 2 þ 2  d ab  ρb Π ab trðd ab Þ þ ρb Rab f ab  d ab 2 b Dt ρa ρb b b 4 X mb k a k b r 1 þ ðT a  T b Þ ab  ∇a Wab ðhÞ þ S a 2 ρa b ρb k a þ k b ρa rab

1437

(54)

and ca

  X K a αth,a T0,a K b αth,b T0,b DTa 3X ¼ ρb þ ρb Π ab trðd ab Þ trðd ab Þ  2 2 2 b Dt ρa ρb b |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ wel,a ¼wvi,a 4 X mb k a k b r 1 þ ðT  Tb Þ ab  ∇ W ð hÞ þ S a ρa b ρb k a þ k b a ρa r2ab a ab |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼wth,a   X 1X σ σ n þ ρb βpl,a βTQ,a 2a þ 2b  d 0ab þ ρb Rab f ab  d ab : (55) 2 b ρa ρb b |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼wpl,a ¼: war,a

While doing so, it is essential that the conservation properties of the original balance equations are retained as is the case with the presented formulations [36].

4.2

Variable Smoothing Length

The original SPH method as it was introduced in [2, 3] includes particles featuring a constant smoothing length h. This invariance regarding the extent of the smoothing areas can result in a loss of contact between neighboring interpolation points, i.e. their decoupling, in case of large strains of the modeled material. This directly affects the densities, the velocities, and other state variables of these and the surrounding particles that are determined by SPH interpolation over their supports. Once a critical degree of decoupling is reached, the simulated solid disintegrates and, with that, ultimately fails due to this effect, what is called numerical or artificial fracture [37, 35]. As such numerical fracture may compete with physics-driven damage and fracture processes (see Sect. 3.7) and further leads to a degradation in the quality of the obtained results, the appearance of a numerical fracture has to be prevented. Basically, there are two approaches to tackle the problem of a numerically induced fracture. On the one hand, additional interpolation points replacing the former neighbors can help to prevent an undesired decoupling when being introduced into the areas of decreasing particle density. On the other hand, it is possible to dynamically adapt the smoothing lengths of the existing SPH particles and, thus, the extents of their smoothing areas in such a way that a constant number of neighbors is maintained throughout the simulation even in case of high relative motions.

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Although both of them work, the former strategy seems not to be the best choice mainly for two reasons. First, it is computationally more expensive and requires a higher effort in terms of preserving the attained conservation properties of the SPH-discretized governing equations of continuum mechanics. Second, the issue of numerical fracture is, in general, a global one and not restricted to distinct regions of the model domain. This may necessitate a large number of additional SPH particles to be inserted in order to be able to effectively prevent numerical fracture. Therefore, the use of the latter approach is recommended. The basic idea of a variable smoothing length, note that the term variable here refers to a variance in both space and time, is not a new one. It has a long tradition in SPH and is frequently used to solve the problem of particle decoupling. However, there are different strategies to appropriately adapt the smoothing lengths of the deployed particles. The extension called Adaptive Smoothed Particle Hydrodynamics (ASPH) [38, 39] addresses the issue of a numerical fracture on a solely methodological basis, i.e. without any direct link to the physics of the considered problem, what gives it a universal character. To achieve the set goal of an invariant number of neighboring interpolation points during the entire simulation within the framework of the ASPH extension, the smoothing length ha of particle a is adjusted according to the change in its kernel-gradient-weighted distance Ψ a from all of its neighbors. To that end, the Euclidean norm rˇab := ||rˇab||2 of the distance vector rˇab := (ra  rb)/ha, which has been normalized with respect to ha, is introduced. Besides, the sum of the weighted distances for the interpolation point a is given as

Ψa :¼

X

!1=d k∇a W ð rˇ ab , 1Þk2

:

(56)

b

At the beginning of the simulation, a table mapping the value of Ψa to the effective number of nodes per smoothing length nh,a must be set up based on the initial particle arrangement. It is defined nh,a := ha/L0, where L0 denotes the initial distance of particle a from its nearest neighbors. If using the Wendland C2 kernel and a uniform initial particle distribution, this mapping results in an almost linear relationship for a wide range of different values of nh,a in the cases of one-, two-, and threedimensional problems as shown in Fig. 3. Therefore, only a few sampling points are needed to accurately describe the relation between Ψa and nh,a when interpolating linearly between them, what allows for a numerically efficient table lookup operation. Here, the very same table can be used for all particles of an SPH simulation if employing the normalized distance rˇab instead of rab. The last step before actually starting the simulation is to specify a reasonable target value for the variable nh,a, which is the value the ASPH extension tries to achieve in case of any particle by adjusting its smoothing length properly. However, this can be a tedious task if not applying a kernel function renormalization technique since then the optimal target value decreases when approaching either of the free surfaces of a model. Therefore, it is suggested not to use a global but a particle-specific target to make the employed algorithm more flexible and convenient. For this purpose, the initial effective

Smoothed Particle Hydrodynamics for Ductile Solid Continua

kernel-gradient-weighted distance (m− (d+1) )

45

8

1439

d=1 d=2 d=3

7 6 5 4 3 2 1 0

1

2

3

4

5

6

7

8

9

10

effective number of nodes per smoothing length (−) Fig. 3 Almost linear relationship between kernel-gradient-weighted distance and effective number of nodes per smoothing length for a varying dimensionality d of the considered problem if using the Wendland C2 kernel and a uniform particle distribution

number of nodes per smoothing length is calculated for each particle in the simulation in a preprocessing step and assigned as target value ntarget h,a . Having specified the target, the ratio sa := ntarget h,a /nh,a is determined at each time step, what causes only a small computational overhead due to the fact that Ψa can be computed together with the right-hand sides of the governing equations of continuum mechanics. In order to obtain a smooth adjustment of ha, not solely the previously calculated ratio sa but, in addition, the auxiliary variable aa, which is defined as [40] ( aa :¼

0:4ð1 þ s2a Þ

if sa < 1

0:4ð1 þ s3 a Þ else,

(57)

is used to scale the smoothing length in both the predictor and the corrector step of the employed leapfrog time integration scheme (see Sect. 5.3). This is done according to the relation hnew ¼ hold a a ð 1  aa þ aa s a Þ

(58)

while this adjustment focuses on the smoothing length of the particle a only, while not affecting its physical properties. Allowing for a particle-specific extent of the smoothing area is accompanied by the need for a decision on how to treat the interactions between interpolation points varying in smoothing length. To be more precise, an advanced strategy specifying how to calculate the spatial gradients of the employed kernel function in such a way that the conservation properties of the previously derived SPH-based fundamental equations of continuum mechanics are preserved is required. In this regard, three

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common approaches that retain symmetry of the interaction even in case of differing smoothing lengths are to [41]: • Take the arithmetic mean of the smoothing lengths ha and hb • Use the mean value of the kernel gradients ∇aWab (ha) and ∇aWab (hb) • Incorporate the spatial gradients of the smoothing lengths ∇aha and ∇ahb into the calculation rule for the kernel gradient, respectively While the former strategy is the most simple, easiest to implement, and numerically cheapest one, the latter requires more efforts in terms of both implementation and computation but promises to give best results. Although using the average hab := 0.5 (ha + hb) is the least sophisticated approach among the mentioned ones, it was shown in [36] for different validation examples that it provides results that are as accurate as the ones obtained with the second strategy and even better than the ones achieved when incorporating the spatial gradients of the smoothing lengths into the kernel gradient calculation. Therefore and due to the fact that none of the three approaches has proven its universal superiority over the others yet, the procedure employing the arithmetic mean of the smoothing lengths for calculating the spatial gradients is chosen in the following. Depending on the specific model setup, the second or third strategy may yield more accurate results.

4.3

Kernel Function Renormalization

In Sect. 2.3, it was explained that the Dirac delta function has to be replaced by the kernel function W(rar, h) in order to enable the SPH interpolation to correctly reproduce continuous spatial distributions. While doing so, the kernel function must adhere to several fundamental properties inherited from the substituted Dirac delta function. One of them is the so-called normalization condition [5]

ð ! W ðra  r, hÞdΩr ¼ 1,

(59)

Ω

which demands that the integral of the kernel function over the entire domain Ω takes a value of one. In discrete form and if employing the arithmetic mean hab of the smoothing lengths of the interacting particles a and b for the evaluation of the kernel function (see Sect. 4.2), the normalization condition reads C a :¼

Xm ! b Wab ðhab Þ ¼ 1: ρb b

(60)

Here, attention has to be paid to the fact that the set of neighboring particles b includes the interpolation center a itself as, otherwise, the afore-described normalization condition will not be met. But even if the reference particle a is considered, a

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

1441

violation of the normalization condition may occur in case the smoothing area of an SPH particle is only partially covered with neighboring interpolation points. Then, the lack of contribution from the non-existing neighbors to the SPH interpolation at the position of particle a results in a sum of the kernel function value over its support that significantly differs from one. As a consequence thereof, the SPH approximation of any field quantity at this specific point yields a too low value. This issue typically arises close to the free surfaces of a model (exterior particle), is termed kernel support incompleteness or, simply, support incompleteness, and needs to be remedied in order to obtain accurate results. To overcome the problem of support incompleteness, it is necessary that the normalization condition (60) is met even in the case of an exterior particle. A simple but effective way to ensure this is to give more weight to the contributions from the existing neighbors to the SPH interpolation at the reference position by scaling the kernel function properly. This approach aimed at restoring the zeroth-order consistency, i.e. completeness, of the SPH interpolation is commonly referred to as kernel function renormalization by kernel sum [37]. For the intended purpose, the actual sum of the kernel function values according to (60) needs to be determined for each particle in the SPH simulation in a first step. In a second step, Wab is normalized by multiplying it with the reciprocal of this sum. The resulting, renormalized kernel function, which is typically called a Shepard function [11], evolves as W ðh Þ Wˇ ab ðhab Þ :¼ ab ab : Ca

(61)

When normalizing, i.e. scaling the kernel function by the factor 1/Ca, the associated kernel gradient reads ∇aWˇ ab ðhab Þ :¼

∇a Wab ðhab Þ : Ca

(62)

The renormalized kernel function and gradient are to be applied instead of the original, uncorrected ones in the case of all kernel-function-based interactions. Then, the normalization condition is met even by the SPH particles suffering from an incomplete support. An exception to this rule is the ASPH algorithm dedicated to the adjustment of the smoothing lengths of the particles (see Sect. 4.2), which still needs to operate on the uncorrected kernel function Wab respectively kernel gradient ∇aWab. At this point, the reader should pay attention to the fact that, as a consequence of the afore-described particle-specific normalization of Wab, the kernel function value and gradient are no longer global but position-dependent quantities, which also vary over time. Therefore, all fundamental equations of continuum mechanics to be evaluated on the basis of the employed kernel function become particle-specific as well. This also means that their conservation properties, which had been guaranteed so far, are no longer valid. One way to ensure exact conservation even in the case of the corrected kernel function is to use the arithmetic means defined as

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P. Eberhard and F. Spreng

ˇ ˇ ab ðhab Þ :¼ Wab ðhab Þ þ Wba ðhab Þ ¼ Wab ðhab ÞðC a þ C b Þ W 2 2C a C b

(63)

ˇ ˇ ab ðhab Þ :¼ ∇aWab ðhab Þ  ∇aWba ðhab Þ ¼ ∇a Wab ðhab ÞðC a þ C b Þ ∇a W 2 2C a C b

(64)

and

for determining the right-hand sides of the governing equations of continuum ab and the averaged mechanics. However, employing the averaged kernel function W  kernel gradient ∇a Wab contradicts the intended particle-specific normalization of the kernel function due to the associated smoothing over neighboring interpolation points. This leads to either an insufficient mitigation (exterior particles) or even an overcompensation (interior neighbors of the exterior particles) of the influence of the support incompleteness on the results. Furthermore, the error introduced if replacing the original kernel function and gradient with their renormalized counterparts is of negligible extent compared with the improvement in results accuracy following from this modification. Therefore, it is recommended to use (61) and (62), and not to adopt the symmetrization approach given in (63) and (64).

5

Numerical Solution of the Discretized Equations

Regardless of the specific computational tool that is employed to solve the spatially discretized governing equations of continuum mechanics, the basic procedure of any SPH simulation is always the same. It consists of three main steps and is illustrated in Fig. 4. After having specified the initial particle configuration, it is necessary to determine for each interpolation point its contacts with both other Fig. 4 Basic computational procedure of any SPH-based particle simulation

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Smoothed Particle Hydrodynamics for Ductile Solid Continua

1443

SPH particles and possible boundary geometries. This provides the particle-related set of neighbors  b and is, therefore, called neighborhood search. Based on the interactions with these neighbors, the right-hand sides of the discretized equations of continuum mechanics can be evaluated. That enables to update the SPH particles’ states and, thereby, to simulate the temporal evolution of the overall system’s behavior using numerical time integration. If the desired end point of the simulation tend has not been reached, the time further advances and the computational procedure starts, again, at the stage of neighborhood search. Although the afore-described scheme provides the basis for any SPH simulation, the single steps can be realized in different ways. How they are actually implemented in Pasimodo is explained in this section.

5.1

Neighborhood Search

Due to the meshless character of the SPH method and the fact that the introduced particles strictly move along with the modeled material, the neighbors of any interpolation point are not known in advance in contrast to mesh-based discretization techniques. Instead, this information must be acquired through an additional operation. But this also means that no extra computational effort has to be made at this stage of the simulation process in order to identify a possible contact between the SPH continuum and a boundary structure. Since the neighborhood search is an integral part of each simulation step, much computation time can be saved if employing a numerically efficient algorithm to that end. Lots of studies concerning neighborhood search have been carried out in the past, and sophisticated procedures have been developed. Typically, they are based on two-step processes including a less effortful pretest on the basis of either simplified particle geometries or a partitioned simulation domain, and a computationally more expensive final test using the exact shapes. Cell-lists-based rough tests [42] are widely used in SPH because of their high efficiency. If boundaries of complex shape are involved, the approach of geometry substitution can be more convenient and is, therefore, taken in Pasimodo. Possible surrogate bodies employed for this kind of pretest are enclosing spheres or cuboids, often referred to as bounding spheres respectively bounding boxes. While the former alternative allows for a simpler check for overlap, the latter one is easier to implement. In the case of bounding boxes, their start and end coordinates in each direction are entered into separate lists, the so-called linked linear lists [43]. Only if an overlap in all directions is detected, adjacent particles may be in contact and the fine test is performed. When using particles with compact support, these lists show only small changes from one time step to the next, and the order of the problem reduces from the one of the naive search (check any particle against any other) to O ( b) [42]. Besides, also the orientation of the bounding boxes replacing the actual geometries influences the achieved numerical efficiency. Reasonable choices are to align the edges of the enclosing cuboids according to either the directions of the inertial coordinate system, what results in axis-aligned minimum bounding boxes

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P. Eberhard and F. Spreng

(AABBs), or the spatial orientation of the related object, leading to oriented minimum bounding boxes oriented minimum bounding boxes (OBBs). Due to the spherical smoothing areas of the SPH particles, the easier-to-handle AABBs are of no disadvantage compared to OBBs and, hence, used for the simulation examples presented in Sect. 6. If the pretest indicates a possible overlap, a fine test for contact is performed. In general, elaborate procedures are necessary to be able to reduce its computational cost to a minimum. However, a simple center distance test is sufficient in the case of SPH particles because of their spherical supports.

5.2

Interactions

The previous identification of the neighbors of an SPH particle makes it possible to evaluate the right-hand sides of the ODEs describing its behavior. In doing so, additional terms besides the expressions resulting from the intra-SPH interactions may arise from contacts with boundary structures. This is the case with the discretized momentum equation (51), the SPH form of the energy equation (54) and the transformed heat equation (55). Accordingly, additional force and heat models are needed to quantify the contributions from the boundary contacts to these balance equations. In order to make it more convenient for the user to describe such interactions with non-SPH structures, solid boundaries are represented by triangular surface meshes in Pasimodo. In terms of the boundary force, a penalty approach is taken. The force value is calculated based on the modified Lennard-Jones potential presented in [44]. Using the Euclidean norm of the distance between a particle’s center and a triangle rtr, the penalty force acting normal to the triangle plane is given as 8  2 4 2 > > < ðH  rtr Þ  H  rtr ,0 ðH  rtr Þ   gra ðrtr Þ :¼ ξ 2 H r 2H  r > tr tr , 0 , 0 > : 0

if rtr < H

(65)

else:

Here, the scalar H represents the maximum distance up to which the particle and the triangle interact. The force function leads, in general, to repulsive and attractive forces while the distance at which the sign of gra changes is denoted by rtr,0. If solely repulsive boundary forces shall act, it has to be defined that rtr,0  H. The maximum force value, which is observed for a vanishing distance between the particle and the triangle, is determined by the stiffness scale factor ξ. In addition to the normal force gra, which acts in the radial direction of a particle, also one in its tangential, i.e. circumferential direction is provided by the employed model. This allows to consider friction resulting from boundary contact. The friction force gta depends on the difference in the partners’ tangential velocities vta since it is defined as

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

  vta g ta ðrtr Þ :¼ n gra ðrtr Þtanh χ kvta k2 , kvta k2

1445

(66)

where n is the friction scale factor, χ determines the gradient of the force function with respect to the difference in tangential velocity for ||vta||2 ! 0, and the hyperbolic tangent tanh() ensures a smooth transition between the regions of opposite sign [36]. The contributions from the friction force to the balances of both internal and thermal energy can be incorporated in the form of external heat sources.

5.3

Numerical Time Integration

Having calculated the rates of temporal change (right-hand sides) of the field quantities ρa, va, ua, Ta, and τ a in the previous step, the state of particle a can be updated using numerical time integration. This yields the values of these state variables at time t + Δt and, with that, provides the basis for the subsequent simulation. To that end, the explicit, second-order predictor-corrector leapfrog method described in [45] is employed. At the beginning of each time step, the new value of any state variable is predicted ensuing from the previous one, i.e. the quantity at time t, according to the relation P ¼ ðt Þ þ Δt

D  ðt Þ : Dt

(67)

When updating the deviatoric stress tensor τ a, caution has to be exercised in order to not confuse the trial stresses (marked with an asterisk) with the plastically corrected ones (carrying no asterisk). The proper prediction rule reads τ a, P ¼ τ a ðt Þ þ Δt τ_ a ðt Þ:

(68)

Based on the results from the predictor step, new rates of change of the field variables are calculated. Those updated rates are then used to correct the predicted values for target time t + Δt based on the formula   1 DP D  ðt Þ ðt þ Δt Þ ¼ P þ Δt  : 2 Dt Dt

(69)

With regard to the deviatoric stress tensor, this means τ a ðt þ Δt Þ ¼ τ a,P þ

 1  Δt τ_a,P  τ_ a ðt Þ : 2

(70)

It is also the integration scheme which determines the relationship between the kinematic quantities ra and va. The position vector experiences the immediate update

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P. Eberhard and F. Spreng

1 Dv ðt Þ ra ðt þ Δt Þ  ra,P ¼ ra ðt Þ þ Δtva ðt Þ þ Δt 2 a : 2 Dt

(71)

Since this prediction rule is already of second order as requested, the value of ra,P is used for the subsequent calculations without any further correction. The presented leapfrog method, which belongs to the group of Verlet integrators, allows for a variable time step size while exhibiting an only minor degradation in energy conservation [46]. The maximum step size is limited by the CourantFriedrichs-Lewy condition. It demands that the temporal resolution of the considered problem must fit its spatial one or, to be more precise, that the time step used for integration must be small enough to be able to capture the effects of a traveling wave within the spatial domain represented by a single discretization element. That is why in the context of SPH simulations the step size is governed by the minimum smoothing length in the way [19] 0 Δt  Υ min @ a

1 ha

 A, cs,a þ 0:6 αv cs,a þ βv maxb μab

(72)

where Υ denotes a safety factor, which is known as Courant number. In [19], a Courant number of Υ = 0.25 is suggested to obtain a stable time integration. However, general statements on the choice of Υ are to be treated with caution as this safety factor depends not only on the applied integration scheme but also the specific arrangement of the SPH particles as well as the employed kernel function W. Thus, to find an optimal value for Υ, i.e. one that avoids instabilities while reducing the computational expense, is a model-dependent problem to be individually solved. Although (72) is a necessary condition for stability, it is not a sufficient one. Therefore, additional limitations on the step size are needed to guarantee the stability of the explicit time integration. One that is concerned with the maximum particle acceleration, i.e. its change in momentum, reads [19] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ha Δt  Υ min : a kDva ðt Þ=Dt k2

(73)

If the discretized energy equation (54) and the SPH-approximated heat equation (55) are explicitly solved, the fractional changes in specific internal energy and temperature per time step are to be limited as well. This results in the two additional requirements  Δt  Υ min a

and

ua ð t Þ Dua ðt Þ=Dt

 (74)

45

Smoothed Particle Hydrodynamics for Ductile Solid Continua

 T a ðt Þ Δt  Υ min : a DT a ðt Þ=Dt

1447



6

(75)

Exemplary Applications

Both the versatility and the accuracy of the presented SPH formulation for ductile solid continua shall be demonstrated in this section. To that end, two challenging scenarios, the industrial applications of friction stir welding and orthogonal metal cutting, are investigated. In either case, the simulated material is engineering steel AISI/SAE 1045. The properties of this kind of steel in reference state are summarized in Table 1, while Table 2 provides the corresponding configuration of the SPH solid model. As in the following the focus is on the general applicability of the presented formulation, the specific numerical setups are not further discussed as part of the present contribution. Detailed information on them is given in [30].

6.1

Friction Stir Welding

Since its invention in 1991, the joining technique of friction stir welding has attracted considerable attention and, even more, is commonly regarded as the most significant development in metal joining in the last decade [47]. This is due to the simple but genius idea that is behind it. A rotating tool made of material of high strength (exemplary geometry shown in Fig. 5) is plunged into the interface area between two adjoining workpieces (blue and green color, respectively, in Fig. 6; measurements according to Table 3) and, subsequently, moved along the abutting surfaces. Since the tool is pushed with a certain force, the so-called axial or downward force [47], against the workpieces while traversing them, they are locally heated based on friction and plastic deformation. This results in a softening of the Table 1 Physical properties of engineering steel AISI/SAE 1045 in reference state Property Mass density Young’s modulus Poisson’s ratio Yield strength Tensile strength Fracture strain Specific heat capacity Thermal conductivity Thermal expansion coefficient Speed of sound

Symbol ρ E μ σy σt ef c k αth cs

Value 7800 2.1  1011 0.3 7.37  108 9.35  108 0.1 460 55 1.2  105 6000

Unit kg/m3 N/m2  N/m2 N/m2  J/(kg K) W/(m K) K1 m/s

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Table 2 Configuration of the SPH model for isotropic ductile solid continua if simulating engineering steel AISI/SAE 1045 Property Initial smoothing length Kernel function Linear artificial viscosity parameter Quadratic artificial viscosity parameter Artificial stress scale factor Artificial stress exponent Artificial stress average particle spacing Courant number EoS for hydrostatic stress Reference pressure Room/reference temperature Reference specific internal energy Grüneisen coefficient Linear Hugoniot slope coefficient Elastic response Yield criterion Flow rule Hardening law JC reference yield strength First JC strain hardening parameter Second JC strain hardening parameter JC strain rate hardening parameter JC temperature softening parameter JC reference equivalent plastic strain rate Melting temperature First JC stress-triaxiality-related damage parameter Second JC stress-triaxiality-related damage parameter Third JC stress-triaxiality-related damage parameter JC strain-rate-related damage parameter JC temperature-related damage parameter TQ coefficient Stiffness scale factor of boundary force model Friction scale factor of boundary force model Friction gradient factor of boundary force model Maximum interaction distance of boundary force model

Symbol h0 W αv βv γ n ΔL Υ  p0 T0 u0 Γ sH     AJC BJC nJC CJC mJC Depl,vM,0/Dt Tm D1,JC D2,JC D3,JC D4,JC D5,JC βTQ ξ n χ H

Value 3.4 L0 Wendland C2 1 2 0.18 4 L0 0.25 Mie-Grüneisen 0 293.15 c T0 = 1.3  105 2.17 1.49 Linear (Hooke) Von Mises Associated JC 5.5  108 9.5  108 0.25 1.34  102 1 1  103 1493 0.15 4 1.5 2  103 0.6 0.9 2  107 1 1 h0 = 3.4 L0

Unit m      m   N/m2 K J/kg       N/m2 N/m2    s1 K       N   m

near-interface regions of the two parts, what allows the tool to mechanically intermix them in order to produce the desired sound weld. In doing so, the temperature of either workpiece does not reach the melting temperature of the base material. Therefore, it is not melted but maintains its solid state throughout the entire process. That is why friction stir welding belongs to the group of solid-state joining techniques.

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1449

Fig. 5 Geometry of the welding tool model

Fig. 6 Initial particle arrangement for the threedimensional SPH model of the friction stir welding process (blue and green color, workpieces; red color, frame)

Although being already successfully applied in industry, the friction stir welding procedure is not yet completely understood. This is, though its basic concept is rather simple, because the underlying physics, which is governed by a variety of different, interdependent phenomena, is complex. Severe plastic deformation, complex flow, high temperatures, and high strain rates of the welded material are only some of the main features of the friction stir welding process that hamper not only its experimental but also its numerical exploration. Therefore, innovative simulation approaches, such as the presented SPH solid formulation, are needed for this purpose.

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Table 3 Model parameters of the simulated friction stir welding process Property Initial length of workpiece Initial width of workpiece Initial thickness of workpiece Initial blind hole centerline distance from front frame Initial blind hole depth Initial blind hole radius Downward force of welding tool Traverse velocity of welding tool Rotational velocity of welding tool

Symbol Gw,s Ww,s Tw,s rh,g,s rh,t,s rh,r,s Fw vw ωw

Value 4  102 1  102 3  103 1  102 3  103 4.5  103 4000 1.2 100

Unit m m m m m m N m/s rad/s

The prerequisite for producing a sound weld in friction stir welding is the sufficient intermixing of the softened workpieces. For this reason, detailed knowledge of the resulting material flow is of utmost interest if intending to optimize the achieved weld quality and/or process productivity. So, an accurate reproduction of the intermixing of material observed for the real joining procedure is critical to the ability of an employed numerical simulation technique to correctly describe the scenario of friction stir welding. Therefore, the material flow that is seen if employing the introduced SPH method for ductile solid continua is evaluated first. Due to the fact that the process of material intermixing is primarily contingent on the thermally induced softening of the two workpieces, the temperature distribution that is predicted by the developed SPH solid model is investigated afterwards.

6.1.1 Material Flow The degree of intermixing of the welded material, to be more precise, the one of the SPH particles representing the workpieces at two different stages of the welding phase of the friction stir welding process, i.e. at two different simulations times t, is illustrated in Fig. 7 from the top. Here, the geometry of the clockwisely rotating, rigid tool is hidden to allow for a direct observation of the current stir zone lying underneath. In order to indicate the area of contact between the tool and the two workpieces, the edge of the tool shoulder is marked by a solid line. As can be seen from Fig. 7a, the sections of the abutting parts (blue and green color, respectively) that interact with the welding tool are moved around its pin (dashed line) and, with that, across the initial interface due to friction. That is why the process of material intermixing mainly focuses on the current stir zone, but also particles that are adjacent to the edge of the tool shoulder are affected by it to a certain extent. Since the rotating tool traverses the workpieces with advancing time, which means that it moves from left to right with a constant translational velocity of vw = 1.2 m/s in the present case, the region of the model featuring a high level of intermixing expands in this direction. In contrast to that, the areas of the workpieces located at some distance in front of the welding tool still exhibit the

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Fig. 7 Top view of the particle arrangement observed at different stages of the SPH simulation of the friction stir welding process (blue and green color, workpieces; red color, frame). (a) t = 1.5 s. (b) t = 2.75 s

initial particle arrangement as depicted in Fig. 7b. Therefore, it can be stated as a first result of the presented simulation of the friction stir welding procedure that, because of its meshless character and due to the fact that the introduced particles move along with the modeled material, the employed SPH model naturally provides a possibility for describing the intermixing of the two parts along a non-predetermined weld seam. However, it is further capable of reproducing the material flow appearing in the vertical direction, i.e. perpendicular to the bottom end of the tool shoulder. The particle distribution along the initial interface between the workpieces that is observed at the beginning of the simulation and the one at time t = 1.5 s are shown in Fig. 8 from the side. Here, all moving particles are colored according to their initial vertical positions. This means that the SPH particles belonging to the initial upper parts of the modeled workpieces are colored blue, whereas the ones representing the

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Fig. 8 Side view of the particle arrangement along the workpiece-workpiece interface observed at different stages of the SPH simulation of the friction stir welding process (blue color, initial upper parts; green color, initial lower parts of the workpieces; red color, frame). (a) t = 0 s. (b) t = 1.5 s

initial lower parts are assigned a green color as illustrated in Fig. 8a. As can be seen from Fig. 8b, also the initially upper and lower parts are intermixed by the rotating tool (solid and dashed line). Correspondingly, the already welded areas are characterized by a high degree of vertical intermixing, while the regions that have not yet been reached by the tool still feature the initial particle arrangement. Besides the level of intermixing of the workpieces, the temperature is the second decisive factor determining the microstructural evolution and, with that, the quality of the produced weld [47]. For this reason, the temperature distribution that results from the employed SPH model of the friction stir welding process is examined in the following.

6.1.2 Temperature Distribution The thermal conditions of the simulated workpieces at the two previously considered process times t = 1.5 s and t = 2.75 s are depicted in Fig. 9 from the top while coloring the particles according to their current temperatures. Again, the welding tool is not visualized, and the edge of its shoulder is marked by a solid line. As can be seen from Fig. 9, the maximum temperature exists close to the tool shoulder and the pin (dashed line) as indicated by the red color of the corresponding SPH particles. This is in agreement with what is reported in the literature (see for example [48]) and seems reasonable since those areas are simultaneously affected by both parts of the welding tool. Furthermore, the interpolation points that are located at some distance from the tool pin but are or formerly had been in direct contact with the tool shoulder show a high temperature due to friction and deformation as well. With an increasing distance from the current weld zone, the material temperature decreases as the color of the depicted particles changes from red over yellow and

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Fig. 9 Top view of the distribution of temperature over the workpieces observed at different stages of the SPH simulation of the friction stir welding process (blue color, low temperature; red color, high temperature). (a) t = 1.5 s. (b) t = 2.75 s

green to, finally, blue. Here, an almost symmetric distribution of temperature normal to the initial interface between the workpieces is identified. This corresponds to the experimental observations of the friction stir welding process presented in [48]. As can be further seen from Fig. 9, such symmetry is not found when studying the temperature distribution in the direction of welding. Because of the traverse motion of the tool, a high temperature is observed for the former stir zones, whereas the sections of the model that have not yet been welded exhibit a low temperature. This is not only the case with the near-surface regions of the workpieces but also the subjacent ones. To illustrate this fact, Fig. 10 provides the side views of the distributions of temperature along the initial interface at the two process times t = 1.5 s and t = 2.75 s. As can be seen from that, the highest temperature exists near the transition between the tool shoulder and the pin. The higher the distance of an SPH particle from this transition area in the direction of either the length or the

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Fig. 10 Side view of the distribution of temperature along the workpiece-workpiece interface observed at different stages of the SPH simulation of the friction stir welding process (blue color, low temperature; red color, high temperature). (a) t = 1.5 s. (b) t = 2.75 s

thickness of the workpieces, the lower its temperature while the interpolation points that are in direct contact with the tool feature a higher temperature than the surrounding ones. Based on the presented results, it can be summarized that the introduced SPH method for ductile solid continua is, at least from a qualitative point of view, wellsuited for accurately describing the process of friction stir welding in the simulation. Nonetheless, further studies on this topic that also include a quantitative assessment need to be carried out in the future.

6.2

Orthogonal Metal Cutting

The procedure of orthogonal metal cutting [49, 50] represents one of the most basic machining techniques used in industry to bring a metal workpiece (blue color in Fig. 11; measurements according to Table 4) into a desired shape. To that end, a wear-resistant tool (gray color) penetrates into the processed workpiece in a controlled way. This penetration results from a unidirectional relative motion between the workpiece and the cutting tool (from left to right in the present case), and causes the machined metal to locally fail in front of the cutting tool edge due to excessive shearing. Hereby, a certain amount of material, the so-called chip, is removed from the workpiece and its required shape is produced. In the specific case of orthogonal metal cutting, the cutting edge of the penetrating tool is perpendicular to the direction of tool respectively workpiece motion, what yields symmetry of the process setup with respect to the width of the workpiece. For this reason, the procedure of orthogonal metal cutting is of rather simple nature compared with other machining methods. Nevertheless, it is of industrial interest because of the several

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Fig. 11 Initial particle arrangement for the two-dimensional SPH model of the process of orthogonal metal cutting (blue color, workpiece; red color, clamp; gray color, tool)

Table 4 Model parameters of the simulated process of orthogonal metal cutting Property Initial length of workpiece Initial thickness of workpiece Depth of cut Cutting velocity

Symbol Gc,s,sim Tc,s,sim Hc vc

Value 1  102 9  104 3  104 1.6

Unit m m m m/s

characteristics it has in common with most other manufacturing processes based on the principal of material removal, e.g. large strains, high strain rates, and high temperatures of the machined workpiece. New insights into the physical mechanisms governing orthogonal metal cutting will, therefore, also enhance understanding of more complex machining processes and, thus, facilitate their optimization. If employing the presented SPH solid model, the behavior of a real machined metal workpiece can be accurately reproduced as will be demonstrated in this section. For this purpose, the results obtained from the SPH simulation of this industrial application are evaluated from both a qualitative and a quantitative point of view. In doing so, the focus is on the most important aspects of the considered problem, which are chip formation and chip morphology, stress distribution, cutting force, and temperature distribution.

6.2.1 Chip Formation and Chip Morphology Since the purpose of the machining technique of orthogonal cutting is to bring a workpiece into a desired shape by removing a certain amount of material from it, the chip formation and the chip morphology that result from the SPH model of this application are of utmost interest and, hence, studied first. To that end, Figs. 12 and 13 show four different stages of the simulated procedure of orthogonal metal cutting. As can be seen from Fig. 12b, the process of material separation and, with that, of chip formation due to local material failure in front of the cutting tool edge starts immediately after the tool has penetrated into the workpiece. With advancing

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Fig. 12 Distribution of the von Mises equivalent stress observed at two early stages of the SPH simulation of the process of orthogonal metal cutting (blue color, low stress; red color, high stress). (a) t = 0 s. (b) t = 7  105 s

simulation time (see Fig. 13a), the removed material first slides along the chip-facing surface of the cutting tool, which is typically called rake face [50]. Once the chip reaches a critical length, however, its tip begins to curl away from the rake face as illustrated in Fig. 13b. Even though this curling of the produced continuous chip is expected for the given set of process parameters and also observed in the experiment, it is not an effect to be easily reproduced in the simulation. For this purpose, a sophisticated material model that takes the strain history, the strain rate, and the temperature into account while predicting the behavior of the machined metal is required. The fact that the phenomenon of chip curling can indeed be simulated using the presented SPH method for ductile solid continua indicates the high quality of the proposed formulation.

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6.2.2 Stress Distribution The basis for the formation of a chip in metal cutting is the mechanism of local material failure due to excessive shearing that occurs in front of the tool edge. Accordingly, the maximum von Mises equivalent stress (red color) manifests in the primary shear zone [49], i.e. the area between the cutting tool edge and the free surface at the transition between the workpiece and the chip. This can be seen from the distribution of equivalent stress depicted in Fig. 13b. The primary shear zone, which is responsible for the separation of material, is surrounded by areas of midlevel stress as indicated by the yellow respectively green color of the corresponding SPH particles. As these regions are small in comparison with the measurements of the workpiece, high gradients in stress are observed in the vicinity of the primary shear zone. Thus, the von Mises equivalent stress decreases with an increasing distance from this area and reaches a low level (blue color) for the far-off parts of both the workpiece and the chip. An exception to this is the tool-averted surface of the produced chip, which keeps its high stress as a result of the effect of chip curling. As can be further seen from the different process stages illustrated in Figs. 12 and 13, the distribution of equivalent stress basically remains the same throughout the entire machining procedure. This means that the orthogonal cutting process reaches a quasi-steady state a short time after the tool has penetrated into the workpiece (at about t = 5.5  104 s in the present case). Therefore, also the cutting force obtained from the SPH simulation is almost constant during this stage as discussed hereinafter.

6.2.3 Cutting Force In Fig. 14, the temporal evolution of the experimentally measured cutting force for the considered setup (solid line) is depicted. As can be clearly seen, the experimental curve features a steep rise throughout the initial process phase, i.e. when the tool begins to penetrate into the workpiece. At a time of approximately t = 5.5  104 s, the orthogonal cutting process reaches the aforementioned quasisteady state. Accordingly, the force transitions into a fluctuation around a constant level of approximately 2250 N. Likewise, the simulated cutting force (dashed line), which has been scaled with respect to the width of the real workpiece (5  103 m for the present example), is characterized by a steep rise and a subsequent fluctuation around an invariant average level of force, too. Here, the curve resulting from the SPH simulation shows a straighter, i.e. more linear initial increase and a short-time overshoot in the force because the cutting tool is assumed to be a rigid body following a prescribed trajectory. So, the influence of the stiffness of the real machine on the evolution of the cutting force that becomes obvious when examining the experimental data is not considered in the simulation. However, this simplification affects the results quality to a significant extent only during the initial phase and is of minor importance as soon as the cutting process has reached its quasi-steady state. Then, a similar mean value and a comparable fluctuation range are found for the experimental and the simulated cutting force.

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Fig. 13 Distribution of the von Mises equivalent stress observed at two late stages of the SPH simulation of the process of orthogonal metal cutting (blue color; low stress; red color, high stress). (a) t = 1.2  103 s. (b) t = 2  103 s

6.2.4 Temperature Distribution Up to now, solely mechanical aspects of the scenario of orthogonal metal cutting, namely, the chip formation and chip morphology, the stress distribution, and the cutting force, have been studied. Whether the presented SPH solid model is able to predict the thermal condition of the experimentally machined workpiece as well is

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3000

cutting force (N)

2500 2000 1500 1000 500 0

experiment SPH 0

0.5

1

1.5

2

2.5

3

process time (s)

3.5

4

4.5

5

×10-3

Fig. 14 Agreement between the experimental and the simulated cutting force for the considered process of orthogonal metal cutting

evaluated in the following. To that end, the distribution of temperature that is found for the simulation of the orthogonal cutting process at time t = 2  103 s is depicted in Fig. 15. As can be seen from that, the maximum temperature (red color) exists neither close to the cutting tool edge nor in the primary shear zone as one could expect. Instead, it occurs at some distance from the tool edge along the rake face. This observation is in agreement with what is reported in the literature (see for instance [50]) and results from the friction between the tool and the workpiece. To explain this fact, the contributions to the thermal energies of the SPH particles that arise from friction are illustrated in Fig. 16. Exclusively those interpolation points that are located at distances smaller than their current smoothing lengths from the triangles representing the tool are in contact with it. Consequently, only these particles are heated due to friction (red color), while the remaining ones are not (blue color). However, also the SPH particles that are not in contact with the boundary structure at time t = 2  103 s exhibit changes in their temperatures, what is the result of an increasing plastic deformation. Since the area of largest increase in plastic strain is the primary shear zone, the highest rise in temperature caused by material plastification is localized there. To support this claim, Fig. 17 depicts the contributions to the thermal energies of the particles arising from the phenomenon of plastification. As expected, the largest increase in temperature induced by the plastic deformation of the metal is identified for the primary shear zone as indicated by the red color of the corresponding SPH particles. In contrast to that, the interpolation points that are located outside this region show a smaller rise in temperature due to plastification (blue color).

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Fig. 15 Distribution of temperature observed at time t = 2  103 s of the SPH simulation of the process of orthogonal metal cutting (blue color, low temperature; red color, high temperature)

Fig. 16 Distribution of increase in thermal energy due to friction observed at time t = 2  103 s of the SPH simulation of the process of orthogonal metal cutting (blue color, small increase; red color, large increase)

7

Conclusions and Future Directions

In this contribution, a numerical simulation model for ductile solid continua was presented. It is based on the meshless SPH spatial discretization technique. Accordingly, the developed formulation is naturally well-suited for the simulation of solids experiencing large deformations, major changes in topology, material failure including structure disintegration and/or a large number of contacts with the environment

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Fig. 17 Distribution of increase in thermal energy due to material plastification observed at time t = 2  103 s of the SPH simulation of the process of orthogonal metal cutting (blue color, small increase; red color, large increase)

occurring at the same time. For this reason, it has the potential to become a beneficial complement to the well-established numerical models, which mainly make use of the FEM. This was demonstrated on the basis of two examples, the industrial applications of friction stir welding and orthogonal metal cutting. In either case, the introduced SPH method for ductile solids was found to be capable of accurately describing both the mechanical and the thermal macroscopic behavior of the real material. To that end, the original SPH discretization scheme was variously extended and modified in the course of the present chapter. The presented enhancements for the SPH method further enable the detailed analysis of even more complex applications of industrial relevance than the manufacturing processes of friction stir welding and orthogonal metal cutting. One such example is the machining technique of oblique metal cutting. Since in this case the cutting tool edge is no longer perpendicular to the tool respectively workpiece motion, three-dimensional models are required for its numerical investigation. Currently, this intention is hampered by the significant increase in necessary computational effort that accompanies the aforementioned transition from two- to threedimensional simulations. One possible solution to this problem could be the use of an adaptive discretization scheme allowing to dynamically adjust the local resolution to the required level. The time- and space-dependent discretization of the model domain would allow to considerably improve the efficiency of SPH solid simulations so that acceptable computation times would be obtained even for such numerically challenging scenarios. Another interesting possibility that follows from the presented application of the SPH spatial discretization technique to cutting problems is the inclusion of coolant lubricants using a coupled SPH fluid/solid simulation. Furthermore, a coupling of the developed SPH solid formulation with mesh-based models, e.g. on the basis of the FEM, seems to be a very promising way of accurately reproducing the macroscopic

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behavior of real ductile solids while requiring a further reduced computational cost. This is due to the fact that such a hybrid approach would allow to combine the advantages of both kinds of modeling techniques, i.e. the mesh-based and the meshless ones, while compensating the disadvantages of the single methods at the same time.

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Simulation of Crack Propagation Under Mixed-Mode Loading

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Importance of Crack Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What Are Mixed-Mode Cracks? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Modeling Cracks and Their Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Crack Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mesh-Conforming Crack Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Extended Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Calculation of Stress Fields and Fracture Mechanics Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Stress-Field Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Accurate Calculation of Stress and Strain Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Calculating Fracture Mechanics Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Crack Propagation Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Maximum Circumferential Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Maximum Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Crack Tip Opening Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Cohesive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Considerations for Different Loading Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Propagating a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Mesh-Conforming Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Extended Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Crack Propagation in Different Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M. Bäker (*) Institut für Werkstoffe, Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] S. Reese Institute of Applied Mechanics, RWTH Aachen University, Aachen, Germany e-mail: [email protected] V. V. Silberschmidt Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, UK e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_29

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6.1 Effects of Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Functionally Graded Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Engineering components frequently contain cracks, either as an unavoidable consequence of their manufacturing (for example, pores in sintering processes or machining flaws) or due to processes occurring in service (cyclic loads, corrosive attacks, wear, etc.). Since it is not possible to completely avoid the formation of cracks, engineering safety requires to ensure that cracks do not lead to failure of a structure.

1

Introduction

1.1

The Importance of Crack Simulation

Engineering components frequently contain cracks, either as an unavoidable consequence of their manufacturing (for example, pores in sintering processes or machining flaws) or due to processes occurring in service (cyclic loads, corrosive attacks, wear, etc.). Since it is not possible to completely avoid the formation of cracks, engineering safety requires to ensure that cracks do not lead to failure of a structure. In some cases, it may be sufficient to know whether a crack will propagate because a component may be considered failed as soon as this happens. In many cases, however, it is necessary to actually compute a crack propagation path and rate. In critical components subjected to fatigue, for example, crack detection methods may be used in maintenance to ensure the absence of detectable cracks. It can then be assumed that a crack is present in the structure with a size equal to the detection limit of the method. The maintenance interval can be chosen so that this crack cannot become critical during this interval. Similar techniques are used in civil engineering to ascertain that cracked concrete structures remain serviceable. In many cases, the safety of a structure with regard to crack propagation can be ensured by employing simple analytical calculations. If, however, loading of the structure is complex, these simple approaches may not be sufficient. Since full-scale testing of components is often impractical or prohibitively expensive, simulation methods are needed to predict whether a crack can propagate leading to the component’s failure. In this review, the most important simulation techniques used in crack propagation analysis are discussed. An overview of this chapter is given at the beginning of Sect. 1.3.

46

1.2

Simulation of Crack Propagation Under Mixed-Mode Loading

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What Are Mixed-Mode Cracks?

During crack propagation, material separates so that new free surfaces are created on both sides of the crack. Three different load cases are distinguished. A crack can be opened (mode I) or the crack surfaces can be sheared in the crack direction (mode II) or perpendicular to it (mode III), see Fig. 1. In many cases, propagation in mode I is dominant and a growing crack reorients itself to propagate in this mode. However, if the local stress field is position-dependent, cracks may change their directions and propagate in mode II or III as well. To safely design parts that contain or may contain cracks, it is therefore necessary to be able to predict the propagation of cracks for different loadings. Except for simple situations, this can only be done numerically. At a crack tip, the stress field becomes singular (see Sect. 3.1). Therefore, it is convenient to use a coordinate system centered at the crack tip. Usually, a cylindrical coordinate system is used with r being the distance from the crack tip and θ the angle relative to the crack tip, see Fig. 2. If a crack is curved in three dimensions, the orientation of this coordinate system is adapted accordingly at each point of the crack tip. In most parts of this chapter, it is assumed that the material is linear-elastic with Young’s modulus E, Poisson’s ratio v, and shear modulus G = E/(2(1 þ v)). In two dimensions, calculations can be done in plane stress or plane strain. To allow for a unified description, the two-dimensional Young’s modulus E0 can be defined with E0 = E for plane stress and E0 = E/(1 þ v2) for plane strain.

(a) Mode I

(b) Mode II

(c) Mode III

Fig. 1 Load cases in fracture mechanics. Taken from [1] with kind permission of Springer-Verlag

Fig. 2 Cylindrical coordinate system at a crack tip

y r θ x

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Modeling Cracks and Their Propagation

The simulation of crack propagation problems encompasses three different aspects. First, the crack itself must be represented in the simulation model (Sect. 2). In standard FEM, for example, this is done by considering the surface of the crack as an external surface of the part under study, ensuring that the mesh conforms to the crack surface. Second, the stress and strain fields in the vicinity of the crack must be calculated correctly (Sect. 3). Due to the stress singularity that occurs at any sharp notch or crack, this calculation may require special techniques to resolve the singularities, especially if a high precision prediction of the local stress or strain field is needed. Third, the propagation of the crack must be calculated. This in itself comprises two tasks, namely, on the one hand, testing some crack propagation criterion to assess whether and in what direction the crack propagates (Sect. 4), and, on the other hand, implementing the crack propagation with the accompanying change in the part’s geometry in the model (Sect. 5). Issues involved in modeling complex materials like composites or concrete are discussed in Sect. 6. Due to the scales involved, crack propagation is usually modeled using continuum methods. In this review, the focus is on the finite element method as the most frequently used technique, not only due to its versatility, but also thanks to a number of commercial codes to model complex crack propagation cases being available. One alternative to standard finite elements are meshfree methods, described, for example, in [2, 3]. Boundary elements provide an alternative to the finite element method, especially in situations where deformations are small and linear elastic fracture mechanics can be applied [4, 5]. Crack propagation calculations can also be performed using a phase field method, see [6] for a review. It is also possible to model crack propagation employing atomistic methods with classical molecular dynamics (MD), see [7] for a review on purely atomistic largescale simulations. Due to the small scale of molecular dynamics simulations, it is often convenient to embed a local molecular dynamics domain at the crack tip in a finite element method on a larger scale [8–10]. Atomistic methods and their coupling to larger length scales are described in detail in [11]. However, these small-scale methods are not well suited for the analysis of actual components and are thus not considered further in this chapter. Crack formation and propagation may also be due to material degradation and damage, leading to a successive loss of load bearing capacity. Such damage models are reviewed in [12] and are not described in detail in this article. Some remarks on damage models can be found in Sects. 5.1.3 and 6.4. A comprehensive review of finite element simulations of crack propagation (in German) can be found in [13], and a short overview of the most important methods in [14]. A detailed overview of fracture mechanics in general is given in [15].

46

Simulation of Crack Propagation Under Mixed-Mode Loading

2

1469

Crack Representation

In this section, different ways to represent a crack in a finite element simulation are discussed. This problem is naturally linked to the implementation of crack propagation in a finite element simulation, but for conceptual clarity, a detailed discussion of the latter question is postponed to Sect. 5. Basically, there are two ways to represent a crack: the crack may either conform to the mesh or be modeled by enriching the shape functions of single elements to allow for arbitrary node separation; Fig. 3 gives a schematic overview of the techniques.

2.1

Mesh-Conforming Crack Representation

In a standard finite element implementation, the crack is treated in the same way as any other external surface, i.e., the mesh conforms to the crack surface. To implement crack propagation, special modeling techniques are needed. In a situation with branching cracks or several cracks growing simultaneously, the associated changes in the geometry and topology of the model may be rather complicated, see [16] for a detailed discussion. There are several possibilities to implement the necessary changes in geometry when a crack propagates. In a node-release scheme, cracks propagate between elements (i.e., nodes initially shared by neighboring elements are doubled and allowed to move independently). In Cohesive models, a similar strategy is used, but an interfacial behavior between the adjacent elements is implemented even before the elements become fully separated (see Sects. 4.4 and 5.1.4). In both cases, the crack is restricted to run between existing elements. To allow for arbitrary propagation directions, remeshing techniques are frequently used (Sect. 5.1). Another possibility to implement crack propagation is element removal. Here, structural elements of the modeled material are removed (their stiffness is set to zero) when a critical condition is fulfilled. Since any element inside a structure may

Node release

Cohesive elements

Element removal

XFEM

Fig. 3 Different methods to represent a crack in a finite element simulation. The first three methods are mesh-conforming, i.e., the crack surfaces coincide with the finite element mesh

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fail in this way, even complex cracks and multiple crack fronts can be modeled. However, the removal of elements is an unphysical process due to the loss of mass and energy (see Sect. 5.1.3 for further discussion). In the FEM implementation, both sides of a crack form a contact pair so that the movement of the nodes is unrestricted when the surfaces move away from each other, but contact forces prevent penetration of one crack surface into the other. If a crack propagates (see Sect. 5), the newly formed surfaces also need to become part of the contact surface definition. Appropriate contact surface definitions are available in all major finite element codes, although implementation details may differ.

2.2

Extended Finite Elements

The extended finite element method (XFEM) augments the standard FEM formulation by using additional shape functions Mji ðxÞ to calculate a displacement function u(x) from the nodal degrees of freedom [17–19]: uð x Þ ¼

X iI

N i ðxÞui þ

m X X j¼1 i  I



Mji ðxÞaji

(1)

j

The first term is the standard FEM formulation, where Ni are the standard FEM shape functions with node displacements ui summed over all nodes i of the model (collected in set I). The second term is the so-called “enrichment.” Additional degrees of freedom aji are added at some (but not necessarily all) nodes collected in node set I* with additional shape functions (enrichment functions) Mji ðxÞ . Frequently, more than one enrichment is used. The number of enrichment functions is m, with j denoting the index of the enrichment function. The XFEM method can be used to model weak and strong discontinuities in a wide range of applications [18–20]. In crack propagation modeling, two different types of enrichment can be used: one to model the crack opening itself and the other to improve the accuracy of the displacement field near the crack tip. The second aspect is covered in Sect. 3.2.2. Mathematically, a crack line can be represented using the level set method [18, 21], where the zero level of a set of functions ϕk(x) is used to determine a geometrical region. Two functions are needed to characterize an unbranched crack, see Fig. 4. The first function (normal level set ϕn) is defined so that its zero level coincides with Fig. 4 Definition of the level set functions ϕn and ϕt to characterize a crack. After [18]

φt0 φn>0 φn 0

(2)

as a global enrichment function. To use this function in Eq. (1), local enrichment functions Mi(x) are defined using the node shape functions Ni: Mi ðxÞ ¼ H ðϕn ðxÞÞN i ðxÞ:

(3)

This approach relies on the fact that these shape functions form a partition of unity. Usually, only those elements that are cut by the crack are enriched. To ensure correct behavior of the adjacent elements (“blending elements”), the enrichment functions may be “shifted.” Details on this method can be found in [18]. In a standard finite element, integration points (also called Gauß or material points) used to numerically integrate the virtual-work equations are situated at fixed positions inside the elements. If an element is cut by a crack or another discontinuity, numerical integration using these points becomes problematic because the underlying assumption in standard integration schemes is that integrated functions are smooth. Therefore, elements containing discontinuities are subdivided to enable correct numerical integration. This will be explained in more detail in Sect. 5.2 in the context of crack propagation.

3

Calculation of Stress Fields and Fracture Mechanics Parameters

3.1

Stress-Field Singularity

An ideal crack tip with plane crack surfaces has no inherent length scale. On a sufficiently small length scale, the externally imposed stress field can thus be considered as homogeneous, so that the stress and deformation state near the crack tip is characterized by a small number of parameters. Because stresses become singular near a crack tip, load components that do not cause deformation in one of the three modes play only a minor role.

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Table 1 Stress and displacements for pure mode I, II, and III cracks in plane strain [23] Mode I

Mode II Mode III     K II pffiffiffiffi 2θ θ r θ 2θ u cos sin 1  2v þ sin ¼ 2  2v þ cos u x = uy = 0 x 2π 2 2 2π 2 2 G     qffiffiffiffi K I pffiffiffiffi K II pffiffiffiffi 2θ r θ 2θ r θ uy ¼ G 2π sin 2 2  2v þ cos 2 uy ¼ G 2π cos 2 1  2v þ sin 2 uz ¼ KIII 2r sin π G ux ¼ KGI

pffiffiffiffi r

uz = 0 I σ xx ¼ pKffiffiffiffiffi cos 2πr

θ 2

I σ yy ¼ pKffiffiffiffiffi cos 2πr

θ 2

 

1  sin

θ 2

sin

3θ 2

1  sin

θ 2

sin

3θ 2

σ zz = v(σ xx + σ yy) I sin σ xy ¼ pKffiffiffiffiffi 2πr

θ 2

cos

θ 2

cos

σ xz = σ yz = 0

(a) Mode I

3θ 2

 

uz = 0 II sin σ xx ¼ pKffiffiffiffiffi 2πr

θ 2

II sin σ yy ¼ pKffiffiffiffiffi 2πr

θ 2



2 þ cos

cos

θ 2

cos

σ zz = v(σ xx + σ yy)  θ 2 1  sin

II σ yy ¼ pKffiffiffiffiffi cos 2πr

θ 2

cos

3θ 2



θ 2

sin

σ xx = σ yy = σ zz = σ xy = 0 III sin θ2 σ xy ¼  pKffiffiffiffiffi 2πr

III σ yz ¼  pKffiffiffiffiffi cos 2πr

3θ 2

3θ 2

θ 2

θ 2



σ xz = σ yz = 0

(b) Mode II

(c) Mode III

Fig. 5 Vector plots of the displacement field near a crack tip for mode I–mode III calculated from Table 1 using v = 0

Hence it is possible to characterize the stress field near a crack tip by three numbers, the stress intensity factors KI, KII, and KIII for the three modes. For the case of linear elastic material behavior and small deformations (linear elastic fracture mechanics, LEFM), the resulting stress and displacement fields are listed in Table 1. Figure 5 shows the displacement fields for all three modes. The stress intensity factors for the three modes depend on the crack length a as pffiffiffiffiffi K i ¼ σ  πaY ðaÞ,

(4)

where i denotes the mode, σ  is the appropriate stress component in the far field for the considered mode, and Y(a) is a factor characterizing the crack geometry. Crack propagation criteria frequently use critical values of the stress intensity factors KIc, KIIc, and KIIIc. These factors are material constants. For most materials, the critical mode for crack propagation is mode I. Instead of the stress intensity factors, the energy release rate can be used to characterize the state of a crack (Griffith criterion). If a crack propagates by an amount da, the stored elastic energyU(el) in the material changes and work W may be done by external forces. The energy release rate is defined as

46

Simulation of Crack Propagation Under Mixed-Mode Loading

g¼

1473

  1 dW dU ðelÞ  , t da da

(5)

where the energy is nomalized to an infinitesimal crack extension and to thickness t [1]. If a crack propagates purely in mode I, II, or III, the energy release rate is denoted by gI, gII , gIII , respectively. For loading in a pure mode, the energy release rate and the stress intensity factor are related by mode I : K 2I ¼ gI E0 ,

(6)

mode II : K 2II ¼ gII E0 ,

(7)

mode III : K 2III ¼ gIII

E0 , 1þv

(8)

For mixed-mode loading, the energy contributions of these modes are added: g ¼ gI þ gII þ gIII ¼

 1 2 1 2 2 K 0 K I þ K II þ E 2G III

(9)

The energy release rate is useful because a crack can only propagate if the energy released upon propagation is larger than the energy needed to create new surface on the two crack faces and the energy dissipated in the vicinity of the crack tip by plasticity or other mechanisms (this region is called the process zone). If this energy is assumed to be independent of the loading mode (as in the case of a purely elastic material where energy is only needed to create new surfaces), there is one critical value gc of the energy release rate, which can be used to predict crack propagation. To determine whether and in which direction a crack propagates, it is necessary to calculate fracture mechanics parameters such as stress intensity factors. This, however, is difficult due to the stress singularity. To alleviate this difficulty, the finite element formulation can be adapted to represent the singularities more accurately than possible with standard elements. Two ways of doing this (using quarter-point elements or the extended finite element method) are discussed in the next section. Alternatively, special techniques like the J-integral (see Sect. 3.3.3) that do not rely on an accurate representation of the stress field close to the crack tip can be used to calculate energy release rates.

3.2

Accurate Calculation of Stress and Strain Fields

3.2.1 Quarter-Point Elements In linear pffiffi elastic fracture mechanics, the displacement near the crack tip behaves as u  r , see Table 1. Second-order elements can represent this behavior [24]. Consider an 8-node element in two dimensions. If the nodes along one edge of the

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7

3

3 7

8

6

4 8 1

6 5

1

5

2

¼L

¾L

2

Fig. 6 Node positions in a collapsed second-order quarter-point element. One edge of the element is collapsed to a point. Mid-side nodes of the adjacent edges are shifted to quarter-point positions. The right-hand figure shows the arrangement of these elements around a crack tip. After [25]

element are collapsed to a single point and if the nodes on the adjacent edges are placed at 1/4 of the edge length towards the collapsed nodes (see Fig. 6), the displacement field behaves in the desired way if the three collapsed nodes are fixed to have the same pffiffi displacement. Strains calculated from this displacement field show the correct 1= r -behavior. Such elements are called quarter-point elements. If the material is ideally plastic, the strain singularity changes to e~r1. This behavior can also be modeled with the collapsed elements, provided that the three collapsed nodes are allowed to displace independently. Quarter-point elements can be easily generalized to three dimensions: One face of the element is collapsed to a line to form a wedge-shaped element [26]. Using these collapsed quarter-point elements, the stress and strain fields near the crack tip can thus be calculated accurately. However, to derive fracture mechanics quantities like the stress intensity factor from local stress fields, it is still necessary to use a rather fine mesh (at least six elements around the crack tip are recommended to resolve the angular dependence), whereas other methods, based, for example, on the J-integral (Sect. 3.3.3), frequently yield good results with coarser meshes. Furthermore, quarter-point elements are not easily suited for crack propagation calculations because the position of the crack tip is fixed by the collapsed nodes. Therefore, to allow for crack propagation, remeshing is necessary. Despite these complications, commercial software tools using collapsed elements and remeshing are available and are widely used to simulate crack propagation problems [16, 27, 28].

3.2.2 XFEM for Crack Tip Displacements It was shown in Sect. 2.2 that a crack can be represented using an enrichment of the finite elements with a globally defined Heaviside function. Although this method allows to represent a discontinuous crack displacement, it cannot represent the actual displacement near the crack tip very well. Therefore, the stress singularity is also not represented correctly. This problem can be remedied by using additional enrichment functions in the vicinity of the crack tip, based on the analytically known displacements close to the crack. A set of four global enrichment functions γ is usually chosen (note that the order of these functions is different in different references, here we follow [21]). In polar coordinates with the origin at the crack tip, these functions are (see Fig. 7)

46

Simulation of Crack Propagation Under Mixed-Mode Loading 1.5 1 0.5 0 -0.5 -1 -1.5

1.5 1 0.5 0 -0.5 -1

1

0.5 y

0

-0.5

-0.5

-1 -1

0

0.5

1.2 1 0.8 0.6 0.4 0.2 0

1.5 1 0.5 1

1

0.5 y

x

0

-0.5

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

0.5

0.5 y

0

-0.5

-0.5

-1 -1

-1 -1

-0.5

0

0.5

1

x

(b) γ2

(a) γ1

01

1475

0

0.5 x

1

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

1 0.5 0 -0.5 1 0.5 y

(c) γ3

0

-0.5

-1 -1

-0.5

0

0.5

1

x

(d) γ4

Fig. 7 Global enrichment functions for the displacement function [29]

 γ ð xÞ ¼

 pffiffi pffiffi θ pffiffi θ pffiffi θ θ r cos , r sin , r sin sin θ, r cos sin θ : 2 2 2 2

(10)

The derivatives of the global enrichment functions have a singularity near the crack tip in order to represent the strain singularity [17]. Local enrichment functions Mji ðxÞ are calculated from γ via Mji ðxÞ ¼ γ j ðxÞN i ðxÞ,

(11)

similar to Eq. (3). The region of enrichment for these functions can be chosen either as the single element that contains the crack tip or as a set of elements surrounding the crack tip within a certain radius. Note that the level set functions employed to describe the crack tip can be directly used to calculate r and θ. A review on crack simulations using the XFEM method can be found in [20].

3.3

Calculating Fracture Mechanics Parameters

To determine whether a crack would propagate and to calculate the direction of its propagation, fracture mechanics parameters such as the energy release rate or the stress intensity factors are needed. These quantities are then compared to corresponding material parameters like the critical stress intensity factor KIc or the

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critical energy release rate gc . (Crack propagation criteria are discussed in Sect. 4.) Stress intensity factors can be calculated directly from the stress field, see Sect. 3.3.1. To determine the energy release rate, the virtual crack-closure technique (VCCT, Sect. 3.3.2) or the J-integral (Sect. 3.3.3) is usually employed. It is possible to extract stress intensity factors from the energy release rate or the J-integral as explained in Sect. 3.3.3. Note that these techniques are independent of the representation methods described above, i.e., VCCT or J-integral can be used with standard finite element meshes, collapsed quarter-point elements, or enriched XFEM elements. In [26], an overview of different methods to extract energy release rates and stress intensity factors from finite element simulations is given.

3.3.1 Calculation from Displacement or Stress Fields Stress intensity factors can be determined by comparing the analytical equations for displacements or stresses from Table 1 to the calculated field in the FE simulation. In the displacement correlation method [30], the displacements are evaluated at different distances r from the crack tip. For mode I, the transversal displacement uy on the crack line is related to KI by (see Table 1) pffiffiffiffiffi 2π E uy pffiffi : KI ¼ 4ð 1  v 2 Þ r

(12)

In a finite element simulation, uy is evaluated at different distances from the crack tip. This results in a series of different values for the stress intensity factor. From this series, a limiting value for r ! 0 can be calculated and used as KI. Similar formulae hold for KII and KIII. Instead of the displacement field, the stress field can also be used to calculate the stress intensity factors in a similar way [30]. If quarter-point elements pffiffi are used, the local displacement field near the crack tip can correctly represent the r-behavior so that KI can be calculated directly from the quarternode elements [31].1 There are different methods to do this (called displacement correlation technique or quarter-point displacement technique), depending on which nodes of the quarter-point elements are used in the calculation. This technique to calculate stress intensity factors is employed in the commercial software FRANC3D [27].

3.3.2 Virtual Crack-Closure Technique The virtual crack-closure technique is frequently used to calculate mode-dependent energy release rates. Its history and implementation are discussed in detail in [25] on which this section is based. Consider a sufficiently finely meshed finite element model of a crack in two dimensions. One conceptually simple way to calculate the energy release rate is to actually release one node in the finite element model at the crack tip and thus to propagate the crack by a finite increment. The energy before and after the crack Note that, somewhat confusingly, the technique called “displacement correlation method” in [30] is termed “displacement extrapolation method” in [31].

1

46

Simulation of Crack Propagation Under Mixed-Mode Loading

Fig. 8 Calculation of energy release rate from nodal quantities near the crack tip in the VCCT method. After [25]

1477

δa

F2 δu1 δu 2

F1

F1

F2

propagation can then be compared. This two-step method has sometimes been used in the past [25, 32]. However, it has two disadvantages. It requires two finite element calculations so that the computational effort is large. Furthermore, it suffers from the dilemma of discretization: since the crack length increases by a finite amount, the calculated change in the model’s energy approximates the true energy release rate only if the crack length increment is sufficiently small. However, the smaller the increase in crack length, the smaller the change in the energy stored in the model. In a large model, care must therefore be taken to ensure that the energy change is not obscured by the finite precision of each of the energy calculations since a small difference between two large numbers is calculated numerically. Instead of actually changing the crack length, it can also be changed virtually. The underlying assumption is that the stress and deformation state near the crack tip does not alter significantly if the crack length changes by a small but finite amount δa. The energy released on extending the crack by δa can thus be considered to be equal to the energy needed to close the crack by the same amount. Consider a pure mode-I loading of the crack. To close the crack by an amount of δa, the required energy will be approximately given by the opening of the crack at distance δa from the crack tip and the force component on the node at the crack tip, see Fig. 8: gI ¼ 

1 F2 δu2 : 2δa

(13)

A similar equation can also be introduced for the mode-II energy release rate. (Note that the implementation is slightly more involved if standard or quarter-point collapsed second-order elements are used, see [25] for details.) Therefore, the nodal forces at the crack tip and the displacements at the opened nodes directly adjacent to the crack tip can be used to calculate the energy needed to close the crack. This method is thus called the virtual crack-closure technique (VCCT). The VCCT relies on the principles of linear elastic fracture mechanics and is thus especially suited for calculations in brittle materials. It allows to directly calculate the

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Fig. 9 Coordinate system and integration contour for the definition of the J-integral. Adapted from [1] with kind permission of Springer-Verlag

C

x2

x1 ds

n

energy release rate and thus the stress intensity factors for the different modes in one simulation. Since the forces and displacements in the direct vicinity of the crack are used, an accurate calculation of these quantities is required.

3.3.3 J-Integral The J-integral is a function of the stress and displacement fields that can be derived from the stress energy tensor [1]. For a two-dimensional crack, it is defined as Z J¼ C

 

@u wdy  σ   nds : @x

(14)

Here w is the energy density, σ is the stress tensor, and u is the displacement field. The J-integral extends over a contour C with line elements ds and a normal vector n, see Fig. 9.2 The value of the J-integral is independent of the geometry of the contour, provided that the following conditions are fulfilled [14]: • there are no time-dependent processes like creep (if creep is present, a generalized concept, the Ct-integral, can be used instead [33]). • there are no volume forces. • deformations are small (the nominal strain can be used as strain measure). • the material is homogeneous and can be described by a hyperelastic material law σ ij = @W/@eij, where W is an energy function and « is the strain tensor (see Sect. 6.3 for the case of a nonhomogeneous material). • the crack runs parallel to the x1-axis. • stresses and strains do not depend on x3. It should be noted that n is perpendicular to the contour, not parallel to it as in standard contour integrals because the integration domain is actually a two-dimensional slice taken out of a threedimensional surface integral.

2

46

Simulation of Crack Propagation Under Mixed-Mode Loading

1479

Under these conditions, the value of the J-integral is identical to the energy release rate upon crack propagation: J ¼ g. Because the J-integral is independent of the geometry of the contour, the energy release rate can be determined from the J-integral using a contour that does not run close to the crack tip. Therefore, inaccuracies in the calculated stress and strain fields due to the singularity at the crack tip that is difficult to represent using standard finite elements exert only a small influence on the calculated value of g. This allows to calculate the energy release rate without the special techniques described in Sect. 3.2 while still getting good accuracy. In a finite element simulation, evaluating results along an arbitrary path may be affected by the mesh structure because variables at the points on the path have to be interpolated from nodes or integration points. Therefore, the contour integral is usually converted into a domain integral using Gauß’ theorem [34]. Generalizations of the J-integral concept for other material laws (e.g., power law creep) can also be made. These concepts are discussed in [33]. Although the J-integral can be compared to a critical energy release rate to determine whether a crack propagates, evaluation of the direction of crack propagation needs additional information. Several of the criteria discussed in Sect. 4 require knowledge of the stress intensity factors. In a linear elastic material, the J-integral is related to the stress intensity factors by the following relation [35, 36]: J¼

1 T 1 K B K, 8π

(15)

where K = (KI, KII, KIII)T is the vector of stress intensity factors and B is the socalled pre-logarithmic energy-factor matrix. For an isotropic linear elastic material, this equation simplifies to Eq. (9): J¼

 1 2 1 2 2 K : 0 K I þ K II þ E 2G III

(16)

Since Eq. (15) is an equation with three unknowns, it is not directly possible to extract the values of the stress intensity factors from the value of J. To do so, an additional load case is considered where an auxiliary deformation field (e.g., a pure mode-I displacement, stress and strain fields) is superimposed on the current solution. Due to linearity, the effect of this solution on the J-integral can then be calculated using the so-called interaction integral method. The procedure is detailed in [17, 35, 36]. As stated above, the J-integral is only path-independent if the material can be described by a hyperelastic material. In elastoplastic materials, the J-integral can be used only if loading is proportional so that the theory of deformation plasticity is valid. If unloading of a structure occurs, for example due to a propagating crack, the J-integral becomes path-dependent. Several generalizations of the concept have been proposed in the past, a brief review is given in [37]. The theory of material forces (also called configurational forces) also allows for generalizations of the method to situations where the standard formulation of the J-integral is problematic, see, for example [38–40].

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4

Crack Propagation Criteria

A crack propagates when a parameter determined by a load on a structure exceeds a critical value, which can be considered as material (or interface) parameter. In linear elastic fracture mechanics, the stress intensity factor or the energy release rate is usually used. If a significant amount of energy is dissipated in a process zone in front of the crack tip, as in the case of plastic deformation, the assumptions of LEFM do not hold. In this case, other criteria may be needed to determine whether a crack can grow. In a crack propagation analysis, it is not sufficient to determine whether a crack will grow, because the direction of propagation has also to be predicted. Therefore, additional criteria are needed. In a two-dimensional analysis, the direction of crack propagation is determined by a single number, the kink angle, whereas in three dimensions, a crack may propagate in a different direction at each point of the crack tip line. In addition, the crack may not only kink but also tilt, see Fig. 10. There are two general ways to determine the direction of crack propagation: Criteria may be based either directly on the local stress or strain field or on quantities like the stress intensity factors that can be determined globally, for example, using the J-integral (see Sect. 3.3.3). Local criteria require a sufficiently good resolution of the stress field (see Sect. 3.2). There is a large number of different criteria, see, for example, [15, 42, 43]. In the following, some of the most frequently used criteria suitable for LEFM are presented. Furthermore, the critical crack opening displacement, suitable for crack calculations in plastically deforming materials, and damage-based cohesive models are discussed.

4.1

Maximum Circumferential Stress

The maximum circumferential stress criterion assumes that a crack (in two dimensions) propagates in the direction where the circumferential stress is maximum. To evaluate this, a small circle can be centered at the crack tip and the stress along the circumference of the circle can be calculated. The difficulty with doing this directly from the calculated stress field is that stresses are only known at the integration points, hence interpolation is required. This leads to a strong mesh dependence. An

crac

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ac kp lan

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Fig. 10 Kinking and tilting of a crack in three dimensions. After [41]

cra

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improved way to calculate local stresses, using maximum principal stresses calculated at the integration points, is presented in [42]. Alternatively, the criterion can be rewritten using the stress intensity factors. The kinking angle η can be calculated from [43] K I sin η þ K II ð3 cos η  1Þ ¼ 0:

(17)

The maximum circumferential stress criterion is not suitable for problems where mode III is relevant because this mode does not affect circumferential stresses [44]. A similar scheme is the criterion of vanishing KII value. Here, it is assumed that a crack propagates in the direction where KII = 0 [45]. This criterion can also not be used in cases with mode-III loading. If mode-III loading occurs, other criteria can be employed as explained in the next section.

4.2

Maximum Energy Release Rate

Physically, it is plausible that a crack will propagate in the direction that maximizes the energy release rate. The energy release calculated, e.g., from the Jintegral, however, does not take the direction of crack propagation into account. To do so, it is possible to actually extend the crack in different directions and measure the energy release rate (similar to the two-step method described in Sect. 3.3.2). In two dimensions, finding the direction of maximum energy release rate is a onedimensional optimization problem, but its solution needs several finite element simulations for each crack propagation step and is thus rather expensive [32]. The method has the advantage that it does not rely on any assumptions on the material behavior. In two-dimensional problems, it is possible to analytically calculate the direction of crack propagation in a linear elastic material and also at an interface between two such materials from the known stress intensity factors KI and KII [46, 47]. For a crack in a homogeneous material, the difference between this analytical solution for the maximum energy release rate and the KII = 0-criterion from the previous section is small [47]. An approach that is suitable also for mode III loading is presented in [44]: There, an approximate equation for the angle-dependent energy release rate was derived and used to calculate the kinking angle η:    

κþ1 2 η 3η 3η 3η η K I sin þ sin  K 2II 3 sin  5 sin þ 4K I K II cos 8 2 2 2 2 2 η (18) þ K 2III sin ¼ 0, 2 where κ = 3  4v for plane strain and κ = (3  v)/(1 þ v) for plane stress. For the case with vanishing mode III, this equation reduces to Eq. (17).

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4.3

Crack Tip Opening Displacement

In ductile materials, the crack tip region plastifies, blunting the initially sharp crack tip. Since the plastic zone is large in such materials, it can be assumed that crack propagation is dominated by plastic deformation. This deformation can be measured by the crack tip opening displacement (CTOD) or the crack tip opening angle (CTOA). The crack tip opening displacement can be calculated geometrically by drawing two lines at the blunted crack tip at 45 to the crack’s symmetry plane, see Fig. 11a. The distance between the intersections of these lines with the crack surfaces is then used as a measure of crack opening [1]. Because of significant plastic deformation near the crack tip, it is important to use nonlinear geometry and large-scale deformations in the calculations. Furthermore, measuring the CTOD accurately depends on a good local resolution of the displacement field; therefore, quarter-point elements are recommended [14]. The crack tip opening angle is calculated as the angle between the crack surfaces near the crack tip, see Fig. 11b. To implement this in a finite element code, the crack opening can be measured at a fixed distance from the crack tip and a critical value for this opening can be given [36]. This technique is frequently used for thin plates [14]. Provided the plastic zone is small compared to the crack length or other geometrical quantities (small-scale yield), the CTOD can be related to the stress intensity factor by the relation δ ¼ K 2I =ðσ Y E0 Þ , where σ Y is the yield stress under the assumption that the material is ideally plastic [13]. It is difficult to experimentally measure the CTOD or CTOA directly. Therefore, the crack opening is usually measured at the end of a specimen (e.g., a CT specimen) with the values at the crack tip being back-calculated from this. Thus, experimental values can be used directly without the need for any simplifying assumption concerning the size of the plastic zone.

a

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CTOD

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Fig. 11 Determination of crack opening using the crack tip opening displacement δt or the crack tip opening angle α

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The CTOD criterion can also be generalized to a mixed-mode situation (mode I and II) by introducing the crack tip shear displacement, measuring the relative movement of points on the crack surface in the local x-direction [48].

4.4

Cohesive Models

Cohesive models provide a different approach to calculate whether a crack propagates. Instead of using a single critical quantity like the energy release rate and propagating the crack if the critical condition is fulfilled, a method based on damage mechanics is employed. The (predefined) crack line is considered as a kind of adhesive material so that the crack can open by a certain amount and still transfer loads. These loads are then ramped down in a way that ensures that the total energy needed to open the crack is equal to the critical energy release rate. Cohesive models therefore are both a method to calculate whether a crack opens and to implement the actual crack propagation (see Sect. 5.1.4). They are suitable for a wide range of applications where energy is dissipated during crack propagation, see Fig. 12. Using a cohesive model thus allows implemention of a process zone in front of the crack tip where the material is damaged. Therefore, this technique is well suited to problems with significant energy dissipation. A further advantage of cohesive surface modeling is that cracks can be initially fully closed – thus it is suited for modeling damage processes of initially crack-free materials.

Fig. 12 Schematic of a cohesive crack opening for various failure phenomena. Taken from [49] with kind permission of Elsevier and the author

1484 Fig. 13 Separation of two surfaces with a cohesive element. A local coordinate system is used to split the separation into its normal and tangential parts. After [14]

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continuum elements

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Two slightly different approaches can be used to model cohesive behavior of the interaction between two surfaces. One possibility is to introduce a layer of elements with finite thickness between the two material parts that are initially bonded (cohesive elements) [50] and the other is to start with surfaces that are initially in contact with no separation (cohesive surfaces, also called cohesive elements with zero thickness) [51]. Both methods are similar in their behavior, some remarks on their differences are made below. A traction-separation law is used to describe the mechanical behavior of the cohesive model. Since the initial distance between the two crack surfaces is small or zero, using strains to quantify the behavior upon separation is not meaningful. Therefore, the separation δ = u+  u is used, where u+ and u are the displacements between adjacent nodes on the two surfaces, see Fig. 13. This separation is calculated in a local coordinate system so that normal and tangential components can be distinguished. A surface traction vector t acts at each point of the cohesive surface. The work of separating the two surfaces by a total separation of δt is thus given by Z W¼

δt

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(19)

0

The function t(δ) is the traction-separation law. In mode I, only the normal components of this law are relevant, so that the traction-separation law is a scalar relation. Usually, two quantities are used to quantify the traction-separation law: The total area under the traction-separation curve, equal to the critical energy release rate, and a critical separation, for which the traction value vanishes. The shape of the traction-separation curve is chosen to be a simple function, see Fig. 14 for examples. More complicated relations with additional parameters can also be used. The maximum traction in a traction-separation law can be considered as a critical stress: if this stress value is reached, damage occurs in the cohesive zone, reducing the load bearing capacity of the interface. In mixed-mode loading, tangential movement of the two surfaces can occur. As long as the traction is below the critical value, the behavior of the interface is linearly

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Fig. 14 Traction-separation laws for cohesive zone modeling. Traction and separation are normalized. After [50, 52–54]

elastic in most of the traction-separation laws (see Fig. 14) and can be specified by a stiffness matrix S: t ¼ Sδ:

(20)

If cohesive methods are used to model crack propagation within a single material, the stiffness of the cohesive interface has to be defined so that correct bulk behavior is recovered in the undamaged regions. In the cohesive surface method, this implies that the stiffness must be large compared to the elastic stiffness [51, 55]. This, however, results in a stiffness matrix with a large condition number, potentially impeding convergence of the calculation. If cohesive elements are used, their thickness also plays a role: very thin elements are needed to ensure small deformation in the elements and correct stress transfer between the regions. The resulting large aspect ratio sometimes also causes numerical problems [51]. These problems can be circumvented with a remeshing technique, with cohesive elements introduced only at an interface after fulfilling the condition for damage initiation [56]. Damage occurs if a critical value of the traction is reached. To calculate this value in a mixed-mode situation, different criteria can be used. [50] uses the separation in tangential and normal directions, δt and δn, as a criterion:

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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  n m  t m ffi δ δ m D¼ þ t , n δ0 δ0

(21)

where δn0 and δt0 are the critical separations in the two directions and m governs the interaction between the modes. For m = 1, their interaction is linear, whereas for m ! 1, the modes do not interact. Alternatively, similar laws can be used for the tractions instead of the separations. Although cohesive elements and cohesive surfaces are similar in their behavior, there are some differences [36]: Cohesive elements are implemented similar to other finite elements. They have a finite thickness even without onset of damage so that there is always a small separation between the two contacting surfaces. This should be considered in defining the traction-separation law. Furthermore, cohesive elements have integration points, usually situated on the edges of the element normal to the crack surface. The calculation of the traction and stresses is performed at these points, whereas it is usually done at surface nodes (on the slave surface) in a cohesive surface.

4.5

Considerations for Different Loading Regimes

4.5.1 Fatigue It is usually impractical to directly model the processes occurring during fatigue failure in a finite element simulation. On the one hand, the number of cycles can be of the order of millions or more; on the other hand, the crack’s advance during each cycle is usually less than 1 μm and is dominated by local plasticity, so that an extremely fine mesh would be needed to correctly resolve the processes at the crack tip. In fatigue loading, the stress intensity factor at the crack tip varies proportionally with the load. Although it is the maximum of the stress intensity factor that governs fatigue crack propagation, usually the stress intensity range ΔK = Kmax  Kmin is used, together with the ratio R = |Kmin|/|Kmax| to characterize fatigue loading [1]. Crack propagation in fatigue loading is described by the crack growth rate da/dN, defined as the crack growth da per cycle. If the maximum stress intensity factor is below a threshold value, there is no fatigue crack propagation. Therefore, a threshold value ΔKth can be defined. On the other hand, if the maximum stress intensity factor exceeds the critical value for static crack propagation, the crack growth rate becomes infinite. A schematic crack-growth-rate curve is shown in Fig. 15. In a wide range of ΔK-values (region II in the figure), the crack growth rate can be described by a power law, called the Paris law [1] da ¼ CΔK n , dN

(22)

where C is a constant, depending on the material and the R ratio, and n is an exponent. Typically, n lies between 2 and 7 in metals, whereas it can be much larger in brittle materials. There is a large number of different crack growth laws to describe the crack growth rate in the full range of ΔK-values [13].

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Fig. 15 Crack-growth-rate curve, plotting da/dN versus the cyclic stress intensity factor ΔK in a double-logarithmic plot. There are three characteristic regions; region II is described by the Paris law Eq. (22). Adapted from [1] with kind permission of Springer-Verlag

For mixed-mode loading, a simple equation to define the mixed-mode stress intensity range is [57] ΔK mixed ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p m m m ΔK m I þ ΔK II þ ΔK III ,

(23)

where m governs the mode interaction, similar to Eq. (21). Fatigue cracks initially grow in the plane of maximum shear stress (i.e., at an angle of 45 to the loading direction in uniaxial loading). If a crack exceeds a critical length (usually between 0.05 and 2 mm [1]), its orientation changes and the crack plane aligns with the maximum normal stress plane so that the crack is loaded in mode I. Still, mixed-mode situations can occur because a growing crack can propagate through a complex stress field. A comprehensive overview of failure criteria for multiaxial fatigue to determine the crack growth direction can be found in [58]. To use fatigue crack growth laws in a finite element analysis, a load cycle is modeled, and the crack propagation speed is calculated at each node of the crack front. Because an increase in the crack length also increases the stress intensity factors at the crack tip, an integral equation can be used to calculate the expected growth rate for a certain number of cycles. For the Paris law, this equation is [1, 13] Z Δa ¼ C

N 0

Z ½ΔK ðaÞm dN ¼ CΔσ m

N pffiffiffiffiffi

m πaY ðaÞ dN:

(24)

0

Here, Δσ is the stress range and Y(a) is the length-dependent geometry factor. The crack is then advanced using the techniques described in Sect. 5 and the next step is calculated. For a precise simulation, the crack increment must be sufficiently small so that the stress distribution at the crack tip does not change significantly.

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4.5.2 Dynamic Effects Dynamic (i.e., inertia) effects have to be considered in a crack propagation analysis if the cracked structure is loaded dynamically or if the change in the kinetic energy of the material during crack propagation is significant. The latter situation occurs mainly in brittle materials where dissipation effects are small. Dynamic crack propagation is frequently unstable so that cracks do not stop. A detailed review of dynamic fracture mechanics can be found in [59] and [60], a good introduction in [13]. In dynamic crack propagation, a dynamic stress intensity factor Kdyn can be defined. It depends on the crack speed v and the static stress intensity factor K(a) via K dyn ¼ kðvÞK ðaÞ:

(25)

Here, a is the crack length and k(v) is a function of the crack velocity that does not depend on the crack length or details of loading [61]. Analytical solutions for displacement, velocity, and stress fields, analogous to those in Table 1, are given in [62]. Crack propagation criteria can be chosen in the same way as in a static simulation; frequently, the maximum circumferential stress criterion is used. If a J-integral is used to calculate the energy release rate or stress intensity factors, the kinetic energy in the integration region has to be taken into account, see [13] for a detailed description of the method. Similar to a fatigue analysis, the crack propagation speed has also to be calculated. There are two ways of doing this: on the one hand, the velocity can be chosen according to Eq. (25). On the other hand, it can be assumed that the crack speed is proportional to the excess in circumferential stress between current and critical values. The latter approach has the advantage that it is valid even if the function k(v) has a small or vanishing slope [62]. One important aspect in dynamic crack propagation is the phenomenon of crack branching. The stress field changes with increasing crack velocity so that the maximum circumferential stress is not located directly in the crack line even in mode-I loading if a critical speed is exceeded [63], see Fig. 16. If crack propagation in the direction of maximum circumferential stress is assumed (see Sect. 4.1), kinking and branching of the crack should be expected. Branching of cracks is indeed frequently observed in this situation, although the excess of kinetic energy also plays an important role. Although the situation is symmetric, inhomogeneities in a real material often cause one of the branches to propagate faster than the other so that the crack tip of the slower branch is partially unloaded and frequently stops. One way to simulate crack branching is to introduce cohesive elements during the simulation at appropriate positions between elements, see, for example, [64–66]. The inhomogeneity can be introduced by randomly varying the properties of the cohesive elements. Since inertia effects are important in dynamic crack propagation and simulated times are usually short, an explicit finite element method can be used. In defining material behavior in a dynamic crack propagation simulation, it should be kept in mind that the strain rate near the crack tip is very large even at moderate propagation speeds. Therefore, it is important to account for rate-dependent behavior, for

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Fig. 16 Polar plot of the circumferential stress for different values of the crack speed v/cs (cs is the wave speed). For small crack speeds, the maximum occurs at kinking angle 0 , and for large values of the speed, the maxima lie at a large angle away from 0 . The small inner loops are due to compressive stresses near polar coordinate θ = π. From [60] with kind permission of the American Physical Society

example rate-dependent hardening in plastic materials. Such hardening can cause brittle behavior, reducing energy dissipation [13]. Extended finite element methods are frequently used for dynamic crack propagation, see, for example, [67, 68]. In [69], the XFEM is used to calculate crack propagation in elastoplastic materials. In [70], crack simulations using a Gurson–Tveergard–Needleman damage model and an XFEM model are compared for static and dynamic cracks.

5

Propagating a Crack

Using the methods described in the previous sections, it can be determined whether a crack will propagate and what the direction of propagation is. Although it is often appropriate to assume failure of a component as soon as a crack starts propagating, this is not always the case. In a complex loading situation, cracks can form in a highly stressed region and then propagate to a region of lower stresses with an insufficient driving force for further propagation. In fatigue problems, service intervals can be calculated based on the crack propagation rate as already explained in the introduction. In some components, it may also be necessary to know how a crack propagates through the structure during failure. In this section, different techniques employed to implement crack propagation are discussed, see also Fig. 3. Cracks can conform to newly formed free surfaces of a

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standard finite element model, either by opening the structure between two elements or by element removal. To accurately represent the stress fields, remeshing procedures may be used to ensure that there is always a fine mesh in the region of the crack tip even when the crack grows through the structure. If cohesive models are used, cracks also propagate between elements. Alternatively, the XFEM can be used to propagate cracks directly through the mesh.

5.1

Mesh-Conforming Cracks

5.1.1 Node Release In a standard finite element simulation (see below for cohesive models) with a meshconforming crack representation (see Sect. 2.1), initially bonded nodes of adjacent elements may be released to propagate the crack. If the crack path is known beforehand, element boundaries can be placed accordingly when the finite element mesh is constructed. Alternatively, the structure can be remeshed so that nodes are placed on the calculated crack line and then released. Consider first the case of a simple node release along a predefined crack path. A finite element mesh can be constructed with element boundaries along this path. The nodes of the elements on both sides of the path initially coincide and are bonded by a constraint condition so that their displacements are identical. If the crack propagation criterion is fulfilled, this constraint can be removed so that the nodes can move independently. Note that at the initial crack position, quarter-point elements may be used, but they can obviously not be used everywhere along the crack line when nodes are released. It is possible to use remeshing techniques to alleviate this problem, see Sect. 5.1.2. Due to the stress singularity, a considerable force acts on each of the crack tip nodes to be released. Simply removing the constraint equation from the system of equations to release the nodes might therefore be problematic because a large change in nodal forces makes the problem highly nonlinear and can lead to convergence problems and unstable behavior. Therefore, the force between the nodes may be “ramped down,” i.e., the force at the beginning of the crack propagation time increment is reduced over a short time interval. If the problem is static and the material behavior is not time-dependent, this technique usually does not cause any problems. However, in a dynamic problem or when material parameters strongly depend on time (as in viscoelastic or viscoplastic materials), the results may be affected by the choice of the ramping-down method and time interval. In these cases, care should be taken to ensure reliability of the results by varying these conditions. If quadratic elements are used, it is not advisable to release only an edge node, see Fig. 17. Due to the quadratic approximation of the element boundary in an isoparametric element, a material overlap can occur, invalidating the results [25]. For quadratic elements, it is therefore better to release two nodes simultaneously, so that the edges of two adjacent elements are either fully bonded or fully released. If first-order elements are used in a propagation analysis, crack propagation criteria that do not rely on the evaluation of the local stress field (like the J-integral,

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b bonded nodes released nodes

c

Fig. 17 Node-release technique. (a) General principle of node release along a predefined crack line (dashed red line). (b) If 8-node elements are used, opening of a corner node only leads to incompatible deformation where the shape function (in blue) penetrates the surface. (c) If both corner and edge nodes are released simultaneously, deformation is compatible. Figure 17b, c after [25]

see Sect. 3.3.3) can be used to avoid inaccuracies stemming from the stress singularity. After crack propagation, the released nodes become part of the crack surfaces. Therefore, they need to be included into the surface definition preventing penetration of the two crack surfaces if crack closure is possible.

5.1.2 Remeshing Techniques For a crack propagating in an arbitrary direction, it is not possible to define an appropriate mesh beforehand. Instead, after the crack propagation direction is calculated, a new mesh has to be defined. Results for stresses and displacements are then transferred from the previous calculation step by interpolating the appropriate values from the old nodes and integration points to the new positions. Care has to be taken in regions of large gradients in the fields or the internal parameters because this mapping procedure introduces slight inaccuracies and may dampen these gradients by numerical diffusion.

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Remeshing the structure has the additional advantage of allowing adaptive mesh refinement. As the crack propagates, regions adjacent to its faces become unloaded so that the mesh density can be reduced there, whereas stresses in a region with low initial stress become large when the crack tip approaches it. As stated above, it is not possible to have quarter-point elements everywhere along a crack line, hence calculation of stress fields can be problematic. One possibility to circumvent this problem is to proceed in the following way: quarter-point elements are situated at the initial crack tip. If a critical condition is fulfilled and the crack propagates, a new mesh is introduced where the crack has advanced by a finite amount (usually several element lengths) along the calculated direction. This new mesh is constructed with quarter-point elements centered around the new crack tip to accurately calculate the stress field for prediction of the next crack gowth increment (see, for example, [42]). A remeshing technique can also be used to determine the direction of crack propagation. To do so, test cracks are propagated in different directions, and the test crack maximizing the energy release rate is accepted as the next crack increment [32]. This technique has the advantage of being capable to model crack propagation near or at interfaces flexibly; however, it involves a high computational cost and is not easily generalized to three dimensions. In a case of complex geometry, especially in three dimensions, remeshing of a full component may not be feasible. In this case, remeshing can be restricted to the vicinity of the crack tip. The commercial software ZENCRACK [28, 71] uses a technique where a structure is first calculated without cracks. Selected hexahedral elements are replaced by predefined “building blocks,” containing a crack. Remeshing is then performed within these blocks so that the overall mesh of the structure does not need to be changed. A slightly different approach is used in the software FRANC3D or FRANC3D/NG [27]: here, a submodeling technique is used with part of the original mesh being replaced by a finely meshed submodel, which is then changed to contain the crack. During crack propagation, the submodel is remeshed, always using quarterpoint elements at the crack tip. Calculations are always performed on the full model so that the effect of the crack on the structure is accounted for [72]. A similar approach is also used in the freely available FE software CalculiX [73].

5.1.3 Element Removal Techniques Another possibility to enable crack propagation is to delete elements from the calculation. In the simplest implementation, standard elements are deleted as soon as a criterion (e.g., critical plastic deformation) is fulfilled and their stiffness is then set to zero. This technique is frequently used in simulations of forming processes like machining. Although it allows for complex cracks with multiple crack fronts, its disadvantage is that removing elements is basically unphysical because mass and energy are removed from the structure. Furthermore, the shape of the crack tip in an element removal scheme is determined by the shape of the removed element, so that the stress state in the vicinity of the crack is not predicted precisely. When elements are removed from the simulation, it is important to ensure that the resulting free surfaces are part of the contact pairs of the crack surface to resolve stresses correctly if the crack closes.

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Element removal is frequently used in the context of damage models, see for example [12, 74]. In a finite element simulation, local damage models frequently exhibit strong mesh dependence because a finer mesh causes earlier failure of an element due to the stress singularity. Therefore, nonlocal damage methods are frequently employed. Alternatively, the mesh size itself may be considered as a model parameter [12].

5.1.4 Failure in Cohesive Models As explained in Sect. 4.4, cohesive models allow for a gradual failure of an interface between two material regions, implemented either with elements of finite thickness or as cohesive surfaces with zero initial nominal thickness. During failure at a point or element, stresses transmitted between the two regions continuously decrease until complete failure and no stress transfer. The newly formed surfaces then have to be treated as contact surfaces similar to those in the node separation or element removal techniques. One advantage of cohesive models in comparison to node-release methods is that there is no need to start the simulation with a directly introduced initial crack as usual for node-release techniques. The two surfaces joined by cohesive elements or cohesive surface definition can fail at any point so that there is no need to know the positions of the most highly stressed regions beforehand. This makes cohesive methods especially suitable for situations with complex stress fields. However, crack propagation is still restricted by the mesh geometry. If the crack direction is not known beforehand, cohesive elements can be added between all volume elements of a structure [75], but the computational cost associated with this is large. Furthermore, it is difficult to define cohesive behavior to ensure correct bulk behavior in the uncracked region. Alternatively, cohesive elements can be introduced into a model only at positions where conditions for damage initiation are fulfilled [65]. Another possibility is to use cohesive methods in the context of the XFEM technique (see the next section).

5.2

Extended Finite Elements

In XFEM, the crack is described by the two level-set functions (see Sect. 2.2) ϕt and ϕn. If a crack propagates, these functions have to be updated. An algorithm for this is described in [22]: Basically, the tangential function ϕt is updated so that its zero line lies at the new position of the crack tip. New values for ϕn are calculated in the region of crack growth to accurately represent the crack position, without any change in ϕn in the region where the crack was already present. If crack tip enrichment functions (see Sect. 3.2.2) are used, the location and orientation of the crack tip are updated in these functions. As mentioned in Sect. 2.2, elements cut by a crack have the problem that numerical integration using standard integration points is not possible. Therefore, these elements have to be subdivided. The position of the crack line is calculated from the values of the level set functions stored at the nodes of enriched elements. In an element with linear shape functions (triangular in two or tetrahedral in three dimensions), the crack line is then always linear. However, in a bi- or trilinear element, the mapping from the

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=

+

Fig. 18 Crack propagation using phantom nodes. When the crack is closed, phantom nodes are constrained to move with the real nodes; upon opening of the crack, the phantom nodes can move independently. After [36]

regular (square or cubic) master element to the actual shape of the isoparametric element causes the calculated crack line to curve in the element. The subdivision of the element for numerical integration has to take this into account. Details on the procedure can be found in [18, 19], an extension of the method to three dimensions is shown in [29]. The subdivision for an element in which the crack tip is located is different [19]. Here the current position of the crack tip is used, and the element is then divided by lines from the crack tip to the element’s corners. Integration points are centered on the crack tip with a distribution that concentrates at the singularity, similar to arrangement of nodes and integration points in a quarter-point element. One problem of the subdivision method is the need to transfer material properties to the newly calculated integration points [18]. An alternative method based on Gauss quadrature without subdivision that avoids these problems is presented in [76]. A simpler possibility to implement crack propagation is to use phantom nodes [77]. In this method, it is assumed that elements can only be cut fully by the crack so that its tip never lies within an element. To represent the crack, phantom nodes are superimposed on the original nodes on both sides of the crack line, see Fig. 18. These nodes are constrained to move with their corresponding real nodes when the crack is still closed. Upon its opening, the constraint is released so that the nodes can move independently. If first-order reduced-integration elements are used, each of the split elements employs its copy of the integration point without the need for subdivision. However, this method does not account for the stress field singularity near the crack tip. The phantom node technique has been implemented in a commercial code [36].

6

Crack Propagation in Different Materials

So far, generic tools to implement crack propagation simulations were described. However, complex materials can require special techniques or additional considerations to correctly model the behavior of cracks. Some of these issues are discussed in this section.

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Effects of Microstructure

To understand material behavior and to design new materials, it is crucial to correlate the microstructure of a material with its mechanical behavior. This can be achieved by simulations at the microscale. Reviews of this topic can be found in [78, 79]. Possibly the simplest approach to simulate microstructures is the unit cell approach [78]. A single inclusion, void, or fiber is simulated in a unit cell of appropriate volume. Periodic boundary conditions are employed to avoid surface effects and simulate bulk behavior. Alternatively, an embedding method is used where the cell is embedded in an “equivalent material” with properties that are determined in a self-consistent manner, see [80] for an example. Although this technique offers some insights into material behavior, it is limited because using a simple unit cell does not account for the random distribution inherent in microstructures. To create more realistic microstructures, micrographs can be digitized so that the randomness of the structure is represented [78]. Digitized images then serve as basis for a computational model as a representative volume element (RVE). There are two problems related to this: standard micrographs only give a two-dimensional view of the structure so that effects of the third dimension, which frequently are important [79], are neglected. This problem is avoided by using three-dimensional images created with a computer tomography analysis or by sampling the three-dimensional structure with a focused ion beam. The second problem is that boundary conditions are difficult to define from such micrographs. If the volume element is supposed to represent a sample of a bulk material, periodic boundary conditions should be used, but real-world images are not periodic. In RVE-based calculations, this is frequently circumvented by performing a statistical analysis of several images and then recreating a realistic microstructure using a numerical model that ensures a correct distribution of the microstructural components. In the context of grain simulations, for example, a Voronoï tesselation can be used [81]. Crack propagation in microstructural simulations is frequently implemented using the element removal method (Sect. 5.1.3), see, for example, [74]. If the focus is on crack propagation and damage at the interface between different phases or at grain boundaries, cohesive elements (Sect. 5.1.4) can be used [82, 83]. Cohesive elements can be introduced throughout the structure to allow for arbitrary crack paths [84]. For interface cracks, simple node debonding (Sect. 5.1.1) can also be used [85]. In [86], remeshing techniques using the software Franc2D are employed. Microstructural simulations of crack propagation using nanoscale techniques like discrete dislocation dynamics or molecular dynamics are reviewed in [79].

6.2

Composites

Predicting delamination is an important factor in the design of composite materials like fiber-reinforced polymers. Often, layered composites are used, in which each layer contains uniaxially arranged fibers, with layers differing in the fiber direction. In such composites, delamination between layers is an important failure mode.

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In principle, all major crack modeling techniques discussed in this chapter can be used to model delamination [87]. Due to the bond between matrix and fiber, progressive damage occurs during crack propagation; hence this failure mode is highly suitable for modeling with cohesive or other damage methods. Reviews of such methods can be found in [88, 89]. To describe delamination behavior, customized constitutive equations can be created, see, for example, [87, 90]. In [91], a representative volume element is used to derive a material law for a composite that can be directly used as a solid-shell element in finite element simulations. In [92], an interface model using a smoothly varying constitutive tensor allowing for a unified treatment of homogeneous and layered materials is presented. Bone is a natural composite material of great interest in medical engineering. It has a complex hierarchical structure, reviewed in [93]. At the nanoscale, bone is a composite of collagen, hydroxyapatite, and water. These components are arranged in mineralized collagen fibrils forming larger fiber bundles. Fiber bundles are arranged in layers with different organizations depending on the type of bone under consideration. In human (or, generally, mammalian) cortical long bone (like humerus, femur, or tibia), the layers are arranged in the longitudinal direction. Blood vessels in the bone are surrounded by circular layers, so-called osteons. During life, bone is remodeled with more osteons forming so that the microstructure mainly consists of overlapping osteons with interstitial lamella between them. Due to the laminated microstructure, crack propagation during bone fracture is highly anisotropic [94]. Crack propagation parallel to the layers has a considerably smaller critical energy release rate. The interface between an osteon and the surrounding matrix (cement line) can debond so that cracks frequently do not run through osteons. If a crack propagates perpendicularly to the layers (the most frequent case for long bones), it is deflected from a straight path due to the anisotropy so that the critical energy release rate is considerably larger. Finite element models of bone fracture can be created at different length scales [94, 95]. At the macroscale, the material is described using a bulk material law. Crack propagation is simulated using cohesive elements [96] or XFEM [95]. In [97], toughening mechanisms due to fiber bridging and crack deflection of a propagating crack were studied and their contribution to the critical stress intensity factor was quantified. Microscale models aim to study the effect of the lamellar structure and the osteons on crack propagation. In [98], a unit cell approach is used to study the interaction of a crack with an osteon. Crack propagation is simulated using a remeshing procedure. [95] and [99] use a more realistic microstructure obtained from micrographs and the XFEM approach to study crack propagation between osteons. Crack growth in the lamellae of a single osteon and the influence of lacunae (micro-pores containing bone cells) is modeled in [100] using the node-release technique.

6.3

Functionally Graded Materials

Cracks in functionally graded materials often propagate in a mixed-mode regime. Even if the external load is perpendicular to the crack line (so that, nominally, loading

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is in mode I), the local change in Young’s modulus effectively causes a mixed-mode loading at the crack tip. Fracture mechanics of such materials is discussed in [101]. One important consideration is that the graded properties add another length scale to the problem so that the standard analytical expressions (Table 1) for the stress field near the crack tip are only valid in regions where material properties do not change considerably. It is important to note that the standard formulation of the J-integral is not valid in a functionally graded material since the value of the J-integral becomes path-dependent [102]. An additional term containing the gradient of the elasticity tensor is needed to restore path independence. A detailed analysis of stress intensity factors for different orientations of a crack and a material gradient is presented in [103]. A method to calculate stress intensity factors using an interaction integral method (see the end of Sect. 3.3.3) for graded materials is described in [104]. Different crack propagation criteria are evaluated in [105]. Displacement correlation methods (Sect. 3.3.1) yield good results to calculate fracture mechanics parameters. Different crack propagation criteria were employed, with good agreement of resulting crack paths. A material property gradient strongly influences the crack path. Functionally graded materials can also be simulated based on their microstructure, see, for example, [106].

6.4

Concrete

As already mentioned, a correct prediction of crack propagation in concrete structures is of great interest in civil engineering. Because of its microstructure, concrete is a material with a complex behavior differing strongly in tension and compression. During loading, microcracks can form and grow in concrete. These microcracks cause a reduction in stiffness, but a microcracked region can still have a load bearing capacity due to crack bridging or interlocking of the crack surfaces. Therefore, modeling techniques based on damage models are well suited to simulate crack propagation in concrete. Cohesive methods (Sect. 4.4), also called “discrete crack models,” have long been used for this problem [107, 108]. These models are suitable when there is only a small number of dominant cracks propagating in the structure so that damage processes in regions away from these cracks are of lesser significance. However, in many cases, failure of concrete structures is due to a large number of distributed cracks instead of being dominated by a single crack. Therefore, continuum damage models are also used. In the context of concrete modeling, these methods are frequently called “smeared-crack models” [109, 110]. Because microcracks are mostly perpendicular to the direction of maximum tensile stress, the damage process is anisotropic with a loss in stiffness occurring mainly in this direction. To account for this, a local coordinate system is introduced and stiffness is reduced only in the appropriate directions. A detailed description of different damage models can be found in [110]. Although discrete and smeared-crack models are conceptually different at first sight, it is actually possible to unite these schemes using a partition-of-unity

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approach as detailed in [111]. This can be done by introducing cohesive segments in a way similar to introducing cracks in an XFEM approach (Sect. 2.2). Although damage processes are important in concrete, they become less relevant at large crack sizes where linear elastic fracture mechanics may be more appropriate to describe crack propagation. As shown experimentally in [108, 112], crack propagation in concrete has a strong size dependence and can be described with LEFM when the size of the structure and the crack are sufficiently large.

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Ye. Ye. Deryugin, G. V. Lasko, and Siegfried Schmauder

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Relaxation Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Relaxation Element as a Specific Defect in Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Connection of Plastic Deformation with the Tensor of Stress Relaxation . . . . . . . . . . 3 Stress-Strain State of the Continuum with the Site of Plastic Deformation . . . . . . . . . . . . . . . 3.1 Examples of the Site of Plastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Construction of Relaxation Elements with Gradients of Plastic Deformation . . . . . . 4 Stresses in a Continuum with Band Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bands of Localized Plastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Plastic Deformation in the Site of Elliptical Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Field of Stresses from the Site of Elliptical Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 REM in the Models of Localization of Plastic Deformation and Fracture . . . . . . . . . . . . . . . . 5.1 Peculiarities of the Simulation by the Relaxation Element Method . . . . . . . . . . . . . . . . 5.2 Modelling of Localization of Deformation in Polycrystals by RE-Method . . . . . . . . 5.3 The Influence of the Rigidity of the Testing Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Accounting for Edge Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Jump-Like Propagation of the Band of Localized Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Self-Organization of Plastic Deformation in Polycrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Application of the Relaxation Element Method to the Investigation of the Mechanisms of Plastic Deformation and Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Stresses of Lüders Band Initiation in Polycrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Modified Model of Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Y. Y. Deryugin (*) Institute of Strength Physics and Materials Science, SB RAS, Tomsk, Russia e-mail: [email protected] G. V. Lasko · S. Schmauder Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_30

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Abstract

In this chapter a new method – the relaxation element method is justified. The definition of the changing of stress fields in solids under loading as a result of the change of elastic energy in a local volume, undergoing plastic deformation, is laid down at the basis of the method. The Relaxation Element Method (REM) solves effectively two problems of a deforming solid (DS): 1. The construction of the different distributions of plastic deformation in local regions of various geometrical shape. 2. Modelling of the consequent involvement of separate structural elements into plastic deformation, operating on the principle of an inverse task of mechanics of deforming solids. With this method a stress-strain state of the elastic plane with the sites of plastic deformation in the form of a circle, rectangle, and a localized shear band is analytically described. Examples of the construction of the sites of plastic deformation with gradients are given. The stress-strain state of a plane with a round inclusion is considered. Examples of the simulations by the REM of the effects of Lüders band formation and interrupted flow in polycrystals are given. The analysis of the influence of rigidity of a testing device on qualitative and quantitative characteristics of the loading diagram is presented. The effect of the gradients of plastic deformation on the stress of Lüders band initiation is analyzed. It is shown that the dependence of the stress of Lüders band initiation on grain sizes is the consequence of the independency of the gradient of plastic deformation under the changing of grain sizes. A modified model of Griffith crack surrounded by a layer of plastically deformed material is proposed. Plastic deformation is shown to eliminate the singularity at the crack tip. The maximum stresses are observed at the boundary of the plastic zone in an elastically deformed matrix. The stress concentration increases as the thickness of the plastic layer decreases. The obtained results testify to high predictable possibilities of the developed method. They are in a good agreement with known experimental data.

1

Introduction

The development of plastic deformation in the solid is known to proceed inhomogeneously in space and irregular in time. This property defines the process of plastic strain localization, developing in the course of loading of structurally

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inhomogeneous materials. The evolution of nonhomogeneous distribution of plastic deformation is realized under the influence of various stress concentrators, caused by the inhomogeneity of the initial structure of the material and changing of stress state of the loaded system under prescribed boundary conditions. In the course of plastic deformation of local volumes, the continuous changing of the fields of internal stresses in the whole volume of the solid takes place. Technological and strength properties of the material depend on the character of the process of plastic strain localization and material degradation. In this connection there is an actual problem of the description and simulation of the process of plastic strain localization and changing of the stressstrain state of a structurally inhomogeneous medium with the site of plastic deformation under the different boundary conditions of loading. The description and simulation of these effects demand accounting for the multilevel character of the development of the processes in the deformed system, where the surface layer and internal interfaces affect the process of plastic deformation and fracture [1–4]. The evolution of an inhomogeneous distribution of plastic deformation is realized under the influence of different stress concentrators, and first of all under the influence of stress concentrators at the free surface of the solid. The pattern of macrolocalization of plastic deformation inherits the character of the distribution in the thin near-surface layer. On the macrolevel one can select three types of macrolocalization of plastic deformation: Lüders band propagation, Portevin–Le Chatelier effect (PLC), and neck formation at the stage of prefracture [5–9]. For the description and simulation of the effect of intermittent flow, the relaxation element method (REM) developed by the present authors is suitable and promising [9–14]. The change in stress field in the solid under loading as a result of the decrease in elastic energy in the local volume, undergoing plastic deformation, lies on the basis of the method. Plastic deformation is the relaxation process in its nature. That means that in the site of plastic deformation a decrease in elastic energy takes place, i.e., the decreased level of stress is observed in comparison to the average stress level beyond the site of plastic deformation. For the fulfillment of this condition in the relaxation element method the notion of “relaxation tensor” is used, which characterizes a decrease in the field of elastic stresses in the given local volume as a result of its plastic deformation. The above approach allows solving effectively two problems of a deformed solid: a) The description of the stress-strain state of the solid with the sites of different geometrical shape and with different distributions of the plastic deformation. Results are obtained using the known technique of continuum mechanics of a deformed solid and are represented in the form of analytical expressions for the components of the tensor of plastic deformation and stresses. b) Simulation of the sequence of the involvement of separate structural elements into plastic deformation. Stress relaxation in local volumes changes the stress state in the whole volume of the solid. Thus, structural elements, having undergone plastic deformation in the surrounding of the elastically deformed matrix, play the role of the defect on a mesoscopic scale with its own field of internal stresses.

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The presence of defects in a continuum is related to the existence of inhomogeneous stress fields in a volume of the material. Plastic deformation is the consequence of the collective movements of defects of different scale level. At the same time, the linear dependence of the components of the stress tensor breaks. There arises a fundamental problem of definition of the unambiguous connection between deformation and stresses. In the linear theory of elasticity, such a connection is represented by the generalized Hooke’s law. In the continuum theory of defects [15, 16, 17], this obstacle is removed by the condition that the tensor of total deformation is equal to the sum of elastic eeij and plastic epij deformation: eij ¼ eeij þ epij :

(1)

The tensor of total deformation eij is assumed to obey the compatibility condition. Thus, in the general case the tensors eeij and epij could be incompatible. But the incompatibility of the tensor of elastic deformation is fully compensated by the incompatibility of the tensor of plastic deformation and vice versa. In particular, both eeij and epij could be compatible. Then, the stress state of the medium will be exclusively defined by the tensor of elastic deformation eeij , connected with stresses through the generalized Hooke’s law: σ ij ¼ С ijkl eeij :

(2)

Hence, according to the accepted assumptions, compatible plastic deformation cannot be connected with the stresses existing in the volume of a deforming solid. In other words, without additional conditions a compatible plastic deformation cannot be represented as an analytical function of the coordinates. In the continuum theory of defects, a compatible and incompatible plastic deformation is calculated only in connection with the presence of the internal stress fields in each volume of the material. Investigations show that the absolute majority of the defects at the stage of developed plastic deformation disappear as a result of annihilation, absorption at the sinks and interfaces, and the exposure at the external surface of the solid [18]. Together with deformation defects the stresses caused by the presence of the latter in the volume of the material disappear as well. Traditional theories of plasticity take the macroscopic diagrams of work-hardening of real materials [19, 20, 21] into account, when defining the relation of the stresses in the local volume of a solid with plastic deformation in it, i.e., the integral (macroscopic) characteristics of mechanical properties are prescribed to the local volumes of the solid. Such an approach doesn’t fully account for the relaxation nature of plastic deformation. On the other hand, continuum theories consider the field of plastic deformation in the local volumes of a solid in the form of defects with corresponding fields of internal stresses [15, 16, 17]. However, at that time, the problem of the connection of these fields

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with the boundary conditions of loading of the solid and with the geometrical parameters of the initial microstructure of the material is not considered. In order to reveal this connection, let us consider the physics of the phenomenon. According to the modern conception of physical mesomechanics [2–4], a physical reason of plastic flow in real materials is the mass transfer under the influence of an external applied stress. The creation and movement of dislocations occurs practically at the very beginning (onset) of loading, when due to the difference in elastic properties and anisotropy of structural constituents of the solid, the peak stresses (stress microconcentrators) arise in the volume of solid. Mass transfer causes continuous stress relaxation in the local volumes of the material. A different strain rate of plastic deformation in the local volumes results in its inhomogeneous distribution over the whole volume of a solid. In many cases with a definite accuracy one can select a local volume, which undergoes plastic form changing. In such a case, one can state that plastic strain localization occurs. Very often such local volumes are called the sites or zones of plastic deformation, or the zones of plastic shearing. Experiments show that there are definite peculiarities of the formation of the zones of localized plastic deformation at each stage of loading [22–27]. A classical example is the Lüders band at the stage of its formation and propagation in polycrystals with highly pronounced yield stresses [28, 29]. The sources of movable deformation defects are, first of all, the free surface of the solid and various interfaces (grains, phases). Therefore, by prescribing the field of plastic deformation within these boundaries, we in principle take into account an initial structure of the material. Taking additionally into account the hierarchy of the structural levels of deformation [2–4], and the notions of physical mesomechanics, the site of plastic deformation should be considered as a volume defect of a higher scale level than microdefects such as dislocations and vacancies. On this level, being commensurable with the elements of the initial structure (on mesolevel) bands of localized plastic deformation (LPD) form. On the background of a quasi-homogeneous distribution of plastic deformation, the bands of localized plastic deformation could be considered as defects of a corresponding mesoscale. In polycrystals, for example, a single grain could be considered as a structural element of mesoscopic scale. The bands of localized plastic deformation can be considered as defects of a higher mesoscale if they embrace a conglomerate of grains. If the zone of localization spans over the region being commensurable with the width of the specimen, such a defect is necessary to be considered as a defect of macroscopic scale. Below we consider the construction of the inhomogeneous distribution of plastic deformation on meso- and macrolevel, when the substitution of the real displacement field (with microscopic discontinuities from the movement of microdefects, such as dislocations) with the continuous one could be justified. This allows one to use fully the known methods of continuum mechanics when constructing an inhomogeneous distribution of plastic deformation in local regions and to calculate the corresponding stress concentrations in the solid at a definite scale level under prescribed boundary conditions of loading.

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In this work, the relaxation element method (REM) is theoretically justified. This method allows one on the data of the sites of localized plastic deformation with a definite degree of accuracy to calculate the components of inhomogeneous stress fields caused by these sites. Besides that, in combination with other methods of analytical and numerical calculations of mechanics of deforming solid, this method allows by variation of structural, geometrical, and physical parameters to model the processes of the development of localized plastic deformation and at that time to check the validity of certain mechanisms of plastic deformation and fracture of real materials. The idea of REM arose due to the authors’ attempt to relate the change in plastic shape of a local volume of continuum with the value of stress drop or decrease in elastic energy in it. The phenomenon of elastic stress relaxation within the solids has long been known [30–33]. Its essence lies in the following: If a loaded sample is deformed up to a definite value, then at a fixed position of clamps the load will decrease in the process of time. The physical reason for stress relaxation, which is especially pronounced at elevated temperature, is the increase of the contribution of plastic deformation to the total deformation of the sample with time. Since the latter keeps constant, then with time the contribution of elastic deformation correspondingly decreases locally. A decrease in elastic deformation in the considered case is possible only under the decrease in applied stress in this volume. Analogous processes, apparently occur in local volumes of a solid, undergoing plastic flow. However, in literature the analysis of the correlation between physical processes, especially the process of plastic deformation and the value of relaxation within the local regions of a solid has received insufficient attention.

2

Relaxation Element Method

2.1

Relaxation Element as a Specific Defect in Continuum

Plastic deformation is the phenomenon of essentially relaxation nature. Hence, plastic (irreversible) form changing of the local volume in a loaded solid should be accompanied by stress relaxation in the given volume and arising of an inhomogeneous stress field in the whole volume of the solid. The process of formation of an inhomogeneous distribution of plastic deformation in continuum can be represented as a result of consequent chain of elementary acts of stress drop in the various local volumes of solid. Since the stress state of a solid is a tensor characteristic, then the value of stress relaxation in the local volume should be characterized by the tensor. In this connection, let us introduce the notion “relaxation element” (RE) as a specific defect in continuum. Tensor of relaxation Δσ r for such a defect should be unambiguously connected with the plastic formchanging of the local volume, i.e., inside this volume the field of plastic deformation, characterized by tensor Δep, obeying the plastic displacement of the points at the external boundary of the given local volume should be assigned. Thus in general case, relaxation element is characterized by the geometrical shape of a local relaxation region and relaxation tensor Δσ r, defining the field of internal stress outside and the field of plastic deformation Δep inside the given region.

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One can imagine that in the process of loading in the volumes of a solid a number of elementary acts of stress relaxation in different local regions Ri occurs. By prescribing the distribution of the relaxation elements in the volume of material, we prescribe a definite distribution of plastic deformation, accumulated in the local volume in a definite time interval. One can imagine that in the process of loading in the volume of a solid a number of elementary acts of stress relaxation dσ ri in different local volumes Ri occurs. By assigning a distribution of relaxation elements in the volume of a material, at that time we prescribe a definite distribution of plastic deformation, accumulated in the local volume in a definite time interval. By summation of the fields from various RE just as for (as well as for) dislocations the superpositional principle is used since the elementary fields (from the elementary tensor of relaxation dσ ri ) depend linearly on the corresponding boundary conditions of loading and on the boundary of region of relaxation Ri. In other words, the characteristics of elementary fields depend linearly on the components of elementary tensor of relaxation dσ ri . It should be noted at that time, that the constancy of the loading conditions and the stress distribution at the external surface are assumed. A temporal sequence of the elementary acts of stress relaxation doesn’t affect the integral result of the stress state of the solid.

2.2

Connection of Plastic Deformation with the Tensor of Stress Relaxation

The same relaxation tensor in principle can meet a variety of options of plastic strain distribution in the local areas of relaxation. In such a situation, a continuum theory suggests a most simple variant of the connection of tensor of relaxation with the tensor of plastic deformation. In principle, it is sufficient to solve the inverse problem: for the known field of internal stresses to define the tensor of plastic deformation in the local volume. Let us consider a simple example. Shown in Fig. 1 is the scheme of tensile loading of the plane with the site of plastic deformation of arbitrary shape. Under the operation of external stress in a local region

Fig. 1 Representation of a solution for the plane with the site of plastic deformation in the form of superposition of two solutions

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S, a stress drop of the value Δσ takes place. The general solution can be represented in the form of the algebraic sum of two solutions. The superposition principle is valid here because two solutions are summed up within an approximation of the linear theory of elasticity and the sum satisfies the required boundary conditions at the fields (places) of applied external stress and at the contour of the relaxation region. As it is seen, without a homogeneous field of stresses σ  Δσ one can obtain the case of full stress relaxation within the regions of given configurations under external stress Δσ. The absence of stresses in the local region testifies to the fact that the formchanging of the local region Ri is characterized solely by the compatible tensor of plastic deformation Δepij [15, 16, 17]. In such a case, the continuity condition of the stress fields and field of plastic deformation are broken only at the boundary of this region, because tensors of elastic deformation eeij in the region R and stresses σ ij beyond this region will be equal to zero. Let us emphasize once more that we consider only the final result but not the kinetics of stress relaxation. A field of stresses fully determines an elastic deformation of the plane. Hence, beyond the region of relaxation, the deformation is fully defined. On the other hand, an inhomogeneous stress field beyond the relaxation region unambiguously defines the displacements of the points of the boundary of relaxation region, independently on the process of mass transfer inside the region of relaxation. The same displacements should be defined by the field of plastic deformation inside the local region of relaxation. Therefore, from the many solutions for the tensor of plastic deformation, satisfying the given formchanging S, it is reasonable to choose the simplest linear one, obeying the equation Cijmn depmn, j ¼ 0, as in the case of a boundary-value problem of the theory of elasticity with prescribed displacements. Thus, within this accepted approach, the plastic form changing of the local region we don’t connect with the specific physical mechanism of plastic deformation. We are interested only in the stress state of the continuum. At the boundary of the site of plastic deformation the compatibility condition of the tensor of plastic and elastic deformation is broken, as the stresses undergo discontinuity. At the same time, the sum of the tensor of elastic and plastic deformation satisfies the compatibility conditions everywhere. Thus, from the standpoint of continuum theory of defects the value Δσ r and the geometrical shape of the local region of relaxation fully determine the stress-strain state of the plane and the distribution of plastic deformation in this region. Let us emphasize that under the condition of full stress relaxation, the boundary conditions at the boundary of plastic regions are unambiguously related to the external applied stress. The changing of the load distribution at the external surface of a solid results in the corresponding changing of point displacements of the contour of ideal site of plastic deformation, i.e., boundary conditions at this contour, defining the tensor of plastic deformation inside this contour. It is clear that there is no sense to speak about stress relaxation without the presence of a definite system of forces at the external surface of the solid, if suppose that there were no stresses in the solid before the testing. Disappearance of the elastic

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field in a given volume is unambiguously connected with the arising of the field of plastic deformation. Thus, a physical law manifests itself explicitly, which points to the connection between the tensor of plastic deformation and stress relaxation tensor in the local region.

3

Stress-Strain State of the Continuum with the Site of Plastic Deformation

3.1

Examples of the Site of Plastic Deformation

Apparently, the stress state of the continuum with the site of plastic deformation depends on which components of stress tensor change as a result of relaxation, i.e., on the component of stress relaxation. However, in each case, a decrease in elastic energy in the site of plastic deformation will be accompanied by increase in stress beyond the site of plastic deformation. As a simple example let us consider a stress state of the plane with the site of plastic deformation of round shape under the operation of external applied tensile stresses along the coordinate axis у. On a mesoscopic level in such an approximation one can imitate the influence of a separate grain of the polycrystal, undergone plastic deformation in surrounding of elastically deformed matrix. For the simplification, let us consider the case of RE the tensor of relaxation of which is characterized by nonzero component Δσ у = Δσ directed along tensile axis у. That means that as a result of plastic deformation only the normal component of the tensor of stresses σ у is relaxed. The scheme of loading is represented in Fig. 2. The general solution (а) can be represented in the form of superposition of two separate solutions: homogeneous stress field: σ  Δσ (b) and stress field for the plane under external tensile stress Δσ (c), in the round region of which there exist no stresses. The last solution is known as Kirsch’s problem for the plate with a round hole [34, 35]. s s +s ∗ s–Δs

s

Δs +s ∗

b

a

=

s– Δs

s–Δs

s

Δs

s–Δs

+

c

s =0

Δs

=

s

e

s∗

d

+

–Δs

s

Fig. 2 Representation of the solution а in the form of superposition of simpler solutions: 1 – homogeneous stress field Δσ  σ (b) and Kirsch’s solution (с); 2 homogeneous field σ (d ) and the field of internal stresses σ* (е)

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In the system of coordinates at the center of the circle and 0у-axis along the tensile axis, beyond the site of plastic deformation, Kirsch’s solution is characterized by the components: 
    2 Δσa2 2y2 8y2 x2 3a 3 1 2 þ 1 4 2 þ σ; Δσ y ¼ 2r 2 r r r2 
    2 Δσa2 2y2 8y2 x2 3a Δσ x ¼ 1   2 ;  1  2r 2 r2 r4 r2   Δσa2 yx 2ð3a2 þ 4y2 Þ 12a2 y2 3  þ Δσ xy ¼ : r4 r2 r4

(3)

Here r2 = x2 + y2, σ – external tensile stress. The considered example testifies to the fact that Kirsch’s solution has a more deeper physical sense than the simple one as a stress in the plane with round hole. This solution defines also the fields of stresses beyond the sites of plastic deformation of round shape under the relaxation of stresses in it in a value Δσ. Theoretically one can imagine the case of full stress relaxation in a round region, i.e., when σ = Δσ (Fig. 2c), then we obtain Kirsch’s solution (3) in its pure form. As it is seen the absence of material in the local region doesn’t mean the absence of material there. In our problem, the stress-strain state is caused by plastic deformation of the material in the local region of round shape and the region without stresses cannot be considered as a region where there is no material. Since the plastic deformation is irreversible, then under unloading the plane will be elastically deformed. Therefore without homogeneous field of external stress σ (Fig. 2d), Kirsch’s solution defines an inhomogeneous field of internal stresses in unloaded material σ* (e), caused by the presence of the site of plastic deformation. Then inside the site the stress is equal to Δσ y = Δσ, i.e., the material is in the state of compression along the tensile axis till the value Δσ. Equation (1) without stresses Δσ for the component Δσ y characterizes the field of internal stresses of a defect in the solid. Such a defect is the result of stress drop σ y by the value Δσ in a local volume of round shape. Shown in Fig. 3 are the patterns of spatial distributions of all the components of the fields of internal stresses σ*. Maximum and minimum values of the component Δσ y are equal to Δσ ymax = 2Δσ and Δσ ymin = Δσ, respectively. As a result of stress relaxation inside the circle stress concentrations are observed at the boundary of the circle. Let us characterize the field of plastic deformation in the site, corresponding to stress relaxation in the value Δσ in the round region for the case of tension. This field ensures the full displacements of the points of the circle in Fig. 2c under the absence of the stresses in the site of plastic deformation. According to the solution of Kirsch’s problem, the components of displacements of an arbitrary point (x0, y0) at the circle are equal: uy ¼ 3y0 Δσ=E,

ux ¼ x0 Δσ=E:

(4)

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Fig. 3 Distribution of the components of the field of internal stresses σ*: Δσ y(а), Δσ x(b), Δσ xy(c)

The point displacements of the contour along the y-axis due to elastic deformation are equal to uye = y0Δσ/E. Hence, the displacements due to plastic deformation are equal to uyp = 2y0Δσ/E and uxp = x0Δσ/E. Thus, the field of plastic deformation should satisfy the following boundary conditions in displacements at the boundary of the contour: upy ¼ 2yΔσ=E,

upx ¼ xΔσ=E:

(5)

The derivatives of the components of displacement field (5) over the corresponding coordinates define the components of the tensor of plastic deformation: epy ¼ 2Δσ=E,

epx ¼ Δσ=E, epxy ¼ 0:

(6)

Under the operation of external stress Δσ under the pure elastic deformation of plane (without plastic deformation), an elastic deformation will be observed inside the circle which is characterized by the components eey ¼ Δσ=E,

eex ¼ νΔσ=E, eexy ¼ 0,

(7)

where ν – is the Poisson’s ratio. Average values ν for solids in most cases lie within the range ν  0.33. After stress relaxation in a Δσ-value, the elastic deformation of the circle decreases. This decrease is defined by the components represented by Eq. (7). A comparison of (6) and (7) shows that plastic deformation not only compensates the disappeared contribution from elastic deformation, but makes additional contribution which two times exceeds the elastic deformation. Herewith, the region of relaxation increases along the у-axis and decreases along х-axis. Thus, the contribution of plastic deformation of the element to the changing of the shape of the circle

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Fig. 4 The scheme for the calculation of stresses in the plane with a relaxation element of pure shear

two times exceeds the contribution of elastic deformation, disappeared as a result of stress relaxation. At the boundary of the site, the components of the tensor of plastic deformation change jumplike from the value (6) down to zero, but the components of the tensor of elastic deformation from the value (7) up to (3) at the contour of the circle. Nevertheless, the displacements are not discontinuous when they transfer through the boundary of the site. A joint action of elastic and plastic deformation satisfies the continuity condition (compatibility) of the material at the boundary of the site of plastic deformation. Let us consider another case, when stress relaxation occurs on the scheme of pure shear in conjugate directions at an angle of 45 with respect to the tensile axis (Fig. 4). Stress relaxation of pure shear can be realized by superposition of two separate solutions, correspondingly for the case of tension along the y-axis and for the case of compression along the x-axis (Fig. 4). The case of tension is considered above and is described by Eq. (3). Analogically, an inhomogeneous stress field for the case of compression along the х-axis is defined. In the latter case, the same Eq. (3) are used where the stress Δσ is taken with the opposite sign. Further in Eq. (3), the variable х will be exchanged into у and у into -x. The solution for the compression along x-axis will be obtained. Summing this solution with the solution (3), the following equations for the components of the stress tensor for the site of plastic deformation of pure shear will be obtained.    2 
 Δσa2 2y2 8y2 x2 3a þ 1  2 1   2 , r2 r2 r4 r2    
   2 2Δσa2 xy 3a2 Δσa2 2y2 8y2 x2 3a Δσ xy ¼  1 Δσ ¼ 2 1   2 :  1  x r2 r2 r2 r4 r2 r4 Δσ y ¼

(8)

The patterns of spatial distribution of all the components of the field of internal stresses in the system of coordinates in Fig. 4 are shown in Fig. 5. The comparison

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Fig. 5 Distributions of the components of the field of internal stresses of an RE of pure shear with respect to the system of coordinates in Fig. 4: Δσ y(а), Δσ x(b), Δσ xy(c)

with Fig. 3 reveals qualitative and quantitative discripancies as well. Let us notice that here the maximum and minimum values of Δσ y components are equal to Δσ ymax = 3Δσ and Δσ ymin = 1.4Δσ, respectively. The Δσ х-distribution is the inverse distribution of the Δσ y-distribution, which is rotated by 90 as well. Extreme values of the Δσ хy-component are equal to 2Δσ. Inside the RE, there exists a homogeneous field of plastic deformation with the components: epy ¼ 4Δσ=E, epx ¼ 4Δσ=E, epxy ¼ 0:

(9)

It is not difficult to define that the plastic deformation of pure shear in conjugate directions at an angle of 45 with respect to the tensile axis in the site is calculated according to the formulae   γp ¼ epy  epx =2 ¼ 4Δσ=E:

(10)

At that time, in the direction at an angle of 45 with respect to tensile axis a shear stress is observed:  τ ðx, yÞ ¼ Δσ y  Δσ x =2:

(11)

The equations obtained (8), (9), (10), and (11) above fully define the stress-strain state of the simplest site of plastic deformation of pure shear of round shape.

3.2

Construction of Relaxation Elements with Gradients of Plastic Deformation

The disadvantage of the considered distributions is the fact that at the boundary of RE the components of stresses and deformations possess jumps. And though formally the condition of compatibility for the total deformation (elastic þ plastic) is obeyed the jumps of stresses mean loose of continuity of material along the boundary. The relaxation element method allows to construct and to find relaxation

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Fig. 6 The scheme of the family of RE and profile of plastic deformation in the family

elements with smooth distributions of plastic deformations. Let us demonstrate it with an example of RE of round shape. By definition, the site of plastic deformation is the relaxation element is the defect in the continuum medium with its own fields of internal stresses. When Δσ tends to a small value dσ, we obtain an elementary defect in the continuum. The presence of such a defect doesn’t change the elastic properties of the medium, i.e., it doesn’t affect the solution of the boundary-value problem of linear theory of elasticity. Therefore, for the internal fields of stresses of similar defect the superposition principle is valid. This defect can be used as the element for the construction of the various fields of localized plastic deformation. Prescribing a definite distribution of RE, one can construct the sites with the gradients of plastic deformation. Let us consider it on the example of the site of plastic deformation of а-radius, constructed by superimposition of the RE of round shape on each other with the common center (Fig. 6). Each RE in the family is characterized by a definite dimension and the value of the elementary tensor of relaxation dσ. The boundaries of the neighboring elements are assumed to lie at the equal distance dа from each other. Quantitative view (representation) of expected profile (smooth, differentiated) of plastic deformation is represented in Fig. 6 (to the right) in the form of envelope line, embracing the steps the height of which characterized the value of plastic deformation dep of a definite RE. Let us select the near-boundary region with the width h in the given family of RE, in which parameters of RE will define in the following manner: aðt00 Þ ¼ a  ht00 ,

dσ ðt00 Þ ¼ ðβ þ 1Þh Δσt00β dt=a,

0  t00  1:

(12)

Here t00 is an integration variable, Δσ is prescribed value, β – is the parameter, defining the value of elementary step when transfer from one element to another, β þ 1 is normalization factor. It is seen that with increase in В t00 the dimension of RE evenly decreases from а till a  t. Herewith, the magnitude of the elementary relaxation tensor dσ(t00 ) is gradually increasing. The growth rate depends on the value of the β-parameter. The more β, the more the growth rate of dσ(t00 ).

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Let the value of steps dep in the central zone gradually decreases down to zero when approaching the center of the site. For that the changing of parameters of relaxation elements can be defined in the following manner. β

aðt0 Þ ¼ ða  hÞð1  t0 Þ, dσ ðt0 Þ ¼ ðβ þ 1Þð1  h=aÞ Δσ ð1  t0 Þ dt, 0  t0  1:

(13)

Here t0 – is the integration variable, which is changing within the interval 0  t0  1, β – is the parameter, defining the change in the value of elementary stress drop dσ(t0 ). Accordingly (13), an increase in t0 defines the gradual decrease in radius RE a(t0 ) and the value dσ(t0 ) as the center of the site is approached. The rate of reduction is controlled by the β-parameter. The higher is β-parameter, the quicker the value dσ(t0 ) decreases. Using parameters (12) and (13) for the corresponding RE in Eqs. (8) and (10), we obtain the dependencies for the elementary stress field dσ i and plastic deformation depi on the variables t0 and t00 . Integration of the elementary fields of plastic deformation from all RE under the prescribed conditions results in the smooth stress fields τðx, yÞ 8 9


 2  ðβ þ 1Þa2 3ðβ þ 3Þa2 > > 2 2 2 > > > >  2 1  8 1  y =r =r  a , y , if r 2 2 < ðβ þ 3Þr2 ðβ þ 5Þr 2 = ¼ Δσ :

2 > >  r βþ1

 2  β 1 > 2 2 2> > > 1  8 1  y 1 þ =r =r  a y þ 1 , if r : ; 2 2 a 2ðβ þ 3Þðβ þ 5Þ

(14) and plastic deformation  8  r βþ1 i h h h > < 1  1  þ , r 2  ða  hÞ2 ah 2Δσ p a a γ ¼ h a  r βþ1 E > : , ða  hÞ2  r 2  a2 : a h

(15)

Here r – is the distance from the center of the family of relaxation elements to the point with the coordinates (х, у). To be short, Eq. (14) is written at the value h = 0. In Eq. (15), the upper equation behind the bracket defines the value of γp-component at each point of central zone (Fig. 6), and lower at each point in the near-boundary zone. Apparently by pointed out method one can construct the smooth fields of plastic deformation and stresses for each RE of any type. A smooth distribution of plastic deformation of pure shear at the value β = 5 is depicted in Fig. 7a. The profiles of plastic deformation for the different values of βparameter are represented in Fig. 7b. At a distance h from the boundary of the site, the maximum gradient of plastic deformation is observed gradepy ¼ 2Δσ ðβ þ 1Þ=Ea:

(16)

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Fig. 7 Distribution (а) and profiles (b) of plastic deformation in the site at h = 0.5а and β = 5. Numbers at the curves mean β-values

It is seen that the value of the gradient is defined by the parameter β. The more the value, the higher is the gradient of plastic deformation. At β ! 1, we yield to the considered above the site of plastic deformation with the jump at the boundary of the site. Shown in Fig. 8 are the parameters of the spatial distribution of plastic deformation and stress τ(x,y), for the various values of the parameter β at h = 0.5а. As the parameter β increases, the maximum gradients of plastic deformation and stresses increase as well. The case considered above with the jump of plastic deformation (and stress) is the limit case at β = 1. Figure 9 demonstrates the stress field (а) and isolines of pure shear (b) of the site of plastic deformation of pure shear at the values β = 8 and h = 0.2а. It is seen that such a site causes elevated stresses of a mesoscopic scale in the direction of 45 with respect to the tensile axis.

4

Stresses in a Continuum with Band Structures

4.1

Introduction

The band of localized deformation defines an extended narrow region on the surface of the material, where the process of stress relaxation occurs intensively and the material undergone the formchanging, additional to elastic one. On microlevel this region is represented as a pack of slip traces. In monocrystals such packs are formed along the closed-packed planes of atoms, favorably oriented for easy slip and spans (intersect) over the whole cross section of the specimen [36–39]. In polycrystals, first of all the conjugate grains are involved into plastic deformation, in which the slip traces are favorably oriented for the realization of shear component of external stress. On mesolevel, the straight bands of localized plastic deformation are formed. As a rule, they are oriented along the planes of maximum shear stress with respect to

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Fig. 8 Spatial distributions of stresses τ(x,y) (a) and plastic deformation of pure shear γp(b), for different values of β-parameter

external applied stresses, independently on the crystallographic orientations of slip systems. As experiment shows [38, 39] the phenomenon of material fragmentation on the mesolevel is, also, the consequence of the intersection and interaction of the forming mesobands, oriented at an angle of 45 or 60 with respect to tensile axis. On

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Fig. 9 Stress field (а) and isolines of RE of pure shear (b): β = 8, h = 0.2а

macrolevel they are incomplete Lüders band at the initial stage of loading, or band structures, being formed under the interaction of the macrobands of localized shear before the onset of fracture. In the case of the alloys with martensitic transformation inelastic formchanging of the local volumes takes place by formation or interaction of martensitic lamellae. In such a manner, for the definition of the stress state of the medium with the band of localized deformation in the first approximation one should select the contours, beyond which the material deforms elastically, but inside undergoes plastic deformation, i.e. the plastic deformation is connected with stresses disappeared as a result of relaxation. Further, for convenience, under common term “Localized deformation” one should understand (mean) any inelastic deformation, obtained as a result of stress relaxation, independently on the physical mechanisms, providing inelastic form changing. Assigning (prescribing) in different manner the tensor of plastic deformation inside the contours, one can perform the analysis of the qualitative pattern of the change in the inhomogeneous field of stresses at the interaction of band defects at the mesoscale level. In the given chapter provides the examples of construction of the bands of localized deformation with very different distribution of plastic deformation. It is shown how the field of stresses from corresponding distributions of RE are integrated. The stress-strain state of the band with localized plastic deformation, depending on the angle of its orientation with respect to tensile axis, is analyzed. Then, the interaction of the bands with each other and the edge effect are considered. The examples of the point displacement fields of continuum with the bands of localized deformation of different orientation and mutual location are considered. Qualitative and quantitative differences in the patterns of inhomogeneous stress field of fragmentated and non-fragmentated material are revealed.

4.2

Bands of Localized Plastic Deformation

One of the most interesting tasks practically and theoretically demanded in mechanics of deformed solid is the task of stress-strain state of the plane with the band of

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Fig. 10 The scheme of the construction of the band of localized deformation from RE of round shape

localized plastic deformation (LPD). Let us consider the simplest case of the construction of the band of localized plastic deformation, being normal to the tensile axis y. Let us compose it from the RE of round shape with the field of internal stresses (3). Let us define for each RE an elementary value of stress drop dσ = Δσdx/ 2a, where 2а is the diameter of the circle and let us place the RE at the equal distance dx between each other along х-axis (Fig. 10). At such a construction in the arbitrary point (х,у), the stress drop will be equal to ndσ, where n–is the number of RE, inside which lies the given point. Stress drop in the value dσ, according to (6), is provided by elementary field of plastic deformation with the components depy ¼ 2dσ=E,

depx ¼ dσ=E,

depxy ¼ 0:

(17)

Integrating these elementary fields (17), we obtain the following equations for the epy -component of plastic deformation in hemi-infinite band: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > x þ a 2  y2 > < , x2 þ y2  a2 ; 2a ffi epy ¼ epymax pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 > > : a  y , x 2 þ y 2  a2 : a

(18)

Here epymax ¼ 2Δσ=E. The spatial distribution of the epy-component of tensor of plastic deformation in the band is represented in Fig. 11. The maximum value of plastic deformation epymax is observed along the axis of the band. As the boundary of plastic region is approached, the degree of plastic deformation decreases down to zero. The same way, integrating elementary stress fields according to Eq. (3), where Δσ is replaced by equation dσ = Δσdx/2a, one can obtain the field of stresses for the given semi-infinite band of localized deformation. In a coordinate system with the origin at the 0-point (see Fig. 10), the components of the tensor of internal stresses of the given semi-infinite band are described by the simple formulas:

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Fig. 11 Semi-infinite band of LPD

Fig. 12 Field of internal stress σ у of semi-infi-nite LPD-band, being normal to tensile axis

8     2 a2 4y < ax 2 þ 1   1 , r 2  a2 , Δσ y ¼ Δσ r 2 r2 r2 : 2 2  r  a 2  2  8 x=a, a 4y < ax 2 1 2 1 , r 2  a2 , Δσ x ¼ Δσ p r2 r2 ffi r : ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2, 8 1 y =a2   r  a2  a 4y < ay 1 2 3 2 , r 2  a2 : Δσ xy ¼ Δσ r 2 r r : 2 2 0, r  a

(19)

Shown in Fig. 12 is the distribution of the stress field σ у (19). The stress field is seen to be perturbated only at the end of the band. There exists a stress concentration in front of the band. At that time the stresses are lower than the average level of external stress in the band itself, near the end. This example shows that the source of the driving force during the formation of the shearband is the concentration and high

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gradients of stresses at the end of the band. One should take into account the following peculiarities of the bands of localized shear: 1. The stress field is highly perturbated only at the end of the band. There the stress concentration is observed always. It contributes to the formation of the LPD-band. 2. The zone of stress concentration at the end of the band is always combined with the zone of anticoncentration of stresses inside the band.

4.3

Plastic Deformation in the Site of Elliptical Shape

Let us consider an example of plane stress state of the medium, where the site of plastic deformation is described by the family of the elliptical regions of relaxation enclosed into each other (Fig. 13). It is supposed that beyond a given site of plastic deformation, the material is homogeneous, isotropic and is deformed elastically under the operation of tensile stress σ, directed along y-axis. For definiteness let us assume that all ellipses have common center at the origin of coordinates. For the simplicity let us consider the case when the length of the semi-axes are equal to a = a(1  t) и b = b(1  t), i.e., when the ratio of the lengths of the semi-axes is the same and is equal to a/b, where the semi-axis b of the site is directed along the tensile y-axis, t – is the integration variable, changing within the interval from 0 to 1. At t = 0 we obtain the largest semi-axis. Hence an increase in t corresponds to the consequent transition from external contour to the center of the family. A specific t-value chooses a definite contour from the family. The point with the coordinates (x,y) will fit the contour of ellipses with the value t = 1  (x/a)2 þ (y/b)2. The value of relaxation tensor for the given ellipse we represent in the form of the function of the integration variable in the following manner: Fig. 13 Graphical representation of the distribution of the relaxation elements in the form of ellipses with the same relation of axis lengths

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dσ r ¼ ðβ þ 1Þσ ð1  tÞβ dt,  1  β  1:

(20)

The parameter β is seen to regulate the change in the value of relaxation under the continuous transition from the contour to contour. The coefficient β þ 1 at the integration of dσ r from 0 to 1 determines the value of external applied stress σ. One can show that the plastic displacements of the ellipse’s contour as a result of stress drop inside the ellipse in the value dσ r are defined by the tensor of deformation with the components: dex ¼ dσ r =E0 , dey ¼ ð1 þ 2a=bÞ dσ r =E0 , dτxy ¼ 0:

(21)

Apparently, the fields which covered the point (x,y) are superposed, i.e., the integral result is obtained by integration of the Eq. (21) within the limits 1 < t < 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðx=aÞ2 þ ðy=bÞ2 . By substituting into Eq. (21) instead of dσ r the Eq. (20) and integrating within the corresponding limits one can obtain the tensor components, describing the smooth field of plastic deformation with the gradients at the boundary of the site: The distribution of the epy -component is characterized by simple equation epy

"  2 ðβþ1Þ=2 # σ a x y2 1þ2 ¼ 1 2þ 2 : E b a b

(22)

In particular, along x-axis we obtain the following equation for the profile of plastic deformation
x βþ1  σ a 1þ2 epy ¼ 1 : (23) E b a An example of spatial distribution of plastic deformation for the different βvalues according to Eq. (23) is represented in Fig. 14. As it is seen, this field is smooth. At the boundary of plastic region there is no breaks, only gradients of plastic deformation are broken. The maximum epy -value at the center of the site is not dependent on β-parameter and is equal epymax ¼

σ a 1þ2 : E b

(24)

A derivative of the Eq. (22) on x variable results in the following equation for the gradient of plastic deformation along x-axis: gradepy ðxÞ ¼ epymax

β þ 1  x β : a a

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Fig. 14 The change in the distribution of plastic deformation ey in the site of elliptical shape with increase in β-parameter in Eq. (23); a/b = 4, σ/E = 4.86  103

At the end of the band a gradepy ðaÞ ¼ epymax

βþ1 : a

(25)

It is seen that with increase in β-parameter the gradients in front of the boundary increase. At the center of the site at β > 1, the gradient of plastic deformation is equal

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Fig. 15 The profiles of plastic deformation along xaxis when β increases according to Eq. (23)

to 0. At β = 0, the gradients along any directions from the center of the site are constant, i.e., plastic deformation decreases linearly from maximum value at the center down to zero at the boundary of plastic region. Then along x-axis gradepy ðxÞ ¼ epymax =a: At β ! 1 at the center of the site the gradient of plastic deformation tends to infinity. Figure 15 reflects all mentioned peculiarities on the profiles of plastic deformation epy along x-axis under increasing the value of β-parameter according to the law, β = 0.95 þ 2j9, where j in order takes the numerical values from 0 till 18. It is clear that the selection of another law for the relaxation value at the same distribution of relaxation elements will result into another distribution of plastic deformation. Shown in Fig. 15 is the analogous example, when the law for the value of relaxation is set by the equation dσ r ¼ ðβ þ 1Þσtβ dt,  1  β  1:

(26)

Then the distribution of epy -component is defined by the equation 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3βþ1   σ a x2 y2 5 1 þ 2 41  epy ¼ þ : E b a2 b2

(27)

And for the profile of plastic deformation the following equation is obtained epy ¼

σ a h x iβþ1 1þ2 : 1 E b a

(28)

It is seen that, unlike the previous case, here, conversely, increase in β-parameter results in decrease in localization of plastic deformation. The peculiarity of the distribution of (28) lies in the fact that at β > 0 at the boundary of the site there is no jump of the gradient of plastic deformation (Fig. 16). It should be noted that the profiles in Fig. 16 exactly copy inversely reflected profiles in Fig. 15. Besides

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Fig. 16 The changing of the profiles of plastic deformation along x axis under increase in β according to Eq. (27)

that, maximum values epy at the prescribed values of relations a/b and σ/E don’t depend on the β-value. In Figs. 15 and 16 the ratio a/b = 4, σ/E = 4.86  103, epymax = 4.43%. By combination of solutions (23) and (27) one can obtain the distribution which will be smooth (differentiable) at all points of the site of plastic deformation including the boundary of the site. For this it is necessary for the RE in the nearboundary zone of the width h along x-axis (Fig. 13) to prescribe the relaxation law by the equation the type (20), and for the rest of relaxation elements in distribution – the Eq. (26), providing the matching of the gradients at the contour, separating the selected zones. Then the continuity condition of the gradients will be fulfilled everywhere and the distribution of plastic deformation will be described by the formulae:

epy p eymax

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 
 qffiffiffiffiffiffiffiffiffiffiffiffi2ffiβþ1  h 2 2 > > h x y h > > 2 y a x > > > > 1 ,  1  þ þ 1  ah þ 2 2 > a b = < a a a> a2 b2 2 3 0 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s βþ1 (29) ¼ > > > > h 4@ x 2 y 2 A a5 a x2 y2 > > > > 1 þ ,1   þ  1: > > ; :a h a2 b2 h a2 b2

The result at h/a = 0.5 and for the rparameters of ellipse is represented in Fig. 17. As should be expected, distribution of plastic deformation is turned out to be smooth. The maximum of plastic deformation is concentrated in the center of the site. The gradient of plastic deformation increases in the direction to the contour, dividing the near-boundary zone. The gradient is maximum at the point on the contour and equal to the value, prescribed by Eq. (25) (Fig. 18). Equation (29) and the results, presented in Figs. 17 and 18, reveal that with increase in β-parameter the gradients of plastic deformation increase at the distance h from the boundary of the site. Along with it, within the interval х < a  h the region of practically homogeneous plastic deformation arises and expanded. The change in

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Fig. 17 The changing of the distribution of plastic deformation ey in the site of elliptical shape with increase in parameter β in Eq. (29)

Fig. 18 The changing of the profile of plastic deformation along x axis according to the Eq. (6.10) when β increases by law β = 20.5j (j = 0, 1, 2, 3, . . . 10). h/a = 0.5

the width of the near-boundary zone h doesn’t influence the maximum gradient, but reduce the area of homogeneous plastic deformation inside the site (Fig. 19). Qualitative pattern of the distribution of the component ex apparently will be analogous to the distributions considered, i.e., the Eq. (21) for dey and dex differ only in the coefficients.

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Fig. 19 The change in the profile of plastic deformation along x at h increase from 0 to a. h/a = 0.5j

4.4

Field of Stresses from the Site of Elliptical Shape

Since there exists an analytical connection between the tensor of relaxation and plastic deformation then by prescribing the distribution of RE in the local volume, the resulting inhomogeneous stress field is defined automatically, in the whole plane, including the region of relaxation. For the relaxation element in the form of ellipse, the components of tensor of elementary stress field along x-axis can be written in the following form:
 b2 x  a a x 1  3 , a  b ða  bÞ c c
 2 a b xða  2bÞ b2 x r þ 3 , þ ¼ dσ a  b ða  bÞ ða  bÞc c

dσ xx ¼ dσ r dσ yy

(30)

dτxy ¼ 0: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Here c ¼ x2  a2  b2 ð1  tÞ2 . The equation for dσ r in Eq. (30) is selected depending on the character of distribution of plastic deformation. Integrating the Eq. (30) over t variable for the general case of distribution (29) we obtain the distributions, depicted in Fig. 20 depending on the value of the parameter β. The results of integration for different h values at β = 4 are represented in Fig. 21 in the form of corresponding profiles of stresses. A comparison of stress fields with corresponding profiles of plastic deformation immediately reveals a clear correlation. The stress fields have no breaking points (turning points) at β > 0. Increasing the gradient of plastic deformation (increase in parameter β) causes corresponding growth of the stress gradient, but with the opposite sign. Maximum of the gradients is observed at the points of the contour, separating the near-boundary zone. Besides that, the increase in the average degree of plastic deformation is accompanied by decrease in the stress field inside the site.

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Fig. 20 Profile of σ y along xaxis under increasing β. σ/ E = 4.8  103, β = 20.5 j. Numbers at the curves relate to the corresponding j-values

Fig. 21 Changing the profile of stress σ y along x-axis with increasing h. σ/E = 4.8  103, h = a j/11. The numbers at the curves correspond to j-values

Increase in the width of the near-boundary zone doesn’t affect the maximum gradient and results in increase in the average stresses inside the site (Fig. 21). At the same time within the limits of the width h the stress continuously goes through the maximum. With increase in β-parameter maximum grows. With increase in h an effect of stress concentration wanes, the maximum is smoothing and displacing to the center of the site of plastic deformation. The zone of localized plastic deformation is reduced. Represented variant of the model of the site of plastic deformation reflects the elastic-plastic properties of real structures in the approximation of continuous medium, when the near-boundary zone is considered to be the physical boundary of the structural elements where the ability to stress relaxation is smoothly decreases as the geometrical boundary with elastically deformed matrix is approached. The results of calculations show that in this case the maximum stress concentration is observed not at the geometrical boundary but in the near-boundary zone with the width not more than h. Increase in the gradients ahead of the physical boundary,

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decrease in its width h, and the increase in the length of plastic region promote the effect of stress concentration.

5

REM in the Models of Localization of Plastic Deformation and Fracture

5.1

Peculiarities of the Simulation by the Relaxation Element Method

Using the relaxation elements as the defects, characterizing the interrelation between plastic deformation and stress allows one to simulate the process of strain localization and to obtain the dependencies of flow stress on the sequence of the involvements of separate structural elements into plastic deformation. A modelled medium is represented in the form of conglomerate of discrete elements, playing the role of elements of the structure. A structural element with the field of plastic deformation is itself a relaxation element. With a certain degree of accuracy this field can be characterized using the different types of RE, considered above. By placing a definite relaxation element in the element of the structure, we at the same time determine the corresponding change in the field of internal stresses in a solid. In such a way, the interaction of the internal stress fields from the elements of structure, undergone plastic deformation, lies on the basis of the modelling with relaxation element method. Models developed on the basis of REM operate on principles of cellular automata [40, 41]. The calculation field is divided into a number of cells, playing the roles of elements of structure (for example, grain in polycrystal). Each element of the modelled medium possesses the ability to switch its state by discrete jumps in plastic deformation, prescribed by a definite relaxation element. In such a manner, an element of structure is able to increase discretely the degree of plastic deformation and a stress concentration to affect the stress state of the whole volume of solid. The involvement of the structural elements into plastic deformation is realized by definite transition rule (for example at the moment of achieving of a critical value of shear stress). The interaction of the stress field from different structural elements undergone plastic deformation occurs automatically. When interpreting the results of a simulation, one should take into account that the stress state of a deformed system is controlled only by incompatible plastic deformation connected with stresses in the volume of a solid. The relaxation elements in the present case play the role of defects, responsible for the field of plastic deformation. However, it was experimentally found that the absolute majority of deformation defects at the stage of developed plastic deformation disappear as a result of annihilation, disappearing at internal interfaces and exposing at the free surface of the solid. The stresses caused by these defects will also disappear. What remains is the corresponding field of plastic deformation, not connected with stresses, which satisfies the compatibility condition. Thus in general case one should not neglect the compatible plastic deformation. The problem lies in the fact that the

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compatible plastic deformation cannot be represented by a definite analytical function of coordinates. The same formchanging of the solid can be realized by a number of variants of mass transfer of the material, not influencing the stress field. That means that the relaxation element method can correctly calculate the change of stress fields and plastic deformation, caused by the presence of the defect of plastic nature in the volume of material with time. For the definition of the total plastic deformation, including the contribution from the defects, exposed on the surface of the material, the additional conditions are necessary. They can be formulated when calculating the specific structures, taking the known quantitative data from the experiment. In other words, it is necessary to know the prehistory of the development of plastic deformation.

5.2

Modelling of Localization of Deformation in Polycrystals by RE-Method

Let us consider an example of the simulation of plastic strain localization in polycrystals under loading, using the relaxation element in the form of a circle. In this case we use an approximation, assuming that on the mesolevel each crystallite involved into plastic deformation in the polycrystalline agglomerate can be considered as a relaxation element of round shape. As a criterion of the crystallite involvement into plastic deformation, serves a critical shear stress τcr in one of two possible directions which were assigned in each crystallite by the generator of random numbers. The direction imitates the orientation of slip planes in crystallite lattice. The second slip system was directed at an angle of π/3 with respect to the first one. Let us define the boundary condition by the assumption that the transition of the crystallite from elastic into plastic deformed state occurs at the minimum external applied tensile stress σ. That means that the probability of the involvement into plastic deformation is excluded for all crystallites except only one. According to this condition, a new element with the coordinates (хп, уп) transfers into the plastic state under external stress: σ n ðx n , y n Þ ¼

h i P 2 τcr  n1 Δτ ð x , y Þ i i i¼1 1 þ 2Δτðxn , yn Þ=Δσ

,

(31)

where the sum represents the contribution of all previous REs to the shear stresses. The contribution of i–RE to the effective shear stress from п-RE is calculated according to the equation h i

 Δτðxi , yi Þ ¼ Δσ iy  Δσ ix sin αn cos αn þ Δσ ixy cos2 αn  sin2 αn , where αn is the angle between the allowed direction and the axis of loading, the components Δσ xi, Δσ yi and Δσ xyi are calculated, using the Eq. (3) for RE of the round form at the values Δσ = 50 MРa and τcr = 50 MРa. кΔσ ix The calculation field

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Fig. 22 Self-organization of the bands of localized plastic deformation in the modelled 2Dpolycrystal under vertical tensile loading

Fig. 23 Dependence of the external stress on the number of crystallites, involved into plastic deformation

consists of 2550 points – centers of virtual crystallites, which in Fig. 22 are depicted in the form of hexagons. Shown in Fig. 22 is the sequence of the patterns of the crystallites involvement into plastic deformation. It is seen that under the present conditions, the model predicts the self-organization of the bands of localized shear. From the very beginning of plastic flow, the mesobands divide the material into fragments. The external applied stress oscillates near some average value (31) (Fig. 23). Each stress drop is associated with the formation of a separate mesoband embracing several grains. A different pattern is observed in the polycrystal with rigid inclusions (Fig. 24). Plastic deformation starts at the boundary of the inclusions and is located near it. The macrobands in the conjugate directions of maximum tangential stresses originate from the inclusion. The contribution of the additional stress fields from the inclusion results into a lower flow stress of the polycrystal. The development of plastic deformation is realized in this case by the formation of mesobands, causing oscillations of the external stress. Thus, modelling by relaxation element method reveals qualitative and quantitative distinctions of the developments of plastic deformation in polycrystal structures without the inclusion and with a rigid one. The model presented predicts the jumplike dependence of the external stress on the number of acts of structural elements (grains of polycrystals) involvements into plastic deformation. Its value (external

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Fig. 24 Self-organization of shear bands in the modelled polycrystal with rigid inclusion

stress) defines the onset of plastic flow in the grain, where a critical value of shear stress is achieved. Since plastic deformation of each new grain changes the field of internal stresses, then the value of external stress oscillates within corresponding limits. The presented results of the simulation by the relaxation element method points out to the necessity of further improvement of the model. A simplified approach doesn’t allow to describe a temporal evolution of strain localization. Besides that, to determine the value of σ(x,y) = σ min, according to the Eq. (31), practically is not possible. That is why any further improvement of the model of plastic strain localization is realized with accounting of the real boundary condition of loading.

5.3

The Influence of the Rigidity of the Testing Device

Theoretical models of plasticity are developed in assumption of a definite and as a rule simple boundary condition of loading. The changing of the applied load with time is controlled by the change of plastic deformation. It is not possible to realize in practice precise theoretical boundary conditions. Therefore, the loading diagram of the same material depends essentially not only on the mode of loading (tension, compression, bending, torsion), geometrical shape, and the dimension of the specimen but on the technical characteristics of the testing device. One of the most important technical characteristics of the machine is the rigidity modulus M. It is defined as a force which is necessary to apply to the punch of the machine in order to shift in it in 1 mm at the rigid coupling of clamps. Such a displacement is possible due to elastic deformations of the parts of the machine from the punch to the clamp. Depending on the M value the machines conditionally are divided in two classes: into rigid with a big value and into soft with a small value of rigidity modulus. Rigid

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Fig. 25 The view of the loading diagram under tensile loading with application of «rigid» (а) and «soft» (b) testing

Fig. 26 The scheme: 1 sample, 2 clamp

machines are very sensitive to the quick change in the rate of plastic deformation and react on it by a drop in the load (Fig. 25a). The stress-strain diagrams obtained on the soft machine have stair-case type (Fig. 25b). Let the rate of movement of the clamps of testing machine v0 is known in unloaded state. In the loaded state in a course of deformation of the specimen, the elastic deformation of the intermediate parts of the machine takes place. Therefore a constant change of the strain rate of the specimen occurs. Shown in Fig. 26, the parts of the machine which contribute to elastic displacement are depicted in the form of springs. According to the definition, the modulus of rigidity of the machine is equal to M ¼ F=Δlm ,

(32)

where Δlm – is the displacement of the clamps and elastic displacement of the springs. Experimentally, the modulus of rigidity can be defined in the following manner. The clamps of the machine are coupled, excluding the possibility to move with respect to each other. Then the loading is switching on till the nominal magnitude of the force Fnom, after which the loading is stopped. Next, it is necessary gradually to uncouple the clamps, to measure the distance Δlm between them and calculate the modulus of rigidity of the machine M, according to Eq. (32). During the time Δt the punch of the machine will move in a distance Δl* = v0Δt. At the rigid coupled clamps, it will match to the force Δf* = MΔl* = v0Δt.

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If the specimen is deformed in the clamps, then due to the deformation the distance between clamps will increase in Δl ¼ ½Δeе ðΔtÞ þ Δep ðΔtÞ l,

(33)

where l is the effective length of the working part, Δeе(Δt) and Δep(Δt) – respectively contribution of elastic and plastic deformation of the specimen. In the same length Δl will decrease «the effective length» of the imaginary spring, representing elastic deformation of the intermediate parts of the machine. As a result, the decrease in the force applied to the clamps will take place in a value Δf = M Δl. The resulting changing of the force will be equal to ΔF ¼ Δf   Δf ¼ MðΔl  ΔlÞ ¼ Mðv0 Δt  ΔlÞ: Then the elastic deformation of the specimen is equal Δee ðΔtÞ ¼ ΔF=ðSEÞ,

(34)

where S – is the area of the cross section of the working part of the specimen and Е is Young’s modulus. Plastic deformation of the specimen is equal to Δep ðΔtÞ ¼ e_p Δt, where e_p is the average rate of plastic deformation in a prescribed time interval Δt. Taking into account the equations (33) and (34), we obtain ΔFðΔtÞ ¼ MΔtðv0  vp Þ=½1 þ Ml=ðSEÞ :

(35)

where vp is the rate of specimen length change due to its plastic deformation. The sign of ΔF(Δt) depends on the difference of the rates of the punch of the machine and the rate of the change of the length of the specimen due to its plastic deformation vp. At vp = v0 the flow plateau will be observed, and at vp > v0 – decrease in external load. The higher the rigidity of the machine М, the higher is the value of the external drop of the load. The Portevin–Le Chatelier Effect is connected with periodical spontaneous arising of the band of localized shear. The accumulation of plastic deformation in the band proceeds so effectively, that due to this time the velocity of the increase in the length of the specimen due to its plastic deformation passes ahead the velocity of the movement of the punch of the testing machine. For example, in Al-Al2O3 alloys the decrease in external load occurs during 1–2 s. The dependence of the change in external load ΔF (35) in that time interval on the average rate of plastic in the given time interval for the given alloy is presented in Fig. 27. The amplitude of oscillations of external stress at the pointed parameters of the experiments is seen to be 4 MPa if the velocity of plastic deformation of specimen two times passes ahead of the rate of movement of the punch of the machine. Equation (36) can be transformed for the calculation of the increment of external stress dσ in time interval dt, during which the stress relaxation in the value Δσ takes place in the crystallite undergone plastic deformation.

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Fig. 27 Dependence of the changing of external load on the relative rate of plastic deformation vp/v0: Е = 70 GPa, l = 18 mm, M = 1.3  103 kN/ mm, Δt = 2 s, v0 = 5  102 mm/s

 M v0 dt  Ka2 Δσ=bE : dσ ¼ b þ Ml0 =ðEhÞ

(36)

Here, v0 – is the velocity of the movement of the punch of the machine, а – is the radius of the crystallite, S, l0, h, and b are the cross section, length, and the width of the working part correspondingly.

5.4

Accounting for Edge Effects

More exact accounting of the boundary conditions of loading in the model is planned to perform because of the following circumstances. The equations represented above for the stress fields are valid for an infinite plane. At that time at the lateral boundaries of the modelled specimen there exist normal and tangential stresses, which should be absent according to the boundary conditions of loading. Removal of the tangential stresses has been performed, using the method of reflection [42] following the scheme, depicted in Fig. 28. A system of 3 RE is considered, one of which is actual and two others are fictitious. The last create the fields of inverse stresses. The proposed scheme of RE-position practically fully compensates the shear components at the edge of the specimen. It is seen from Fig. 29 that RE causes at the edge of the specimens the components of shear stresses of the significant value (Fig. 29a). Application of technique proposed allows easily and simply to compensate for these stresses up to practically negligible values (Fig. 29b). At that time, as seen, the stress distribution near the given edge undergoes essential change. RE causes at the edge of the semi-plane also the normal stresses. For the removal of normal stresses at the edge of the specimen, the following technique has been applied. Fictitious RE, located symmetrically with respect to the boundary of semiplane increases these stresses twice. This distribution is symmetrical with respect to the line, connecting the centers of the real and imaginary relaxation element. In mechanics of deformed solid, a problem on the concentrated force applied normally with respect to the surface of isotropic elastic half-plane is known as the Flahman problem [42]. According to the known formula of Flahman, in the system

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Fig. 28 The scheme of compensation of the shear stresses at the edges of the specimen

Fig. 29 The distribution of shear stresses in the specimen σ ху before (a) and after (b) compensation of them at the edge of the specimen

of coordinates, depicted in Fig. 30, under the influence of an elementary force dFx = Δσ xdl, operating at the boundary of the half-plane, in the volume of the material the field of stresses with the components arise: dσ x ðlÞ ¼

2 Δσ x ðlÞx3 2 Δσ x ðlÞxl2 dt, dσ y ðlÞ ¼   dt, 2 π x2 þ l 2 π x2 þ l2 2 dσ xy ðlÞ ¼

2 Δσ x ðlÞx2 l  dt: π x2 þ l 2 2

(37)

Here dl is the elementary segment of the half-plane, in which the stress Δσ x operates.

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Fig. 30 The scheme of compensation of normal stresses at the edges of the specimen

In order to compensate the normal stresses, created by RE at the edge of halfplane, it is enough to prescribe exactly the same stress distribution in the plane but with the sign reversed. In the system of coordinates in Fig. 30, according to the second equation in the system (38) RE creates the following distribution of normal stresses at х = 0, which we will write down with opposite sign: Δσa2 Δσ x ðlÞ ¼ 2 p þ t2


2 1

# !  " 2t2 8p2 t2 3a2 2 ,  1 p2 þ t2 ½p2 þ t2 2 p2 þ t2

(38)

where l is the variable, defining at the edge of the half-plane a location of a specific point in the distribution Δσ x(l ) (Fig. 30), Δσ is the value, defining the degree of shear stress relaxation in the RE, and p is the distance from the edge of the specimen till the fictitious RE. Substitution of the function Δσ x(l) in Eq. (37) and their integrations within the limits –L till L, where l is the length of the specimen, defines with enough accuracy the changing of the corresponding components of the stress field in the plane specimen and the compensation of the normal stress at the edge of the specimen. This is very well illustrated in Fig. 31, where the distribution of the normal σ х stress is represented in the specimen before (а) and after (b) their compensation at the edge of the specimen. It is seen that the approach applied really results in releasing of normal stresses at the edge of the specimen. At that time, an essential changing of the character of distribution takes place. The technique of the releasing of normal and shear stresses at the edge of the specimen will be further used in modelling.

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Fig. 31 Distribution of normal stresses σ х in the specimen before (a) and after (b) their compensation at the edge of the specimen

5.5

Jump-Like Propagation of the Band of Localized Shear

Equation (8) was put on the basis of algorithm, for the distribution of the stress of pure shear in the conjugate directions at an angle of 45 with respect to the tensile axis without accounting for the different orientations of the slip systems. Such a simplification is justified by the fact that in any case the front of localized shear is usually oriented at an angle of maximum tangential stresses. This match the case when the crystallites have a large number of slip systems. The calculation field is represented in the form of a matrix with 10  50 points the centers of crystallite. Onset of plastic deformation was initiated at the edge of the calculation field by placing the RE, represented in Fig. 7 and by Eqs. (14) and (15) at the values h = 0 and β = 6. This grain created a nonhomogeneous field of shear stresses (14) in the volume of the polycrystal. The field of plastic deformation in the crystallite at h = 0 is characterized by the tensor with the components
 r βþ1  Δσ ey ðx, yÞ ¼ 2 1 , ex ðx, yÞ ¼ ey ðx, yÞ, exy ðx, yÞ ¼ 0: E a

(39)

By the formula (31), the minimum external stress was calculated, at which at the center of any crystallite the shear stress achieves its critical value τcr = 50 MPa. The sum in the numerator defines the contributions from the previous relaxation elements in shear stress. The minimum values of this function correspond to the minimum external stress, at which there is a possibility of involvement of a single crystallite into plastic deformation. The coordinates of the point in which σ = σ min have been found and a new relaxation element of considered type was put there. In such a manner, such a crystallite possesses a discrete value of plastic deformation (40) and creates around it a corresponding field of internal stresses (14). Thus, the crystallite effects the stress field in the whole volume of solid. The interaction of the fields of internal stresses from the crystallites undergone plastic deformation together with the external applied stress results in the formation of meso- and macrobands of localized plastic deformation.

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Fig. 32 Consequent formation of meso- and macrobands of localized deformation

Figure 32 illustrates the results of the simulation of the process of strain localization under influence of external applied tensile stress. It is seen that under the operation of the changing inhomogeneous stress field in the volume of the polycrystal a consequent development of the band structure of type B occurs, being typical for a clearly pronounced PLC bands. The bands of localized plastic deformation, consequently formed as a result of jump-like displacement of the process of strain localization along the working part of the specimen belongs to such type of bands [8]. In the considered case the macroband consists of three mesobands with the width of the diameter of the grain. In front of the formed macroband, a fragment of new mesoband is formed (see the frames 1, 6, 9, 10). It spans over the whole specimen cross section (frames 2, 4, 5, 7) at angle of 45 with respect to tensile axis. The development of the mesoband occurs spontaneously without increasing the external stress. Further the expansion of such mesoband takes place on the mechanism of the front of the Lüders band. (frames 1–5) by consequent involvement of the grains into plastic deformation along the front of initially formed band. A pack (bunch) from three mesobands represents a fully formed macroband (frame 5). At a definite distance from the macroband at the edge of the specimen at that time the zone of increasing shear stresses arises (dark background at the edge of the specimen in the frame 8). Achieving of the critical value of τcr defines the initiation and development of new macroband of localized shear at the edge of the specimen (frames 6, 9, 10). The process of initiation and propagation of BLD is repeated periodically. After the process of strain localization achieves the opposite end of the specimen (frame 11), the repeated formation of the macrobands takes place, but already in conjugate direction of the maximum shear stresses (frame 12). Shown in Fig. 33 is the dependence of external stress on the number of grain involvement into plastic deformation. In fact, this dependence represents by itself a loading diagram of the modelled polycrystal, i.e., each n-act defines a definite quantum of plastic deformation (40).

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s, MPa

Δs = 50MPa

100

90

80

70

0

40

80

120

N

Fig. 33 Effect of interrupted flow at the curve of the dependency of external stress on the number of grains, involved into plastic deformation

Each peak is connected with the initiation of a mesoband of localized deformation at the edge of the specimen. The formation of mesobands occurs spontaneously at the decreasing external stress. The first mesoband from three, composing the macroband requires for its initiation the highest external stresses than others. The lowest external stress corresponds to the initiation of the second mesoband. In the course of the development of deformation, the onset of the formation of the new BLD occurs at higher external stresses in comparison with the previous band. Thus the change in the field of internal stresses in the chosen mode of intermittent flow results in the effect of work-hardening. A similar result is obtained for a polycrystal with a noticeably large number of grains (50  200) (Fig. 34). Besides the high-frequent oscillations of external stress, here it is observed a long-periodical modulation of stress, connected with arising and formation of new macrobands of localized shear. It is seen that in the process of the formation of the band, work-hardening of the material takes place. The evolution of the development of strain localization in the modelled polycrystal qualitatively repeats the regularities of the BLD-formation in the alloys with pronounced PLC-effect at the stage of jump-like propagation of the band of localized shear along the working part of the specimen [8, 9].

5.6

Self-Organization of Plastic Deformation in Polycrystals

All the problems of the simulations in mechanics of deformed solid (MDS) is divided into two classes: direct problems (those in which the boundary loading conditions are prescribed and it is necessary to find the stress-strain state) and inverse (stress-strain state is prescribed and it is necessary to define boundary conditions of loading). In MDS there is an actual problem of the solving of inverse problems for

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Fig. 34 Jump-like propagation of the process of localized deformation (a–d)

the case of the plastic deformation in structurally-inhomogeneous media with external and internal interphases. In the given section, an example of the simulations of self-organization of plastic deformation in polycrystal on mesolevel is given, on the example of the solution of inverse problem of MDS yielding to the macroscopic loading diagrams «σ-e». A variant of the model in the approximation of plane stress condition of deformed polycrystal is given. On the mesolevel each crystallite involved into plastic deformation is represented as the relaxation element of the round shape. Figure 3 illustrates the perturbation of the homogeneous stress field σ in the volume of polycrystal under the stress relaxation in crystallite in Δσ. The stress concentration of crystallite beyond the crystallite reaches the value σ þ 2Δσ. The same effect resutls in plastic deformation of the crystallite, characterized by the components epy ¼ 2Δσ=Ε, epx ¼ Δσ=Ε, epxy ¼ 0, where Е is the Young’s modulus and epy component is directed along the tensile axis. The onset and development of plastic deformation in a separate grain is assumed to be characterized by two values of shear stress. The first value τmax defines the onset of grain involvement into plastic deformation at τmax  τcr, where τcr, according to the concepts of physical mesomechanics [2–4, 43], is controlled by the processes in grain boundary planar systems. In grain boundaries the movable deformation defects arise and move in the volume of crystallite. In a definite time step dt the relaxation of shear stress occurs in the crystallite till the level τ0, below

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which the plastic deformation in the grain stops. A relaxation measure in the volume of crystallite is defined by the value Δσ = 2(τcr  τ0). The time interval is defined from physical reasoning. For example, the macroband of polycrystals Al-Al2O3 with the dimension of crystallite 40 μm contains not less than 4  103 grains. The band is formed in 1–2с. Hence, a separate grain is involved into plastic deformation during the time 5  104 s. Localization of deformation is developed by the consequent involvement of crystallite into plastic deformation under the operation of concentrators of inhomogeneous field of internal stresses. The involvement of the crystallites into plastic deformation is accompanied by the geometrical formchanging. A force sensor of the test device responds to the change in the length of the specimen. This allows to obtain the dependence of the flow stress on the sequence of separate structural element involvement into plastic deformation. In this way REM allows to realize a transition from the processes of deformation on mesolevel to the loading diagrams of macroscale level. From the Eq. (37) it is seen that the sign of the increment dσ depends on the difference of the rates of velocities of the free movements of the clamps and the rate of change in the specimen length due to plastic deformation vp. At vp = v0 a flow plateau is observed, and at vp > v0 – a decrease in external load. The bigger is the rigidity of the machine М, the more will be the value of drop of the external load. The equations for the stress fields, derived by REM are valid for the infinite plane. At that time, at the lateral boundaries of the modelled specimen the normal and tangential stresses arise, which should not exist according to the real boundary conditions of loading. During loading these stresses were deleted following the procedure described in 5.4. The σ-e was constructed by summation of the increment dσ, using the obtained Eq. (37). The effect of rigidity of the tensile testing machine on the loading diagrams of polycrystals of low-carbon steel illustrates Fig. 35 with E = 210 GPa [44]. The curve 1 for «soft» mode of loading (M = 1.3  102 kN/mm) has stair-case type. As the rigidity of the machine M increases from 1.3  102 to 1.3  108 kN/ mm, the curve takes a more saw-tooth shape. The yield drop appears and grows. Fig. 35 The effect of the rigidity modulus of the machine M on the type of loading diagramm: М, kN/ mm = 1.3  102 (1), 1.3  103 (2), 1.3  105(3), 1.3  108(4)

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Fig. 36 The effect of the velocity of movement of clamps of loading device v0 on the shape of loading diagrams: v0, μm/s = 1 (1), 10 (2), 20 (3), 30 (4), 40 (5), 50 (6), 80 (7), 110 (8)

Fig. 37 The influence of τcr on the loading diagrams:τcr = 200 (1), 100 (2), 80 (3), 60 (4), 40 (5)

The amplitude of external stress oscillations increases. At all curves the flow plateau is observed, after which the stage of work-hardening follows, caused by internal stress field from relaxation elements. The flow plateau is formed on the mechanism of the Lüders band propagation, when the crystallites is involved into plastic deformation, consequently filling up the working part of the specimen. Repeated involvement of the grains into plastic deformation occurs at higher external applied stresses. The rate of loading exerts much influence upon the σ–e curve. The less the rate of loading, the more pronounced is the effect of intermittent flow (Fig. 36). The increase in the loading rate results in a decrease in an amplitude of oscillations of the external curve. Starting from the definite rate of loading there exist no oscillations of external load at the curves (curve 5). A further increase in loading rate results in disappearing of the yield drop and the flow plateau (curves 6–8). The flow stress increases and the effect of a sharp yield stress disappears. Along with the influence of the rigidity of the machine and the rate of loading, defining the boundary conditions of loading, the effect of the characteristics τcr of material itself has been considered at the same other parameter of the model. Shown in Fig. 37 are the σ–e curves for different values of τcr. If the dislocations are not

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blocked by the atoms of admixture (when τcr = τ0), then the phenomenon of intermittent flow is not observed (low curve). As the τcr increases, the Portevin–Le Chatelier effect arises and enhanced. At that time, the flow stress, flow plateau, and the amplitude of the peaks at the loading diagram increase. The performed simulations allowed us to reveal qualitative and quantitative changing curves of loading, depending on the characteristics of the material itself and on the boundary conditions of loading. The obtained characteristics of the changes of the qualitative and quantitative loading diagrams when varying the parameters of the model are in agreement with known experimental findings [5–9, 36, 37, 45].

6

The Application of the Relaxation Element Method to the Investigation of the Mechanisms of Plastic Deformation and Fracture

6.1

Stresses of Lüders Band Initiation in Polycrystals

It is known that plastic deformation onset far before the yielding of the material [22, 23, 43]. The grains being exposed on the free surface of the polycrystal are involved into plastic deformation first of all. That is why Lüders band embryo can be represented as the site of plastic deformation at the edge of plane polycrystal. The zone of plastic deformation is assumed to propagate inside the polycrystal at the distance, being equal to the average diameter of polycrystal’s grain. For the definiteness let us represent the grain in the form of a proper hexagon. Let the large diagonal of hexagon is directed normal to with respect to tensile y-axis (Fig. 38). An embryo of Lüders band is represented in the form of a number of relaxation elements enclosed one into another (RE) with the common center and the symmetry axes (Fig. 38). Let us construct the form of the Lüders band embryo in the following manner. The boundary of the embryo, starting from the edge of the polycrystal till the straight line ВВ0 , passes along the half-ellipse’s contour with the big half-axes, touching the side of the hexagon in the points В and В0 . Further, from the hand-side from the straight line ВВ0 , the boundary passes now along the side of the hexagon till the junction. The slope of the tangent of the ellipse in the points of tangency B and B' is equal to  
2 h i pffiffiffi dy   ¼ b x0 ¼ tg π ¼ 3, dx a y0 3 where b is the small half-axis of the given ellipse. From here we obtain the connection: y0 ¼


2 b x0 pffiffiffi : a 3

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Fig. 38 Scheme of the embryo of the Lüders band at the edge of plane polycrystal

From pffiffiffi the other hand, from the equation of the straight line AB it follows that y0 ¼ 3ða0  x0 Þ: By equating these two last formulas, we obtain the following equations: 3a2 a0 x0 ¼ 2 and 3a þ b2

pffiffiffi 2 3b a0 y0 ¼ 2 : 3a þ b2

(40)

The point with the coordinates (x0, y0) belongs to ellipse. Let us present the value of the half-axis of ellipse in the form a = f a0, where a0 is the distance from the edge of polycrystal to the point А (Fig. 38), f is the coefficient of proportionality. By substitution of the value (40) in the equation of ellipses, we obtain the following equation: b¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi  3 a20  a2 ¼ a0 3 1  f 2

(41)

For the definiteness let us assume that the radius of curvature of the given halfellipse at the end of its big half-axis, directed along the x-axis, is equal to the half of the average of the dimension of the grain of the polycrystal. The connection d with the big diagonal of the hexagon a0 we will define by the condition that the area of this grain in the form of hexagon is equal to the area of hypothetical grain q in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the formffi of pffiffiffi the round region with diameter d. According to this condition d ¼ a0 3 3=2π : Let us construct with the help of ellipses enclosed in each other the continuous distribution of plastic deformation in the embryo of Lüders band. Let us give to the relaxation elements, not intersecting the straight line BB0 (Fig. 38), the shape of halfellipses with the same relation of the half-axes, being equal to a/b. Within the interval 0 < x < x0 the variable t = 1  x/a will select from the many RE the

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specific contour of the given family. For the simplicity let us assume that the length of the half-axes are equal aðtÞ ¼ ð1  tÞdt and bðtÞ ¼ ð1  tÞdt:

(42)

Elliptical contours, interacting the straight line ВВ0 , smoothly transforms into hyperbolas with the common center at the junction point А. The value of relaxation for the arbitrary RE from the given distribution we will assign in the form of the function on the variable t: dσ ¼ ðβ þ 1Þσ ð1  tÞβ dt,

 1  β  1,

(43)

where σ – is the external applied stress. It is assumed that before the loading, there were no fields of internal stresses in polycrystal and that the matrix beyond the embryo is deformed elastically and isotropically. From general reasoning it is clear that the maximum stress concentration is expected in the vicinity of the junction A of three grains. The instant of Luders band initiation we will define by critical stress σ cr y . According to Saint Venant principle, essential influence of the geometrical shape of the grain boundary on the field of stresses will be essential only in the vicinity of the point A. Let us show that in the points of contours intersection of the contours of RE with the straight line ВВ0 the continuous transition of the contours from ellipses to hyperbolas takes place. Let us find the coordinates of the points of intersection of ellipses with the straight line ВВ0 . According to the equation for ellipses xðtÞ2

yðtÞ2

aðtÞ

bðtÞ2

þ 2

¼ 1,

(44)

at x(t) = x0 we will obtain the equation f2 ð1  tÞ

þ 2

yðtÞ2  ¼ 1: 3a20 ð1  tÞ2 1  f 2

From here we will find the coordinates of the points of intersection:    yðtÞ2 ¼ 3a20 1  f 2 1  f 2  tð2  tÞ :

(45)

On the other hand, hyperbolas in our system of coordinates will be described by the equation ða0  xðtÞÞ2 aг ðtÞ

2



yðtÞ2 bг ðtÞ2

¼ 1:

(46)

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Relaxation Element Method in Mechanics of Deformable Solid

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For all the hyperbolas, theprelationship of the half-axis is the same and equal to ffiffiffiffi bг ðtÞ=aг ðtÞ ¼ bg ðtÞ=ag ðtÞ ¼ 3: From here we will find the small half-axis of the hyperbolas: aг ðtÞ ¼ a0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  f 2 tð2  tÞ:

(47)

Differentiation of (46) with accounting of (47) for each value of the variable t gives the equation of the tangential for the hyperbola at the point of intersection with the straight line ВВ0 (Fig. 38): pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 3 1f dy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : dx ð1-tÞ2  f 2

(48)

By differentiation of Eq. (44), with accounting of the Eqs. (40) and (41), we assure that the tangential line for the arbitrary ellipse in the point of interaction with the straight line ВВ0 is also described by Eq. (49), i.e., the smooth transition of the contours of ellipses into contours of hyperbolas. Thus, parameter t, changing from 0 to 1, unambiguously defined the location, shape, and the sizes of the local regions of relaxation in the embryo of the Lüders band. Now we have enough parameters, in order by variation of the β-parameter to regulate the gradients of plastic deformation between the grain boundary and to analyze their influence on the stress concentration in the vicinity of triple grain junction. Further we will neglect the edge effects at the free boundary of plane polycrystal and to consider the stress fields for the case of unlimited medium. According to Saint Venant principle, the effect of the latter on the stress concentration in the point A is insignificant. That is why instead of the half-contours at the edge of the half-plane one can consider the symmetric full contours in unlimited plane. Before prescribing the distribution of plastic deformation, let us divide the embryo into three zones (Fig. 39) according to the different shape of RE. In the zone I there are RE of proper elliptical shape. The contours intersecting the straight line ВВ0 fit into zone II, after then the elliptical shape transfer into hyperbolic one. Zone III selects the region, where the contours of the given RE have hyperbolic shape (Fig. 39). Displacements of the contour of RE of elliptical shape as a result of stress drop in it in the value dσ provides the homogeneous field of plastic deformation dεy = dσ (1 þ 2b/a) [14]. With accounting of (43) we obtain the following equation: dεy ¼ ð1  tÞβ ð1 þ 2b=aÞdt:

(49)

Integration of elementary fields (50) within the interval of changing of the variable t from 1 to 1  (x2/a2 – y2/b2) defines the following distribution in zone I: eIpy

"  2 ðβþ1Þ=2 # σr  a βþ1 x y2 1þ2 ¼  2þ 2 f : b E a b

(50)

1550

Y. Y. Deryugin et al.

Fig. 39 Zone of plastic deformation in the embryo of a Lüders band

The contribution to plastic deformation from RE not having proper elliptical shape is not possible to define exactly, because there is no exact analytical solution for the displacement of the contour of ellipses, transferring into hyperbola under stress relaxation in it in a definite value. Nevertheless, in simplified variant one can calculate the distribution of plastic deformation in zone II exactly enough. The results of numerical calculations of the boundary-value problem of the theory of cut-outs reveal, that essential influence on the displacements of the points of the contour and on the stress concentration under tensile loading only two geometrical parameters: the length of the cut-out in the direction normal to tensile axis and the minimum radius of the curvature at the edge of the cut-out along this direction [46–48]. Hence, it is possible to substitute the contour of RE with hyperbolic trajectory at the end of the big semi-axis a(t) with the elliptical contour with the half-axis of the same length and with the radius of the curvature at the end of the band r(t), being equal to those for the corresponding hyperbola. The point displacement of the contour of such an ellipse is easy to define. They will exactly match the displacements of the points of real contour of RE. Then, neglecting the influence of the substitution on the transverse component depx , one can easily find an expression sffiffiffiffiffiffiffiffi) ( σ ð β þ 1 Þ aðtÞ y0 1þ2 ð1  tÞβ dt depy ¼ E rðtÞ y ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) σ ð β þ 1Þ 2 1 y0 1 þ pffiffiffi  ð1  t Þβ , ¼ 2 E y 1  f t ð 2  t Þ 3

(51)

where y and y0 are the coordinates of corresponding points at the real and substituted contours. r(t) = [a0  a(t)] is the curvature radius at the end of the big semi-axis of hyperbola.

47

Relaxation Element Method in Mechanics of Deformable Solid

1551

Integration of (51) over the variable t with accounting for (44) defines the contributions of RE into plastic deformation, intersecting the straight line ВВ0 . For RE in the Zone II the relationship y0 /y is equal rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 4 f tð2  tÞ a20 1  1  f 2 tð2  tÞ  x2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðy0 =yÞII ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ffiffiffiffiffiffiffiffiffiffiffiffi h p  4 2 2 1f 1 1  f tð2  tÞ a20 f 2 ð1-tÞ2  x2 And contribution of distorted RE in the distribution of plastic deformation within the zone II is defined by the equation qffiffiffiffiffiffiffiffi 1

ðβ þ 1Þσ r eIIpy ¼ E

x2 y2 2 þ b2

ða

" 2 1 þ pffiffiffi 3

0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #
 1 y0  ð1  tÞβ dt: 1  f 2 tð2  tÞ y II

(52)

In zone I at that time will be observed the distribution of plastic deformation eIp y , which is defined by the integration of the Eq. (49) from 0 to 1  f. Thus, in the region, limited by elliptical contour, touching it in the point x0, this contribution is summed with the component (50). In the zone III RE contours have the shape of hyperbola. At that time the relationship y0 /y is equal

ðy0 =yÞIII

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qp h pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 ffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 1  f tð2  tÞ a0 1  1  f tð2  tÞ  x ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : h pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  1  f 2 tð2  tÞ ða0  xÞ2  a20 1  f 2 tð2  tÞ

The contribution of RE in the distribution of plastic deformation in the given zone is defined by the equation eIIIpy ¼

ðβ þ 1Þσ r E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 1

1

3ða0 xÞ y2 2Þ

ð3a0 ð1f

 0

"

2 1 þ pffiffiffi 3

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
 1 y0  ð1  tÞβ dt: 2 1  f tð2  tÞ y III

(53)

The sum of distributions epy ¼ eIpy þ eIpy  þ eIIpy þ eIIIpy fully describes the distribution of plastic deformation in the nuclei of the Luders band. Shown in Fig. 40 are the examples of the spatial distributions of plastic deformation in the Lüders band embryo, obtained with the application of the derived above equations. Quantitative and qualitative characteristics of the plastic

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Y. Y. Deryugin et al.

Fig. 40 Examples of the distribution of plastic deformation in the embryo of Lüder’s band

deformation distribution depends are seen to depend essentially on the value βparameter. At increasing β the gradients of plastic deformation at the junction of the embryo increase too. At this time, at a definite instance of time, the zone of elevated concentration of strain arises at the junction. A maximum value of plastic deformation epy gradually increases, approaching to the junction.

47

Relaxation Element Method in Mechanics of Deformable Solid

1553

Fig. 41 Profiles of plastic deformation epy(a) and stresses σ y(b) along x-axis in the embryo of Lüders band. β = 2m  1. The numbers at the curves indicate m values

The profiles along x-axis of the epy component for the various values of β are presented in Fig. 41a. At the small values of β a smooth decrease of epy is observed as the junction is approaching. The data are obtained at the value а0 = 10 of the conditional units, Е = 210 GPa and high external applied stress σ = 1 GPa. Within creasing β the gradient of plastic deformation at the junction and the maximum is arising and developing, more and more displacing to the junction. At the high gradient the plastic deformation at the junction exceeds the one in the interior of the grain. The calculations show that the contribution of distributions (51) and (53) into plastic deformation at the high values of β-parameter become negligible.

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Y. Y. Deryugin et al.

Unfortunately, up to now there is no quantitative experimental data on the distribution of plastic deformation in a separate grain of a polycrystal at the stage of Lüders band initiation. This doesn’t allow to define the specific value β, characterizing the degree of heterogeneity of the distribution of plastic deformation for the given material. Nevertheless, given model allows to analyze a stress state of a material in the vicinity of the triple grain junction depending on the gradients of plastic deformation. About the value of stress state around the embryo one can judge by the profile of the σ y component along the х-axis. Stress field from the stress relaxation in the local area in the local value of elliptical shape in the value Δσ corresponds to the stress distribution for the case of the plane with elliptical cut-out at the uniaxial stress with intensity Δσ. In the latter case the solution is usually given in the elliptical coordinates [49]. The transition to the Cartesian coordinates results to the following equation for the profile of the component σ y of the stress tensor along х-axis
2  Δσa b xða  2bÞ b2 x þ 3 , þ Δσ y ðxÞ ¼ (54) a  b a  b ða  bÞc c where с = (x2  a2 + b2)1/2. For the arbitrary contour from the family of RE Δσ = dσ(t), with accounting of (42) and (43) the integration of the Eq. (54) will result to the following equation for the σ y-component along x-axis: σIy ¼ ðβ þ 1Þ  σ2 0 2

B 6 b þ@ 4 ab

ð1

ð1  tÞβ A ab

1

3 2

axða  2bÞ b xað1  tÞ C 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ  3=2 5  dt, 2 2 2 2 2 2 ða  bÞ x  ða  b Þð1  tÞ x2  ða2  b Þð1  tÞ 2

(55) where А = 1  x/a within the interval 0  x  x0 and А = 1  f at x > x0. For the definition of the contribution to the stress fields from the rest of RE with hyperbolic shape beyond the limits x > x0, i.e., intersecting the straight line ВВ0 , let us use the same Eq. (54), where instead of a and b half-axis of ellipses we substitute the corresponding values af and bf of half-axis of fictitious ellipses. Radius of curvature of these ellipses at the end of semimajor axis is equal to the radius of curvature of corresponding hyperbolas. By construction, the ratio of semi-axis of all hyperbolas satisfies the equations b'2/a'2 = 3. From the equations of hyperbolas we can find the distance from the point А to the corresponding hyperbola a0 ðtÞ ¼ a0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  f 2 tð2  tÞ::

(56)

47

Relaxation Element Method in Mechanics of Deformable Solid

1555

Hence, the length of the semimajor axis of fictitious ellipses is equal to
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  af ¼ а0 1  a0 1  f 2 tð2  tÞ

(57)

Corresponding radii of curvature at the end of the major half-axis of fictitious ellipses are equal 0

r ðtÞ ¼ 3a ¼ 3a0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  f 2 tð2  tÞ:

(58)

The length of the small semi-axis of fictitious ellipses with accounting of (57) and (58) are defined according to the formula rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi   4 f 1  f 2 tð2  tÞ 1  f 2 tð2  tÞ: b ¼ a ðtÞr ðtÞ ¼ a0 3 1  f

(59)

Integrating the Eq. (54) within the corresponding limits, where a and b are replaced by af and bf, correspondingly, we obtain the values σ y along x-axis. At x > a0 the limits of integration are the same. Within the intervals x0 < x < а0 limits of integration have the following values, the lower limit is t1 = 1  f and upper limit sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1 x 2 : 1  t2 ¼ 1  1  a0 1  f2

(60)

The profiles of the components σ y along x-axes for the different β values are depicted in Fig. 41b. It is seen that the stress starting from zero value passes through the maximum near the junction, then falls, asymptotically approaching the level of external applied stress. In the grain interior one can always select the area where the level of σ y is lower than σ r. With increase in the gradients (β-parameter) stress concentration in the vicinity of junction increases. The resulting equations define gradient part of stress fields. Besides that, under external applied stress there exist a homogeneous stress field σ 0 = σ  σ r. At increasing β the maximum σ y is getting closer to the junction. The area at σ y < σ expands. At the high gradients of stress concentration may exceed one order of magnitude and more the external applied stress σ. In a similar way, one can obtain the equation for the component σ х. The component σ хy along х-axis is equal to 0. From general considerations it is clear that stress-strain state in the vicinity of triple grain junction at the moment of Lüders band initiation should not be essentially dependent on the grain size. That means that within the wide range of grain sizes the stress gradients in the vicinity of tripple grain junction should be practically constant. Analysis shows that the maximum (in absolute value) gradient of stress σ y is always observed in the point of junction А.

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Y. Y. Deryugin et al.

Fig. 42 Stress dependence of the Lüders band initiation on the grain size of polycrystal

From the presented distributions of plastic deformation and stresses it is seen that at the constant values β and σ r with increase in the grain diameter d the gradients of the field decrease under the law of inverse proportionality. In order to fulfill the condition of constancy of the critical stress gradient, it is necessary to decreasing of the gradient to compensate due to the corresponding increase in β parameter and vice versa. To be specific, it is necessary the specific value d0 (arbitrary unit of grain diameter) to connect with the specific value of β0 parameter. Shown in Fig. 42 is the dependence of the stress initiation σ 00 of the Luders band on the grain size in the embryo at τcr = 1000 MPa in the coordinates σ 00  d1/2 at d0 = 1 and β0 = 0. It is seen that with decrease in the grain size the stress of Luders band initiation increases. At β < 1 σ 00 tends to the value of 2τcr asymptotically. Within a wide range of grain sizes the dependence σ 00 on d1/2 obeys almost linear dependence, known as Hall–Petch effect. Experimental verification of the dependence of flow stress on the grain sizes within a wide interval of d confirms the justice the Hall–Petch law. In this way, the dependence of mechanical properties of polycrystals on the grain sizes, observed in experiments, is explained by the stress-strain state of the material in the vicinity of the junction of three grains, in which the gradient of plastic deformation plays a principal role. In this connection the attention of experimentalists should be concentrated on the obtaining of the information about the gradients of plastic deformation at the grain boundaries at different stages of loading of structurally inhomogeneous materials. The method presented extends the possibility on the experiment data to analyze and check the mechanisms, being responsible for the arising and development of plastic flow in the local volumes of corresponding scale. In particular, at the given example of continuum model of Luders band initiation it is shown that specific peculiarities of loading diagrams of real polycrystals are not possible to explain without the accounting of the gradients of plastic deformation in the local volumes and the geometry of elements of structure of the material. At the microconcentrators in structurally inhomogeneous materials, stress concentration can one order of

47

Relaxation Element Method in Mechanics of Deformable Solid

1557

magnitude exceed the external applied stress. The prediction of the model is very good consistent with the typical regularities, observed in experiments: – within a wide range of grain sizes the flow stress obeys the Hall–Petch law; – the average degree of plastic deformation in the grain decreases with decrease in the grain size [50]; – in coarse-grain polycrystals the plastic deformation at the grain junction exceeds essentially those in the interior of the grain [51].

6.2

Modified Model of Crack

The process of the formation of substructures of deformation is often accompanied by the discontinuities and microcracks. In this connection there is an actual problem of the description of the field of stresses at the crack tip [52–54] and the investigation of the influence of plastic deformation of the stress state of solid with crack. On the basis of the engineering calculations on strength lifetime of the elements of constructions as a rule, the principles and criteria of linear mechanics of fracture are used [52–54]. The latter are obtained, based on the properties of classical model of Griffith crack, in spite of its essential drawback, connected with the presence of singularity at the crack tip, approaching of which results in very quick growth of stress. In this connection in the mechanics of fracture the physical notion of coefficient of stress concentration at the crack tip is not used, but as a characteristic of inhomogeneous stress field a stress intensity factor in the vicinity of singular point is used. An assumption on the nonlinearity at the crack tips allows to take into account a number of known models for the calculations [55, 56]. However, their application is essentially restricted. They are valid only in the case when the zone of plastic deformation is extremely small in comparison with the length of the crack itself. At the description of the stress state near the crack with significant zone of plastic deformation, there are great mathematical and calculational difficulties. Critical values of the stress intensity factor КIс (SIF), characterizing inhomogeneous stress field in the vicinity of crack tip in quasi-brittle material, cannot serve as a parameter, characterizing the ability of the material to resist the crack propagation in metals and alloys, where large plastic deformation precedes the crack nucleation and propagation. That means that the characteristic KIс for plastic material lose its previous physical sense, especially in the case of testing of small-scaled specimens where the plane strain condition is clearly violated. It is known that a definite degree of plastic deformation always preceeds the moment of fracture. That means that crack propagation occurs in the layer of plastically deformed material. In other words, the crack from the moment of its nucleation is always surrounded by the layer of plastically deformed material. The thickness of the layer and the distribution of plastic deformation in it apparently depend on the ability of the material to plastic deformation.

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Fig. 43 The scheme of the crack in the surrounding of plastically deformed material

Let us imagine the crack in the form of an elliptical hole with the layer of plastic deformation along the ellipse’s contour (Fig. 1). In fact, our task is reduced to the definition of the field of internal stresses of the given system. The continuity condition of material beyond the hole requires smooth changing of stresses from the values in the volume of the material to zero value at the boundary of the hollow. The matrix beyond the selected layer is assumed to be isotropic and is elastically deformed under the operation of tensile stress σ, directed along y-axis. Shown in Fig. 1 is the layer around the crack in the form of relaxation elements of elliptical shape, enclosed in each other. For the definiteness it is necessary to characterize the geometrical parameters of each RE from the family. Let us do it in the following manner. Inside each RE from the given family let us prescribe the value of elementary stress relaxation (elementary stress drop dσ) in such a manner, that under the external σ an integral sum of stress drop from all the RE eliminates the all stresses in the hollow at the external stress σ (Fig. 43). Assume that all ellipses have the common center at the origin of coordinates and half-axis coinsiding with the coordinate axis. The length of the semi-axes we define by equations a(t) = a0 + h(1 – t), b(t) = b0 + h(1 – t), where a0 and b0 are the big and small semi-axes of the hole, half-axis b of the site is directed along the tensile axis y (Fig. 43), h is the thickness of the length around the hollow, t is the variable, changing within the limits from 0 to 1. At t = 0 semi-axes are maximal. Increase in t corresponds to the consequent (gradual) transition from the external ellipses to the geometrical line of the hollow. Specific value t chooses a definite contour from the family. At b!0 the hollow transforms into a crack. Let a0 >> h, then the point at the axis x corresponds to the RE contour with the variable t ¼ 1  ðx  a0 Þ=h:

(61)

The elementary value (infinitely small value) of relaxation tensor for every relaxation element can be written as a function of the variable t: dσ r ¼ ðβ þ 1Þσtβ dt,  1  β  1:

(62)

Relaxation Element Method in Mechanics of Deformable Solid

Fig. 44 Normalized relaxation stress in the surface layer for crack with β = 0 (1), 1(2), 3(3), 7(4), 15(5)

σr /σ

47

1559

a0 =9 h=1

0.8 1 0.6 2 0.4 3 0.2

4 5

0

0

0.2

0.4

0.6

0.8

x-a0

The parameter β in the Eq. (62) regulates (controls) the degree of relaxation with continuous transition from one contour to another. Note that the relaxation stress increases with t. Superposition holds for the relaxation element as in the linear theory of elasticity. When integrating Eq. (62) from 0 to 1, the normalization factor βþ1 ensures inside the hole a full absence of stresses. When integrating Eq. (56) from the value t = 0 to the value according to Eq. (61) the value of stress drop increases as the point at the hole is approached. The parameter β regulates the speed of change in the given value. Larger β corresponds to increase in the speed of relaxation. Figure 44 plots the normalized relaxation stress σ r/σ as a function of distance along x-axis within the thickness of the layer. It is seen that the resulting relaxation tensor ensures smooth decrease in the value of stress relaxation. Increase in the rate of stress relaxation (parameter β) results in the fact that a significant stress drop starts at small distances from the crack tip. In other words, increase in β results in the effect of decrease of physical width of the zone of plastic deformation. Since there is an analytical connection between the elementary relaxation tensor inside and elementary stress field outside [11–14], then by prescribing with the help of the distribution of RE the value of relaxation in the local regions, a resulting inhomogeneous stress field in the whole plane as well as in the layer itself is defined automatically. In the system of coordinates, depicted in Fig. 43, for the components of elementary stress field of arbitrary RE along the x-axis, according to conditions (61) and (62), the following equations can be written: "

# h2 ð1  tÞ2 x dσ xx ¼ σ ðβ þ 1Þt (63)  3=2 dt, x2  a20 " # 2 2 x h2 ð1  tÞ2 x β h ð1  tÞ dσ yy ¼ σ ðβ þ 1Þt þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi þ  3=2 dt:, dτxy ¼ 0: a20 x 2  a0 x2  a20 β

x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 2 x  a20

!

1560

Y. Y. Deryugin et al.

In equation the small semi-axis (64) of arbitrary ЭР in the family is equal to b = h (1  t), as the hollow transfer to the crack at b0 = 0. By integrating Eq. (63) on the variable t, it is necessary to take into account that beyond the layer the limits of integration are taken from 0 to 1, and at the points, fit in the layer from t to t = 1  (x  a0)/h. For the profiles of the σ y component along the x-axes we obtain the following equations: ! σy 2h2 1 x x þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi ¼ AðxÞ, if x  a0 þ h and ¼ þ 2 3∕ 2 2 σ ð β þ 2Þ ð β þ 3Þ a0 x 2  a0 x2 
 a0  σy x  a0 βþ1 ¼ Að x Þ 1  1  σ h !  β þ 1 h2 xh2 β þ 4 x  a0  x  a0  þ þ þ 1  ,  3∕ 2 β þ 3 a20 βþ2 h h x 2  a2 0

(64) if a0  x  a0+h. The changing of the stress distribution σ y, according to (64), at the variation of βparameter is demonstrated in Fig. 45. It is seen that contrary to the Griffith solution (curve 7), there is no singularity at the crack tip (Fig. 45). Within the surface layer, the stress continuously increases, starting from 0 at the end of the crack, passes through the maximum, then decreases asymptotically approaching the external applied stress σ. Beyond the layer, a qualitative and quantitative discrepancy of the curves is practically disappears. The increase in β parameter results in growth of stress concentration and to the displacement of the maximum towards the boundary of the hollow. In limiting case at β ! 1, we yield a Griffith curve (Curve7). The decrease in the physical width h of the surface results in analogous effect (Fig. 46). The increase in the length of the crack affects greatly the stress concentration (Fig. 47). At that time, contrary to the case in Fig. 46 with increase in the length of

20

7 h=5 a0 = 95

6

15 σy/σ

Fig. 45 Profiles of stress σ y at the crack tip. The numbers at the curves identify the degree of j for β = 2j

β = 2j

5

10 4 3

5

2 1 0

0

0

1

2

3

4

5

6

x - a0

47

Relaxation Element Method in Mechanics of Deformable Solid 20 a0 = 95 β=1

15 σy /σ

Fig. 46 Profiles σ y at the crack tip. The arrows indicate the boundary of the zone of plastic deformation

1561

10

5

0

1

2

40

3

4

5

6

x - a0

h=5 a0 = 5.2 j β=2

30 11

σy/σ

Fig. 47 Profiled of stressσ y at the crack tip.The numbers at the curves point out the degree of j for а0 = 5  2j

0

10

20 9 8 7 6

10

4 0

0

0

10

20

x - a0

30

the crack, the stress increases not only in the near-boundary layer, but also beyond it, i.e., the area of elevated concentration is expanded essentially. The advantage of the considered description of stress-strain state of solid with crack is apparent, because the singular solution follows from it as a particular case, when the thickness of the layer of plastically deformed material tends to zero, or when the parameter β tends to infinity. The proposed model allows to analyze an effect of the value of the gradient of plastic deformation on the concentration and stress distribution at the crack tip.

7

Conclusion

In this chapter, the possibilities of new methods of the relaxation elements are elucidated. Examples of the construction of sites with gradients of plastic deformation have been given. A stress-strain state of the plane with a round inclusion is

1562

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considered. The analytical description of the plane with the band of localized shear is performed. The possibilities of the modelling by such a method of the effect of the localization of plastic deformation are considered. An analysis of the influence of the rigidity of the testing device on the effect of interrupted flow is presented. Important results testify to the high predictive possibilities and perspectives of developed method. They are in agreement with known experimental data: • Effect of intermittent flow is a sequence of the formation of mesobands and macrobands of localized deformation. • In the changing field of stresses the jump-like propagation of the bands of plastic deformation along the working part of the specimen takes place. • The increasing of the loading rate suppresses the effect of interrupted. • A “saw”-type of the σ–e curve is typical for the rigid mode of loading (device with the a high value of rigidity), in the soft mode of loading σ–e curves have stair shape. The following peculiarities of the development of localized deformation have been elucidated: – The formation of mesobands occurs by the mechanism of the Lüders band propagation and is accompanied by the relaxation of internal stresses. – The structure of a separate macroband consists of the number of mesobands, being oriented along the direction of maximum tangential stresses. – Accumulation of the fields of internal stresses in the volume of a polycrystal results in the effect of strain hardening. – A necessary condition of the arising of Portevin–Le Chatelier effect is the fact that the stress of the onset of plastic deformation of the element of structure is essentially higher than the stress at which the following plastic deformation can proceed. A sufficient condition is the condition at which the velocity of increase in the length of the specimen due to plastic deformation periodically outpaces the velocity of the movement of the clamps of the machine.

References 1. Sutton AP, Balluffi RW. Interfaces in crystalline materials. Oxford: Clarendon Press; 1995. 2. Panin VE. Modern problems of physical mesomechanics. In: Proceedings of the International Conference of MESO-MECHANICS 2000: Role of Mechanics for Development of Science and Technology, June 13–16, 2000, vol. 1. Beijing: Tsinghua University Press; 2000. p. 127–142. 3. Panin VE, Grinyaev YV. Physical mesomechanics – new paradigm at the junction of physics and mechanics. Phys Mesomech. 2003;6(4):9–36. 4. Panin VE, Egorushkin ВЕ, Panin AV. Physical mechanics of deformed solid as multi-level system. Phys Mesomech. 2006;9(3):9–22. 5. Estrin Y, Kubin LP. Spatial coupling and propagative plastic instabilities, Chapter 2. In: Mühlhaus H-B, editor. Continuum models of materials with microstructure. Chichester: John Wiley & Sons Ltd; 1995. p. 395–450. 6. Klose FB, Ziegenbein A, Weidenmüller J, Neuhäuser H, Hähner P. Portevin-Le Chatelier effect in strain and stress controlled tensile tests. Comput Mater Sci. 2003;26:80–6.

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7. Zhang Q, Jiang Z, Jiang H, Chen Z, Wu X. On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital specle pattern metrology. Int J Plast. 2005;21:2150–73. 8. Krishtal ММ. Instability and mesoscopic inhomogeneity of plastic deformation. Phys Mesomech. 2004;7(5):5–45. 9. Deryugin YY, Panin VE, Schmauder S, Storozhenko IV. Effects of strain localization in the composites on the basis of Al with Al2O3. Phys Mesomech. 2001;4(3):35–47. 10. Deryugin YY. Relaxation Element method. Novosibirsk: Nauka, Siberian publishing House RAN; 1998. 11. Deryugin YY, Lasko GV. Relaxation element method in the problems of mesomechanics and calculations of band structures. In: Panin VE, editor. Physical Mesomechanics and computeraided design of materials, vol. 1. Novosibirsk: Nauka; 1995. p. 130–61. 12. Deryugin YY, Lasko GV, Schmauder S. Relaxation element method. Comput Mater Sci. 1998;11(3):189–203. 13. Deryugin YY. Relaxation element method in calculations of stress state of elastic plane with the plastic deformation band. Comput Mater Sci. 1999;19:53–68. 14. Deryugin YY, Lasko G, Schmauder S. Relaxation Element Method In Mechanics of Deformed Solid. In: Oster WU, editor. Computational Materials. Hauppauge: Nova Science Publishers, Inc; 2009. p. 479–545. 15. Eshelby JD. Continuum theory of defects. In: Seitz F, Turnbull D, editors. Solid State Physics, vol. 3. New York: Academic Press; 1956. p. 79. 16. de Wit R. (1970) Linear theory of static disclinations. In: Fundamental aspects of dislocation. Ed. by J.A. Simons, R. de Wit, R. Bullough, Nat. Bur. Stand. (US) Spec. Publ. 317, vol. I 651-673. 17. Likhachev VА, Volkov АЕ, Shudegov VЕ. Continuum theory of defects. Leningrad: Leningrad University Publishing House; 1986. 18. Rybin VV. Large plastic deformations and damage of metalls. Moscow: Mettallurgia; 1986. 224 p. 19. McHugh PE, Asaro RJ, Sich CF. Cristal plasticity models. In: Suresh S, editor. Fundamental of metal matrix composites. Boston: Butherworth-Heinman; 1993. 20. Harder J. Simulation lokaler Fliessvorgänge in Polykristallen. Braunschweiger Schriften zur Mechanik. B. 28. Braunschweig; 1997. 21. Balokhonov RR, Romanova VА. Numerical simulation of thermo-mechanical behaviour of steels with accounting of Luders band propagation. Appl Mech Tech Phys. 2007;48(5):146–55. 22. Dudarev EF. Microplastic deformation and yield stress of polycrystals. Tomsk: Publishing House of Tomsk University; 1988. 256p. 23. Dudarev EF, Deryugin YeYe. Microplastic deformation and yield stress of polycrystals. Izv Vyzov Fizika. 1982;(6):43–56. 24. Iricibar R, Mazza I, Cabo A. The microscopic strain profile of a propagating. Lüders band front in mild steel. Scr Metall. 1975;9(10):1051–8. 25. Andronov VM, Gvozdikov AM. Stress state at the front of Chernov-Lüders band and unstable plastic flow of crystals. Physica Metals Metallovedenie. 1987;63(6):1212–9. 26. Danilov VI, Zuev LB, Letahova EV, Orlova DV, Оhrimenko IА. Types of localization of plastic deformation and the stages of loading diagrams of metallic materials with different crystalline structure. Appl Mech Tech Phys. 2006;47(2):176–84. 27. Barannikova SА, Danilov VI, Zuev LB. Localization of plastic deformation in mono- and polycrystals of Fe3%Si –alloy under tension. J Tech Phys. 2004;74(10):52–6. 28. Iricibar R, Mazza I, Cabo A. The microscopic strain profile of a propagating Lüders band front in mild steel. Scr Metall. 1975;9(10):1051–8. 29. Sakui S, Sakai Т. The effect of strain rate, temperature and grain size on the lower yield stress and flow stress of pure iron. Proc Int Conf Sci Technol Iron Steel, Tokyo Part 2 Tokyo (1971). 1970:989–91. 30. Bell JF. In: Filin AP, editor. Experimental basis of mechanics of deformed solids. М.: Nauka; 1984. 600 p. 31. Wirtman J, Wirtman JР. Mechanical properties, insignificantly depending on the temperature. In: Han RU, Haasen P-T, editors. Phizicheskoe metallovedenie. Мoscow: Metallurgiya; 1987. 663 p.

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32. Stress relaxation testing. In: Fox A, editor. Symposium on Mechanical Testing, American Society for Testing and Materials, Kansas City, Mo, 24, 25 May 1978. Philadelphia: American Society for Testing and Materials; 1979. p. 216S. 33. Oding IA, editor. Creep and stress relaxation in metals. Academy of Sciences of the U.S.S.R. All-Union Institute of Scientific and Technical Information. Edinburgh: Oliver & Boyd, X; 1965. p. 377S. 34. Kirsch G. Die Theorie der Elastizitat und die Bedurfnisse der Festigkeitslehre. Zantralblatt: Berlin Deutscher Ingenieure. 1898;42:797–807. 35. Timoshenko SP, Goodier JN. Theory of elasticity. 3rd ed. New York: McGraw Hill; 1970. 36. Ziegenbein А, Plessing, Näuhauser X. Investigation of the mesolevel of deformation at the formation of Luder’s band in monocrystals of concentrated alloys on the basis of Iron. Phys Mesomech. 1998;1(2):5–20. 37. Malin A, Hubert J, Hatherley M. The microstructure of rolled copper single crystals. Zs Metallk. 1981;72(5):310–7. 38. Nakayama Y, Morii K. Microstructure and shear band formation in cold single crystals of AlMg alloy. Acta Metall. 1987;35(7 (2)):1747–56. 39. Zasimchuk EЭ, Markashova LI. 1998. Microbands in nickel monocrystals, deformed by rolling. Kiev: АNUkr SSR, Institute of Metallophysics, Preparing No. 23. 36p. 40. Gould H, Tobochnik Y. An introduction to computer simulation methods. Pt. 2: Application to physical systems. Moscow: Mir; 1990. 41. Smolin АY. Development of the method of movable cellular automata for the simulaton of the deformation and fracture of the media with accounting to their structure. Habilitation Thesis, Томск, 2009, 285 p. 42. Crouch SL, Starfield AM. Boundary element methods in solid mechanics. London: George Allen & Unwin. 43. Panin VЕ. Physical mesomechanics of the surface layers in solids. 1999;2(6): 5–23. 44. Physical values: handbook. Moscow: Energoatomizdat; 1991. 45. Koneva NА, Kozlov EV. Physical nature of the stages of plastic deformation. Izv Vyzov, Filika. 1990;33(2):89–106. 46. Kelly А. High-strength materials. Мoscow: Mir; 1976. 264p. 47. Peterson IV. Stress intensity factors. Мoscow: Mir. 1977. 304p. 48. Thompson A. Substructure strengthening mechanisms. Met Trans. 1977;6:833–42. 49. Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity. Moscow: Nauka. 708 p. 50. Utyashev FZ. Compatibility condition of plastic deformation and the ultimate refinement of the grains in metals. Fizika i technika vysokih davlenii. 2010;20(4):10–20. 51. Rybin VV. Regularities of Mesostructures Development in Metals in the Course of Plastic Deformation. Prob Mater Sci. 2003;33(1):9–28. 52. Hertzberg RW. Deformation and fracture mechanics of engineering materials. New York: Wiley; 1976. 53. Broek D. Elementary engineering fracture mechanics. Leiden: Noordhoff; 1974. 54. Stepanova LV. Mathematical methods of fracture mechanics. Moscow: FIZMATLIT; 2009. 336 p. 55. Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids. 1960;8(2):100–8. 56. Slepyan LI. Mechanics of cracks. Leningrad: Sudostroenie; 1990. 296 p.

Damping Characteristics of Shape Memory Alloys on Their Inherent and Intrinsic Internal Friction

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Shih-Hang Chang and Shyi-Kaan Wu

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Ti50Ni50 SMA (B2 ! R, R ! B190 , and B2 ! B190 Martensitic Transformations) . . . . 2.1 Specimens Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Inherent and Intrinsic Internal Friction Under Isothermal Conditions . . . . . . . . . . . . . . 3 Ti50Ni50-xCux SMAs (B2 ! B19, B19 ! B190 , and B2 ! B190 Martensitic Transformations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Specimens Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Inherent and Intrinsic Internal Friction Under Isothermal Conditions . . . . . . . . . . . . . . 4 Ti50Ni50-xFex SMAs (B2 ! R and R ! B190 Martensitic Transformations) . . . . . . . . . . . . 4.1 Specimens Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Inherent and Intrinsic Internal Friction Under Isothermal Conditions . . . . . . . . . . . . . . 5 Groups I–III Ni2MnGa SMAs (P ! I, I ! M, and P ! M Martensitic Transformations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Specimens Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Inherent and Intrinsic Internal Friction Under Isothermal Conditions . . . . . . . . . . . . . . 6 Ni-Mn-Ti SMA (β ! θ0 Martensitic Transformation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Specimen Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Inherent and Intrinsic Internal Friction Under Isothermal Conditions . . . . . . . . . . . . . . 7 Cu-Al-Ni SMAs (β1 ! β10 and β1 ! γ10 Martensitic Transformations) . . . . . . . . . . . . . . . . . . 7.1 Specimens Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Inherent and Intrinsic Internal Friction Under Isothermal Conditions . . . . . . . . . . . . . . 8 Concluding Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S.-H. Chang Department of Chemical and Materials Engineering, National I-Lan University, I-Lan, Taiwan S.-K. Wu (*) Department of Materials Science and Engineering, National Taiwan University, Taipei, Taiwan Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_31

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Abstract

In this chapter, damping characteristics of the inherent and intrinsic internal friction (IFPT þ IFI) peaks for Ti50Ni50, Ti50Ni50-xCux, Ti50Ni50-xFex, Ni2MnGa, Ni-Mn-Ti, and Cu-Al-Ni shape memory alloys (SMAs) are reviewed. Ti50Ni50 SMA exhibits obvious (IFPT þ IFI) peaks with tan δ above 0.02 during martensitic transformations, but they only exist in a narrow and low temperature range. Ti50Ni50-xCux (x  10) SMAs show higher (IFPT þ IFI) peaks than Ti50Ni50 SMA because B19 martensite in Ti50Ni50-xCux SMAs is originated by substituting Ni with Cu atoms while R-phase in Ti50Ni50 SMA is caused by the introduction of abundant defects/dislocations. Ti50Ni48Fe2 and Ti50Ni47Fe3 SMAs also exhibit higher (IFPT þ IFI) peaks than Ti50Ni50 SMA because R-phase formation is due to the substitution of Ni by Fe atoms rather than induced by introduced dislocations. However, the martensitic transformation temperatures of Ti50Ni50-xFex SMAs are suppressed to lower temperatures simultaneously by the addition of Fe atoms. NiMn-Ti and Ni-Mn-Ga magnetic SMAs both exhibit relatively high martensitic transformation temperatures. Unfortunately, the undesirable brittle nature of NiMn-Ti and Ni-Mn-Ga SMAs critically limits their workability and high-damping applications. Cu-Al-Ni SMAs exhibit acceptable martensitic transformation temperatures and good workability; however, their (IFPT þ IFI) peaks are relatively low. Among these various types of SMAs, Ti50Ni40Cu10 SMA is more suitable for high-damping applications because it possesses the advantages of high (IFPT þ IFI) peaks, adequate workability, and an acceptable martensitic transformation temperature near room temperature. Keywords

Shape memory alloys (SMAs) · Martensitic transformation · Internal friction · Dynamic mechanical analyzer · Isothermal treatment

1

Introduction

When materials are subjected to an oscillatory variable loading, most of the energy of the loading is dissipated as heat but some of which is stored by the materials as potential energy. The processes that dissipate the energy as heat in materials are termed as damping or internal friction. The internal friction behaviors of materials are associated with a wide range of physical mechanisms, including applied loading strain, frequency, temperature, and the microstructure of the materials [1, 2]. The loss coefficient Q1 is typically used for describing the damping behavior of materials, which is defined as: Q1 ¼ tan δ

(1)

where δ is the phase angle and the Q is the quality factor [1]. The damping level of the materials is a critical criterion for engineering design. Normally, metals and alloys with Q1 above 102 can be considered as high-damping materials [3].

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Although particular polymers and rubbers also possess good damping capacity, metals and alloys with high internal friction are still indispensable for many engineering designs, especially for high-damping applications where sufficient mechanical strength is also required. The origin of the damping capacity of high-damping metals and alloys is associated with several mechanisms, including the hysteretic movement of defects and the interaction of defects, dislocations, interfaces between different phases, magnetic domain walls, or between martensite variants [3]. Shape memory alloys (SMAs) are classified as high-damping materials because they typically exhibit an obvious internal friction (IF) peak during martensitic transformation concurrently with a significant storage modulus drop. Generally, the martensitic transformation internal friction peak of SMAs can be decomposed into three terms: IFTr, IFPT, and IFI [3–5]. IF ¼ IFTr þ IFPT þ IFI

(2)

The first term IFTr is a transitory internal friction which appears only at lowfrequency and non-zero heating/cooling rates. The second term IFPT is the internal friction due to the phase transformation, but it does not depend on the heating/ cooling rates. The third term IFI is the internal friction of austenitic or martensitic phase. At low-frequency range, the transient peak IFTr is the dominate contribution to the IF peak during martensitic transformation. Delorme et al. [4] reported that the internal friction of the transient peak IFTr can be described by the following model: IFTr ¼

1 dψ ðV m Þ dV m dT    ω dV m dT dt

(3)

where Vm is the volume fraction of transformed martensite, ω is the angular frequency of applied stress, and ψ(Vm) is a monotonic function associated with the dψ is constant transformation volume change and/or shape strain. In supposing that dV m for all thermoelastic martensite, the equation indicates that the damping is inversely proportional to the frequency and proportional to the amount of martensite formed and to the cooling or heating rate. Dejonghe et al. [5] assumed that there is a critical stress σ cnecessary to reorient the already existing variants or to stress-induced new variants; therefore, the model of IFTr is improved as:

IFTr

( " •
3 #) k @V m T 2 @V m σc þ σ0 1 ¼ J @T ω 3π @σ σ0

(4)

where k is a constant, J is the elastic compliance, σ 0 is the stress amplitude, and σ c is the critical stress to move the intervariant boundaries. Zhu et al. [6] reported that the IFTr during martensitic transformation in TiNi SMA is not a linear function of the cooling or heating rate. They proposed that IFTr not only is due to the change of the volumetric fraction of the new phase but also is bound up with the variations of the mobility of structural defects associated with the transformation. Bidaux et al. [7, 8] also reported that IFTr of pure cobalt is a function

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of temperature cooling rate/frequency. But contrary to Delorme’s model, they proposed that IFTr is not linearly proportional to cooling rate/frequency. Dejonghe et al. [5] stated that, when SMAs are isothermal at temperature during martensitic transformation, the transient IFTr approaches to zero and only the IFPT term lingers. The internal friction of the IFPT can be described as: IFPT

"
3 # 2k @n σc σ0 1 ¼ 3πJ @σ σ0

(5)

@n where @σ is the transformed volume fraction per stress unit. Mercier et al. [9] proposed a dislocation mechanism, which takes into account the effect of anisotropic on the elastic energy of dislocations. They showed that the dislocation damping in a crystal is inversely proportional to the crystallographic anisotropic factor (A), and IFPT can be expressed as:

λl4 Bω IFPT  h pffiffiffiffiffiffiffiffii2 F A1

(6)

where λ is the dislocation density, l the average dislocation length, B the damping constant, and ω the angular frequency and the function F depends on the crystallographic directions of the dislocations, by assuming that the anisotropic factor A is maximum at the Ms temperature. Finally, IFI is the internal friction of austenitic parent phase or martensitic phase and is strongly dependent on microstructural properties such as interface density, vacancies, and twin boundaries. Generally, the value of IFI in martensite is much higher than that in parent phase because of the reversible motion of the interfaces between the different variants in martensite. Gotthardt and Mercier [10] reported that there are some interface dislocations in SMAs forming an ordered dislocation network and producing a higher internal friction during damping. These dislocations are more mobile in martensite state than that in parent phase and therefore contribute to higher internal friction. For most SMAs, the IF peak observed during martensitic transformation is mainly ascribed to the IFTr. However, damping capacities of IFPT and IFI are more important than that of IFTr because high-damping materials are generally used at a steady temperature rather than at a constant temperature rate. Therefore, in this chapter, the inherent and intrinsic internal friction of several types of SMAs are reviewed, including cold-rolled and annealed Ti50Ni50 SMAs (B2 ! R, R ! B190 , and B2 ! B190 martensitic transformations), Ti50Ni50-xCux SMAs (B2 ! B19, B19 ! B190 , and B2 ! B190 martensitic transformations), Ti50Ni50-xFex SMAs (B2 ! R and R ! B190 martensitic transformations), Ni-Mn-Ga SMAs (P ! I, I ! M, and P ! M martensitic transformations), Ni-Mn-Ti SMA (β ! θ0 martensitic transformation), and Cu-xAl-4Ni SMAs (β1 ! β10 and β1 ! γ10 martensitic transformations). The damping capacity of these SMAs was measured by dynamic mechanical analyzer (DMA) under isothermal treatment at different temperatures.

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Each specimen was cooled starting from parent phase at a constant cooling rate of 3  C/min and was kept isothermally for 30 min at the set temperature. After being isothermal for 30 min, the specimen was heated up to parent phase, and then the specimen was cooled to another temperature and kept isothermally at that temperature for 30 min and so on. During the isothermal treatment, the set temperature was chosen in which all the martensitic transformations can be covered. DMA is very suitable to determine the inherent and intrinsic internal friction of SMAs because it is absolute accuracy and repeatability to measure the tan δ and storage modulus values of SMAs by precisely controlling the experimental parameters such as strain amplitude, frequency, and temperature [11]. Except of DMA, the martensitic transformation sequence, transformation temperatures, and transformation enthalpy (ΔH) of each SMA specimen were also determined by differential scanning calorimetry (TA Q10 DSC) under a constant cooling/heating rate of 10  C/min.

2

Ti50Ni50 SMA (B2 ! R, R ! B190 , and B2 ! B190 Martensitic Transformations)

TiNi SMAs are known as the most important SMAs because they possess good shape memory effect, superelasticity, corrosion resistance, biocompatibility, and recoverable strain [12]. TiNi SMAs also perform a high mechanical damping and are propitious for the energy dissipation applications [13–16]. TiNi SMAs normally exhibit a one-stage B2 ! B190 martensitic transformation during cooling. However, TiNi SMAs can also show a two-stage B2 ! R ! B190 martensitic transformation by conducting a thermochemical treatment, adding a third element, or appropriately aging a Ni-rich TiNi SMA to produce Ti3Ni4 precipitates [12]. Cold-rolled Ti50Ni50 SMA annealed at different temperatures can exhibit two-stage B2 ! R ! B190 or one-stage B2 ! B190 martensitic transformation in cooling without Ti3Ni4 precipitates formation. Ti50Ni50 SMA with one- or two-stage martensitic transformations both show significant internal friction peaks during transformations. Besides, Ti50Ni50 SMA also possesses good damping capacity in R-phase and B190 martensite because there are abundant movable twin boundaries in these phases [13]. Therefore, the damping properties of the inherent and intrinsic internal friction of Ti50Ni50 SMA are very important to investigate in detail.

2.1

Specimens Preparation

The Ti50Ni50 SMA was prepared using pure raw titanium (purity 99.9 wt.%) and nickel (purity 99.9 wt.%) by the conventional vacuum arc remelter. The as-melted ingot was hot rolled at 850  C into 2-mm-thick plate, and then the plate was solution treated at 850  C for 2 h followed by quenching in water. The surface oxide layer of solution-treated plate was removed by a solution of HF : HNO3 : H2O = 1 : 5 : 20 in volume. Then, the plate was cold rolled at room temperature along the hot-rolling direction to reach a 30% thickness reduction. Subsequently, the cold-rolled plate was

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cut into test specimens, sealed in an evacuated quartz tube, and annealed at 650  C in salt bath for different time intervals. Specimens for DMA experiments were cut to the dimension of 40  5  1.26 mm along the rolling direction to eliminate the influence of rolling texture. The specimens used in DSC experiments were cut to the segment of approximately 30 mg, and the heating and cooling rates were set at 10  C/min. Figure 1a, b shows the DSC results of the cold-rolled Ti50Ni50 SMA after annealing at 650  C for 2 and 30 min, respectively [17–19]. As shown in Fig. 1a, the

a

0.6 o

Heat Flow (W/g)

0.4 0.2

Ti50Ni50 annealed at 650 C for 2min B19'←R o R←B2 -1.2 C o 24.4 C

Cooling ΔH=24.3 J/g

0.0

ΔH=24.6 J/g

-0.2

Heating

-0.4 o

50.4 C

-0.6 -200

-150

-100

-50

0

B19'→B2 50

100

150

200

Temperature (oC)

b

0.8

o

Ti50Ni50 annealed at 650 C for 30min B19'←B2

0.6

o

Heat Flow (W/g)

24.3 C

0.4 0.2

Cooling

ΔH=24.9 J/g

0.0 -0.2

ΔH=25.1 J/g

Heating o

-0.4

61.2 C

B19'→ B2 -0.6 -200

-150

-100

-50

0

50

100

150

200

Temperature (oC) Fig. 1 The DSC curves of the Ti50Ni50 SMA annealed at 650  C for (a) 2 min and (b) 30 min

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cold-rolled Ti50Ni50 SMA annealed at 650  C for 2 min possesses a two-stage B2 ! R ! B190 martensitic transformation with ΔH = 24.3 J/g in cooling and a one-stage B190 ! B2 transformation with ΔH = 24.6 J/g in heating. Figure 1b shows that the cold-rolled Ti50Ni50 SMA annealed at 650  C for 30 min exhibits a B2 ! B190 transformation peak with ΔH = 24.9 J/g in cooling and a B190 ! B2 one with ΔH = 25.1 J/g in heating.

2.2

Inherent and Intrinsic Internal Friction Under Isothermal Conditions

Figure 2a, b plots the DMA tan δ curve of the cold-rolled Ti50Ni50 SMA after annealing at 650  C for 2 and 30 min, respectively [17–19]. The solid line represents the tan δ values measured at a constant cooling rate of 3  C/min, and the empty circles symbolize the tan δ values measured after 30 min of isothermal treatment at different temperatures. Figure 2a shows that, when Ti50Ni50 SMA was measured at a constant cooling rate, there are B2 ! R and R ! B190 internal friction peaks appearing during martensitic transformations. Except for these internal friction peaks, there is an extra broad peak appearing at approximately 65  C, which is known as the relaxation peak. Figure 2a also shows that, when Ti50Ni50 SMA was measured under isothermal conditions, an inherent and intrinsic internal friction peak corresponding to B2 ! R transformation, say (IFPT þ IFI)B2 ! R, appears with a tan δ value of 0.018 at approximately 30  C. The temperature shift between the (IFPT þ IFI)B2 ! R peak and the B2 ! R transformation internal friction peak measured at a constant cooling rate of 3  C/min is due to the cooling rate effect [20, 21]. Besides, there is another inherent and intrinsic internal friction peak corresponding to R ! B190 transformation, say (IFPT þ IFI)R ! B190 , appears with a tan δ value of 0.024 at approximately 5  C. When the damping capacities of (IFPT þ IFI)B2 ! R and (IFPT þ IFI)R ! B190 are determined at various experimental parameters, it is found that the tan δ values of (IFPT þ IFI)B2 ! R and (IFPT þ IFI)R ! B190 are both linearly related to applied strain amplitude and inversely proportional to the square root of the frequency [17]. This feature is due to the fact that the damping mechanism associated with (IFPT þ IFI)B2 ! R and (IFPT þ IFI)R ! B190 is mainly generated from stress-assisted martensitic transformation and stress-assisted motions of twin boundaries during martensitic transformation. It is well known that a lot of dislocations are associated with martensitic transformation. During the damping, the stress-induced motions of twin boundaries/phase interfaces also force these dislocations to move. However, the lattice resistance hinders the dislocation glide. Therefore, when the frequency is higher, the dislocation glide becomes more difficult to keep up with the frequency pace due to the hindrance of lattice resistance, which can result in a lower increment of energy dissipation. Besides, the amplitude of stress-induced motions can increasingly correspond with increasing applied strain and hence lead to a higher energy dissipation of IFPT þ IFI. This characteristic corresponds with Dejonghe’s model [5]

1572

S.-H. Chang and S.-K. Wu

a 0.12

o

Ti50Ni50 annealed at 650 C for 2min 0.10

o

cooling at 3 C/min isothermal 30 min B19'←R

0.08

Tan δ

R←B2 0.06 0.04

(IFPT+IFI)R→B19'

(IFPT+IFI)B2→R

0.02 0.00 -100

-50

0

50

100

150

Temperature (oC)

b 0.12

o

Ti50Ni50 annealed at 650 C for 30min 0.10

o

cooling at 3 C/min isothermal 30 min

B19'←B2

Tan δ

0.08 0.06 0.04

(IFPT+IFI)B2→B19'

0.02 0.00 -100

-50

0

50

100

150

Temperature (oC) Fig. 2 The tan δ values vs. temperature of the Ti50Ni50 SMA annealed at 650  C for (a) 2 min and (b) 30 min. The solid curves were measured at a constant cooling rate of 3  C/min, and the empty circles were the inherent tan δ curves measured after 30 min isotherm treatment at different temperatures. The testing frequency and strain amplitude are 1 Hz and 5 μm, respectively

which proposed that the tan δ value of inherent internal friction is linearly proportional to the applied strain when the measurement is under isothermal condition. Meanwhile, as shown in Fig. 2a, the tan δ values of (IFPT þ IFI)R ! B190 are larger than those of (IFPT þ IFI)B2 ! R. This is because the transformation strain of R ! B190 is larger than that of B2 ! R transformation [12, 13]. Besides, the

48

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

1573

abundant twin boundaries in the R-phase and B190 martensite can self-accommodate the strain which comes from the stress-induced movement of twin boundaries between the variants of R-phase or B190 martensite. Both R-phase and transformed B190 martensite subsist during R ! B190 transformation, while only transformed R-phase appears during B2 ! R transformation. Accordingly, more twin boundaries result in a greater dissipation of energy and a higher (IFPT þ IFI)R ! B190 peak during R ! B190 transformation. Figure 2b shows that the cold-rolled Ti50Ni50 SMA annealed at 650  C for 30 min exhibits an inherent and intrinsic internal friction peak of (IFPT þ IFI)B2 ! B190 with a tan δ value of 0.013 at approximately 30  C. Comparing to Fig. 2a, the tan δ value of (IFPT þ IFI)B2 ! B190 is lower than those of (IFPT þ IFI)B2 ! R and (IFPT þ IFI)R ! B190 . This is because Ti50Ni50 SMA becomes more softened as the R-phase is formed and can induce easier movement of twin boundaries and more dissipated energy under constant stress amplitude [17–19]. Consequently, Ti50Ni50 SMA with R-phase formed during martensitic transformation should possess good damping capacities of the inherent and intrinsic internal friction peaks. However, the formation of Rphase for Ti50Ni50 SMA generally accompanies with the depression of martensitic transformation temperatures, which is unfavorable for practical high-damping applications.

3

Ti50Ni50-xCux SMAs (B2 ! B19, B19 ! B190 , and B2 ! B190 Martensitic Transformations)

The martensitic transformation sequence of Ti50Ni50-xCux SMAs with x < 30 at.% are complex and closely related to the Cu content [22–26]. For Ti50Ni50-xCux SMAs with a Cu content below 7.5 at %, the transformation sequence of the alloys is B2 ! B190 in cooling. When the Cu content of the alloys is between 7.5 at % and 16 at %, the transformation sequence of the alloys is a two-stage B2 ! B19 ! B190 in cooling. For Ti50Ni50-xCux SMAs with a Cu content above 16 at %, the transformation sequence is B2 ! B19 in cooling. Several literatures reported that substituting Cu for Ni in the Ti50Ni50 SMA can improve the damping capacity of martensitic transformation internal friction peak. In addition, adding Cu atoms into TiNi SMAs can also elevate the martensitic transformation temperatures of the alloys [27–29]. Therefore, Ti50Ni50-xCux SMAs are expected to show better damping capacity of (IFPT þ IFI) and higher martensitic transformation temperatures than Ti50Ni50 SMA does.

3.1

Specimens Preparation

The Ti50Ni50-xCux (x = 5, 10, and 20) SMAs were prepared using pure raw titanium (purity 99.9 wt %), nickel (purity 99.9 wt %), and copper (purity 99.9 wt.%). The raw materials were remelted using a conventional vacuum arc remelter to form Ti50Ni50-xCux SMA ingots. Each as-melted ingot was solution treated at 900  C

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S.-H. Chang and S.-K. Wu

for 6 h and subsequently quenched in water. Each ingot was then cut into bulks with the dimension of 30.0  4.0  2.0 mm for the DMA tests and the segment of approximately 30 mg for the DSC tests. Figure 3a–c shows the DSC results of Ti50Ni45Cu5, Ti50Ni40Cu10, and Ti50Ni30Cu20 SMAs, respectively [30]. As shown in Fig. 3, The Ti50Ni45Cu5 and Ti50Ni40Cu10 SMAs possess a one-stage B2$B190 and a two-stage B2$B19$B190 martensitic transformation sequence, respectively. The transformation sequence of Ti50Ni30Cu20 SMA is a one-stage B2$B19 transformation in cooling.

3.2

Inherent and Intrinsic Internal Friction Under Isothermal Conditions

Figure 4a–c plots the tan δ curves of Ti50Ni45Cu5, Ti50Ni40Cu10, and Ti50Ni30Cu20 SMAs, respectively [30]. Figure 4a shows that Ti50Ni45Cu5 SMA exhibits a (IFPT þ IFI)B2 ! B190 peak corresponding to the B2 ! B190 transformation with tan δ = 0.019 at approximately 30  C. Figure 4b reveals that Ti50Ni40Cu10 SMA exhibits a (IFPT þ IFI)B2 ! B19 peak with tan δ = 0.034 at approximately 30  C and a (IFPT þ IFI)B19 ! B190 peak with tan δ = 0.036 at approximately 15  C. Figure 4c shows that Ti50Ni30Cu20 SMA exhibits a single (IFPT þ IFI)B2 ! B19 peak with tan δ = 0.033 at approximately 40  C. According to Fig. 4, the damping capacities of the inherent and intrinsic internal friction peaks of Ti50Ni40Cu10 and Ti50Ni30Cu20 SMAs are higher than that of Ti50Ni45Cu5 SMA, i.e., the tan δ values of the (IFPT þ IFI)B2 ! B19 and (IFPT þ IFI)B19 ! B190 peaks (tan δ > 0.03) are larger than that of the (IFPT þ IFI)B2 ! B190 peak (tan δ < 0.02). Moreover, the tan δ value of (IFI)B19 is approximately twice that of (IFI)B190 . These characteristics reveal that Ti50Ni50-xCux SMAs in which B19 martensite appeared during the transformation sequence have more effective damping properties. The Ti50Ni50-xCux SMAs with B19 martensite exhibits higher damping capacity which can be explained by the fact that the main twinning modes of B19 martensite are {011} compound twin and {111} type I twin, in which their twinning shear values are s = 0.110 and s = 0.170, respectively [30], whereas the major twinning mode of B190 martensite is type II twin with a higher twinning shear value of s = 0.280 [31, 32]. The lower twinning shear values of B19 martensite indicate that the twin boundaries move more easily under applied external stress, leading to a higher damping capacity. Comparing with the (IFPT þ IFI) peaks of Ti50Ni50 SMA shown in Fig. 2, the tan δ values of the (IFPT þ IFI)B2 ! R and (IFPT þ IFI)R ! B190 peaks of Ti50Ni50 SMA are both lower than those of the (IFPT þ IFI)B2 ! B19 and (IFPT þ IFI)B19 ! B190 peaks of Ti50Ni40Cu10 SMA. However,   the twinning modes of the R-phase of Ti50Ni50 SMA are 1121 R and 1122 R , having the same twinning shear value of only s = 0.0265, which is much lower than those of the {011} compound twin (s = 0.110)

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

a

Ti50Ni45Cu5

0.2

Heat Flow (W/g)

1575

0.4 0.3

B19'

B2o

24.7 C

Cooling

0.1

ΔH=22.1 J/g

0.0 ΔH=22.1 J/g

-0.1 Heating

-0.2

o

54.7 C

-0.3 -0.4 -200

B19' -150

-100

-50

0

B2

50

100

150

200

150

200

150

200

Temperature (oC)

b

0.4 0.3

Ti50Ni40Cu10

B19

B2 o

30.1 C

0.2

Heat Flow (W/g)

Fig. 3 The DSC curves of the (a) Ti50Ni45Cu5, (b) Ti50Ni40Cu10, and (c) Ti50Ni30Cu20 SMAs

0.1

B19'

B19 o

Cooling

-8.3 C ΔH=17.0 J/g

0.0 ΔH=17.1 J/g

-0.1

o

6.0 C

Heating B19'

-0.2

B19 o

-0.3

50.3 C

B19 -0.4 -200

-150

-100

-50

0

B2 50

100

Temperature (oC)

c

0.4 0.3

Ti50Ni30Cu20

B19

B2 o

45.5 C

0.2

Heat Flow (W/g)

48

Cooling

0.1

ΔH=13.5 J/g

0.0

ΔH=13.6 J/g

-0.1 Heating -0.2 -0.3

o

61.6 C

B19 -0.4 -200

-150

-100

-50

0

B2

50

Temperature (oC)

100

1576

a

0.20 o

Ti50Ni45Cu5

cooling at 3 C/min isothermal 30 min

Tan δ

0.15

0.10

B19'←B2

0.05

0.00 -100

(IFPT+IFI)B2→B19'

-50

0

50

100

150

o

Temperature ( C)

b

0.20

Ti50Ni40Cu10

o

cooling at 3 C/min isothermal 30 min

0.15 B19←B2

Tan δ

B19'←B19 0.10

0.05

0.00 -100

(IFPT+IFI)B19→B19'

-50

0

(IFPT+IFI)B2→B19

50

100

150

o

Temperature ( C)

c

0.20

Ti50Ni30Cu20 0.15

Tan δ

Fig. 4 The tan δ values vs. temperature of the (a) Ti50Ni45Cu5, (b) Ti50Ni40Cu10, and (c) Ti50Ni30Cu20 SMAs. The testing frequency and strain amplitude are 1 Hz and 5 μm, respectively

S.-H. Chang and S.-K. Wu

B19←B2

o

cooling at 3 C/min isothermal 30 min

0.10 (IFPT+IFI)B2ÆB19

0.05

0.00 -100

-50

0

50

Temperature (oC)

100

150

48

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

1577

and {111} type I twin (s = 0.170) of the B19 martensite. This unexpected result can be explained by the fact that the origin of the R-phase of TiNi SMAs typically corresponds to either heat treated after cold working to create rearranged dislocation structures or Ni-rich TiNi alloys aged at proper temperatures to cause the precipitation of Ti3Ni4 phase [12]. These dislocations or Ti3Ni4 precipitates normally impede the movements of the twin boundaries during damping and, therefore, deteriorate the inherent internal friction of TiNi SMAs. By contrast, the B19 martensite in TiNiCu SMAs originated by substituting Ni atoms with Cu atoms instead of introducing dislocations or Ti3Ni4 precipitates. Consequently, although the R-phase has a lower shear twining value than the B19 martensite does, Ti50Ni40Cu10 SMA still exhibits higher (IFPT þ IFI) peaks than Ti50Ni50 SMA does. In addition, as shown in Figs. 2 and 4, the peak temperature of the (IFPT þ IFI)B2 ! B19 for Ti50Ni50-xCux SMAs (approximately 30  C) is relatively higher than that of the (IFPT þ IFI)R ! B190 for Ti50Ni50 SMA (approximately 5  C). Possessing higher (IFPT þ IFI) peak temperature is vital for Ti50Ni50-xCux SMAs because high-damping materials are normally conducted at temperatures near or above room temperature. However, it should be noted that Ti50Ni50-xCux SMAs typically become more brittle and lose their workability and shape recoverable strain as the Cu content increases [33].

4

Ti50Ni50-xFex SMAs (B2 ! R and R ! B190 Martensitic Transformations)

Several literatures reported that replacing some of the Ni in Ti50Ni50 SMA with Fe to form Ti50Ni50-xFex SMAs can influence the transformation sequence of the alloys [34–36]. Similar to cold-rolled and annealed Ti50Ni50 SMA, Ti50Ni50-xFex SMAs also exhibit a two-stage B2 ! R ! B190 martensitic transformation during cooling but do not develop abundant defects/dislocations or precipitates. Therefore, Ti50Ni50-xFex SMAs are expected to exhibit greater (IFPT þ IFI) peaks than coldrolled and annealed Ti50Ni50 SMA does.

4.1

Specimens Preparation

The Ti50Ni50-xFex (x = 2, 3, and 4, in at %) SMAs were prepared using pure raw titanium (purity 99.9 wt %), nickel (purity 99.9 wt.%), and iron (purity 99.98 wt.%). The raw materials were remelted using a conventional vacuum arc remelter to form Ti50Ni50-xFex SMA ingots. Each as-melted ingot was hot rolled at 900  C and then solution heat-treated at 900  C for 1 h, followed by quenching in water. Each plate was then cut with a diamond saw into specimens with the dimension of 30.0  4.0  2.0 mm for DMA tests and into the segment of approximately 30 mg for DSC tests.

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S.-H. Chang and S.-K. Wu

Figure 5a–c shows the DSC results of Ti50Ni48Fe2, Ti50Ni47Fe3, and Ti50Ni46Fe4 SMAs, respectively [37]. As shown in Fig. 5, both Ti50Ni48Fe2 and Ti50Ni47Fe3 SMAs exhibit a two-stage B2$R$B190 martensitic transformation sequence, while Ti50Ni46Fe4 SMA only shows a one-stage B2$R transformation.

4.2

Inherent and Intrinsic Internal Friction Under Isothermal Conditions

Figure 6a–c plots the tan δ curves of Ti50Ni48Fe2, Ti50Ni47Fe3, and Ti50Ni46Fe4 SMAs, respectively [37]. Figure 6a shows that Ti50Ni48Fe2 SMA exhibits a (IFPT þ IFI)B2 ! R peak with tan δ = 0.019 at approximately 0  C and a (IFPT þ IFI)R ! B190 peak with tan δ = 0.036 at approximately 50  C. Figure 6b shows that Ti50Ni47Fe3 SMA exhibits a (IFPT þ IFI)B2 ! R peak with tan δ = 0.026 at approximately 20  C and a (IFPT þ IFI)R ! B190 peak with tan δ = 0.035 at approximately 65  C. In Fig. 6c, Ti50Ni46Fe4 SMA has only a (IFPT þ IFI)B2 ! R peak with tan δ = 0.022 at approximately 50  C. According to Fig. 6, the tan δ values of the (IFPT þ IFI)R ! B190 peak are larger than 0.035 for both Ti50Ni48Fe2 and Ti50Ni47Fe3 SMAs, and they are also larger than that of the (IFPT þ IFI)B2 ! R peak (approximately 0.02) for Ti50Ni50-xFex (x = 2, 3, and 4) SMAs. This feature comes from the fact that the transformation strain associated with R ! B190 transformation is much greater than that associated with B2 ! R transformation [14]. In addition, the (IFPT þ IFI)R ! B190 peak exhibited a higher damping capacity than the (IFPT þ IFI)B2 ! R peak is also due to the appearance of both R-phase and transformed B190 martensite during R ! B190 transformation, but only transformed R-phase can be obtained during B2 ! R transformation. Comparing to Ti50Ni50 SMA shown in Fig. 2a, which also exhibits (IFPT þ IFI)B2 ! R and (IFPT þ IFI)R ! B190 peaks with tan δ values of approximately 0.02, Ti50Ni50-xFex SMAs show a much higher tan δ value of the (IFPT þ IFI)R ! B190 peak (approximately 0.04) than that of Ti50Ni50 SMA (approximately 0.02). This characteristic can be explained by the fact that the origin of the R-phase in TiNibased SMAs is typically either the rearranged dislocation structures in cold-rolled and annealed alloys or the Ti3Ni4 precipitates formed in Ni-rich TiNi SMAs aged at the proper temperature. These dislocations or Ti3Ni4 precipitates normally impede the movements of twin boundaries during damping and therefore reduce the (IFPT þ IFI) internal friction of TiNi SMAs. In contrast, the R-phase formation in Ti50Ni50-xFex SMAs originated from the replacement of Ni atoms with Fe atoms, instead of the introduction of dislocations or Ti3Ni4 precipitates. As a result, Ti50Ni50-xFex SMAs exhibit higher (IFPT þ IFI)R ! B190 peak than Ti50Ni50 SMA does. However, the (IFPT þ IFI)R ! B190 peaks of Ti50Ni50-xFex SMAs (x = 2 and 3) are depressed to a lower temperature than that of Ti50Ni50 SMA, in which Ti50Ni48Fe2 and Ti50Ni47Fe3 SMAs exhibit this peak at temperature below 40  C.

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

a

1579

0.3

Ti 50Ni48Fe2

Heat Flow (W/g)

0.2

R←B2 B19'←R

-3.7oC

-61.4oC

Cooling

0.1 ΔH=18.0 J/g

0.0 ΔH=18.1 J/g

-0.1

B19'→R R→B2

-0.2 -200 -150 -100

-50

0

50

150

200

0.3

Ti50Ni47Fe3 R←B2

0.2

B19'←R 0.1

-14.3oC

Cooling

o

-121.1 C ΔH=14.3 J/g

0.0 ΔH=17.3 J/g

-0.1

Heating

-59.0oC

1.0oC

B19'→R -0.2 -200 -150 -100

R→B2

-50

0

50

Temperature

c

100

(oC)

Temperature

b

Heating

14.0oC

-6.3oC

Heat Flow (W/g)

Fig. 5 The DSC curves of the (a) Ti50Ni48Fe2, (b) Ti50Ni47Fe3, and (c) Ti50Ni46Fe4 SMAs

100

150

200

150

200

(oC)

0.3

Ti50Ni46Fe4 0.2

Heat Flow (W/g)

48

R←B2 -48.1oC

Cooling

0.1 ΔH=5.8 J/g

0.0 ΔH=6.3 J/g

-0.1

Heating

-32.0oC

R→B2 -0.2 -200 -150 -100

-50

0

Temperature

50

(oC)

100

1580

a

0.20

Ti50Ni48Fe2

o

cooling at 3 C/min isothermal 30 min

B19' ← R

Tan δ

0.15

0.10 (IFPT+IFI)R→B19'

R ← B2

0.05

(IFPT+IFI)B2→R

0.00 -150

-100

-50

0

50

100

150

Temperature (oC)

b

0.20

Ti50Ni47Fe3

Tan δ

0.15

o

cooling at 3 C/min isothermal 30 min

B19' ← R

0.10 R ← B2 (IFPT+IFI)B2→R

0.05

0.00 -150

(IFPT+IFI)R→B19'

-100

-50

0

50

100

150

o

Temperature ( C)

c

0.20

Ti50Ni46Fe4

o

cooling at 3 C/min isothermal 30 min

0.15

Tan δ

Fig. 6 The tan δ values vs. temperature of the (a) Ti50Ni48Fe2, (b) Ti50Ni47Fe3, and (c) Ti50Ni46Fe4 SMAs. The testing frequency and strain amplitude are 1 Hz and 5 μm, respectively

S.-H. Chang and S.-K. Wu

0.10 R ← B2 0.05

0.00 -150

(IFPT+IFI)B2→R

-100

-50

0

50 o

Temperature ( C)

100

150

48

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

1581

Therefore, although Ti50Ni50-xFex SMAs possess good damping capacity of the (IFPT þ IFI) peaks, the low martensitic transformation temperatures far below room temperature restrict the use of Ti50Ni50-xFex SMAs in practical high-damping applications.

5

Groups I–III Ni2MnGa SMAs (P ! I, I ! M, and P ! M Martensitic Transformations)

Similar to TiNi-based SMAs, Ni2MnGa SMAs also undergo a martensitic transformation during cooling but show a much higher ferromagnetic transition temperature near 100  C than normal TiNi-based SMAs [38–40]. Chernenko et al. [40] indicated that the martensitic transformation temperature of Ni2MnGa SMAs is highly sensitive to their chemical composition and can exhibit a wide range of martensitic transformation temperature by controlling the chemical composition of the alloys. Therefore, it can be expected that Ni2MnGa SMAs could perform good inherent and intrinsic internal friction at temperatures much higher than normal TiNi-based SMAs, which is advantages for practical high-damping applications.

5.1

Specimens Preparation

Ni2MnGa SMAs were prepared by vacuum arc remelter from the raw materials of nickel (purity 99.9 wt.%), Mn55-Ni45 mother alloy (in wt.%), and gallium (purity 99.9 wt.%) followed by homogenizing at 850  C for 48 h. The homogenized ingots were then cut to the dimension of 20.0  3.3  2.8 mm using a low-speed diamond saw for DMA tests. The precise chemical compositions of groups I–III Ni2MnGa SMAs were determined as Ni50.66Mn26.89Ga22.30, Ni52.04Mn26.51Ga21.35, and Ni53.72Mn26.80Ga19.32, respectively. Group I Ni2MnGa SMA possesses a two-stage P ! I and I ! M martensitic transformation in cooling, while groups II and III Ni2MnGa SMAs both exhibit a one-stage P ! I transformation [41].

5.2

Inherent and Intrinsic Internal Friction Under Isothermal Conditions

Figure 7a–c plots the tan δ curves of groups I to III Ni2MnGa SMAs, respectively [42]. In Fig. 7a, group I Ni2MnGa SMA shows a higher (IFPT þ IFI)I ! M peak with a tan δ value of 0.027 and has a lower (IFPT þ IFI)P ! I peak with a tan δ value of 0.011. In Fig. 7b, c, groups II and III Ni2MnGa SMAs show a (IFPT þ IFI)P ! I peak with a tan δ value of 0.029 and 0.016, respectively. Böhm et al. [43] revealed that Ni2MnGa SMAs can exhibit either 5 M or 7 M martensite after plastic deformation

1582

a

0.15

Group I Ni2MNGa

o

cooling at 3 C/min isothermal 30 min

M←I

Tan δ

0.10

(IFPT+IFI)I→M

0.05 I←P

0.00 -150

-100

-50

0

(IFPT+IFI)P→I

50

100

150

200

Temperature (oC)

b

0.20

Group II Ni2MNGa

Tan δ

0.15

o

cooling at 3 C/min isothermal 30 min

I←P

0.10 (IFPT+IFI)P→I

0.05

0.00 -150

-100

-50

0

50

100

150

200

o

Temperature ( C)

c

0.20

Group III Ni2MNGa

o

cooling at 3 C/min isothermal 30 min

0.15

Tan δ

Fig. 7 The tan δ values vs. temperature of the (a) group I Ni50.66Mn26.89Ga22.30, (b) group II Ni52.04Mn26.51Ga21.35, and (c) group III Ni53.72Mn26.80Ga19.32 SMAs. The testing frequency and strain amplitude are 1 Hz and 5 μm, respectively

S.-H. Chang and S.-K. Wu

0.10

I←P

(IFPT+IFI)P→I

0.05

0.00 -150

-100

-50

0

50

100 o

Temperature ( C)

150

200

48

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

1583

and heat treatment. In addition, it has been reported that the twin boundaries of 5 M martensite possess higher mobility and can dissipate more energy during damping [44]. Therefore, according to Fig. 7, 5 M martensite should be more dominant in group II Ni2MnGa SMA after martensitic transformation. Compared with Ti50Ni50 SMA shown in Fig. 2, groups I–III Ni2MnGa SMAs process good inherent damping capacities (tan δ > 0.02) in a wider temperature range (from 100  C to 100  C). This feature comes from the fact that groups I–III Ni2MnGa SMAs exhibit conspicuous damping capacities not only during martensitic transformation but also in martensite state. Nevertheless, the undesirable brittle characteristic of Ni2MnGa SMAs critically limits their workability and should be overcome before serving as high-damping materials.

6

Ni-Mn-Ti SMA (b ! u0 Martensitic Transformation)

Ni-Mn alloys with proper Ti content undergo transformations from B2 parent phase to different martensitic structure at temperatures up to 400  C. These transformations are thermoelastic with small thermal hysteresis; therefore, Ni-Mn-Ti alloys are classified as high-temperature shape memory alloys [45]. Similar to Ni2MnGa SMAs, Ni-Mn-Ti SMAs are also potential candidates for practical high-damping alloys, because they can exhibit a wide martensitic transformation temperature range by properly adjusting their chemical compositions. A Ni-Mn-Ti SMA with chemical composition of Ni41.6Mn48.7Ti9.7 is chosen to study its damping properties because it exhibits a significant β ! θ0 martensitic transformation at temperatures between 100  C and 200  C and is promising for high-damping SMAs applied at elevated temperatures.

6.1

Specimen Preparation

The Ni-Mn-Ti SMA was prepared from pure raw materials of nickel (purity 99.9 wt %), Ni45Mn55 mother alloy (purity 99.9 wt %), and titanium (purity 99.9 wt %). The raw materials were melted at 1100  C in an evacuated quartz to form a Ni-Mn-Ti SMA ingot. The Ni-Mn-Ti SMA ingot was then homogenized at 900 C for 10 h, followed by quenching in ice water. The homogenized Ni41.6Mn48.7Ti9.7 SMA ingot was cut into the dimension of 35.0  10.0  2.0 mm using a low-speed diamond saw for the following DMA tests and into the segment of approximately 30 mg for DSC tests. The precise chemical composition of the alloy was determined as Ni41.6Mn48.7Ti9.7, and it possesses a β ! θ0 martensitic transformation in cooling [46]. Figure 8a shows the DSC result of Ni41.6Mn48.7Ti9.7 SMA. As shown in Fig. 8a, the Ni41.6Mn48.7Ti9.7 SMA exhibits a β ! θ0 martensitic transformation peak in cooling and a θ0 ! β one in heating. The peak temperatures of β ! θ0 and θ0 ! β transformation peaks are 122.5  C and 143.9  C, respectively. The martensitic transformation enthalpies (ΔHs) of the β ! θ0 and θ0 ! β transformation peaks are approximately 24 J/g.

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S.-H. Chang and S.-K. Wu

a

0.4

Ni 41.6Mn48.7Ti9.7 SMA θ' ← β o

122.5 C

Heat Flow (W/g)

0.2 Cooling

∆H=24.0 J/g

0.0 ∆H=24.8 J/g

Heating

-0.2 o

143.9 C

θ' → β -0.4 -100

-50

0

50

100

150

200

250

300

o

Temperature ( C)

b 0.15 o

Ni 41.6Mn48.7Ti9.7 SMA

0.10

cooling at 3 C/min isothermal 30 min

Tan δ

θ' ← β

0.05 (IFPT+IFI)β→θ'

0.00 0

50

100

150

200

250

o

Temperature ( C) Fig. 8 The (a) DSC and (b) tan δ values vs. temperature curves of the Ni41.6Mn48.7Ti9.7 SMA. The testing frequency and strain amplitude used in DMA tests are 1 Hz and 5 μm, respectively

6.2

Inherent and Intrinsic Internal Friction Under Isothermal Conditions

Figure 8b shows the tan δ curves of Ni41.6Mn48.7Ti9.7 SMA [46]. In Fig. 8b, Ni41.6Mn48.7Ti9.7 SMA exhibits an inherent (IFPT þ IFI)β ! θ0 during martensitic transformation with tan δ = 0.01 at approximately 150  C. Although the

48

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

1585

(IFPT þ IFI)β ! θ0 peak of Ni41.6Mn48.7Ti9.7 SMA exhibits a much higher inherent internal friction peak temperature than those of other SMAs do, the relatively lower damping capacity of the (IFPT þ IFI)β ! θ0 peak and poorer workability may limit their practical high-damping applications.

7

Cu-Al-Ni SMAs (b1 ! b10 and b1 ! g10 Martensitic Transformations)

Cu-Al-Ni SMAs have been widely studied because they possess good shape memory effect, acceptable workability, favorable heat and electricity conductivity, and low cost [47]. However, the martensitic transformation sequence of Cu-Al-Ni SMAs is complicated, in which the β1 (DO3) parent phase can be transformed into β10 (18R), γ 10 (2H), or α10 (6R) martensite. Cu-Al-Ni SMAs possess an obvious internal friction peak during martensitic transformation and a wide martensitic transformation temperature range approaching to 200  C, which is much higher than that of typical TiNi-based SMAs. Therefore, Cu-Al-Ni SMAs are candidates for practical high-damping applications because they should possess high (IFPT þ IFI) peaks at elevated martensitic transformation temperatures.

7.1

Specimens Preparation

Polycrystalline samples of Cu-13.5Al-4Ni (in wt.%) and Cu-14.0Al-4Ni (in wt.%) SMAs were prepared from pure raw copper (purity 99.99 wt.%), aluminum (purity 99.99 wt.%), and nickel (purity 99.9 wt.%). These raw materials were melted at 1100  C in evacuated quartz tubes followed by quenching in ice water. The quenched ingots were annealed at 900  C for 30 min and slowly cooled in the furnace to room temperature. The annealed Cu-xAl-4Ni SMA ingots were cut into bulks with the dimension of 30.0  6.5  1.3 mm for the DMA tests and the segment of approximately 30 mg for DSC tests. Cu-13.5Al-4Ni and Cu-14.0Al-4Ni SMAs were selected to investigate here because they possess a β1 (DO3) ! β10 (18R) and a β1 (DO3) ! γ 10 (2H) martensitic transformation in cooling, respectively, at temperatures above room temperature. Figure 9a, b shows the DSC results of Cu-13.5Al-4Ni and Cu-14.0Al-4Ni SMAs, respectively. Figure 9a indicates that the Cu-13.5Al-4Ni SMA exhibits a significant β1 (DO3) ! β10 (18R) martensitic transformation peak in cooling and a β10 (18R) ! β1 (DO3) one in heating. The peak temperatures of β1 (DO3) ! β10 (18R) and β10 (18R) ! β1 (DO3) transformation peaks are 164.9  C and 178.1  C, respectively. The martensitic transformation enthalpies (ΔHs) of the β1 (DO3) ! β10 (18R) and β10 (18R) ! β1 (DO3) transformation peaks are approximately 10 J/g. Comparing to Cu-13.5Al-4Ni SMA, Cu14.0Al-4Ni SMA only shows a relatively lower β1 (DO3)$γ 10 (2H) martensitic transformation peaks at lower temperatures below 100  C with a lower ΔHs of approximately 5.3 J/g.

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a

0.4

Cu-13.5Al-4Ni

18R ← DO3

0.3

o

164.9 C

Heat Flow (W/g)

0.2 Cooling

0.1

ΔH=9.6 J/g

0.0 -0.1

ΔH=10.1 J/g

Heating

-0.2 -0.3 -0.4 -100

o

178.1 C

18R → DO3 -50

0

50

100

150

200

250

300

200

250

300

Temperature (oC)

b

0.2

Cu-14.0Al-4Ni 2H ← DO3

Heat Flow (W/g)

0.1

o

Cooling

52.2 C ΔH=5.3 J/g

0.0 ΔH=5.3 J/g

-0.1

o

Heating

98.9 C

2H → DO3 -0.2 -100

-50

0

50

100

150 o

Temperature ( C) Fig. 9 The DSC curves of the (a) Cu-13.5Al-4Ni (in wt.%) and (b) Cu-14.0Al-4Ni (in wt.%) SMAs

7.2

Inherent and Intrinsic Internal Friction Under Isothermal Conditions

Figure 10a, b shows the tan δ curves of Cu-13.5Al-4Ni and Cu-14.0Al-4Ni SMAs, respectively [48, 49]. Figure 10c shows the magnified empty circle curves

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

a

1587

0.25

Cu-13.5Al-4Ni

18R ← DO3 o

cooling at 3 C/min isothermal 30 min

0.20

Tan δ

0.15

0.10

0.05

(IFPT+IFI)DO

3→18R

0.00 -100

-50

0

50

100

150

200

250

o

Temperature ( C)

b

0.20

Cu-14.0Al-4Ni

o

cooling at 3 C/min isothermal 30 min

0.15

Tan δ

Fig. 10 The tan δ values vs. temperature of the (a) Cu13.5Al-4Ni and (b) Cu14.0Al-4Ni SMAs. (c) The magnification of the empty circle curves shown in (a) and (b). The testing frequency and strain amplitude are 1 Hz and 5 μm, respectively

0.10 2H ← DO3 0.05

(IFPT+IFI)DO

3→2H

0.00 -100

-50

0

50

100

150

200

250

o

Temperature ( C)

c

0.015 Cu-13.5Al-4Ni Cu-14.0Al-4Ni (IFPT+IFI)DO

3→2H

0.010

Tan δ

48

(IFPT+IFI)DO

3→18R

0.005

0.000 -50

0

50

100

150 o

Temperature ( C)

200

250

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S.-H. Chang and S.-K. Wu

in Fig. 10a, b for more clarity. Figure 10 reveals that there is an inconspicuous (IFPT þ IFI)DO3 ! 18R peak with tan δ = 0.004 at approximately 180  C for Cu13.5Al-4Ni SMA and a more significant (IFPT þ IFI)DO3 ! 2H peak with tan δ = 0.010 at approximately 70  C for Cu-14.0Al-4Ni SMA. In addition, the intrinsic internal friction of Cu-14.0Al-4Ni SMA in γ 10 (2H) martensite phase, (IFI)2H, also possesses a greater damping capacity than that of Cu-13.5Al-4Ni SMA in β10 (18R) martensite phase, (IFI)18R. According to Fig. 10, Cu-13.5Al4Ni SMA possesses a much higher IF peak than Cu-14.0Al-4Ni SMA when measured at a constant cooling rate; however, the damping capacity of (IFPT þ IFI)DO3 ! 18R for Cu-13.5Al-4Ni SMA is much lower than that of (IFPT þ IFI)DO3 ! 2H for Cu-14.0Al-4Ni SMA. This unexpected result is due to the different microstructure exhibited in the transformed martensites of the alloys. The transformed γ 10 (2H) martensite in Cu-14.0Al-4Ni SMA possesses a 2H-type structure with internal twins. These internal twins comprise abundant twin boundaries, which are easily moveable to dissipate more energy during damping, leading to higher tan δ values of (IFPT þ IFI)DO3 ! 2H and (IFI)2H. In contrast, the β1 (DO3) parent phase of Cu-13.5Al-4Ni SMA transforms to β10 (18R) martensite, which possesses an ordered 9R structure with stacking faults rather than twinning with movable twin boundaries, causing an extremely low tan δ values of (IFPT þ IFI)DO3 ! 18R and (IFI)18R [50]. Comparing to other TiNi-based SMAs, the (IFPT þ IFI)DO3 ! 2H peak for Cu-14.0Al-4Ni SMA exhibits the advantage of a higher martensitic transformation temperature (approaching to 70  C) than those of the (IFPT þ IFI) peaks for Ti50Ni50, Ti50Ni50-xCux, and Ti50Ni50-xFex SMAs (below 50  C). However, the (IFPT þ IFI)DO3 ! 2H peak for Cu-14.0Al-4Ni SMA shows a lower tan δ value (approximately 0.01) than those of the (IFPT þ IFI) peaks for TiNi-based SMAs (above 0.02).

8

Concluding Summary

The internal friction (IF) peak associated with the martensitic transformation of SMAs can be decomposed into three terms: IF = IFTr þ IFPT þ IFI. Here IFTr is a transitory IF which appears only in low-frequency and non-zero heating/cooling rates, IFPT is the IF due to the phase transformation, and IFI is the intrinsic IF of austenitic or martensitic phase. For most SMAs, the IF peak observed during martensitic transformation under cooling/heating rates is mainly ascribed to the IFTr. However, damping capacities of IFPT and IFI are more important than that of IFTr because high-damping materials are generally used at a steady temperature rather than at a constant temperature rate. Therefore, in this chapter, the tan δ values of the (IFPT and IFI) peaks exhibited in several types of SMAs measured by DMA under isothermal conditions are reviewed, included Ti50Ni50,

48

Damping Characteristics of Shape Memory Alloys on Their Inherent and. . .

0.04

Ti50Ni50 B19'←R

B19'← B19

Ti50Ni40Cu10

B19← B2

Ti50Ni47Fe3 Group III Ni2MnGa

R←B2

0.03

Tan δ

1589

Cu-14.0Al-4Ni

B19'← R

0.02

R← B2

I←P

2H← DO3

0.01

0.00 -100

-50

0

50

100

150

200

o

Temperature ( C) Fig. 11 The tan δ curves measured after 30 min isotherm treatment at different temperatures of Ti50Ni50, Ti50Ni40Cu10, Ti50Ni47Fe3, group III Ni2MnGa, and Cu-14.0Al-4Ni SMAs. The testing frequency and strain amplitude are 1 Hz and 5 μm, respectively

Ti50Ni50-xCux (x = 5, 10, 20 at.%), Ti50Ni50-xFex (x = 2, 3, 4 at.%), Ni2MnGa/NiMn-Ti and Cu-xAl-4Zn (x = 13.5, 14.0 wt.%), etc. Figure 11 shows the tan δ curves measured after 30 min isotherm treatment at different temperatures of some representative SMAs listed above. The experimental results of each (IFPT þ IFI) peak associated with the martensitic transformation shown in Fig. 11 are summarized in Table 1. From Fig. 11 and Table 1, one can see that the (IFPT þ IFI) peak tan δ values of premartensitic transformation are usually less than those of martensitic transformation, and only those of B2 ! B19/B19 ! B190 in Ti50Ni40Cu10 SMA and R ! B190 in Ti50Ni47Fe3 SMA have the tan δ value >0.03 which is usually regarded as the criterion for a high-damping material. However, R ! B190 transformation temperature exhibited in Ti50Ni47Fe3 SMA is about 65  C, but B2 ! B19 and B19 ! B190 ones in Ti50Ni40Cu10 SMA are 30  C and 15  C, respectively, and the tan δ value keeps around 0.030 ~ 0.036 in the temperature range of 30  C ~ 15  C. Although group III Ni2MnGa SMA shows an acceptable tan δ value >0.015 in a wide temperature range approaching to 150  C, its intrinsic brittle nature is undesirable for processing. In addition, according to our experience, the workability of Ti50Ni40Cu10 SMA is better than that of Ti50Ni40Fe3 SMA and Ni2MnGa-based SMAs. Therefore, among these various types of SMAs, Ti50Ni40Cu10 SMA is more suitable for high-damping applications for its high (IFPT þ IFI) tan δ values exhibited near room temperature.

Ti50Ni50a B2 ! R 30 0.018 1 5

R ! B19 5 0.024 1 5

b

a

Cold rolled and 650  C  2min annealed Solution treated at 900  C  6h, water quenched c Solution treated at 900  C  1h, water quenched d Homogenized at 850  C  48h e Melt at 1100  C  30min, water quenched

SMA Transformation Peak temperature ( C) Tan δ value Frequency (Hz) Amplitude (μm) 0

Ti50Ni40Cu10b B2 ! B19 30 0.034 1 5 B19 ! B19 15 0.036 1 5

0

Ti50Ni47Fe3c B2 ! R R ! B190 20 65 0.026 0.035 1 1 5 5

Group III Ni2MnGad P!I 150 0.016 1 5

Cu-14.0Al-4Nie DO3 ! 2H 70 0.010 1 5

Table 1 The (IFPT þ IFI) tan d values associated with the martensitic transformation of several typical SMAs measured by DMA under isothermal conditions

1590 S.-H. Chang and S.-K. Wu

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Acknowledgments The authors gratefully acknowledge the financial support for this research provided by the Ministry of Science and Technology (MOST) and National Taiwan University (NTU), Taiwan, under Grant Nos. MOST 103-2221-E-197-007, MOST 104-2221-E-002-004, and NTU 105R891803.

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Application of Homogenization of Material Properties

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Ming Dong and Siegfried Schmauder

Contents 1 Modeling of Metal Matrix Composites by a Self-Consistent Embedded Cell Model . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Unit Cell Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Embedded Cell Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Iterative Modeling Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Comparison with Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Geometrical Shape of Embedded Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Systematic Studies with Self-Consistent Embedded Cell Models . . . . . . . . . . . . . . . . . 1.9 Strengthening Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Self-Consistent Matricity Model to Simulate the Mechanical Behavior of Interpenetrating Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Matricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Matricity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Adjusting Matricity in the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Realization of the Adjustability of Matricity by Weighting Factors . . . . . . . . . . . . . . . 2.6 Calculation of Stress-Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Mechanical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Yield Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1596 1596 1599 1601 1603 1604 1605 1607 1607 1614 1615 1616 1616 1617 1618 1619 1620 1621 1622 1622 1622 1626 1627

M. Dong (*) Eberspächer Exhaust Technology GmbH & Co. KG, Esslingen, Germany e-mail: [email protected] S. Schmauder Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_32

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Abstract

The limit flow stresses for transverse loading of metal matrix composites reinforced with continuous fibers and for uniaxial loading of spherical particlereinforced metal matrix composites are investigated by recently developed embedded cell models in conjunction with the finite element method. A fiber of circular cross section or a spherical particle is surrounded by a metal matrix, which is again embedded in the composite material with the mechanical behavior to be determined iteratively in a self-consistent manner. Good agreement has been obtained between experiment and calculation, and the embedded cell model is thus found to represent well metal matrix composites with randomly arranged inclusions. Systematic studies of the mechanical behavior of fiber and particle-reinforced composites with plane strain and axisymmetric embedded cell models are carried out to determine the influence of fiber or particle volume fraction and matrix strain-hardening ability on composite strengthening levels. Results for random inclusion arrangements obtained with self-consistent embedded cell models are compared with strengthening levels for regular inclusion arrangements from conventional unit cell models. Based on the self-consistent embedded cell models, a self-consistent matricity model has been developed to simulate the mechanical behavior of composites with two randomly distributed phases of interpenetrating microstructures. The model is an extension of the self-consistent model for matrices with randomly distributed inclusions. In addition to the volume fraction of the phases, the matricity model allows a further parameter of the microstructure, the matricity M of each phase, to be included into the simulation of the mechanical behavior of composites with interpenetrating microstructures. Good agreement has been obtained between experiment and calculation with respect to the composites’ mechanical behavior, and the matricity model is thus found to represent well metal matrix composites with interpenetrating microstructures. The matricity model can be applied to describe the mechanical behavior of arbitrary microstructures as observed in two-phase functionally graded materials, where the volume fraction as well as the matricity of the phases varies between the extreme values of 0 and 1. Keywords

Composite materials · Micromechanics · Mechanical behavior · Self-consistent embedded cell models · Matricity

1

Modeling of Metal Matrix Composites by a Self-Consistent Embedded Cell Model

1.1

Introduction

Metal matrix composites (MMCs) with strong inclusions are a new class of materials which, due to high strength and light weight, are potentially valuable in aerospace and transportation applications [1].

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MMCs are defined as ductile matrix materials reinforced by brittle fibers or particles. Under external loading conditions, the overall response of MMCs is elastic-plastic. MMCs are frequently reinforced by continuous fibers which are aligned in order to make use of the high axial fiber strength. However, the mechanical behavior of these composites under transverse loading is well behind their axial performance [2–7]. On the other hand, it is well known, both from experiment and calculations [8], that details of transverse strengthening in uniaxially fiber-reinforced MMCs are a strong function of fiber arrangement. To derive the mechanical behavior of MMCs, a micromechanical approach is usually employed using cell models, representing regular inclusion arrangements. As regular fiber spacings are difficult to achieve in practice, most of the present fiber-reinforced MMCs contain aligned but randomly arranged continuous fibers. Thus, the accurate modeling of the mechanical behavior of actual MMCs is very complicated in practice even if the fibers are aligned. Initially, the transverse mechanical behavior of a unidirectionally continuous fiber-reinforced composite (Al-B) with fibers of circular cross section was studied by Adams [9] adopting finite element cell models under plane strain conditions: a simple geometrical cell composed of matrix and inclusion material is repeated by appropriate boundary conditions to represent a composite with a periodic microstructure. Good agreement was achieved between calculated and experimental stress-strain curves for a rectangular fiber arrangement. The influence of different regular fiber arrangements on the strength of transversely loaded boron fiberreinforced Al was analyzed in Refs. [2, 5, 7]. It was found that the same square arrangement of fibers represents two extremes of strengthening: high strength levels are achieved if the composite is loaded in a 0
direction to nearest neighbors while the 45
loading direction is found to be very weak for the same fiber arrangement. A regular hexagonal fiber arrangement lies somewhere between these limits [2, 5, 7, 10, 11]. The transverse mechanical behavior of a realistic fiber-reinforced composite containing about 30 randomly arranged fibers was found to be best described but significantly underestimated by the hexagonal fiber model [5]. One reason for the superiority of the hexagonal over the square arrangement in describing random fiber arrangements is due to the fact that the elastic stress invariants of the square arrangement agree only to the first order with the invariants of the transversely isotropic material, while the hexagonal arrangement agrees up to the second order [10]. Dietrich [6] found a transversely isotropic square fiber-reinforced Ag-Ni composite material using fibers of different diameters. A systematic study in which the fiber volume fraction and the fiber arrangement effects have been investigated was founded into a simple model in Ref. [11]. The influence of fiber shape and clustering was numerically examined in some detail by Llorca et al. [12], Dietrich [6], and Sautter [13]. It was observed that facetted fiber cross sections lead to higher strengths compared to circular cross sections except for such fibers which possess predominantly facets oriented at 45
with respect to the loading axis in close agreement with findings in particlereinforced MMCs [14]. Thus, hindering of shear band formation within the matrix was found to be responsible for strengthening with respect to fiber arrangement and

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fiber shape [12]. In Refs. [8, 11, 15, 16] local distributions of stresses and strains within the microstructure have also been identified to be strongly influenced by the arrangement of fibers. However, no agreement was found between the mechanical behavior of composites based on cell models with differently arranged fibers and experiments with randomly arranged fibers loaded in the transverse direction. The overall mechanical behavior of a particle-reinforced composite was studied with axisymmetric finite element cell models by Bao et al. [17] to represent a uniform particle distribution within an elastic-plastic matrix. Tvergaard [18] introduced a modified cylindrical unit cell containing one half of a single fiber to model the axial performance of a periodic square arrangement of staggered short fibers. Hom [19] and Weissenbek [20] used three-dimensional finite elements to model different regular arrangements of short fibers and spherical as well as cylindrical particles with relatively small volume fractions ( f < 0.2). It was generally found that the arrangement of fibers strongly influences the different overall behavior of the composites. When short fibers are arranged in a side-by-side manner, they constrain the plastic flow in the matrix, and the computed stress-strain response of the composite in the fiber direction is stiffer than observed in experiments. If the fibers in the model are overlapping, strong plastic shearing can develop in the ligament between neighboring fibers, and the predicted load carrying capacity of the composite is closer to the experimental measurements. In Ref. [21] the stress-strain curves based on FE-numerical solutions of axisymmetric unit cell models of MMCs are given in a closed form as a function of the most important control parameters, namely, volume fraction, aspect ratio, and shape (cylindrical or spherical) of the reinforcement as well as the matrix hardening parameter. One reason for the discrepancy between experiments and calculations based on simple cell models in the case of particle, whisker, and fiber-reinforced metals is believed to be the unnatural constraint governing the matrix material between inclusion and simulation cell border [5, 15, 17–19, 22–24] resulting in an unrealistic strength increase. The influence of thermal residual stresses in fiber-reinforced MMCs under transverse tension was studied in Ref. [7] and found to lead to significant strengthening elevations in contrast to findings in particulate reinforced MMCs where strength reductions were calculated [25]. A limited study on the overall limit flow stress for composites with randomly oriented disk- or needle-shaped particles arranged in a packet-like morphology is reported by Bao et al. [17]. In Refs. [26, 27] a modified Oldroyd model has been proposed to investigate analytically-numerically the overall behavior of MMCs with randomly arranged brittle particles. Duva [28] introduced an analytical model to represent a random distribution of non-interacting rigid spherical particles perfectly bonded in a power law matrix. The Duva model is a self-consistent model and should be valid particularly in the dilute regime of volume fractions, f < 0.2. In this chapter, cell models are applied to simulate, for a number of relevant parameters, the transverse behavior of MMCs containing fibers in a regular square or hexagonal arrangement as well as the mechanical behavior of MMCs containing particles in a regular arrangement. MMCs with randomly arranged inclusions are modeled by a recently introduced self-consistent procedure with embedded cell models. This method of surrounding a simulation cell by additional “equivalent

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composite material” was introduced in Ref. [6] for structures which are periodical in the loading direction and was recently extended to non-periodic two-dimensional [29–31] and three-dimensional composites [13, 27]. The method is known to remove the above-described unrealistic constraints of cell models. An initial comparison of two- and three-dimensional embedded cell models in the case of perfectly plastic matrix material depicts elevated strength levels for the three-dimensional case [27] – similar to composites with regularly arranged fibers [11]. The purpose of the this work is to investigate the mechanical behavior of MMCs reinforced with regular or random arranged continuous fibers under transverse loading, as well as the mechanical behavior of MMCs reinforced with regular or random arranged particles under uniaxial loading, and to systematically study composite strengthening as a function of inclusion volume fraction and matrix hardening ability. The finite element method (FEM) is employed within the framework of continuum mechanics to carry out the calculations.

1.2

Model Formulation

A continuum mechanics approach is used to model the composite behavior. The inclusion behaves elastically in all cases considered here, and its stiffness is much higher than that of the matrix, so that the inclusion can be regarded as being rigid. In addition, the continuous fibers of circular cross section and spherical particles are assumed to be well bonded to the matrix so that debonding or sliding at the inclusion-matrix interface is not permitted. The uniaxial matrix stress-strain behavior is described by a Ramberg-Osgood type of power law: σ ¼ Ee h in σ ¼ σ 0 ee0

e  e0 e > e0

(1)

where σ and E are the uniaxial stress and strain of the matrix, respectively, σ 0 is the tensile flow stress, the matrix yield strain is given as e0 = σ 0/E, E is Young’s modulus, and N = l/n is the strain-hardening exponent. Thus, N = 0 corresponds to a nonhardening matrix. J2 flow theory of plasticity with isotropic hardening is employed with a von Mises yield criterion to characterize the rate-independent matrix material. The von Mises equivalent stress and strain are given as rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3 sij sij σ v ¼ 3J 2 ¼ 2 rffiffiffiffiffiffiffiffiffiffiffiffi 1 3 eij eij ev ¼ 1þμ 2

(2)

where sij = σ ij  σ kk/3, eii = eij  ekk/3 and υ is Poisson’s ratio. In the analytical approach, the metal matrix is considered incompressible, so Poisson’s ratio of the matrix will become 0.5 after reaching the yield stress. However, in reality Poisson’s

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ratio of the composite remains below 0.5 as the matrix starts yielding. It changes from the elastic value υ to the limit value 0.5 with increasing yielding zone in the matrix. The Ramberg-Osgood type of matrix power law hardening is assumed to be valid for the matrix described in terms of von Mises equivalent stress and strain  σ v ¼ σ v0

ev ev0

N

(3)

with the following relations between stress and strain under uniaxial loading and von Mises equivalent stress and strain. (a) In the case of a two-dimensional (2D) plane strain condition for continuous fiber-reinforced composites (√3/2  0.866, 2/√3  1.1547): pffiffiffi 3 σ¼ σv 2 2 e ¼ pffiffiffi ev 3

with with

pffiffiffi 3 σ v0 σ0 ¼ 2 2 e0 ¼ pffiffiffi ev0 3

(4)

(b) In the case of three-dimensional (3D) axisymmetrical condition for particlereinforced composites: σ ¼ σv e ¼ ev

with with

σ 0 ¼ σ v0 e0 ¼ ev0

(5)

The global mechanical response of the composite under external loading is characterized by the overall stress σ as a function of the overall strain e. Moreover, to describe the results in a consistent way, the reference axial yield stress σ 0 and yield strain e0 of the matrix, as defined in Eq. (4) for the 2D case and in Eq. (5) for the 3D case, will be taken to normalize the overall stress and strain of the composite, respectively. Following Bao et al. [17] the composite containing hard inclusions will necessarily harden with the same strain-hardening exponent, N, as the matrix for the case of hard inclusions, when strains are in the regime of fully developed plastic flow. At sufficiently large strains, the composite behavior is then described by:  N e σ ¼ σN e0

(6)

where σ N is called the asymptotic reference stress of the composite which can be determined by normalizing the composite stress by the stress in the matrix at the same overall strain e (Eq. (1)), as indicated in Fig. 1:   σ ðeÞ σN ¼ σ0 for e  e0 σ ðeÞ

(7)

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Fig. 1 Composite strengthening

For a matrix of strain-hardening capability N, the limit value σ N =σ 0 is defined as the composite strengthening level, which is an important value to describe the mechanical behavior of composites and which depends only on fiber and particle arrangement, inclusion volume fraction, and matrix strain-hardening exponent.

1.3

Unit Cell Models

Before introducing the self-consistent embedded cell model, two regular aligned continuous fiber arrangements, namely, square and hexagonal arrangements, and two regular particle arrangements, namely, primitive cubic and hexagonal arrangements, are at first considered, as shown in Fig. 2a–d. It is well known that the repeating unit cells 1 (or 3) and 2 in Fig. 2a can be extracted from the regular array of uniform continuous fibers to model exactly the composite with square fiber arrangement under 0
and 45
transverse loading, respectively, whereas the repeating unit cells 6 and 7 (or 8) can be taken from Fig. 2b to model exactly a hexagonal fiber arrangement under 0
and 30
transverse loading, respectively, if the appropriate boundary conditions are introduced making use of the symmetry conditions. Moreover, the unit cells 4, 9, and 10 can also represent geometrically regular arrangements. For further simplification, the modified unit cells 5, 11, and 12 (circular and elliptically shaped cell models) may be derived from the cells 3, 4, 8, 9, and 10. An ellipsoidal unit cell has been used previously in Ref. [32] and shown to possess very complicated boundary conditions. As can be seen later, these unit cells can be

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a

45°

0°

0° 1

3

5 45°

2

4

0°

45°

b

0° 0°

11

9 6

90°

90°

5 8

90° 7 0° 10

12

c

3 1

2

d

3 1

Fig. 2 Modeling of unidirectional continuous fiber-reinforced composites with (a) square and (b) hexagonal fiber arrangements under transverse loading conditions and particle-reinforced composites with (c) primitive cubic and (d) hexagonal particle arrangements

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employed, however, in the embedding method to model the mechanical behavior of composites with random fiber arrangement. The results of the unit cell models (as illustrated in Fig. 7b as an example for three different regular fiber arrangements with nonhardening matrix) show that the composite strengthening levels are quite different, especially at high volume fractions of fibers, with the composite strengthening for square arrangements under 0
loading being highest and for square arrangements under 45
loading being lowest. Further results regarding these three different regular fiber arrangements using unit cell models are given in Ref. [11]. A comparison of the stress-strain curves for the composite Al-46 vol.% B (in Fig. 5a) shows that the stress-strain curve from random fiber packing given in Ref. [5] lies between the curves from square unit cell modeling under 0
loading and hexagonal unit cell modeling. The curves from square unit cell modeling under 45
loading are even lower. As mentioned above, the strength of composites with randomly arranged inclusions cannot be described by modeling regular inclusion arrangements. Rather, a new model must be employed, which describes approximately the geometry and the mechanical behavior of real composites with randomly arranged inclusions, if models with many inclusions have to be avoided for modeling and computational reasons. For particle-reinforced composites, the primitive cubic particle arrangements can be modeled exactly by a conventional cell model with appropriate symmetry and boundary conditions (Fig. 2c); however, the hexagonal arrangement has to be simulated by approximate cell models (Fig. 2d) [17]. The primitive cubic and axisymmetric unit cells have been used in this chapter to model two representatives of regular particle arrangements. Further simplifications for 3D modeling are spherical unit cells shown in Fig. 2c, d which will be employed, however, for embedded cell modeling.

1.4

Embedded Cell Models

In this chapter, 2D and 3D self-consistent embedded cell models will be applied to model the mechanical behavior of composites with random continuous fiber and particle arrangements. Figure 3a describes schematically a typical plane strain (2D) embedded cell model with a volume fraction of f = (d/D)2 or axisymmetric (3D) embedded cell model with a volume fraction of f = (d/D)3, where instead of using fixed or symmetry boundary conditions around the fiber-matrix or particlematrix cell, the inclusion-matrix cell is rather embedded in an equivalent composite material with the mechanical behavior to be determined iteratively in a selfconsistent manner. If the dimension of the embedding composite is sufficiently large compared with that of the embedded cell, e.g., L/D = 5, as used in this chapter, the external geometry boundary conditions introduced around the embedding composite are almost without influence on the composite behavior of the inner embedded cell. Indeed, there exists no difference in the calculated results whether the vertical surfaces are kept unconstrained or remain straight during uniaxial loading.

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a

b uy (Displacement)

Matrix

d

Fiber/ Particle

D

Embedded Inclusion/Matrix Cell

Composite

Straight boundary

Symmetry line

Embedding Composite

Matrix inclusion Symmetry line

L (L >> D) Interface

Cell boundary

Fig. 3 (a) Embedded cell model and (b) finite element mesh for an embedded cell model

To investigate the influence of the geometrical shape of the matrix phase on the overall behavior of the composite, different shapes of cross section of the embedded cell were chosen with a circular shaped continuous fiber surrounded by a circular, square, elliptical, or rectangular shaped metal matrix (Fig. 6). A typical FE mesh and corresponding symmetry and boundary conditions are given in Fig. 3b, where a circular fiber or a spherical particle is surrounded by a circular (for 2D) or spherical (for 3D) shaped metal matrix, which is again embedded in the composite material with the mechanical behavior to be determined.

1.5

Iterative Modeling Procedure

Under axial displacement loading at the external boundary of the embedding composite (Fig. 3), the overall response of the inner embedded cell can be obtained by averaging the stresses and strains in the embedded cell or alternatively the reaction forces and displacements at the boundary between the embedded cell and the surrounding volume. The embedding method is a self-consistent procedure, which requires several iterations as shown in Fig. 4. An initially assumed stress-strain curve (iteration 0 in Fig. 4) is first assigned to the embedding composite, in order to perform the first iteration step. An improved stress-strain curve of the composite (iteration 1) will be obtained by analyzing the average mechanical response of the embedded cell. This procedure is repeated until the calculated stress-strain curve from the embedded cell is almost identical to that from the previous iteration. The convergence of the iteration occurs typically at the fifth iteration step, as illustrated in Fig. 4. It has been found from systematic studies that convergence of the iteration to the final stress-strain curve of the composite is independent of the initial mechanical

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Fig. 4 Iterative modeling procedure: stress-strain curves for different iteration steps

behavior of the embedding composite (iteration 0). From an arbitrary initial stressstrain curve of the embedding composite, the required composite response can be reached after 4–5 iterations for all cases.

1.6

Comparison with Experiments

An example of the composite Al-46vol.%B with random fiber packing taken from Ref. [5] has been selected to verify the embedded cell model. This MMC is a 6061-O aluminum alloy reinforced with unidirectional cylindrical boron fibers of 46% volume fraction. The room temperature elastic properties of the fibers are Young’s modulus, E(B) = 410 GPa, and Poisson’s ratio, υ(B) = 0.2. The experimentally determined mechanical properties of the 6061-O aluminum matrix are Young’s modulus, E(Al) = 69 GPa; Poisson’s ratio, υ(Al) = 0.33; 0.2% offset tensile yield strength, σ0 = 43 MPa; and strain-hardening exponent, N = l/n = l/3. Figure 5a shows a comparison of the stress-strain curves of the composite Al-46vol.%B under transverse loading from simulations of a real microstructure together with results from different cell models. The stress-strain curve from the embedded cell model employed in this chapter shows good agreement with that from the calculated random fiber packing in the elastic and plastic regime, which lies between the curves from square unit cell modeling under 0
loading and hexagonal unit cell modeling. Furthermore, the stress-strain curve from another experiment [27] on the composite Ag-58vol.%Ni with random particle arrangement (Young’s modulus,

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Fig. 5 Comparison of the mechanical behavior (a) of an Al-46vol.%B fiber-reinforced composite (N = l/3, f = 0.46) under transverse loading from different models and (b) of a Ag-58vol.%Ni particulate composite from embedded cell model and experiment

E(Ni) = 199.5 GPa, E(Ag) = 82.7 GPa; Poisson’s ratio, υ(Ni) = 0.312, υ(Ag) = 0.367; and yield strength, σ 0(Ni) = 193 MPa, σ 0(Ag) = 64 MPa) has been compared in Fig. 5b with that from the self-consistent embedded cell model. Good agreement in the regime of plastic response is obtained, although the Ni-particles in the experiment were not perfectly spherical.

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These results indicate that the embedded cell model can be used to successfully simulate composites with random inclusion arrangements and to predict the elasticplastic composite behavior.

1.7

Geometrical Shape of Embedded Cell

As mentioned above, different shapes of cross section of the embedded cell model with a circular shaped fiber, as shown in Fig. 6a, are also taken into account to investigate the influence of the geometrical shape of the embedded cells on the overall behavior of the composite. The stress-strain curves of all embedded cell models with different geometrical shapes are plotted in Fig. 6b. With an exception of the square-45
embedded cell model, the stress-strain curves are very close for all embedded cell shapes, namely, square-0
, circular, rectangular-0
, rectangular-90
, elliptic-0
, and elliptic-90
. From the calculated results of the embedded cell models, localized flows have been found around the hard fiber with preferred yielding at 45
. Because of the special geometry of the square-45
embedded cell model with the cell boundary parallel to the preferred yielding at 45
, the overall stresses of the composite with such a geometrical cell shape are therefore reduced, such that a relative lower stressstrain curve has been obtained from the modeling. The almost identical responses of all other embedded cell models indicate that, besides the special shape of matrix with 45
cell boundaries, the predicted mechanical behavior of fiber-reinforced composites under transverse loading is independent of the modeling shape of the embedded composite cell. That allows us to employ any embedded cell shape to model the mechanical behavior of the composite with random fiber arrangement. In the following systematic studies, the circular shaped embedded cell model with circular cross section of rigid fiber will thus be taken to predict the general transverse mechanical behavior of the composite with random fiber arrangement. In the same way, a spherical shaped embedded cell model containing a spherical particle will be used to predict the axial mechanical behavior of composites with random particle arrangement.

1.8

Systematic Studies with Self-Consistent Embedded Cell Models

Composite strengthening is dependent on fiber and particle arrangement, inclusion volume fraction, and the matrix strain-hardening exponent. The effect of fiber and particle arrangement is best demonstrated by considering different cell models. The effect of the inclusion volume fraction has been taken into account by applying different ratios of the circular (2D) and spherical (3D) matrix and the inclusion in the embedded cell model. The effect of the matrix strain-hardening exponent has been investigated by changing the parameter, N, of the material hardening law for the

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a

Square - 0°

Rectangular - 0°

Square - 45°

Circular

Rectangular - 90°

Elliptic - 0°

Elliptic - 90°

b 300.0 Al/46vol.%B

Square - 0°

Stress (MPa)

Circular 200.0

Square - 45°

100.0 Ellipse - 0° Ellipse - 90° Rectangular - 0° Rectangular - 90° 0.0 0.0

1.0

2.0

3.0

4.0

5.0

Strain (%) Fig. 6 Embedded cell models: influence of (a) different matrix shapes on (b) stress-strain curves for an Al-46vol.%B (N = l/3, f = 0.46) composite

matrix in Eq. (1). Some results from the systematic studies with embedded cell models are given in Figs. 7, 8, 9, and 10. The influence of matrix strain-hardening is shown in Figs. 7a and 8a. The predicted transverse (2D) and axial (3D) overall stress-strain curves (normalized by yield stress and yield strain of the matrix, respectively) for the case of strainhardening exponents between N = 0.0 and 0.5 are depicted in Fig. 7a (2D) for a fiber volume fraction of f = 0.5 and in Fig. 8a (3D) for a particle volume fraction of f = 0.4. At sufficiently large strains (e.g., at e = 10e0), the normalized overall stresses

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σΝ/σ0

a

N=

(f=0.5) Normalized Stress σ(ε)/σ(ε)

0.5 0.4

2.0

0.3 0.2 0.1 0.0

1.0

0.0 0.0

2.0

4.0

6.0

8.0

10.0

Normalized Strain ε/ε0

b

3.0

N=0.0 Square Unit Cell 0°

Strengthening σΝ/σ0

2.5

Embedded Cell 2.0

1.5

Hexagonal Unit Cell

1.0 0.0

0.2

0.4

0.6

0.8

1.0

Volume fraction f Fig. 7 (a) Normalized stress-strain curves from embedded cell models (2D) for different N-values ( f = 0.5) and (b) composite strengthening from different cell models (2D, N = 0)

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Fig. 8 (a) Normalized overall composite stress-strain curves from axisymmetric embedded cell models (3D) for different N-values (f = 0.4) and (b) composite strengthening for matrix strain-hardening N = 0.2 from different cell models (3D)

approach constant values, i.e., composite strengthening levels, as illustrated on the right-hand side of Figs. 7b and 8b for e = 10e0. The strength of the composite is seen to increase with N, and similar trends of nearly linear increase with N are found for all particle volume fractions, f, in Figs. 8a and 10. The dependence of composite strengthening on inclusion volume fraction obtained by the self-consistent embedded cell modeling is shown in Fig. 7b for a nonhardening matrix with continuous fiber reinforcement and in Fig. 8b for a strainhardening matrix (N = 0.2) with particle reinforcement. For comparison, the corresponding values of composite strengthening for regular fiber and particle arrangements taken from unit cell modeling and from two approximate models, namely, Duva’s model [28] and the modified Oldroyd model [26, 27], are also drawn

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Fig. 9 Comparison of composite strengthening by continuously aligned fibers from different 2D cell models: embedded cell model and square and hexagonal unit cell model

as a function of fiber and particle volume fraction in these figures, respectively. A detailed discussion of this comparison is given in Ref. [27]. Duva’s model is a self-consistent model and valid particularly in the dilute regime, f < 0.2. Composite strengthening by the Duva model is given as σN ¼ ð1  f Þð2:11Nþ0:39Þ σ0

(8)

Composite strengthening levels of the modified Oldroyd model are taken from Ref. [27]. A summary of the dependence of composite strengthening on inclusion volume fraction and matrix strain-hardening for randomly arranged continuous fibers and particles is depicted in Figs. 9 and 10a for matrix strain-hardening exponents N = 0.0–0.5 and compared with predictions from 2D and 3D unit cell models. In addition, a comparison with predictions from Duva’s model and the modified Oldroyd model for the case of particle-reinforced composites is shown in Fig. 10b. For continuous fiber-reinforced composites with a nonhardening matrix, the hexagonal arrangement provides slightly higher transverse composite strengthening over the square arrangement at volume fractions f < 0.5, whereas the random arrangement from the embedded cell model supplies an intermediate value between the two regular arrangements except at volume fraction 0.38 < f < 0.5 (Fig. 9). At volume fractions 0.38 < f < 0.5, the random arrangement from the embedded cell

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a

6.0

Strengthening σΝ/σ0

5.0

Primitive Cubic Unit Cell Axisymmetric Unit Cell Axisymmetric Embedded Cell

N=0.5 N=0.5

4.0 N=0.5 N=0.0

3.0

2.0

1.0 0.0

0.2

0.4

0.6

b

N=0.5

6.0

5.0

Strengthening σΝ/σ0

0.8

Volume fraction f

Axisymmetric Embedded Cell Duva Modified Oldroyd N=0.5

4.0

N=0.5 N=0 3.0

N=0.0

2.0 N=0.0 1.0 0.0

0.2

0.4

0.6

0.8

Volume fraction f Fig. 10 Comparison of composite strengthening by spherical particles from different 3D cell models: (a) embedded cell model, primitive cubic, and axisymmetric unit cell model and (b) embedded cell model, Duva model, and Oldroyd model

model possesses the lowest composite strengthening level. On the contrary, the composite strengthening of the square arrangement is higher than that of the hexagonal arrangement for volume fractions f > 0.4. For the composite with a small exponent of strain-hardening matrix, N < 0.2, similar relations among the

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Application of Homogenization of Material Properties

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three arrangements were found, with the cross points of any two curves being closer to f = 0 with increasing N. For composites with a higher strain-hardening exponent of the matrix, N > 0.2, the strengthening values are almost the same for the three arrangements considered at volume fractions of f < 0.2. At volume fractions of f > 0.2, the square arrangement provides again the highest strengthening and the hexagonal arrangement the lowest strengthening. A comparison of composite strengthening in Fig. 9 for three continuous uniaxial fiber arrangements can be summarized as follows: in the case of low strain-hardening of the matrix, composites with hexagonal fiber arrangement behave stronger than those with random fiber arrangement, which behave, however, at low volume fractions stronger than those with random arrangement. With increasing strain-hardening ability of the matrix, the composites of all three arrangements behave similarly at low fiber volume fractions. At higher inclusion volume fractions, the difference in composite strengthening is becoming larger, with the square arrangement possessing the highest, the random arrangement the intermediate, and the hexagonal arrangement the lowest level. From comparisons for particle-reinforced composites in Fig. 10a, b, it can be seen that the composite strengthening levels from the primitive cubic, the axisymmetric unit cell models, and the self-consistent axisymmetric embedded cell models are very close at low particle volume fractions, f, and low matrix strain-hardening exponents, N. With increasing particle volume fraction, f, and matrix strainhardening exponent, N, the strengthening level of the composites increases for all the models considered. However, the primitive cubic unit cell provides the highest composite strength, while the self-consistent axisymmetric embedded cell predicts the softest composite response among these three models. This effect can be explained by considering the constraint of neighboring particles through the necessary strict boundary conditions of the models with regularly arranged particles. As the particle volume fraction increases, the distance between the particles decreases until they touch at a volume fraction of f = 0.5236 for the primitive cubic cell model (primitive cubic particle arrangement) and a volume fraction of f = 2/3 for the axisymmetric cell model (hexagonal particle arrangement). In the self-consistent axisymmetric embedded cell model, the particles remain surrounded by the matrix up to an extreme volume fraction of f = 1, as seen in Fig. 3. The equivalent composite surrounding the embedded inclusion-matrix cell provides thus less constraint on the inclusion-matrix cell compared to conventional unit cell models of the same volume fraction. In most practical relevant composites, the inclusions are randomly arranged and on average, no such restricting constraints exist as in the regular primitive cubic and hexagonal arrangements. For this reason, the selfconsistent axisymmetric embedded cell model is believed to be the most realistic approximation to the geometry of real composites containing randomly arranged spherical particles. At low particle volume fractions ( f < 0.2) and low matrix strain-hardening ability (N < 0.2), the composite strengthening levels from the present self-consistent axisymmetric embedded cell model and the Duva model are very similar. With increasing particle volume fraction, f, the differences in composite strengthenings between these two models are significant for low matrix strain-hardening ability; however, for N ~ 0.5, the two strength predictions are comparable. The modified

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Oldroyd model provides a very narrow distribution of composite strengthening with respect to N of matrix strain-hardening and an almost linear strength dependence on particle volume fraction, f. Good agreement between the self-consistent axisymmetric embedded cell model and the modified Oldroyd model exists only at low volume fractions and high strain-hardening rates (N ~ 0.5) of the matrix. From the above comparisons between the models, it is clear that at low particle volume fractions ( f < 20%), the strengthening of metal matrix composites reinforced with randomly or regularly arranged particles can be predicted with either the axisymmetric unit cell model, the embedded cell model, or the simple Duva model. However, for higher particle volume fractions, the self-consistent embedded cell model should be applied to obtain the overall mechanical response of technically relevant composites reinforced by randomly arranged spherical particles.

1.9

Strengthening Model

The strength of MMCs reinforced by hard inclusions under external mechanical loading has been shown to increase with inclusion volume fraction and strainhardening ability of the matrix for all inclusion arrangements investigated. From the presented numerical predictions, a strengthening model for aligned continuous fiber-reinforced MMCs with random, square (0
) and hexagonal arrangements, as well as for spherical particle-reinforced MMCs with random, primitive cubic and hexagonal arrangements, can be derived as a function of the inclusion volume fraction, f, and the strain-hardening exponent, N, of the matrix [31]: " σN ¼ σ0

f 1 c 1 ð2 þ N Þ

ðc2 Nþc3 Þ

 # N  c4 f þ 5

(9)

where σ 0 is the matrix yield stress, and c1, c2, c3, and c4 are constants summarized in Table 1. Equation (9) represents best fits to the calculated composite strengthening values σ N for matrix strain-hardening exponents N in the limit of 0.0 < N < 0.5 for square 0
, hexagonal, and random fiber arrangements, respectively, and fiber volume fractions f in the range of 0.0 < f < 0.7. A comparison of this strengthening model Table 1 Constants for strengthening models 2D Self-consistent embedding cell Quadratic unit cell (0
) Hexagonal unit cell 3D Self-consistent embedding cell Cubic unit cell Hexagonal unit cell

c1 0.361 0.405 0.305

c2 1.590 2.350 1.300

c3 0.290 0.650 0.050

c4 0.100 0.220 0.000

0.450 0.340 0.380

2.190 2.300 2.500

0.840 0.650 0.700

0.530 0.500 0.660

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Application of Homogenization of Material Properties

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5.0

Embedded Cell Models (2D) Approximate Expression

N=0.5 N=0.4

4.0

Strengthening σΝ/σ0

N=0.3 N=0.2 3.0 N=0.1 N=0.0 2.0

1.0 0.0

0.2

0.4

0.6

0.8

Volume fraction f Fig. 11 Comparison of composite strengthening by 2D cell models: embedded cell models approximate expression (Eq. 9)

(Eq. (9)) for random fiber arrangements with the values calculated by using selfconsistent embedded cell models shows close agreement with an average error of 1.25% and a maximum error of 6.95% [31]. Equation (9) is also available for matrix strain-hardening exponents N in the limits of 0.0 < N < 0.5 for self-consistent axisymmetric embedded cell models (particle volume fractions f in the range of 0.05 < f < 0.65 with an average error of 1.59% and a maximum error of 6.68% for the extreme case f = 0.05, N = 0.5), for axisymmetric unit cell models (particle volume fractions f in the range of 0.05 < f < 0.55 with an average error of 1.22% and a maximum error of 6.18% for the extreme case f = 0.55, N = 0.5), and for primitive cubic unit cell models (particle volume fractions f in the range of 0.05 < f < 0.45 with an average error of 1.43% and a maximum error of 6.38% for the extreme case f = 0.05, N = 0.5) (see Fig. 11).

1.10

Conclusion

The transverse elastic-plastic response of metal matrix composites reinforced with unidirectional continuous fibers and the overall elastic-plastic response of metal matrix composites reinforced with spherical particles have been shown to depend on the arrangement of reinforcing inclusions as well as on inclusion volume fraction, f, and matrix strain-hardening exponent, N. Self-consistent plane strain and axisymmetric embedded cell models have been employed to predict the overall mechanical

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behavior of metal matrix composites reinforced with randomly arranged continuous fibers and spherical particles perfectly bonded in a power law matrix. The embedded cell method is a self-consistent scheme, which typically requires 4–5 iterations to obtain consistency of the mechanical behavior in the embedded cell and the surrounding equivalent composite. Experimental findings on an aluminum matrix reinforced with aligned but randomly arranged boron fibers (Al-46vol.%B) as well as a silver matrix reinforced with randomly arranged nickel inclusions (Al-58vol.%Ni) and the overall response of the same composites predicted by embedded cell models are found to be in close agreement. The strength of composites with aligned but randomly arranged fibers cannot be properly described by conventional fiber-matrix unit cell models, which simulate the strength of composites with regular fiber arrangements. Systematic studies were carried out for predicting composite limit flow stresses for a wide range of parameters, f and N. The results for random 3D particle arrangements were then compared with regular 3D particle arrangements by using axisymmetric unit cell models as well as primitive cubic unit cell models. The numerical results were also compared with those from the Duva model and from the modified Oldroyd model. The strength of composites at low particle volume fractions has been shown to be in very close agreement except for the modified Oldroyd model. With increasing particle volume fractions, f, and strain hardening of the matrix, N, the strength of composites with randomly arranged particles cannot be properly described by conventional particle-matrix unit cell models, as those are only able to predict the strength of composites with regular particle arrangements. Finally, a strengthening model for randomly or regularly arranged continuous fiber-reinforced composites under transverse loading and particle-reinforced composites under axial loading is derived, providing a simple guidance for designing the mechanical properties of technically relevant metal matrix composites: for any required strength level, Eq. (9) will provide the possible combinations of particle volume fraction, f, and matrix hardening ability, N [33]. Thus, strong impact of this work on the development of new particle-reinforced metal matrix composites is expected.

2

Self-Consistent Matricity Model to Simulate the Mechanical Behavior of Interpenetrating Microstructures

2.1

Introduction

Coarse isotropic interpenetrating microstructures are often found in powder metallurgically fabricated composites in the regime of 25–75% volume fraction or result from the infiltration of a porous material with a molten metal of a lower melting point. The arrangement of the phases in most of these technical composites is usually random, resulting in an isotropic overall mechanical behavior of these composites. Besides the volume fraction, such a material requires at least one further parameter to

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Application of Homogenization of Material Properties

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describe the microstructure more closely. In this chapter the selected further parameter is the “matricity,” which was first introduced by Poech [34] for two-phase steels and later used by Soppa [35] for the characterization of Ag/Ni composites. This parameter can be measured from a representative micrograph of a microstructure via an image analyzing system and also be included in a finite element model which was developed for this kind of microstructure [36–40]. There the model is applied to examine the influence of microscopical residual stresses on the macroscopical behavior of composites and the impact of matricity on these residual stresses.

2.2

Matricity

In the following, composites of two phases α and β are considered. Matricity is defined as the fraction of the length of the skeleton lines of one phase Sα, and the length of the skeleton lines of the participating phases Mα ¼

Sα Sα þ Sα

(10)

By definition, the sum of the matricities of all phases equals to one Mα þ Mβ ¼ 1

(11)

To obtain the skeleton lines of a certain phase, we must select this phase within an image analyzing system and reduce the detected structure to a typically non-connecting line by maintaining the topology. In Fig. 12 the matricities have been determined for a ZrO2/NiCr 80 20 cermet which was powder metallurgically

Fig. 12 Micrograph of a ZrO2/NiCr 80 20 composite with 30% volume fraction of ZrO2 (left, greyscale picture; right, binary image with skeleton lines)

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M. Dong and S. Schmauder

fabricated at the University of Dortmund [41]. The structure parameters volume fraction f and matricity M have been determined to be fNiCr = 0.3 and MNiCr = 0.2.

2.3

Matricity Model

To take both the parameters into account for the calculation of the mechanical behavior of the composite, an extension of the embedded cell model [31, 33] has been developed. The embedded cell model has been introduced to simulate the mechanical behavior of composites with randomly distributed inclusions, where the volume fraction of the inclusions is the main parameter in the model. To take the matricity as second microstructural parameter into account, the self-consistent embedded cell model has been extended by a second self-consistent embedded cell model (Fig. 13). In this “matricity model” we are able to define the matricity of the model in the same manner as the matricity is defined in a real microstructure. First the single phases are reduced to skeleton lines. The lengths of the skeleton lines of the inclusions (Fig. 13, left, β; right, α) are zero as the inclusions are spherical and are, therefore, reduced to a point in the process of obtaining the matricity of the phase. The lengths of the skeleton lines Sα and Sβ in the matrices are given as the circumference of a circle with a diameter which is calculated as the arithmetic average of the diameter of the embedded cell and the diameter of the inclusion phase (Fig. 13, left, Sα; right, Sβ). The diameters of the embedded cells are denominated as W1 and W2. The diameters of the inclusion part of the embedded cells depend on the volume fraction of the inclusions and the corresponding factors W1 and W2.

Fig. 13 Matricity model (schematic) with skeleton lines to adjust the measured parameter “matricity” in the model via the factors W1 and W2

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Application of Homogenization of Material Properties

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For the 3D case, the diameter of the inclusion b (Fig. 13, left) is derived as a function of W1 and the volume fraction fβ of β in this cell as Dβ ¼ W 1

qffiffiffiffiffi 3 fβ

(12)

and analogous for inclusion α (Fig. 13, right) as Dα ¼ W 2

pffiffiffiffiffi 3 fα

(13)

Therefore, we derive the skeleton line lengths as

Sα ¼ πW 1

pffiffiffiffiffi  3 fβ þ 1 2

(14)

and Sβ ¼ πW 2

1þ

p ffiffiffiffiffi

3 fα 2

(15)

By taking into account that fα þ fβ ¼ 1

(16)

the matricity M can be calculated as a function of the sizes of the embedded cells and the volume fraction of one of the two phases, as the volume fraction of the phases is held constant in both parts of the matricity model ! p ffiffiffiffiffi 3 fβ þ 1 W1π 2 Sα ! ! Mα ¼ ¼ p ffiffiffiffiffi p ffiffiffiffiffi 3 3 Sα þ Sβ fβ þ 1 fα þ 1 W1π þ W2π 2 2

(17)

or pffiffiffiffiffiffiffiffiffiffiffiffiffi

W1 3 1  f α þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffi

Mα ¼ W1 3 1  f α þ 1 þ W2 3 f α þ 1

2.4

(18)

Adjusting Matricity in the Model

As can be seen in Fig. 13, the volume fractions of the phases as well as the diameters W1 and W2 of the embedded cells are adjustable. To achieve a matricity M in the

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M. Dong and S. Schmauder

model, we first realize the measured volume fraction of the phases in the model and then calculate the diameters W1 and W2. The corresponding diameters W1 and W2 are obtained by rearranging Eq. (18): W1 ¼ W2

qffiffiffiffiffiffiffiffiffiffiffiffiffi  3 1  fβ þ 1

1M pffiffiffiffiffi β  , Mβ 3 f β þ 1

W 2 ¼ 1:0 for Mβ  0:5

(19)

1  Mα ffiffiffiffiffi p

, 3 fα þ 1

W 1 ¼ 1:0 for Mα  0:5

(20)

or W2 ¼ W1

2.5

pffiffiffiffiffiffiffiffiffiffiffiffiffi  3 1  fα þ 1

Mα

Realization of the Adjustability of Matricity by Weighting Factors

If the geometrical boundary conditions are modeled at a distance of about five times the radius of the embedded cell, they are of almost no influence on the models’ mechanical behavior. If we take care that the boundary conditions keep remote, we can model the embedded cell with the surrounding composite in different manners (Fig. 14). As the remote boundary conditions have almost no influence on the mechanical behavior of the embedded cell, it is assumed that the continuum mechanical stress-strain state in the embedded cell is hardly influenced as well. Taking this into account, a unit cell for a specific volume fraction can be used for each part of the matricity model. Further we can see from the almost independency from remote boundaries that it is not necessary to model the matricity as a parameter of the FE mesh, but it is possible to introduce the matricity adjusting weighting factors W1 and W2 only in the evaluation of the results from the inclusion-type geometries. As the results have to be determined by an iterative calculation in about

Fig. 14 Independence of mechanical behavior from size variations of embedded and embedding medium (schematic)

49

Application of Homogenization of Material Properties

1621

Fig. 15 Realization (schematic) of the adjustability of matricity by weighting factors

3~5 iterations, the adjusting weighting factors W1 and W2 must be introduced in the evaluation of all iteration steps (Fig. 15).

2.6

Calculation of Stress-Strain Curves

In principle, stress-strain curves of the two-phase composite are determined from the matricity model in the same iterative manner as it is done for the simple selfconsistent embedded cell model. In each increment the components for stress and strain are determined. This is done by a weighted averaging of the stress and strain values over all integration points of both embedded cells. The three-dimensional weighting is done by the “integration point volume” Vk0 of each corresponding Gaussian integration point, which must be multiplied by W 31 and W 32, respectively, to account for the matricity effects as described above. The factors Wi are in the power of three as the length of the skeleton lines depend linearly on Wi, but the embedded cell volumes depend by the power of three (for the 3D case) on Wi. With these considerations the stress and strain components can be calculated as P k σ ij V k σ ij ¼ P , Vk P k eij V k , eij ¼ P Vk

i, j ¼ ðx, y, zÞ

or

ðr, z, ϕÞ

(21)

i, j ¼ ðx, y, zÞ

or

ðr, z, ϕÞ

(22)

or more detailed (only for the stress components) P σ ij ¼

 P  σ kij V k0 W 32 σ kij V k0 W 31 þ 1 2 P P V k0 W 31 þ V k0 W 32

(23)

where k is the index of summation and 1 or 2 are the part of matricity model whose embedded cell is weighted by W1 or W2. From the stress components, the von Mises equivalent stress at each strain increment is calculated as (Eqs. (24) and (25) are only valid for Cartesian coordinates)

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M. Dong and S. Schmauder

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  

(24) σ v ¼ σ 2xx þ σ 2yy þ σ 2zz  σ xx σ yy þ σ yy σ zz þ σ zz σ xx þ 3 σ 2xy þ σ 2yz þ σ 2zx and the equivalent strain is given as 1 ev ¼ 1þμ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi

2 2 2 2 2 2 exx þ eyy þ ezz  exx eyy þ eyy ezz þ ezz exx þ 3 γ xy þ γ yz þ γ zx (25)

where μ is the elastic-plastic Poisson’s ratio of the composite.

2.7

Mechanical Constants

Poisson’s ratio μ and tangent modulus T0 at zero strain are calculated from the obtained stress and strain components σij and εij. Poisson’s ratio μ is calculated from the context of elastic constants [42]: μ¼

3K  E 6K

(26)

where K is the bulk modulus and E is the Young’s modulus. K is defined as [42] K¼



σ xx þ σ yy þ σ zz σH

¼ 3eH 3 exx þ eyy þ ezz

(27)

If we consider a material where the elastic modulus E equals the gradient of the stress-strain curve at zero strain, then we can determine the constants K and E from the calculated stress-strain curve. As the moduli change with changing strains, we have to extrapolate the bulk modulus K and the tangent modulus T from values near zero strain to a value at zero strain (Fig. 16).

2.8

Yield Stress

In this chapter the yield stress is calculated as the stress belonging to the cross point of the stress-strain curve and a straight line through the 0.2% strain point with the gradient that equals to the above calculated tangent modulus. The model does not include any damage parameter or failure criterion. This might lead to unrealistic high yield stresses for composites with dominating linear-elastic material behavior.

2.9

Results and Discussion

The quality of the matricity model to simulate stress-strain curves and strain distribution frequencies especially for two ductile phases has been demonstrated in

49

Application of Homogenization of Material Properties

1623

Fig. 16 Calculation of tangent modulus T0 by extrapolation of secant modulus values at zero strain

[36–40]. In this chapter we compare the matricity model with other models and examine the influence of the matricity parameter on stress-strain curves and on the influence of residual stresses.

2.9.1 Comparison to Cluster Parameter r The cluster parameter rγ (γ = α; β) is defined in [43] as rγ ¼

Nγ Nα þ Nβ

(28)

where γ = α, β and Nγ is the number of clusters of phase γ. Figure 17 shows some model microstructures which have been calculated with distinguishable assumptions. To compare with the numerical yield stress results based on Eq. (28) [43], we modeled the microstructures also with the matricity model. To do this the matricity of the microstructures was derived by phase reduction (cf. Matricity) which is shown in Fig. 18. The comparison of the matricity model results with the results in [43] shows good agreement with real structure calculations for both models (Fig. 19). The advantage of the matricity model might be that differently sized clusters provide a different contribution to the influence of this cluster to the overall mechanical behavior, whereas the cluster parameter model assumes that all clusters, independent of the cluster sizes, are equally weighted. An obvious advantage of the matricity model is the good agreement of calculated strain distribution frequencies to experimentally obtained results [36–40].

2.9.2 Matricity and Stress-Strain Curves The influence of the matricity on the overall mechanical behavior can be shown by comparing the calculated stress-strain curves while keeping the volume fraction f of the phases constant and by varying only the matricity M of the phases (this means to

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Fig. 17 Model microstructures of regular hexagons by Siegmund et al. [43]

vary Wi correspondingly for evaluating the results). A parameter study was made for a ZrO2/NiCr 80 20 composite where the volume fraction f of the phases was kept constant and the matricity M was varied between 0 and 1. This means that the microstructure lies in a range between a pure inclusion microstructure and a pure

49

Application of Homogenization of Material Properties

1625

Fig. 18 Determination of matricity M for the hexagonal model microstructure with fb = 0.5 [43] (left, original microstructure; right, binary image with skeleton lines)

Fig. 19 Yield stress of α/β composite (* in [43], ** in [43] calculated from [44])

SC-results **

5

FE-results * SC with fraction of clusters * Matricity model

α

3

α,β

R p0,2 / R p0,2 [ ]

4

2

upper bound lower bound

1 0 0.0

0.2

0.4

0.6

0.8

1.0

Volume Fraction of b [ ]

matrix microstructure for one phase and vice versa for the other phase. In Fig. 20, stress-strain curves are shown for two different matricities MZrO2 (= 0.3 and 0.4) with and without residual stresses caused by cooling down the material from processing to environmental temperature. Figure 21 shows the influence of the matricity on the yield stress. The influence of matricity is decisive in the range between M = 0.3 and M = 0.7. Taking residual stresses into account, we can recognize that the influence of matricity on the yield stress diminishes. This can be explained with the modeled isotropic hardening of the ductile NiCr 80 20 phase, which is plastically deformed by residual stresses when acting as a matrix. However, no plastification is obtained when this phase acts as a pure inclusion and, therefore, hardly any hardening of the composite will be expected.

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Fig. 20 Stress-strain curves of ZrO2/NiCr 80 20 composite with fZrO2 = 0.3 and two different matricities (EZrO2 = 206 GPa, ENiCr 80 20 = 214 GPa, αZrO2 = 10  106 K1, αNiCr 6 1 K ) 80 20 = 14  10

Fig. 21 Yield stress of ZrO2/NiCr 80 20 composite with fZrO2 = 0.3 and different matricities M (EZrO2 = 206 GPa, ENiCr 80 20 = 214 GPa, αZrO2 = 10  106 K1, αNiCr 80 20 = 14  106 K1, ΔT = 750 K)

2.10

Conclusion

The influence of the parameter matricity M besides the parameter volume fraction f on the overall behavior of two-phase composites with coarse interpenetrating microstructures is clearly shown. Especially in cases, when the composite consists of a ductile and a linear-elastic phase, a significant influence of the matricity on stress-strain curves has been found. It is also found that the influence of residual stresses on yield stress depends strongly on the volume fraction and the matricity of the ductile phase. The comparison with model structure calculations shows the high quality of the matricity model. The matricity model must be built once for each volume fraction and the matricity is then considered in the evaluation of the result

49

Application of Homogenization of Material Properties

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during the iterative calculation process. The model is, therefore, very simple with respect to FE meshing, and can be quite coarse, as the stress-strain components were evaluated as arithmetical averages over the elements of both embedded cells.

References 1. Harrington WC Jr. Metal matrix composite applications. In: Ochiai S, editor. Mechanical properties of metallurgical composites. New York: Marcel Dekker; 1993. 2. Brockenbrough JR, Suresh S. Plastic deformation of continuous fiber reinforced MMCs: effects of fiber shape and distribution. Scripta Metall Mater. 1990;24:325. 3. Rammerstorfer FG, Fischer FD, Böhm HJ. Treatment of micro-mechanical phenomena by finite elements. In: Kuhn G, et al. editors. Discretization methods in structural mechanics. IUTAM/ IACM Symposium, Vienna; 1989. Berlin/Heidelberg: Springer; 1990. 4. Evans AG. The mechanical properties of reinforced ceramic, metal and intermetallic matrix composites. Mater Sci Eng. 1991; A143:63. 5. Brockenbrough JR, Suresh S, Wienecke HA. Deformation of metal-matrix composites with continuous fibers: geometrical effects of fiber distribution and shape. Acta Metall Mater. 1991;39:735. 6. Dietrich C. Mechanisches Verhalten von Zweiphasengefügen: numerische und experimentelle Untersuchungen zum Einfluß der Gefügegeometrie. VDI-Fortschrittsberichte, Reihe 18, Nr. 128. Düsseldorf: VDI-Verlag; 1993. 7. Nakamura IT, Suresh S. Effects of thermal residual stresses and fiber packing on deformation of metal-matrix composites. Acta Metall Mater. 1993;41:1665. 8. Dietrich CM, Poech H, Schmauder S, Fischmeister HF. Numerische Modellierung des mechanischen Verhaltens von Faserverbundwerkstoffen unter transversaler Belastung. In: Leonhardt G, et al. editors. Verbundwerkstoffe und Werkstoffverbunde. DGM-lnformationsgesellschaft mbH. Oberursel; 1993. 9. Adams D. Inelastic analysis of a unidirectional composite subjected to transverse normal loading. J Compos Mater. 1970;4:310. 10. Jansson S. Homogenized nonlinear constitutive properties and local stress concentrations for composites with periodic internal structure. Int J Solids Struct. 1992;29:2181. 11. Zahl DB, Schmauder S, McMeeking RM. Transverse strength of metal matrix composites reinforced with strongly bonded continuous fibers in regular arrangements. Acta Metall Mater. 1994;42:2983. 12. Llorca J, Needleman A, Suresh S. An analysis of the effects of matrix void growth on deformation and ductility in metal-ceramic composites. Acta Metall Mater. 1991;39:2317. 13. Sautter M. Modellierung des Verformungsverhaltens mehrphasiger Werkstoffe mit der Methode der Finiten Elemente. PhD. Dissertation, University of Stuttgart; 1995. 14. Povirk GL, Stout MG, Bourke M, Goldstone JA, Lawson AC, Lovato M, Macewen SR, Nutt SR, Needleman A. Thermally and mechanically induced residual strains in Al-SiC composites. Acta Metall Mater. 1992;40:2391. 15. Böhm HJ, Rammerstorfer FG, Weissenbek E. Some simple models for micromechanical investigations of fiber arrangement effects in MMCs. Comput Mater Sci. 1993;1:177. 16. Böhm HJ, Rammerstorfer FG, Fischer FD, Siegmund T. Microscale arrangement effects on the thermomechanical behavior of advanced two-phase materials. ASME J Eng Mater Technol. 1994;116:268. 17. Bao G, Hutchinson JW, McMeeking RM. Particle reinforcement of ductile matrices against plastic flow and creep. Acta Metall Mater. 1991;39:1871. 18. Tvergaard V. Analysis of tensile properties for a whisker-reinforced metal-matrix composite. Acta Metall Mater. 1990;38:185. 19. Hom CL. Three-dimensional finite element analysis of plastic deformation in a whiskerreinforced metal matrix composite. J Mech Phys Solids. 1992;40:991. 20. Weissenbek E. Finite element modelling of discontinuously reinforced metal matrix composites. PhD. Dissertation. PhD. Dissertation, Technical University of Vienna; 1993.

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21. Li Z, Schmauder S, Wanner A, Dong M. Transverse strength of metal matrix composites reinforced with strongly bonded continuous fibers in regular arrangements. Scripta Metall Mater. 1995;33:1289. 22. Christman T, Needleman A, Suresh Y. An experimental and numerical study of deformation in metal-ceramic composites. Acta Metall Mater. 1989;37:3029. 23. Suquet PM. Bounds and estimates for the overall properties of nonlinear composites. MECAMAT 93, Int. Seminar on Micromechanics of Materials. Paris: Editions Eyrolles; 1993;361:382. 24. Thebaud F. Vers l’introduction de l’endommagement dans la prévision globale du comportement de composites á matrice métallique. PhD. Dissertation, Université de Paris Sud; 1993. 25. Zahl DB, McMeeking RM. The influence of residual stress on the yielding of metal matrix composites. Acta Metall Mater. 1991;39:1171. 26. Poech MH. Deformation of two-phase materials: application of analytical elastic solutions to plasticity. Scripta Metall Mater. 1992;27:1027. 27. Farrissey L, Schmauder S, Dong M, Soppa E, Poech MH. Investigation of the strengthening of particulate reinforced composites using different analytical and finite element models. Comput Mater Sci. 1999;15:1. 28. Duva JM. A self-consistent analysis of the stiffening effect of rigid inclusions on a power-law material. Trans ASME Series H J Eng Mater Technol. 1984;106:317. 29. Sautter M, Dietrich C, Poech MH, Schmauder S, Fischmeister HF. Finite element modelling of a transverse-loaded fibre composite: effects of section size and net density. Comput Mater Sci. 1993;1:225. 30. Zahl DB, Schmauder S. Transverse strength of continuous fiber metal matrix composites. Comput Mater Sci. 1994;3:293. 31. Dong M, Schmauder S. Transverse mechanical behaviour of fiber reinforced composites – FE modeling with embedded cell models. Comput Mater Sci. 1996;5:53. 32. Järvstråt N. An ellipsoidal unit cell for the calculation of micro-stresses in short fibre composites. Comput Mater Sci. 1993;1:203. 33. Dong M, Schmauder S. Modeling of metal matrix composites by a self-consistent embedded cell model. Acta Mater. 1996;44:2465. 34. Poech MH, Ruhr D. Die quantitative Charakterisierung der Gefügeanordnung. Prakt Metallogr Sonderband. 1993;24:391. 35. Soppa E. Experimentelle Untersuchung des Verformungsverhaltens zweiphasiger Werkstoffe. Fortschr. Ber., VDI Reihe 5, 408. Düsseldorf: VDI-Verlag; 1995. 36. Lessle P, Dong M, Soppa E, Schmauder S. Selbstkonsistente Matrizitätsmodelle zur Simulation des mechanischen Verhaltens von Verbundwerkstoffen. In. Friedrich K, editor. Hrsg. DGM Informationsgesellschaft Verlag; 1997. 37. Schmauder S, Dong M, Lessle P. Verbundwerkstoffe mikromechanisch simuliert. Metall. 1997;51(7/8):404. 38. Schmauder S, Dong M, Lessle P. Simulation von Verbundwerkstoffen mit Teilchen und Fasern. Tagungsband XXIV. FEM Kongreß in Baden Baden, 17–18 Nov 1997. 39. Lessle P, Dong M, Soppa E, Schmauder S. Simulation of interpenetrating microstructures by self consistent matricity models. Scripta Mater. 1998;38:1327. 40. Schmauder S, Dong M, Lessle P. Self-consistent matricity model to simulate the mechanical behaviour of interpenetrating microstructures. Comput Mater Sci. 1999;15:455. 41. Willert-Porada M, Gerdes T, Rödiger K, Kolaska H. Einsatz von Mikrowellen zum Sintern pulvermetallurgischer Produkte. Metall. 1997;51(1/2):57. 42. Sautter M. Modellierung des Verformungsverhaltens mehrphasiger Werkstoffe mit der Methode der Finiten Elemente. Fortschritt Bericht. VDI Reihe 5, 398. Düsseldorf: VDI-Verlag; 1995. 43. Siegmund T, Werner E, Fischer FD. Structure–property relations in duplex materials. Comput Mater Sci. 1993;1:234. 44. Bao G, Hutchinson JW, McMeeking RM. The flow stress of dual-phase, non-hardening solids. Mech Mater. 1991;12:85.

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Fatigue Behavior of 9–12% Cr Ferritic-Martensitic Steel Zhen Zhang, Zhengfei Hu, and Siegfried Schmauder

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9–12%Cr Ferritic-Martensitic Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Chemical Composition and Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Strengthening Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Low Cycle Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Cyclic Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hysteresis Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cyclic Stress-Strain Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Fatigue Damage Accumulation and Life Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fatigue Damage Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Total Life Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nucleation and Growth of Fatigue Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Nucleation of Fatigue Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Growth of Fatigue Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Evolution of Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Z. Zhang (*) School of Material Science and Engineering, Nanjing Institute of Technology, Nanjing, China e-mail: [email protected] Z. Hu School of Material Science and Engineering, Tongji University, Shanghai, China e-mail: [email protected] S. Schmauder Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_34

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Abstract

In this chapter, the cyclic fatigue behavior of the 9–12%Cr ferritic-martensitic steel used in power plants is systematically summarized. At first, the application background and the basic information of these kinds of materials are discussed. The 9–12%Cr martensitic steel has been extensively used in the supercritical steam turbine components in the thermal power plants. The fully understanding of the effects of materials properties and softening behavior on low cycle fatigue (LCF) are important in the improvement of structural design and the reliability assessment for its safety operation. Subsequently, the LCF properties, the nucleation and growth of fatigue crack, and the evolution of microstructure are further discussed. The previous research results and the factors that affect the LCF behavior are also summarized in these sections. Lastly, a brief summary of the chapter and the new research method related to the fatigue behavior of 9–12%Cr ferritic-martensitic steel are introduced for further investigating. This chapter serves as a quick reference of entering the fatigue behavior of materials for researchers, engineers, and students in the mechanical and materials engineering field. Keywords

9–12%Cr ferritic-martensitic steel · Low-fatigue property · Life prediction · Crack nucleation and growth · Microstructure evolution

1

Introduction

The green energy has become an important issue all over the world, which has led to two big changes in power generation industry in the past decades. The first change is the technology of ultrasuper critical (USC) unit generators, as shown in Fig. 1, which have been quickly developed in fossil power plants with the purpose of improving the generating efficiency and reducing emission of CO2. In power plant, great efficiency means a saving in fuel and less emission of carbon dioxide in a given electricity output, which consequentially reduces the rate at which damage is done to the globule environment. The efficiency of steam turbines can be obviously improved by increasing the maximum operating pressure and temperature. The relationship between the efficiency and steam parameters in power plant is shown in Table 1. That is why there is a tendency that operating parameters become much higher to promote efficiency. This tendency of technology will adopt more severe steam working parameters, i.e., higher temperature and/or higher pressure, which bring more extremely cruel service condition for materials. Several grades of 9–12% Cr ferritic-martensitic heat resistant steels are candidate materials for structural components for Gen IV nuclear power plants, due to their high strength, low thermal expansion, good corrosion resistance, and good mechanical properties at elevated temperature. These characteristics permit to consider them being extensively used as structural materials for many components in steam generators and fusion reactors.

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Fig. 1 Ultrasuper critical (USC) unit generator Table 1 Relationship between the efficiency and steam parameters in power plant

Type of unit Medium High Super high Subcritical Supercritical Ultrasupercritical 700
C Ultrasupercritical

Steam pressure /MPa 3.5 9.0 13.0 17.0 25.5 30.0 35.0

Steam temperature /
C 435 510 535 540 600 700 >700

Efficiency /% 27 33 35 38 44 57 60

Coal consumption /g (kWh)1 460 390 360 324 278 215 205

In addition, the components made of these kinds of steels, such as rotor, cylinder, and pipe, are often subjected to repeated thermal stress owing to temperature gradients that occur on heating and cooling during start-ups and shutdowns. This, in turn, may give rise to plastic deformation over a period of time, especially under high working temperatures. Therefore, the deep understanding of the low cycle fatigue (LCF) behavior is an important concern in the life design and the reliability assessment of the structure. In the Sect. 2 the effect of alloy elements and the strengthening mechanism during deformation of these kinds of steels are addressed. In the local critical area of components, limited plastic flow phenomena like the cyclic ratcheting (under the stress-control condition) or the cyclic stress relaxation (under the strain-control condition) may arise and need to be described accurately in terms of cyclic plastic or viscoplastic constitutive equations. Thus the

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low cycle strain- and stress-controlled fatigue properties of the steels are reviewed in the Sect. 3. The cyclic fatigue is a process of damage accumulation and when such damage increase to a certain limit, the nucleation and propagation of fatigue crack will lead to the final fatigue failure. Although safe design and careful condition monitoring have always been of great concern for the power industry, high temperature components loaded with steam pressure in power plants have a high damage potential during long-term service. So in the Sect. 4 the damage evolution during fatigue process and the life prediction using the traditional Basquin and MasonCoffin model are summarized for these kinds of steels at different temperature. Moreover, a new hysteresis energy model to predict the fatigue life is also presented. Microcrack initiation, microcrack growth, and final failure comprise damage accumulation under fatigue loading. The microstructure and mechanical properties of the materials strongly affect crack initiation and its subsequent propagation. The nucleation and propagation of the fatigue crack are summarized in the Sect. 5. Under cyclic loading over a wide range of strains and temperatures, these kinds of steels exhibit continuous softening as a result of microstructural instability, like the annihilation of high initial dislocation density and the breakdown of martensite lath structure. Finally, in the Sect. 6 these microstructural instabilities during the fatigue process are discussed.

2

9–12%Cr Ferritic-Martensitic Steel

2.1

Chemical Composition and Microstructure

9–12%Cr ferritic/martensitic steels have the same body-centered cubic crystal structure as iron. They are simply iron containing with relatively small addition alloy elements, such as the main element chromium added from 9% to about 12%. These ferritic/martensitic grades also have a little manganese, molybdenum, silicon, carbon, and nitrogen, mostly included for their benefits in precipitation strengthening and prompting high temperature behavior. Ferritic grades are used broadly because they are much economic for their low content of alloy additions. They also have some resistance to oxidization at red heat, and which is in direct proportion to chromium content. The chemical compositions of typical 9–12%Cr steels and the creep rupture strength are listed in Table 2. No matter the tiny difference in the chemical compositions, all these 9–12%Cr ferrite-martensite heat-resistant steels share the same microstructure of tempered martensite and display the microstructural instability during the long-term high- temperature service. In general, 9–12%Cr steels are also used in the normalized-and-tempered condition. Martensite forms when products are normalizing by austenitizing above A1 and then air cooling. During this process, the formation of δ-ferrite should be avoided as this may cause embrittlement, resulting in fabrication problems. Subsequent tempering above 700
C for hours, the subgrains are formed in the interior of the martensite laths and the size of these subgrains is in the order of about 500 nm. The initial microstructures of typical 9–12%Cr steels are shown in Fig. 2.

a

stand for the comparison austenitic steels

C Mn P S Si Cr W Mo V Nb N B Ni Al Creep rupture strength

P9

ij ij ij : τp r ij

(28)

Rheological properties of the material   of automaton i are determined by defining i a unified hardening curve σ eq ¼ Φ eieq (here eieq is the equivalent strain; definitions of average strain tensor eiαβ and equivalent strain eieq are analogous to that of average stress [20, 21]). Note that such a formulation corresponds to rate-independent plasticity.

2.2.3 Fracture Model and Criteria A fundamental feature of the MCA formalism is an inherent ability of automata to change surroundings (interacting neighbors). This allows considering MCA as an attractive numerical technique to be used for direct modeling multiple fracture accompanied by formation and mixing of a large number of fragments. Coupling (adhesion) of fragments can be included in the model as well. These abilities are taken into account by means of a change of the state of the pair of movable automata (“linked” pair $ “unlinked” pair, Fig. 4a). Potentialities of the presented approach to building many-body interaction between automata make it possible to apply various parametric “force” fracture criteria known in continuum mechanics (Huber-Mises-Hencky, Mohr-Coulomb, Drucker-Prager, and so on) within the MCA formalism. In MCA fracture is modeled by means of transition of a pair of automata from “linked” state to “unlinked” state with a possibility of further contact interaction of the automata (Fig. 4a). To apply a

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Fig. 4 Schematic of switching between linked (at the left) and unlinked (at the right) states of the pair of movable cellular automata i and j (a) and definition of the local coordinate system X0 Y0 associated with the current spatial orientation of the interacting pair i–j (b)

conventional fracture criterion as a criterion of pair bond breaking, one has to determine the local stress tensor at the area of interaction of the considered pair i–j (hereinafter referred to as σ ijα0 β0 ). For simplicity, the definition of this tensor is shown below for 2D case (extension to general 3D case is trivial). In the local coordinate system X0 Y0 (Fig. 4b), the components σ ijy0 y0 and σ ijx0 y0 for the pair i–j are numerically equal to the specific forces of central (σij) and tangential (τij) interaction of the automata (these forces are applied to the contact area Sij). The other components (σ ijx0 x0 and σ ijz0 z0 ) of local stress tensor in the coordinate system X0 Y0 are defined on the basis of linear interpolation of the corresponding values (σ ix0 x0 and σ xj0 x0 , σ iz0 z0 and σ zj0 z0 ) for automata i and j to the area of interaction:   8 < σ ij0 0 ¼ σ ix0 x0 qji þ σ j0 0 qij =r ij xx xx   , : σ ij0 0 ¼ σ i 0 0 q þ σ j0 0 q =r ij z z ji zz z z ij

(29)

where σ iα0 β0 and σ αj 0 β0 are the components of average stress tensor in the volume of automata i and j in the local coordinate system of the pair. Thus, defined components σ ijα0 β0 can be used to calculate the necessary tensor invariants which then can be used for calculating the current value of the fracture criterion applied. For example, unbinding condition for the pair i–j based on the fracture criterion of Drucker and Prager is as follows: σ ijeq 0:5ða þ 1Þ þ σ ijmean 1:5ða  1Þ ¼ σ c ,

(30)

where σc is the corresponding threshold value for the pair (strength of cohesion/adhesion), a = σ c/σ t is the ratio of compressive strength (σc) to tensile strength (σt) of the pair bond, and σ ijeq and σ ijmean are the corresponding invariants of the stress tensor σ ijα0 β0 . When using the explicit scheme of integration of motion equations, the value of time step Δt is limited above by a quantity associated with the time of sound

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propagation through the bulk of element. Normally the time step is equal to or less than a quarter of this limit. In such a situation, the conventional model of bond breaking during one time step Δt is an idealized condition because it implicitly implies that the spatial separation of atomic layers occurs uniformly over the whole surface of element interaction. Hence, the following approach could be suggested for more accurate description of the crack growth dynamics. It is assumed that breaking of a bond (linked!unlinked transition of the pair state) is a process distributed in time and space. To express it numerically, the dimensionless factor kijlink (0  kijlink  1) which has the meaning of the portion of linked part of the contact area Sij is used. In this case, the linked part of the contact area in the pair i-j is expressed as Sijlink ¼ Sij kijlink . Thus, the dynamics of bond breaking is determined by the dependence kijlink ðtÞ, where kijlink decreases from the initial value 1 (totally linked pair) to the final value 0 (totally unlinked pair). Depending on the automaton size and features of the internal structure of the material, the stable or unstable crack growth models can be applied to describe the pair bond breaking. In the first model (stable crack growth), the process of  fracture develops  in ij ij ij ij ij accordance with the predefined dependence dklink =deeq ¼ f eeq , klink , e_eq . . . < 0, where eijeq is the pair equivalent strain which can be calculated, for example, using components of the strain tensor eijα0 β0 defined by analogy with σ ijα0 β0 . In the simplest case, this dependence can be considered as a constant. In the second model (unstable crack growth), it is assumed that if the fracture criterion threshold is exceeded, then a crack  begins to grow  spontaneously according ij ij to the somehow specified law dklink =dt ¼ f t, klink . . . < 0, where t is the time. In

the simplest case, this dependence can be considered as a constant, which means that the crack advances through the pair interaction area with the constant velocity Vcrack. The value of Vcrack is a predefined (model) parameter, which reflects rheology of the interface material between interacting automata. In particular, for brittle materials, Vcrack could be close to Raleigh wave speed, while for ductile materials, its value should obviously be significantly smaller. Distinctive features of interaction of “unlinked” (i.e., contacting) elements i and j, among others, are (i) lack of resistance to tension (a pair is considered as interacting one if σij  0 only) and (ii) a limited magnitude of the force of tangential interaction. The maximum value of the tangential force (τij) allowed between “unlinked” elements is determined by the model of friction applied (e.g., Amonton’s law, model of Dieterich, and so on).

2.3

Coupling of the Continuum and Discrete Methods

The description of the presented methods demonstrates that in both of them equations of motion can be written for certain masses through stresses or forces acting on the masses. These features suggest that the methods can be combined quite correctly. To do this the simulated medium is considered as consisting of continuum and

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Coupling of Discrete and Continuum Approaches in Modeling the Behavior of. . .

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MCA domain Nodes and automata on the boundary of conjugation of two methods Finite-difference grid domain

a

b

Fig. 5 Variants of conjugation of the continuous and discrete domains: movable automaton size is less than the grid spacing (a); movable automaton size is equal to the grid spacing (b)

discrete regions. The latter are taken to have a boundary of conjugation that is assumed to belong to both regions. In this case, each node of the calculation grid lying on the boundary is associated with a certain automaton (an element of the MCA method). Let us consider the boundary between the regions described by different methods. In the general case, several cellular automata may be located between two neighboring grid nodes lying on the boundary (Fig. 5a). The automaton size should then be multiple of the grid spacing and not exceed it: d = D/n, where d is the automaton size, D is the grid spacing, and n is an integral number. For this case, the displacement of boundary nodes together with neighboring automata located in them takes place in the grid method. Displacement of automata located between two neighboring boundary nodes can be calculated using interpolation of displacements of appropriate grid nodes. In the simplest case, the automaton size coincides with the grid spacing. The automata associated with nodes are then neighboring, i.e., there are no additional automata between them (Fig. 5b). To describe adequately the combined behavior of the continuous and discrete regions, it is necessary to specify such conditions in the conjugation region that would provide a continuity of state parameters in transition through the boundary of region conjugation. To provide motion continuity on the boundary surface, two approaches are possible. In the first approach, for each region, proper boundary conditions are written, for example, the free surface conditions. Then, after a separate calculation of motion rate for boundary nodes and automata, they are adjusted using the momentum conservation equation: mV = maVa + mcVc. Here, mV corresponds to a superposed node-automaton and maVa and mcVc to the node and automaton, respectively. When the velocities and coordinates of nodes-automata are calculated and combined in this way, the other parameters may be computed. The second approach implies that matching of motion of the continuous and discrete regions is realized at the stage of calculating the boundary node velocity in the continuous region. In order to show the features of numerical realization of conditions in the zone of continuous and discrete region conjugation, write

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appropriate relationships for the equations of continuous region motion in the following form:
 III IV IV ð2φ x€ÞO ¼  σ Ixx yI24 þ σ IIxx yII31 þ σ III xx y42 þ σ xx y13 þ

III IV IV þ σ Ixy xI24 þ σ IIxy xII31 þ σ III xy x42 þ σ xy x13 ,   III IV IV ð2φ y€ÞO ¼ σ Iyy xI24 þ σ IIyy xII31 þ σ III yy x42 þ σ yy x13    III IV IV y þ σ y  σ Ixy yI24 þ σ IIxy yII31 þ σ III xy 42 xy 13 ,

(31)

where σ kxx , σ kyy , σ kxy are the stress components in corresponding cells; xkmn ¼ xkm  xkn , ykmn ¼ ykm  ykn , etc. The upper subscript denotes the cell number, and the lower subscript for coordinates denotes the node number (Fig. 1). The interpretation of state in terms of stresses and strains for the MCA region differs from the conventional interpretation for continuous approaches. In this connection for the continuous region, rewrite the difference equation of motion using the notion of force that can be applied to both regions. Whatever the interpretation and approach used – the finite-difference or finite-element one – when constructing the difference analogue of equations, their form and the meaning of components remain the same [17, 24–26]. The left side incorporates the product of mass and component of grid node acceleration, and the right side includes the component of resultant force applied to the node. X X m x€ ¼ Fix , m y€ ¼ Fiy , i¼1, N i¼1, N  i i i i Fx ¼ σ xx Δy þ σ xy Δxi ,   Fiy ¼ σ iyy Δxi  σ ixy Δyi :

(32)

For all internal nodes of the continuous region, the expression for the total force acting on the center of the stencil (Fig. 1) is written in the form: X


 III IV IV Fix ¼  σ Ixx yI24 þ σ IIxx yII31 þ σ III xx y42 þ σ xx y13 i¼I, IV   III IV IV þ σ Ixy xI24 þ σ IIxy xII31 þ σ III xy x42 þ σ xy x13 ,   X III IV IV Fiy ¼ σ Iyy xI24 þ σ IIyy xII31 þ σ III yy x42 þ σ yy x13 i¼I, IV   III IV IV  σ Ixy yI24 þ σ IIxy yII31 þ σ III xy y42 þ σ xy y13 :

(33)

For all boundary nodes, components for the fictitious cells are changed for forces acting on appropriate automata from the МСА region. The motion of such nodes is calculated based on Eq. 32, which includes forces obtained in the MCA region depending on the boundary surface geometry. After the displacement of the

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Fig. 6 Scheme of superposition of the continuous and discrete regions and position of grid nodes and automata in the finite-difference stencil

II

I

III

IV

automaton indissolubly related to the grid node, the corresponding action from the cell region is transferred to the internal MCA region. For example, in case if the boundary of region conjugation is horizontal as is shown in Fig. 6, where the continuous region is in the lower part of the figure and the discrete region in the upper one, the following relationships can be obtained:     III III III IV IV IV IV 0 Fix ¼ σ III y þ σ x y þ σ x þ σ xx 42 xy 42 xx 13 xy 13 þ Fx , i     X III III III IV IV IV IV 0 Fiy ¼ σ III x  σ y x  σ y þ σ yy 42 xy 42 yy 13 xy 13 þ Fy :

X

(34)

i

where F0x , F0y are the components of the force vector acting from the МСА region on the automaton lying on the boundary. Note that since such an approach does not consider the interaction between an arbitrary grid node and automata, when simulating several bodies or medium regions by the grid method, all their boundaries should be “covered” with automata. In case if the automaton size and grid spacing are not equal, the time integration steps turn to be almost identical for both methods. When the automaton size is less than the grid spacing, the node masses will be different from the automaton masses, and the integration time step in the МСА method will be less than the corresponding value for the grid. Hence, generally speaking, the time step should be correlated, and the most minimum value recommended by each method should be used. Schematically, the combined calculation algorithm is thus the following: 1. In the continuous region, a set of Eqs. 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is solved, the stress-strain state is calculated, and velocities and coordinates are determined. 2. Before the velocities of nodes lying on the boundary are calculated, the subprogram realizing the MCA method is called. The subprogram uses the coordinates and velocities of the superposed nodes-automata from the previous step as well as the integration step. 3. Using the data obtained, an integration step of MCA is taken (or several steps for small automata). As a result, it calculates new positions and velocities of

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all automata including the forces acting on the automata superposed with the grid nodes. 4. New data about the boundary automata (nodes) are returned to the grid method. After that, the cooperative motion of the boundary nodes-automata is calculated, and their new coordinates are determined in the method. 5. A new integration time step is correlated.

2.4

Testing the Coupling Procedure

The used methods and programs of their numerical realization have been often applied to describe dynamic processes including elastic wave propagation [10–12, 27–29]. To verify the possibilities of combining the above-described approaches in the framework of a unified method and testing of the developed algorithms, usually problems on the elastic wave propagation in a region with the free surface are solved. One part of the simulated region was considered as continuous and the other as discrete. Their mechanical characteristics were assumed to be identical. Since an almost homogeneous medium was specified in this way, the boundary of conjugation should not behave as an interface between different media. Below it is considered the case when there is only one linear boundary of conjugation of two methods (Fig. 6a). The automaton size coincided with the grid cell sizes. Matching of motion of nodes and automata along the boundary was carried out at the stage of velocity calculation with regard to the forces acting on the automata superposed with these nodes. Variants differing by the types of automaton packing in the discrete region were studied. In the first case, a close automaton packing was used (Fig. 5) and in the second case a square one (Fig. 6). At the first stage, a problem of the propagation of an elastic plane wave with the front parallel to the boundary of region conjugation (Fig. 7a) was solved. The wave was excited from the surface of the continuous (GRID) region. The boundary conditions on the lateral faces imitated an infinite extension along the Y-axis. Figure 7b shows that wave passage through the boundary was not accompanied by the reflected wave generation. No distortion of the pulse shape after its propagation to the discrete region is seen as well. A similar behavior was also observed after the wave was reflected from the fixed back surface of the discrete (МСА) region and passed through the boundary from the discrete region into the continuous one. The calculations have thus shown that if the identical mechanical characteristics of the medium are used in both methods, the plane wave passes through the boundary of conjugation without any distortions. This testifies that the algorithm of a combination of two methods provides the full momentum transfer in the absence of shear deformations. At the second stage, a more complex problem, namely, the problem of the generation and propagation of waves of all types in a medium with the free surface, was solved. For this purpose, a part of the elastic half-space surface was subjected to the short-time action of local vertical load. Two cases were considered: (i) when the source was located symmetrically (directly on the boundary between the GRID and

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60 50

10 5

40

0

30

MCA

-5 -10

y

0

2

4

6

8

10

x

10

м

5

x,с

MCA

U, м/с

GRID

30

20

10

0

0

GRID

-5

-10 0

2

4

6

t, мс

a

8

10

b

Fig. 7 Scheme of the calculation area with loading conditions (a) and velocity changing in successive specimen points at plane wave propagation (b) in the specimen consisting of two parts

Fig. 8 Scheme of the calculation area with loading conditions (a) and velocity changing in successive specimen points at plane wave propagation (b) in the specimen consisting of two parts

MCA regions) and (ii) when the source was located non-symmetrically (dislocated relative to the line of conjugation of the GRID and MCA regions). In all cases of source location relative to the conjugation boundary, the details of longitudinal, transverse, and Rayleigh wave propagation, as well as the symmetry of the displacement velocity field, were analyzed. Testing was carried out for both square and close (hexagonal) packing of automata in the discrete region. As a result of the action, longitudinal (tension-compression) and transverse (shear) waves, P and S, propagating with different velocities are generated in the medium at a distance from the source (Figs. 8 and 9). The longitudinal and transverse 4 Kþ μ 2 3 and V 2 ¼ μ, respectively. Recall that wave velocities can be calculated as V ¼ P

ρ

S

ρ

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Fig. 9 Wave fields for the non-symmetrical location of the source

the wave types are distinguished not by their velocities but, first of all, by the direction of particle motion in the wave front and, correspondingly, by the deformation type. In the longitudinal wave front, particle motion occurs in the direction of the ray, i.e., in the direction of wave propagation. In the transverse wave front, particles move perpendicular to the direction of wave propagation. The presence of the free surface brings about the appearance of the so-called conical and surface waves. Figures 8 and 9 clearly demonstrate that in the considered cases, the conical wave shows up only in the region of the longitudinal wave and free surface interaction. It joins the longitudinal and

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transverse wave fronts, and its own front extends from the place of longitudinal wave arrival on the surface tangentially to the transverse wave front. A difference in the displacement direction causes vortex motion of particles between the conical and transverse wave fronts. Close to the free surface and slightly behind the transverse wave, the surface Rayleigh wave propagates, which has an elliptical polarization and rapidly decays with depth. It is known that in a passing of the wave through the surface of the interface between media with different mechanical characteristics or in case if it is a surface of displacement discontinuity (e.g., as is shown in [27]), a series of reflected and refracted waves is generated. In all cases considered, the calculation results have not revealed considerable wave front distortion on the boundary of conjugation (Figs. 8 and 9). Only a slight difference in the shape of pulses is observed, and no visible secondary (reflected, refracted, and conical) waves are generated. It should be noted that the effect of automaton packing in the МСА region manifests itself in a little change of the pulse shape. The presented numerical results on the elastic wave propagation in a combined region using the continuous and discrete methods allow a conclusion about the possibility of the combined application of the mentioned methods to the description of the elastic behavior of complex media. The prospects of the developed approach and algorithms of its realization are substantiated by the test calculation results, which have shown that even for small and complex elastic displacements in a region with the free surface, the proposed approach does not lead to any artificial or induced effects. It should be especially emphasized that the symmetry of automaton packing in the МСА region does not exert a strong influence on the generation and propagation of waves of different types observed in the work. The obtained results have shown that the proposed approach holds much promise for the solution of problems related to the numerical investigation of the behavior of complex media with strongly varying physical-mechanical properties. This is especially urgent for many problems of geomechanics, soil and rock mechanics. Below the two examples of using the presented coupled approach are presented. The first example concerns simulation of target penetration by long rod at the macroscale. The second one deals with simulation of sliding friction at the scale of contact patch (mesoscale).

3

Simulation of Target Penetration on the Basis of a Combined Discrete-Continuum Approach

3.1

Introduction

It is no doubt that the penetration of target plates by high-velocity impact is a quite complicated physical phenomenon. One of the main reasons for the complexity of this process is that it simultaneously combines many different physical aspects

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such as plasticity, fracture, heating and heat transfer, plastification of target material due to heating, diffusion of penetrator and target material, etc. For the higher velocities (above 2.5–3.0 km/s), there exists a so-called hydrodynamic model that predicts the penetration depth of the impacting rod fairly well. This model is based on the assumption that at very high velocity, the spherical part of stress tensor (or hydrodynamic pressure) prevails over components of stress deviator, and, hence, the transversal stiffness of both the rod and target material can be neglected. At the same time, for the intermediate velocity range, there are numerous phenomenological and dislocation models trying to describe the material response under dynamic loading. These models exhibit different properties concerning their precision at different strain rates, computational expenses required, and to which extent underlying physical mechanisms of plasticity are accounted for. However, there is still a demand to build a model with a reasonably small number of theory constants, which can adequately describe the gradual loosing of transverse stiffness by the material in the whole velocity range. It is very important for the simulation of penetration process because the whole range of strain rates from zero to maximum at the channel occurs in the specimen at the same time. Computer simulation of high-rate deformation processes provides the invaluable possibility to have the insight of process dynamics with unlimited temporal resolution, which is impossible in the experiment; it considerably reduces the number of experiments as well as expenses allowing the extrapolation of experimental data. The existing grid-based computational codes using finite-element or finite-difference methods give numerical results [30, 31] that match reasonably well with the experimental data. However, Lagrange grid codes used for simulation of solids have some well-known problems like high cell distortions, which demand complicated remeshing algorithms, resulting in accuracy losses. Many problems of grid codes have been overcome in meshless particle methods such as smoothed-particle hydrodynamics (SPH) [32], generalized particle algorithm (GPA) [33], etc. These methods, especially SPH, became very popular during the last decade. One of the modern discrete methods used for simulating various aspects of the mechanical behavior of heterogeneous materials and structures is the movable cellular automaton method. Unlike SPH, which is a special approach for discretization of differential equations of plastic flow, MCA is formulated in terms of particle interaction, each particle representing a small but finite volume of material. The main advantage of the MCA method is its ability to naturally describe fracture of material up to its fragmentation and further mass mixing of fragments. The method has proved itself to be a very effective and prominent instrument for simulation of materials and structures under rather small elasticplastic deformations [8–13, 19–21]. To enable a computational study of high-rate processes such as shock loading, high-velocity impact, etc., the MCA method has been improved by several modifications [34], description of which is included below.

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Coupling of Discrete and Continuum Approaches in Modeling the Behavior of. . .

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Modification of the MCA Method for Simulation of High-Rate Processes

As shown above, MCA formulation explicitly accounts for the response caused ! by the volumetric changes. The volume-dependent force F iΩ acts due to the pressure from all the neighboring automata (15). The automaton pressure is determined by the change in the automation volume. In the simplest (linear) case, this dependence takes the form: Pj ¼ K j e j ,

(35)  where ej ¼ Ωj  Ω0j =Ω0j is the bulk strain, Ω0j is the initial (equilibrium) volume of 

automaton j, Ωj is the current volume, and Kj is the bulk modulus. The change in the automaton volume in a time Δt corresponds to computation based on the corresponding changes in the distances Δqij from the center of this automaton to the points of its contact with neighbors, i.e.: Ω ΔΩ ¼

j

Nj P

Δqjk

k¼0

j

Nj P

,

(36)

Δqjk =N j

k¼0

where N j is the number of neighbors of the automaton j. Attempts to simulate high-rate processes such as shock loading, high-velocity impact, etc. using the linear equation for pressure (35) have resulted in incorrect behavior due to large automaton overlapping. It is well known both from experimental data and from the theory of interatomic interactions that response of the material is linear just only in the region of small deformations near equilibrium. When compression gets higher, interaction becomes essentially nonlinear because of nonlinearity of interatomic potentials and, in the case of shock loading, because of adiabatic heating of the material. To avoid nonphysical automaton, overlapping pressure in (35) must grow much faster than the linear dependence of bulk strain. It is well known that experimental data on thermodynamic properties of materials is often described by the shock adiabat, which in wide range of pressure can be represented by the linear relation [35]: D ¼ c0 þ bU:

(37)

Here D is the shock-wave velocity and U is the mass velocity of the substance beyond the shock-wave front. Material constants c0 and b have been measured for a wide range of materials and can be easily found in the literature. From (37) and from the Hugoniot relations for conservation of mass and momentum

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e ¼ U=D, P ¼ ρDU,

(38)

the following relation can be derived: P ¼ ρc20

e ð1 þ beÞ

2

¼K

e ð1 þ beÞ2

:

(39)

It is the simplest nonlinear dependence of the pressure on the bulk strain. To use it in the MCA method, it is natural to redefine the bulk elastic modulus K in (35) as: K0 ¼

dP 1  be ¼K : de ð1 þ beÞ3

(40)

Thus, as far as parameter c0 is assumed to be defined by the quasistatic bulk modulus K ¼ ρc20 , the introduced nonlinear equation of state requires only one additional parameter b that determines the inclination of D-U diagram. It can be found from the analysis of experimental shock adiabats or directly from the literature. It is no doubt that for more accurate simulation of high-rate deformations, the introduction of nonlinear response function is not the only improvement required. It is also necessary to account for strain rate influence on the stress-strain diagram of the material. Heating due to adiabatic compression, which is now neglected, can have an effect on mechanical properties as well. However, the introduction of nonlinear response function under high bulk compression is the most important step to be done. Note that the improvement discussed is also useful for simulation of quasistatic processes where high bulk compression occurs, for example, deformation under constrained boundary conditions, which is a characteristic for processes in geological media. The combined discrete-continuum approach presented above allows uniting advantages provided by the grid and MCA methods. The MCA method is more suitable for the description of intensive damage generation and mass mixing, while the grid method is several times faster than the particle method. So, it is natural to simulate the parts of the structure, where deformations are small with the grid method to shorten calculation time and use the particle method in critical zones to increase space resolution. This approach makes an easier calculation of full-size objects; without that, it would be necessary to simulate only fragments of the structure or increase the discrete element size. For simulation of penetration process, the possibility to consider full-size object is quite important because boundary conditions of the armor plate and properties of the fixers can have a strong influence on the overall penetration pattern. It is also important that the alignment of the grid with particles must be physically correct.

3.3

Computation Results

As an illustration of applicability of the combined MCA and grid method, the results of simulation of steel plate penetration by long tungsten rod under normal impact in

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plane strain statement are presented. Plate thickness was varied from 15 to 40 mm with the step of 5 mm. The dimensions of the impacting rod were 10  100 mm, and the impacting velocity was 1520 m/s. Data on task geometry and loading conditions were taken to correspond to calculations described in the work [34]. The width of the area filled with particles is 80 mm; according to the calculation results, such width is enough because in this case, the damaged zones don’t reach the interface boundary between grid and particles. Unlike [34], where a special type of viscous boundary conditions is applied to the left and right side of the plate, here the plate ends are fixed. Figure 10 shows initial, intermediate, and final states of the simulated structure. One can see the intensive “mushrooming” of the impacting rod and formation of lips near the entrance of the penetration channel. The deformation and damage distribution presented here is not absolutely symmetric, because of stochastic distribution of strength properties of automata, which is added for more realistic fracture pattern. Table 1 summarizes data on residual rod length and velocity after penetration in comparison with calculations from [34] and other sources [30, 36]. The comparison indicates that rod deceleration slightly decreased with regard to [34]. A possible reason for that is the modified boundary conditions. Thus, the MCA method modified for the description of high-rate deformation processes in combination with the continuum grid method is applied for simulation of thin plate penetration. The calculated dependences of residual velocity and erosion of the rod as functions of target plate thickness are compared with other sources. It is shown that results of the coupled discrete-continuum approach are in good agreement with data obtained elsewhere. Overall penetration pattern seems quite realistic, and there is no spurious behavior on the boundary between grid and particles. It can be concluded that the new approach is acceptable for simulation of penetration.

4

Modeling Sliding Friction Using Discrete-Continuum Approach

4.1

Feasibility of Spectral Analysis of Elastic Waves for Identification of the Processes in Contact Patches in Friction

The acoustic vibrations generated in friction carry information on differently scaled complex processes occurring in surface layers of contacting solids. Sources of acoustic waves in friction are deemed to be elastic interactions of local contact zones, plastic deformation, and fracture in interaction regions of micro-irregularities, formation and separation of wear particles, etc. Thus, the spectrum of acoustic vibrations is much diversified and is difficult to analyze. The advantage of the used method of computer simulation is the possibility to reveal and analyze the contributions of individual factors to the generation of elastic waves and to compare them with peaks in acoustic spectra.

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Fig. 10 Rod and target plate structure at 0, 20, and 40 μs

Evidently, the most important processes representative of the mechanisms of sliding friction occur in actual contact patches. Therefore, of primary importance in the developed approach is to analyze the processes occurring in these regions.

4.2

Model of Mechanical Interaction in a Contact Patch

Sliding friction in an actual contact patch was modeled by the method proposed in [37, 38]. Figure 11a shows a schematic of the friction zone. The actual contact

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Table 1 Relative decrease of rod velocity and length versus plate thickness

15 20

MCA þ Grid ΔU/U0 0.06 0.07

Δl/l0 0.25 0.26

25

0.07

0.28

30 35 40

0.09 0.11 0.12

0.29 0.32 0.41

Height, mm

a

MCA ΔU/U0 0.04 0.05 0.06 0.05a 0.02b 0.08 0.10 0.12

Δl/l0 0.20 0.21 0.23 0.24a 0.19b 0.26 0.30 0.39

Experimental data from [36] Computer simulation results from [30]

b

Fig. 11 Schematic of the friction zone (a) and fragment of the central region of the model specimen (b)

regions that make up a few percent of the nominal surface of interacting solids are marked by dotted lines. According to [37, 39], an efficient tool for modeling of contact interaction is the discrete-continuum approach in which a narrow contact zone of interacting solids is described by the discrete method of movable cellular automata and the rest of the material, which experiences elastic deformation, by numerical methods of continuum mechanics. With this approach, we modeled part of the actual contact patch between the vertical lines shown in Fig. 11a. Taking into account the characteristic scale on which the mechanisms responsible for wear and friction develop in surface layers of contacting steel specimens, the computational domain was chosen to be 450 nm along the horizontal and 500 nm along the vertical. The diameter of movable cellular automata was 2.5 nm and the grid step was 10 nm. A structural fragment of the computational domain is shown in Fig. 11b. In the figure, region I was modeled by the discrete movable cellular automaton method and regions II by the finite-difference method through solving dynamic equations of solid mechanics. For decreasing the vertical dimensions of the figure, only part of the difference grid is shown. On the outer surfaces of the blocks (the upper and lower

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Table 2 Parameters of the model material Density, ρ, kg/m3 Young’s modulus E, GPa Poisson’s ratio ν Elasticity limit σy1, MPa Yield strength σy2, MPa Strength σs, MPa Yield strain εy2 Ultimate (rupture) strain εs

7800 206 0.3 510 800 920 0.04 0.112

σ σs σy2 σy1

εy1

εy2

εs

ε

Schematic of the response function of the model material

Fig. 12 Examples of the initial friction pairs (the region shown is that modeled by movable cellular automata)

surfaces of the upper and lower blocks, respectively), a horizontal velocity V increasing gradually with time t from 0 to Vm = 10 m/s was specified. At the same time, the pressure P increased to a maximum value Pm on the upper surface. The lower surface was fixed along the axis Y. The periodic boundary conditions used on the lateral faces along the horizontal ensured specific repetition of the model arrangement such that an automaton leaving the model arrangement, e.g., from the side of the right boundary, passed instantaneously to the left of the arrangement and arrived in the system from the left boundary. The parameters of the model material are presented in Table 2. The simulation took into account the possibility of both the disruption of the existing bonds between the automata according to Mises criterion and the formation of new ones according to relation σ ij < σ link ij ,

(41)

where σ link ij is the absolute value of the ultimate pressure. This allows simulation of adhesion (micro-welding) in contact patches. The initial roughness of the model surfaces was specified in an explicit form (Fig. 12).

4.3

Elastic Waves Generated by the Model Contact Patch in Friction

In the simulation, elastic waves were recorded using a gauge in the form of a witness automaton whose position is indicated by a black circle in Fig. 11b. The

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characteristics of the automaton to be analyzed were the velocity components VX and VY along the axes X and Y, respectively, the stress intensity I, and the pressure P. In the Fourier transform, the first 50,000 records before the establishment of steadystate loading conditions were excluded from the sampling. At this stage, the lapping of the contacting surfaces occurred with attendant considerable noise recorded by the gauge. Characteristic Fourier spectra of the recorded quantities for the specimens shown in Fig. 12 are presented in Fig. 13a. Figure 13b shows the Fourier transform of the acoustic signal produced in experiments [38]. Comparison of the data in Fig. 13 allows the conclusion that the experimental spectrum of the acoustic signal corresponds qualitatively to the pressure spectrum obtained in the model calculations. Note that in the experiments, a microphone fixes acoustic waves in the air that are actually pressure fluctuations (longitudinal waves). In quantitative terms, these spectra correspond to different frequency bands. In the experiment, the band is up to 20 kHz, whereas in the simulation, it is up to 500 kHz. The so high calculated values are due to the small size of the model contact patch. Note that the qualitative agreement observed in the results is indicative of possible similarity of the processes occurring on different scales and responsible for the generation of elastic waves. Indirect evidence for this fact is the much used and experimentally proven the concept of self-similarity (fractality) of rough surfaces of solids on different scales [40]. For a detailed analysis of the obtained spectra, let us first determine the eigenfrequencies of the model system. For the flat specimen, we actually have two bands with edges of which one moves with constant velocity and the other is free. The frequency of the lower band fixed along the axis Y can be determined by analogy with a beam with a fixed end by the formula pk = v(k  0.5)/2 l, where v is the wave velocity and l is the band height [41]. The transverse wave velocity vS = 3.19 km/s and the height l = 2.578107 m. Hence, the first harmonic corresponds to 3.09 GHz, the second harmonic to 9.28 GHz, and the third harmonic to 15.47 GHz. It is these harmonics that are visible in the spectrum of the horizontal velocity component. Longitudinal elastic waves also propagate in the specimen. The longitudinal wave velocity vP is 5.81 km/s. Hence, for these waves, the first harmonic corresponds 5.63 GHz, the second harmonic to 16.9 GHz, and the third harmonic to 28.17 GHz. At these frequencies, peaks of the spectrum of the vertical velocity component are found. On the outer surface of the upper block, we specified only the pressure and this surface has no rigid fixation along the axis Y; therefore, the eigenfrequencies for the band is determined by analogy with a free beam using the formula pk = vk/2 l [41]. Hence, for longitudinal waves, the first eigenfrequency is 11.27 GHz, the second eigenfrequency is 22.54 GHz, and the third eigenfrequency is 33.81 GHz. The higher-order harmonics have small amplitude and are poorly identified against the noise background. Note that the peaks on the presented spectra are not fully coincident with the calculated ones since the interacting specimens are worn (a change occurs in the quasi-liquid layer thickness [42], which is a region of automata linked with none of the blocks) and hence their geometric dimensions are changed.

Fig. 13 Fourier transforms of calculation (a) and experimental (b) data [38]

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As the specimen dimensions are changed, the eigenfrequencies vary in an appropriate manner; this is confirmed by the calculations made in [37]. The change in the size of the automata produces a change only in those peaks that correspond to artificial roughness (automaton size). The only peak (smeared) that remains invariant under changes of the specimen dimensions and automaton size corresponds to about 2 GHz and is present in the spectra of pressure, stress intensity, and vertical velocity component. Further analysis shows that this frequency characterizes the mean time for which linked pairs of automata exist in the quasi-liquid layer near the interacting surfaces. The frequency can be considered as a characteristic frequency of microscale motion in the intermittent mode. The frequency varies as the criterion of recovery of the link between unlinked automata is varied, and hence it is determined by adhesion interaction in contact patches. In general, the spectra shown in Fig. 13 give some average characteristics over the entire sampling of recorded signals. The time variation of frequencies, as a rule, is analyzed by methods like the windowed Fourier transform and wavelet transform [43]. In the present work, we used the wavelet transform (last wave signal processing package [44]). The wavelets widely used were Mexican hat and Morlet functions [43]. Let us analyze the results of the wavelet transform for the data obtained in the simulation of friction (Fig. 14). In the wavelet images, the ordinate is represented in dimensionless quantities, which are termed a scale, inversely proportional to frequency. Therefore, the range of low frequencies is at the top and that of high frequencies is at the bottom of the images. The abscissa corresponds to time (the number of a discrete signal record). The data presented in Fig. 14 allows the conclusion that the elastic waves generated in friction are frequency- and amplitudemodulated.

Fig. 14 Wavelet transforms of the data for the specimen shown in Fig. 12b

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Actually, the eigenfrequencies of the system under consideration depend on the height of the blocks, and this height varies stochastically in friction (even under steady-state conditions) due to wear of the rubbing surfaces and sticking of particles to them (adhesion processes). In the case in question, this is rather clearly defined because of the small block size such that small variations in the quasi-liquid layer thickness considerably affect the eigenfrequencies. As the specimen dimensions increase, this effect becomes less pronounced. The amplitude of the produced elastic waves is associated with the rate and amplitude of change of the interaction force of the upper and lower blocks. As noted above, the friction coefficient, which is determined by the relation of the resistance and pressing forces of rubbing solids, is changing constantly even under steady-state friction conditions. Thus, both the periodically varying friction coefficient and the amplitude modulation of the recorded signals under the specified loading conditions are governed by the same properties of the tribological system under study, namely, by the elastic, strength, and adhesion characteristics and by the surface profile in the contact patch. The foregoing considerations make it possible to identify the high-frequency spectral component as that corresponding to the eigenfrequencies of the system. Analysis of the low-frequency spectral range shows that the position and the height of certain peaks in the range vary greatly with time (Fig. 15). Further analysis of the simulation results shows that the surface profile separating the quasi-liquid layer and the solid part of the specimens change in friction. This makes it possible to suppose that the presence of the peaks may owe to the characteristic parameters of the current roughness of the interacting surfaces. The current surface profile was determined using a modified version of the algorithm proposed in [39]. The algorithm was modified such that the highfrequency components associated with the discrete character of the model were “cut off.” Figure 16 shows the profile of the lower surface of the model friction pair at a certain point in time. Figure 17 shows the functions plotted by the above algorithm for the surface profile of the friction pairs at different points in time. It can be seen in the figure that with time the interaction causes an increase in friction layer thickness and the surfaces in the contact patch are smoothed. A natural question arises of whether the change in the surface roughness parameters in friction can be revealed in spectral analysis. The periodic character of the profiles of the interacting surfaces must show itself up in the frequency spectrum of the elastic waves produced in relative displacements of the solids. Let us introduce an expected signal frequency f (e.g., for a certain characteristic of elastic waves) associated with a distance passed by the solids in relative displacement and equal to a roughness period l. This frequency can be estimated as f = v/l, where v is the relative displacement velocity of surfaces (for the system under study, v = 2Vm). The Fourier transform of the surface profile functions in terms of the spatial variable and expression of the result in terms of v/l can give a spectrum of expected frequencies due to the current surface roughness. Figure 18 shows spectra corresponding to the surface profiles presented in Fig. 17b.

Fig. 15 Fourier spectra of the interaction for the specimen shown in Fig. 12b at different time intervals (in μs): 0.025–1.33 (I), 1.33–2.64 (II), and 2.64–3.95

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Fig. 16 Current profile of the lower surface of the model friction pair

Fig. 17 Surface profiles of the upper and lower blocks at different moments in time t; (a) and (b) correspond to the specimens shown in Fig. 12

The most intense component in the spectra (Fig. 18) is the dominant periodic component of the surface profile. This component corresponds to a signal of frequency 70 MHz due to a characteristic period of the initial surface roughness of ~280 nm (Figs. 12b and 17b at t = 0 μs). In the course of friction with attendant rearrangement of the profile and smoothening of the initial surfaces, the prevailing contribution of the signal of frequency 70 MHz to the spectrum is retained over the entire time interval under study. As this takes place, the peak height can vary and can even exceed in some cases the initial height at the stage of lapping, as is the case at t = 0.3 μs in Fig. 18. However, on completion of lapping of the interacting surfaces, the contribution of the dominant periodic component of the surface profile decreases (t = 4.5 μs in Fig. 18), and the contribution of high-frequency components levels off and becomes moderate, compared to the dominant one. This suggests that steadystate friction conditions are attained and a surface profile whose spectral distribution changes little with time is formed.

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Fig. 18 Signal spectra corresponding to the surface profile of the specimen shown in Fig. 12 at different moments in time t

Turning back to the spectra of elastic waves shown in Fig. 15, we can conclude that the dominant frequency is most pronounced for pressure and stress intensity. Thus, the obtained results suggest that knowing the initial roughness and corresponding expected frequencies, it is possible to analyze the dynamics of variation in roughness using acoustic spectra.

5

Future Directions and Open Problems

This chapter presents coupling of movable cellular automaton method and finitedifference method for 2D problems. As mentioned above, coupling the MCA method with explicit finite-element method does not face any additional troubles. Peculiarities of the coupling procedure in 3D are also out of the discussion. They are connected with close packing of automata and corresponding positions of the FEM grid nodes at the coupling interface plane. Fortunately, the finite element can have practically arbitrary shape, and hence one can put special elements in the coupling plane to place their nodes at the positions of conjugated automata. The second solution consists in placing the automata at the edges or faces of the finite element and use interpolation as was proposed herein for automaton size smaller than grid size.

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In the presented coupling procedure, the boundary between the grid and particle regions can be only linear (a plane in 3D) and have to be assigned before computation. The main advantage of using coupled grid and particle approach is the increased computation speed with the same accuracy. One promising development of the described coupling approach consists in providing an ability to replace continuum grid cell by particle directly during simulation in the case of severe distortion of this cell. This would additionally increase calculation speed and make it possible to simulate a wider range of problems with more complicated and unknown a priori distribution of regions with small and intensive deformations. The next direction of the future study and development is the use of different discrete methods. Now, most popular of them are discrete element method (DEM) and smoothed-particle hydrodynamics (SPH). An example of coupling FEM and DEM for dynamic analysis of geomechanics problems can be found in [45]. A short overview of the grid and meshless methods used for high-velocity impact computations is presented in [46], where some algorithms for conversion of finite element into particles are also used.

References 1. Darrigol O. Between hydrodynamics and elasticity theory: the first five births of the Navierstokes equation. Arch Hist Exact Sci. 2002;56:95–150. 2. Cundall PA, Strack ODL. A discrete numerical model for granular assemblies. Geotechnique. 1979;29(1):47–65. 3. Herrmann HJ. Simulating granular media on the computer. In: Garrido PL, Marro J, editors. 3rd Granada lectures in computational physics. Heidelberg: Springer; 1995. 4. Luding S. Granular materials under vibration: simulation of rotating spheres. Phys Rev E. 1995;52(4):4442–57. 5. Poschel T. Granular material flowing down an inclined chute: a molecular dynamics simulation. J Phys II. 1993;3(1):27–40. 6. Herrmann HJ, Luding S. Modeling granular media on the computer. Contin Mech Thermodyn. 1998;10(4):189–231. 7. Psakhie SG, Horie Y, Korostelev SY, Smolin AY, Dmitriev AI, Shilko EV, Alekseev SV. Method of movable cellular automata as a tool for simulation within the framework of physical mesomechanics. Russ Phys J. 1995;38(11):1157–68. 8. Psakhie SG, Ostermeyer GP, Dmitriev AI, Shilko EV, Smolin AY, Korostelev SY. Movable cellular automata method as a new trend in computational mechanics. I. Theoretical description. Phys Mesomech. 2000;3(2):5–12. 9. Psakhie SG, Chertov MA, Shilko EV. Interpretation of the parameters of the method of movable cellular automata on the basis of continuum description. Phys Mesomech. 2000;3(3):89–92. 10. Psakhie SG, Horie Y, Ostermeyer GP, Korostelev SY, Smolin AY, Shilko EV, Dmitriev AI, Blatnik S, Spegel M, Zavsek S. Movable cellular automata method for simulating materials with mesostructure. Theor Appl Fract Mech. 2001;37(1–3):311–34. 11. Psakhie SG, Ruzhich VV, Smekalin OP, Shilko EV. Response of the geological media to dynamic loading. Phys Mesomech. 2001;4(1):63–6. 12. Goldin SV, Psakhie SG, Dmitriev AI, Yushin VI. Structure rearrangement and “lifting” force phenomenon in granular soil under dynamic loading. Phys Mesomech. 2001;4(3):91–7. 13. Smolin AY, Shilko EV, Astafurov SV, Konovalenko IS, Buyakova SP, Psakhie SG. Modeling mechanical behaviors of composites with various ratios of matrix–inclusion properties using movable cellular automaton method. Def Technol. 2015;11(1):18–34.

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14. Psakhie SG, Smolin AY, YuP S, Makarov PV, Shilko EV, Chertov MA, Evtushenko EP. Simulation of behavior of complex media on the basis of a discrete-continuous approach. Phys Mesomech. 2003;6(5–6):47–56. 15. Psakhie SG, Smolin AY, Stefanov YP, Makarov PV, Chertov MA. Modeling the behavior of complex media by jointly using discrete and continuum approaches. Tech Phys Lett. 2004;30(9):712–4. 16. Wilkins ML. Calculation of elastic-plastic flow. In: Alder B, Fernbach S, Rotenberg M, editors. Methods of computational physics, Fundamental methods in hydrodynamics, vol. 3. New York: Academic Press; 1964. 17. Wilkins ML. Computer simulation of dynamic phenomena. Berlin: Springer; 1999. 18. Daw MS, Foiles SM, Baskes MI. The embedded-atom method: a review of theory and applications. Materials Science Reports. 1993;9:251–310. 19. Psakhie SG, Shilko EV, Smolin AY, Dimaki AV, Dmitriev AI, Konovalenko IS, Astafurov SV, Zavshek S. Approach to simulation of deformation and fracture of hierarchically organized heterogeneous media, including contrast media. Phys Mesomech. 2011;14(5–6):224–48. 20. Psakhie SG, Shilko EV, Grigoriev AS, Astafurov SV, Dimaki AV, Smolin AY. A mathematical model of particle-particle interaction for discrete element based modeling of deformation and fracture of heterogeneous elastic-plastic materials. Eng Fract Mech. 2014;130:96–115. 21. Shilko EV, Psakhie SG, Schmauder S, Popov VL, Astafurov SV, Smolin AY. Overcoming the limitations of distinct element method for multiscale modeling of materials with multimodal internal structure. Comput Mater Sci. 2015;102:267–85. 22. Potyondy DO, Cundall PA. A bonded-particle model for rock. Int J Rock Mech Min Sci. 2004;41(8):1329–64. 23. Smolin AY, Roman NV, Dobrynin SA, Psakhie SG. On rotation in the movable cellular automaton method. Phys Mesomech. 2009;12(3–4):124–9. 24. Noh VF. CEL: a time-dependent two-space-dimensional coupled Eulerian-Lagrangian code. In: Alder B, Fernbach S, Rotenberg M, editors. Methods of computational physics, Fundamental methods in hydrodynamics, vol. 3. New York: Academic Press; 1964. 25. Johnson GR. Dynamic response of axisymmetric solids subjected to impact and spin. AIAA J. 1979;17(9):975–9. 26. Gulidov AI, Fomin VM, Shabalin II. Mathematical simulation of fracture in impact problems with formation of fragments. International Journal of Fracture. 1999;100(2):183–96. 27. Stefanov YP. Wave dynamics of cracks and multiple contact surface interaction. Theor Appl Fract Mech. 2000;34(2):101–8. 28. Stefanov YP, Bakeev RA, Rebetsky YL, Kontorovich VA. Structure and formation stages of a fault zone in a geomedium layer in strike-slip displacement of the basement. Phys Mesomech. 2014;17(3):204–15. 29. Stefanov Y, Bakeev RA. Deformation and fracture structures in strike-slip faulting. Eng Fract Mech. 2014;129:102–11. 30. Johnson GR Stryk RA, Holmquist TR, Souka OA. Recent EPIC code developments for high velocity impact: 3D element arrangements and 2D fragment distributions. International Journal of Impact Engineering. 1990;10:281–94. 31. Beissel SR, Johnson GR. Large-deformation triangular and tetrahedral element formulations for unstructured meshes. Comput Methods Appl Mech Eng. 2000;187:469–82. 32. Johnson GR, Stryk RA, Beissel SR. SPH for high velocity impact computations. Comput Methods Appl Mech Eng. 1996;139:347–73. 33. Johnson GR, Beissel SR, Stryk RA. An improved generalized particle algorithm that includes boundaries and interfaces. Int J Numer Methods Eng. 2002;53:875–904. 34. Chertov MA, Smolin AY, Shilko EV, Psakhie SG. Simulation of complex plane targets penetration by long rod using MCA method. In: Sih GC, Kermanidis TB, Pantelakis SG, editors. Multiscaling in applied science and emerging technology, fundamentals and applications in mesomechanics: proceedings of the sixth international conference for mesomechanics. Patras; 2004. 35. Gust WH. High impact deformation of metal cylinders at elevated temperatures. J Appl Phys. 1982;53(5):3566–75.

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36. Fomin VM, Gulidov AI, Sapozhnikov GA, et al. High-velocity interaction of solids. Novosibirsk: Izdatelstvo SO RAN; 1999. (in Russian) 37. Smolin AY, Konovalenko IS, Psakhie SG. Identification of elastic waves generated in the contact zone of a friction couple. Tech Phys Lett. 2007;33(7):600–3. 38. Psakhie SG, Popov VL, Shilko EV, Smolin AY, Dmitriev AI. Spectral analysis of the behavior and properties of solid surface layers. Nanotribospectroscopy. Phys Mesomech. 2009;12(5–6): 221–34. 39. Dmitriev AI, Smolin AY, Popov VL, Psakhie SG. A multilevel computer simulation of friction and wear by numerical methods of discrete mechanics and a phenomenological theory. Phys Mesomech. 2009;12(1–2):11–9. 40. Blackmore D, Zhou J. A general fractal distribution function for rough surface profiles. SIAM J Appl Math. 1996;56(6):1694–719. 41. Venkatachalam R. Mechanical vibrations. Delhi: PHI Learning Private Limited; 2014. 42. Dmitriev AI, Popov VL, Psakhie SG. Simulation of surface topography with the method of movable cellular automata. Tribol Int. 2006;39(5):444–9. 43. Mallat S. A wavelet tour of signal processing. San Diego: Academic Press; 1999. 44. Bacry E (2002) LastWave home page [Internet]. Updated 20 Jun 2002; Accessed 19 Sept 2016. Available from: http://www.cmap.polytechnique.fr/~lastwave 45. Oñate E, Rojek J. Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems. Comput Methods Appl Mech Eng. 2004;193(27–29): 3087–128. 46. Johnson GR. Numerical algorithms and material models for high-velocity impact computations. International Journal of Impact Engineering. 2011;38(6):456–72.

Numerical Simulation of Material Separation Using Cohesive Zone Models

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mechanical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Constitutive Behavior for Monotonous Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Effect of the Shape of the Traction-Separation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Initial Stiffness: Extrinsic Versus Intrinsic Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Unloading Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 General Framework for Constitutive Laws of Cohesive Zone Models . . . . . . . . . . . . . 3.5 Constitutive Laws Using a Predefined Shape of the Traction-Separation Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Constitutive Laws Adopted from Plasticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Additional Dependencies of the Constitutive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Considerations for Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Practical Material Modeling: Identification of Traction-Separation Laws for Specific Material Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1716 1719 1721 1722 1723 1725 1725 1729 1736 1739 1745 1749 1752 1752

Abstract

This chapter will very briefly summarize the general foundations and developments of the cohesive zone model, which have been reported in literature during the past 50 years. According to the scope of the book, only material separation due to mechanical loading modeled by a cohesive interface will be considered, even though additional environmental effects may be taken into account. The reduction of the load bearing capacity of a cracked (or damaged) region by a separating interface is a strong simplification, but it reduces the complexity of the I. Scheider (*) Institute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_36

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crack models significantly such that the model gained high interest in research and also in industry. The use of cohesive zone models for the simulation of crack propagation in engineering structures has first been reported in 1976. The increase in computer power, which promoted the use of the finite element method as well, has increased the possibilities in the application of the cohesive zone model. It is now used for almost all kinds of materials and many different processes, and it has undergone many extensions and improvements with respect to materials, loading conditions, numerical techniques, and mechanical formulations. After a short historical overview starting with the very beginning of the cohesive interface model, the chapter addresses several recent developments such as identification and use of traction-separation laws, mixed-mode behavior, unloading issues, and cyclic loading.

1

Introduction

The cohesive zone model is one of the most widely used models for the numerical simulation of crack extension in materials and structures. While the foundations go back to the 1960s, the first realization in the framework of finite elements has been reported in 1976. With increasing computer power and variety of materials under consideration, the model has been extended to capture various aspects of their behavior. Until now, a variety of decohesion behaviors have been developed for almost every material class under any mechanical and environmental loading conditions. In addition, the mechanical theory of cohesive zones has been extended further from various points of view, based on the type of material under consideration, loading condition, complexity of the structure, and, last but not least, education of the researcher. It turns out that groups of scientists from different fields in material science, physics, and engineering use similar models for their specific kind of material separation without knowing each other, for example, the debonding of glued materials (process technology), crack extension in ductile metals (material science), and theoretical considerations regarding stresses at internal faces in arbitrary bodies (usually drawn by continuum mechanics people with a shape of a potato). This short review cannot present all and every detail of the variants of the cohesive zone model developed over time, but it will show many aspects of the model and give a broad overview of the applications of the model from different points of view. Considering the large variety of applications, it is reasonable not to talk about crack propagation in general when discussing the features of the model, but rather of material separation, since this is the general focus of the model and the lowest common denominator of all applications discussed here.1

1

There might also be other applications for the cohesive zone model, which do not refer to a separation, that is, a displacement jump, but to other discontinuities in field quantities. However, those jumps are not considered if they do not come along with a displacement jump.

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In order to focus the scope of this chapter, it must be emphasized that the cohesive zone model described here is an interface model. This means that the region where crack extension (called material separation) takes place is confined to a region of zero height. This might not be correct under all circumstances, for example, during the crazing process in polymers, which is a particularly widespread damage region, however, for numerical reasons, the reduction of the damage zone to one plane may also make sense for these processes, if the damage region is still small compared to the structural scale. While the type of damage process does not limit the application of the cohesive zone model, the concept of an interface already promotes specific classes of applications: The cohesive zone model is useful whenever damage, material degradation, or separation occurs in (crack propagation) planes or at material interfaces. Those are cases where the crack propagation direction is known in advance, or is at least limited to a small number of possible crack paths. The interfaces can also be purely numerical planes, that is, the crack may separate an originally continuous material. For those cases, a crack path is commonly presumed in practical applications. In the literature, the year 1962 is taken as the year of birth for the cohesive zone model, with Barenblatt [1]2 being mentioned as the father of the model. The mathematical foundation developed in the publication was called the “theory of equilibrium cracks.” Sometimes the year of birth is preponed to 1960, when Dugdale [2] invented the so-called strip-yield model, which limits the stress ahead of the crack tip to the yield stress. Both researchers were investigating the plausibility and mechanical treatment of crack tip stresses, which based on classical fracture mechanics rise to infinity. Barenblatt’s argument is the atomic bond force, which cannot become infinite, and thus he proposed a cohesive tensile stress in front of the crack tip as a function of the distance from it, see Fig. 1(b). While Barenblatt was interested in brittle materials, which behave purely elastically until fracture, Dugdale’s intention was a stress limit due to plasticity ahead of the crack tip, exemplified on steel sheets, so he kept the crack opening stress in front of the crack tip constant, Fig. 1(c). Another approach that relates the crack propagation to stresses in a zone in front of a crack is even much older: In 1933, Prandtl [3] published “A thought model for the fracture of brittle solids,”3 in which he substituted the proposed crack path by an elastic layer with a specified strength, Fig. 1(d). A similar model has been used by Kanninen [4]. Interestingly, none of these authors used the term “cohesive zone model.” This specific term is documented at least since 1974, when Strifors [5] solved the balance of energy over cohesive zones analytically. In his calculations, however, he avoided

Preliminary works on the topic have also been published by Barenblatt in 1959, starting with “The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axiallysymmetric cracks.” J Appl Math Mech. 1959; 23(3):622–36. 3 Translated by W. Knauss: A thought model for the fracture of brittle solids. Int J Fract. 2011;171:105–9. 2

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Fig. 1 Early models of a cracked configuration shown in (a), Barenblatt (b), Dugdale (c), Prandtl (d), and Wnuk (e)

the identification of stresses in front of the crack tip, but only used energy formulations of the surrounding material. The success story of the cohesive zone model started when it was first used in combination with the finite element method by Hillerborg and coworkers in 1976 [6]. This group used the cohesive model for crack propagation in concrete structures. They used (and still use) the term “fictitious crack model” instead of “cohesive zone model” in order to emphasize that according to the model there is an additional portion of the crack, which is not yet fully open, but still transfers stress. Less often cited but about the same age is the work of Wnuk [7, 8], who similar to Dugdale introduced constant stresses, but he divided the zone into three parts: The process zone, the damage zone, and the cohesive zone, each of them having its specific constant stress level, see Fig. 1(e). The cohesive zone in this model is the one furthest away from the crack tip, where the crack is actually completely closed, and the stress level is equal to the theoretical strength of the material. After this short historical abstract of the starting points of the cohesive zone model, an overview of the foundations, extensions, and applications will be given in the following without spending much attention to the chronological order.

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Mechanical Preliminaries

At first, the kinematics of the cohesive zone model must be introduced. For this, Fig. 2 is showing the traditional continuum mechanics potato (the body ) with an interface sectioning the potato into two virtual halves. Any material point within the body has the reference coordinates X,4 the coordinates at the interface will be denoted as X. On the left, the interface is still intact, that is, it is connecting both sides of the body, and , completely. On the right, the interface is partly open; three regions can be distinguished: In region I, and are still connected, in region II the parts are connected, but there is some distance between the original faces of the body that belong to the interface. In region III, the two faces are completely disconnected. Throughout this chapter, this kind of separable interface will be regarded as cohesive zone. From Fig. 2, one can see that the main characteristic of the cohesive zone is its dimensional reduction: The “zone” is reduced to an interface with initially zero thickness; however, this is not a necessary condition, thus in general one can also define the coordinates of the two faces of the interface separately, X+ ¼ 6 X. Such an interface with finite volume is useful, if structural interfaces like adhesives are considered, which a) indeed have a finite thickness, and b) are characterized not only by their decohesion behavior but also by their deformation (e.g., elastic) properties.

Fig. 2 Body with an interface opening upon loading. The cohesive zone is commonly understood as region II where stresses are still transferred from one part of the body to the other

4

It is common to designate quantities in the reference configuration with capital letters and in the current configuration with lowercase letters. This convention will be used throughout the chapter.

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Fig. 3 Definition of the mid face of the separated interface and the normal and tangential component of the opening. Note that the normal opening is parallel to the vector n

However, if there is a finite distance between the faces, this distance has no effect on the constitutive behavior. The separable faces may be called “upper” and “lower” in the text, respectively, and they are denoted by index “þ” and “” symbols. The definition of top/bottom or þ/, respectively, is arbitrary and depends on the choice of the normal of the interface, which is denoted by N. While in the reference configuration, the normal vector is unique, since even if the faces have an initial distance they are most probably parallel; the normal vector can be different on the upper and lower faces in the current configuration. Mosler and Scheider [9] discussed this issue and proved by condition of stress continuity, which must also be fulfilled across an interface, that only the midsurface between upper and lower faces can be used as current coordinate system, see Fig. 3. The material separation is given by the difference in the displacements, that is, the displacement jump, δ ¼ ½½u :¼ uþ  u :

(1)

It is useful to define a natural coordinate system with normal and tangential components, since the model parameters often distinguish the behavior in these directions. However, it must be kept in mind that the components differ in the reference and actual coordinate system, respectively, that is, Δn ¼ N  δ, Δt ¼ ð1  N  NÞ  δ δn ¼ n  δ, δt ¼ ð1  n  nÞ  δ

(2)

Usually, the current configuration is used for the numerical treatment of cohesive zones, since the nodal forces in the framework of finite elements are difficult to be transferred back into the reference system. However, large deformation/large rotation theories have been introduced for cohesive zone models as well, which use the separation in the undeformed configuration, see for example, [10, 11]. This type is not used in the following, since the vast majority of researchers use the current configuration, which also seems a more descriptive, that is, material-based configuration (in contrast to a more mechanics-based one).

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While the independent variable of the constitutive behavior is the displacement jump, a strain vector can be defined for convenience. For this, the displacement jump is divided by a thickness, t0. If the initial thickness is finite, this may be the initial difference between upper and lower surfaces, otherwise a length-scale parameter is introduced: εn=δn/t0, et=δt/t0. In general, the constitutive behavior of a cohesive zone model relates the displacement jump to the stress vector t, usually called “traction,” acting on the interface. This is calculated from the stress tensor σ in the current configuration by t ¼ σ  n. Alternatively, the undeformed configuration can be used as proposed. For example, in [10]: T = S  N.

3

Constitutive Behavior for Monotonous Loading

To start with, the constitutive behavior of a separating interface is a nonlinear vectorvalued vector relation of the form t(δ), which is naturally called the tractionseparation law (TSL). Whenever damage is modeled by such an interface, the main feature of this law is that it contains a softening behavior, that is, the derivative @t/@δ has a negative slope and the traction will reach 0 after some separation. Purely elastic interfaces, that is, those without softening behavior, have also been investigated intensively, but those are out of scope in this chapter. Remark: Barenblatt, who is often cited as the founder of the cohesive zone model (see above, [1]), did not use a traction-separation law, but he defined the traction as a function of the crack tip distance. Already in 1979, Gurtin defined conditions, when a cohesive zone is what he called “Barenblatt zone,” see [12].

The components of the equation t(δ) can be coupled or uncoupled, which is important for mixed-mode fracture, where normal and tangential separation may occur simultaneously. There are a few characteristics, which are common to most implementations of the cohesive zone model and can be shown even in the most simple 1D formulation of the model, that is, only a single component of material separation (either tangential or normal) is taken into account. For this case, the relation simplifies to a scalar one, t(δ). The only difference is that only tensile traction leads to material separation for normal loading, whereas tangential tractions act in both directions. In general, the cohesive zone model contains three model parameters for each separation direction: – The cohesive strength T0, which is the maximum stress to be transferred across the interface, hence T0 = sup (t(δ)) – The critical displacement jump, δc, denoting the opening, at which the cohesive interface loses its cohesion completely, hence t(δc) = 0 – The fracture energy, Γ0, which is the surface energy of the material and thus the energy dissipated within the cohesive zone due to crack extension.

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For a given shape of the function t(δ), only two of these three parameters are independent, the third parameters can be calculated. Particularly, the fracture energy is given by: Γ0 ¼

ð δc

tðδÞ dδ

(3)

0

For a vector-valued separation, those three scalar parameters can of course be different for normal and tangential material separation, and there may be, in particular, relations for mixed-mode fracture, as will be shown later.

3.1

Effect of the Shape of the Traction-Separation Law

The traction-separation law is a locally acting constitutive law, and, furthermore, in a highly stressed and/or deformed region. Therefore, a direct measurement even of the separation (which anyway is only a model quantity as it usually contains the damage of a small layer) is not possible. Three questions should be answered: 1. Is the shape of such a function t(δ) important for the results of a crack propagation analysis? 2. Can a general shape be qualitatively assigned to each material class? 3. Do methods exist, which allow the identification of the TSL, either experimentally, theoretically, numerically, or with a hybrid technique? The first two questions shall be investigated here; the identification methods will be discussed in the Sect. 5. As early as 1981, Petersson [13] already motivated the use of a number of shapes for the TSL, see Fig. 4, namely a bilinearly decreasing law for reinforced materials (a and b) and a plateau-type law for plastically deformable materials (c). The reason for these shapes is obvious: Heterogeneous materials usually consist of a high strength–low toughness component and a tough component with lower load-carrying

Fig. 4 Different traction-separation laws according to [13]

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capacity. Therefore, the TSL represents a superposition of both constituents: It has a high peak stress with a subsequent sharp drop of the traction, which models the failure of the reinforcing component. The second part adds to the fracture energy, but at low traction values. In the case of concrete, Fig. 4a, which has a low tensile strength, it is first the cementite that breaks, while the aggregates are interlocking and prevent the structure from failing. The Dugdale type, Fig. 4c, is often used for strain hardening materials, since those materials do not show a distinct stress peak at the crack tip, but rather a longer region of high crack opening stresses within the plastic zone in front of the crack tip. The discussion whether the effect of the shape is large enough to be considered at all as long as the fracture energy and the cohesive strength are kept constant was first raised by Petersson [13], who already came to the conclusion: As can be seen, the shape of the σ  w curve [author’s note: the traction-separation law] considerably affects the calculation result. This means that the shape of the σ  w curve is an important parameter that has to be considered when analyzing the fracture process of a material [p. 59].

This question was taken up again in the 2000s, first by Chandra and coworkers in [14] for a fiber pushout test, then by the author of this chapter in [15] for ductile materials, later also by others, see for example, Alfano for various “elastic” materials (mode I, but also mode II) [16], Volokh for the simulation of Peel tests [17], and Campilho [18] for adhesives. A more general review on TSLs is also given by Park and Paulino [19].

3.2

Initial Stiffness: Extrinsic Versus Intrinsic Laws

Two general types of TSLs can be distinguished, which are shown in Fig. 5. The difference between these two is their initial stiffness. On the left, material separation occurs only when a specific cohesive stress has been reached, that is, the initial stiffness is infinitely high, and thus contact conditions apply in this regime rather than a constitutive interface behavior. In [20], it is stated that the initial stress where separation starts must be the cohesive strength, T0, due to the fact that a hardening portion of the TSL after damage nucleation means that neighboring points exceed the damage nucleation stress as well and one gets multiple cracks in a finite Fig. 5 Types of tractionseparation laws. (a) Extrinsic (no separation occurs before cohesive strength is reached, i.e., initially infinitely stiff), (b) intrinsic law with initial compliance

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region, which should rather be included in a single cohesive zone. Models with such kind of purely decreasing TSLs are called extrinsic laws. Intrinsic TSLs, on the other hand, start separation upon loading right from the beginning, that is, they have an initial compliance. Structural or numerical reasons may be responsible for such a behavior. If the interface contains some elasticity before damage nucleation, this can be modeled by a finite elasticity. Futhermore, with respect to the argument for extrinsic laws, damage nucleation at several places in a finite region of the structure might be a desirable feature if widespread damage is considered as, for example, in polymer crazing. A more common reason is the numerical one: If the cohesive zone model is used in the framework of the finite element method as an interface element, the implementation is much easier if the stiffness is finite. In [21], Kubair and Geubelle conducted a comparison of the two types of laws for dynamic crack extension analyses. They concluded that under steady-state and spontaneous dynamic crack propagation, intrinsic models can be less stable than extrinsic ones. Another way how to distinguish between intrinsic and extrinsic cohesive interfaces is one emphasized, for example, in [22, 23] among many others: Extrinsic interfaces are adaptively inserted into the bulk structure whenever they are needed, that is, the initial structure does not contain cohesive interfaces at all. This is the reason why this type is called extrinsic. On the other hand, intrinsic interfaces are included in the structure from the very beginning.5 From the mechanical point of view, there is an important difference between these two: The initial compliance of an intrinsic TSL adds to the overall compliance of a structure, see for example, [24]. Let the initial stiffness be denoted as ktsl, such that the traction is t = ktslδ. Then the total strain of a bar of length L under tension, σ containing one cohesive interface, is etot ¼ econt þ Lδ ¼ Econt þ ktslσ L þ. The term econt is the strain of the pure continuum without cohesive elements. The second term is an artificial straining due to the compliance of the cohesive interface. If the structure contains a series of n cohesive interfaces in the bar, the total compliance is: 1 1 n ¼ þ Etot Econt ktsl L

(4)

It is clear that the second term should be as small as possible, n EktslcontL  1, in order to not affect the elastic behavior. However, as shown in Eq. (4), the compliance not only depends on the stiffness ktsl, but also on the length of the structure and the number of serial cohesive interfaces. Examples for the use of a high number of cohesive elements with an intrinsic TSL in structural simulations are, for example, given in [25]. The effect of the finite stiffness is also discussed in [100]. For mode I fracture, the behavior under compression is also different between intrinsic and extrinsic laws. While the extrinsic law acts like a boundary condition

Notice, however, that inserting cohesive elements in commercial finite element codes “on the fly” during a simulation is usually complicated.

5

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1725

and thus does not allow any displacement difference under compression, the intrinsic law has a finite stiffness also under compression. This stiffness is commonly set to a very high value; nevertheless, it may lead to a penetration of material.

3.3

Unloading Behavior

Another issue, which is particularly important for non-monotonous loading, is the behavior of a cohesive interface, when the applied stress at the interface decreases. As in all models of dissipative processes, taking the loading path in the reverse direction is not an option, since energy is not returned into the material. In the literature, two types of unloading behavior can be found: elastic and plastic, see Fig. 6. The two graphs on the left show the behavior for normal separation under load case elastic loading ① – dissipative loading ② – unloading ③ – elastic reloading ④– dissipative reloading ⑤. Tangential separation is visualized on the right under load case loading ① ② – unloading ③ – reverse elastic loading ④ – reverse dissipative loading ⑤ (right). One can see that for the elastic unloading type (upper graphs), the separation reduces to zero upon complete unloading. Contrarily, the inelastic separation remains upon complete unloading for the plastic unloading type, compare particularly points C after complete unloading in the graphs. For this type, the unloading slope is usually set equal to the initial loading slope. While these characteristics are shown for an intrinsic law in Fig. 6, either mechanism is possible for extrinsic laws as well, just the elastic regions ① are rigid, and ③, ④ are also rigid for the plastic unloading type. However, the author is not aware of any extrinsic model using a plastic unloading type. For the modeling of unloading, it is necessary to introduce an internal variable in order to distinguish between dissipative separation and elastic unloading/reloading cases. For the elastic unloading, it is sufficient to compare the separation to the maximum separation reached so far (or its absolute value for tangential separation), but for a plastic unloading type, the internal variable has the meaning of a back separation storing a shift in the origin of the TSL (see lower right graph in Fig. 6). The majority of cohesive zone models in the literature uses an elastic unloading type, but the plastic unloading type can be found as well, for example, in [25–28]. The unloading and reloading behavior is even more important for fatigue loading. This particular field will be considered in a later section, but at this point it can be stated that the schemes shown in Fig. 6 are not suitable for fatigue loading, since a saturation is easily reached for the linear unloading plotted in all four graphs therein.

3.4

General Framework for Constitutive Laws of Cohesive Zone Models

In the following, some physical principles are given for a consistent formulation of TSLs. However, for most of the applications of cohesive zone models to real

1726

I. Scheider

Fig. 6 Two different types of interface behavior for unloading and reloading in case of normal material separation (left), and unloading plus reverse loading for tangential separation (right); Top: elastic, bottom: plastic cycle

structures a very simple formulation of the TSL is sufficient, since the damage occurs in a single plane under monotonous loading in a specified direction resp. fracture mode. For those cases, it is sufficient to define a simple curve t(δ), enter it into commercial finite element codes (most of them have cohesive zone models included nowadays), and run the simulation. For more complex loading states with combined normal and tangential separation, a vector-valued TSL must be defined. Two approaches can be found in the literature for intrinsic cohesive zone models: • So-called ad-hoc models, which define the traction-separation behavior directly, for example, by a scalar function for each direction: t = (tn(δ), tt(δ)), and

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1727

• Models obeying thermodynamics principles, where the traction must be at least a potential of an energy function, t ¼ @ϕ @δ . The latter is necessary to fulfill nonnegative dissipation at any time. As mentioned in the previous section, an internal variable must be introduced to distinguish between loading and unloading cases. For arbitrary shapes of a cohesive zone model, two simple ways can be found to introduce a potential-based TSL. The first approach has been considered by Tvergaard and Hutchinson [29, 30], starting with an arbitrary scalar law of an effective traction, b t , depending on an effective separation, κ, which must be monotonously increasing: 0

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  n 2  t  2 δ δ k ¼ max@kðτÞ ^ τ  ½0, t, þ t A δnc δc

(5)

with hxi ¼ 1=2 ðx þ jxjÞ. The potential can be written as: Ψ ¼ δnc

ðk

b t ðk0 Þ dk0

(6)

0

The tractions are then derived as tn ¼

dΨ dk b t ðkÞ δn ¼ , dk dδn k δn0

tt ¼

dΨ dk b t ðkÞ δn0 δt ¼ dk dδt k δt0 δt0

(7)

For this approach, it is assumed that the energy consumed during the damage process does not depend on the mode of failure, which is a rather strong assumption. At least it is possible to have different cohesive strengths for normal and tangential separation, given by the ratio of the critical separation, δnc =δtc . For the second approach, the potential is formulated in accordance with continuum damage mechanics laws introduced originally by Kachanov: 1 Ψ ¼ ½1  dðkÞ k j δ j 2 1 Ψ ¼ ½1  dðkÞ δ  K  δ 2 1 1 t t ðnÞ ðtÞ n 2 ðnÞ Ψ ¼ ½1  d n ðkn Þ½1  dðtÞ n ðk t Þ k n ½δ  þ ½1  d t ðkn Þ ½1  d t ðkt Þ kt δ  δ 2 2 (8) In these potentials the loss in load-carrying capacity is introduced by a damage variable, which in contrast to Eq. (5) can be an arbitrary monotonous function of an (also monotonously increasing) internal variable, κ or κn, κt, respectively (anyway, Eq. (5) is a special case for a valid function of the internal variable). While the codomain of κ is κ  [0, 1), the damage range is d  [0, 1]. All models in Eq. (8)

1728

I. Scheider

have in common that the shape of the TSL is determined purely by the shape of the damage function(s), d(κ). In Eq. (8), it is sufficient to introduce the damage evolution equation directly rather than in rate form. The simple form leading to a linearly decreasing softening function (see Fig. 4a) is exemplarily given by 8 0 >   k < knucl < knucl kini  k knucl  k < kini d ¼ 1 > k kini  knucl : 1 kini  k

(9)

Different versions of this kind of cohesive laws have been introduced by various authors, for example, Allix et al. [31], Scheider et al. [9], or Camanho et al. [32]. The term [1  d] in Eq. (8) is the reduction factor between the stress including damage, t, and the fictitious undamaged configuration, ~t , for example, for Eq. (8)1: 9 1 = t ¼ ½1  dðkÞ kδ > 2 d ¼ 1  ½tðiÞ = et ðiÞ  (10) > ; et ¼ 1 kδ 2 The damaged and the fictitious undamaged tractions are shown in Fig. 7a. Due to the high elastic stiffness in the beginning, the ratio ~t =t increases quickly, thus the damage is a highly nonlinear function of the separation, see Fig. 7b. Particularly, the damage reaches 0.9 at a separation less than 0.1δc. This clearly indicates that the damage has no physical meaning. Two other quantities are given in the figure for comparison, which can serve as damage indicators, namely, the relative separation δ/δc (dashed line) and the relative fracture energy Γ/Γ0 (dotted line). For further

a

b

8

1,0 t t^

0,8 damage

T/T0

6

4

0,6 0,4 damage indicators

2

d , eq. (9) G/G0

0,2

d/dc

0

0,0 0,0

0,2

0,4

0,6 d/dc

0,8

1,0

0,0

0,2

0,4

0,6

0,8

1,0

d/dc

Fig. 7 (a) Normalized traction-separation law, and fictitious undamaged traction (dashed line). (b) Damage for this traction-separation law, according to Eq. (9) and alternative damage indicators

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1729

information on the restrictions and possibilities for this kind of potential-based TSL, the interested reader is referred to the respective references. The three versions in Eq. (8) have different versatility (and, of course, different numbers of parameters) regarding the elastic behavior and the damage evolution, which in (8)3 can be given separately for normal and tangential separation as well as ðnÞ for mixed mode, where the interaction terms dðntÞ , dt are important. Besides that, Eq. (8)1 contains a single elastic stiffness value, k, and the stress will always be in the direction of the separation. The other two can have different stress values in normal and tangential directions, and the directions for traction and separation may differ. A popular definition of a stiffness tensor as given in Eq. (8)2 is: K ¼ kn n  n þ kt ½1  n  n

(11)

that is, no interaction exists for the normal and tangential elastic separation.

3.5

Constitutive Laws Using a Predefined Shape of the TractionSeparation Relation

In the previous section, a framework has been set up to provide constitutive laws for cohesive zones, which can be used for various shapes of traction-separation relations and thus classes of materials. Often, specific laws are defined in the literature rather than defining the damage evolution within a general framework. Some of them have a number of so-called shape parameters, which can be used to adjust the shape to conform to the material’s requirement. While the number of TSLs is very large, there are some laws, which have attracted great attention and are used by many groups around the world. A few of these laws, summarized in Table 1, will be detailed in the following. Of course, this is a limited selection, but it shows the broad variability of defining such laws. Often such laws have not only been used, but also extended by others, and a few important extensions are given in the table together with the original version. Some characteristics of the TSLs are given in the table already, for example, the principal shape of the curves and the numbers of parameters. In particular, one can see that some of the laws have four independent parameters (for normal and shear separation together), whereas others have three, and the fourth parameter is derived from the others. A special case is the law originally derived by Xu and Needleman, which follows an exponential function. For this function, the traction only approaches zero at infinite separation, and hence there is no critical separation. Instead, they use the separation at maximum cohesive traction, here denoted by δn0 = δt0, as a characteristic length parameter. Due to the lack of δnc , δtc, the fracture energy is 1 Ð calculated by Γ0 ¼ tðδÞ dδ instead of Eq. (3). 0

In a first comparison between these laws, a pure normal and a pure tangential separation will be applied in order to see the differences of the TSLs under uniaxial loading. The parameters for the calculation will be T n0 ¼ 300 MPa, T t0 ¼ 300 MPa,

3

2

1

Tvergaard and Hutchinson (1992, 1993) [29, 30] Xu and Needleman (1994) [34]

Author (Year) and reference Tvergaard (1990) [33] Chaboche et al. (1997) [26]

Atomic binding, elastic

Shape, unloading behavior Polynomial, elastic Polynomial (plastic under compression) Trapezoidal, elastic

T n0

δ1, δ2 T t0 ¼ T n0 δn0 =δt0

T n0

r,q T t0 ¼ T n0 δn0 =δt0

δn0 , δt0

Cohesive strength (norm., tang.) Additional shape parameters T n0 T t0 ¼ αT n0

δnc , δtc

Critical separat. (charact. Length) δnc , δtc

Table 1 Selection of constitutive laws frequently used in the literature

σ(δeff) is a trapezoidal function of the normal traction

 n Γn δ T n ¼  n exp  n δ δ 0 " "  t 2 ! 0  t 2 !#  # δn δ 1q δ δn 1  exp  t exp  t þ r n r1 δn0 δ0 δ0 δ0     h i n t n n Γ δ δ r  q δ T t ¼  n 2 0t t q þ n  δn0 δ0 δ0h i  r  1 δ0 2 δ t exp  n exp  δδt 0 δ0

δn σ ðδeff Þ δn0 δeff n t δ δ σ ðδeff Þ T t ¼ 0t t δ0 δ0 δeff Tn ¼

Traction-separation law for monotonic loading qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n 2 t t 2ffi δn δeff ¼ δ =δ0 þ δ =δ0 T n ¼ n Fðδeff Þ δ0 27= n Fðδeff Þ ¼ 4 T 0 ½1  δeff 2 δt T t ¼ α t Fðδeff Þ δ0

1730 I. Scheider

Scheider and brocks (2003) [25]

5

T n0 , T t0

T n0 , T t0

δnc , δtc

δnc , δtc

T n0 , T t0

δ1, δ2, p

α, β, δnc , δtc λn ¼ δnc =δn0 , λt ¼ δtc =δt0 α½α  1λ2n β½β  1λ2t m¼ ,n ¼ 1  αλ2n 1  βλ2t  m  n Gn ¼ Γn0 mα , Gt ¼ Γt0 βn

–

 n n  t 2 ! Γn δ δ δ exp  exp  δn0 δn0 δn0 δt0    n  t 2 ! Γn δt δn δ δ t T ¼ 2 n t 1 þ n exp  n exp  t δ0 δ0 δ0 δ0 δ0

f ðδnÞgðδt Þ, T n ¼ T n0 f ðδt Þgðδn Þ T n ¼ T n08 h i2 δ > δ > δ < δ1 > < 2 δ1  δ1 f ðδÞ ¼ 1 δ1 < δ < δ2 > h i3 h i2 > > : 2 δδ2  3 δδ2 þ 1 δ2 < δ < δ0 δ0 δ2 δ0 δ2 h h ip i1=p δ gðδÞ ¼ 1  δ0

"         # Gn δn α m δn m1 δn α1 m δn m þ n þ n Tn ¼ n m 1  n α 1 n α δ0 α δ0 δ δ0 δ0  h0  i h in t β t m n δt Gt 1  δδt þ þ h G  G i t β δ0 " 0 β  n1  β1  n # Gt δt n δt δt n δt t T ¼ t n 1 t β 1 t þ t þ t δ0 δ0 δ0 β δ0 β δ0 h h i δt iα h im n m t δn m δn G 1  δn α þ δn þ hG  G i t 0 0 δ0

Tn ¼

The equations are shown for one tangential direction only, and without unloading or reverse loading cases

Piecewise polynomial, plastic

Park, Polynomial, Paulino and elastic Roesler (2009) [36]

4

Van den Bosch et al. (2006) [35]

52 Numerical Simulation of Material Separation Using Cohesive Zone Models 1731

1732

I. Scheider

and the fracture energy in normal and tangential directions is Γn0 ¼ Γt0 ¼ 20kJ=m2 . The corresponding critical displacements in normal and tangential directions are given in Table 2. The shape parameters for model 5 (Scheider and Brocks) and model 2 (Tvergaard and Hutchinson) are chosen such that a pronounced regime with constant cohesive stress occurs. The parameters for model 4 (Park et al.) are then adjusted in a way to yield the same critical separation as in the other two models (2 and 5). Since the initial stiffness can also be prescribed in these three models, it is set to a rather high value. The shapes of the TSLs are shown in Fig. 8. The area under the curve (the fracture energy, see Eq. (3)) is the same for each law, both for normal and the tangential Table 2 Critical displacements of the specific models for the parameter sets given in Table 1 based on a cohesive strength T n0 ¼ 300 MPa, T t0 ¼ 100 MPa, and a fracture energy Γn0 ¼ Γt0 ¼ 20 kJ=m2. For the Xu/Needleman model, the separation at 99% loss of strength is given

δnc

Tvergaard 0.11852

Tvergaard/ Hutchinson 0.09195

Xu/Needleman, van den Bosch δn0 ¼ 0:02453 δn99% ¼ 0:18733

Park, Paulino, Roesler 0.0908

Scheider/ Brocks 0.09091

δtc

0.35556

0.27586

δt0 ¼ 0:17155 δn99% ¼ 0:43326

0.27238

0.27273

Add. param.

–

δ1 ¼ 0:05δ0 δ2 ¼ 0:5δ0

–

λn ¼ λt ¼ 0:05 m ¼ n ¼ 1:41

δ1 ¼ 0:05δ0 δ2 ¼ 0:5δ0 p ¼ 0:82

Initial stiffness

K n ¼ 17086 K t ¼ 1900

K n ¼ 66000 K t ¼ 7333

K n ¼ 32590 K t ¼ 1360

K n ¼ 1133290 kt ¼ 125724

K n ¼ 132000 kt ¼ 14667

Fig. 8 Traction-separation laws of the models summarized in Table 1; (a) for pure normal, and (b) for pure tangential separation

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1733

separation. The main differences of the five curves can be summarized in the following keywords: Shape: According to the algebraic function, the traction at any specific separation differs between the curves. The question of the effect of the TSL on the simulation of crack propagation has been addressed by several authors as already mentioned above. Due to the additional shape parameters included in some of the laws, their shape may vary significantly. For example, the bilinear shape is the special case for the trapezoidal law with δ1 = δ2, and the parameters of the model by Park et al. (4) can be adjusted so that the shape is similar to the cubic polynomial of Tvergaard (1). Critical displacement: As shown in Table 2, the critical displacements differ, that is, the opening at which the cohesive zone loses its stress-carrying capacity is different. Of course, this closely connected to the shape of the curve. For the Xu and Needleman model, where the critical displacement is not defined, one can only compare the shapes by inspecting Fig. 8 and judging the trailing part of the TSL qualitatively. In Table 2, the separation at 99% loss of strength is given. Initial stiffness: One can see that the initial stiffness is different for the curves. Here it is particularly important that the TSLs 2, 4, and 5 have additional parameters to specify the initial stiffness of the cohesive zone, whereas the stiffness for law 1 scales with the critical separation δc, law 3 with δ0. The stiffness values for the specific laws are also given in Table 2. Notice that they vary by two orders of magnitude. The model with a cubic function by Tvergaard (1) is particularly soft and should be used with care if several cohesive layers are introduced into a structure. From the preceding paragraphs, one can see that many choices are possible and the number of parameters can already be large for a pure normal or tangential material separation. The problem becomes more challenging when mixed mode is included, that is, when a combination of normal and tangential separation occurs in the simulation. In such situations, the total fracture energy is calculated by a sum of the energy due to normal separation and tangential separation: Γtot ¼

Γn0

þ

Γt0

¼

ð δnc 0

t dδ þ n

n

ð δtc

tt dδt

(12)

0

Since in fracture mechanics it is generally assumed that the critical energy release rate GIc is unequal GIIc, one may also argue that the cohesive fracture energies should be different. Four out of the five presented TSLs may have different fracture energies for normal and tangential separation, only the Tvergaard/Hutchinson model has been developed for equal energies in normal and tangential directions. Whenever two different fracture energies are given, the question arises what the value of the total fracture energy under mixed-mode separation is. In order to investigate this issue, various monotonous separation paths as shown in Fig. 9 have been prescribed to all TSLs. The separation loading is applied in two steps. Up to 50% of the maximum separation (which is δn δt δn ¼ δt ¼ 1), a prescribed mix ratio c

c

1734

I. Scheider

Fig. 9 Various loading paths to be applied to the tractionseparation laws shown in Table 1.

1.0 0.2 r mix=

rel. normal. sep., d n/d n0

0.8 .5

=0

0.6

ix

rm

Step 2

Step 1

0.4

0.2 0.8 r mix=

0.0 0.0

0.2

0.4

0.6

0.8

1.0

rel. tang. sep, d t/d t0

Table 3 Scenarios of fracture parameters investigated during the mixed mode study Parameter set 1 2

(Γt0 (Γt0

< >

ΓN0 ) Γn0 )

T n0

T t0

Γn0

Γt0

300

100

30

20

300

100

20

30

r mix ¼

δ

t

δt =δtc þ δn =δnc

=δtc

(13)

is applied. In Step 2, both separations increase linearly up to the maximum separation. The loading paths are simulated by all models using two sets of fracture parameters: In a first scenario, the mode II fracture toughness (corresponding to Γt0 ) is smaller than the mode I toughness, Γn0. For the second scenario, the fracture energies are changed. The parameters for the two scenarios are shown in Table 3. The results for this study are shown in Fig. 10. It turns out that the pure mode energies are achieved for the extreme cases, but the transition is different for the considered models. In general, the transition is nonlinear and not even symmetric (with the exception of the Tvergaard model). For the Xu and Needleman model, the mixed-mode energy has a clear tendency toward the tangential fracture energy, while the mixed-mode energy for the Park et al. model drops rather quickly to the respective lower fracture energy value. Two of the models, namely, the Scheider and Brocks model and the Park et al. model, have extra parameters to adjust the mixed-mode effect. The experimental identification of mixed-mode fracture toughness values is provided for many different classes of materials. For brittle materials, the fracture toughness is identified in terms of critical stress intensity factor, while for ductile

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1735

Fig. 10 Total fracture energy calculated by Eq. (12) vs. mixed-mode ratio rmix, Eq. (13), for simulations using the models given in Table 1 under varying loading paths as shown in Fig. 9 for the two-parameter scenarios given in Table 3: (a) Γt0 < Γn0 , (b) Γt0 > Γn0

materials, the toughness is usually expressed by the J-integral. Anyway, it turns out that the toughness indeed depends on the mixed-mode ratio, see for example, [37]. Even if the fracture toughness in mode I and mode II is the same, the toughness for the mixed-mode case may be different as shown, for example, in [38] and the examples therein. Numerically, this behavior can only be reproduced by the Scheider and Brocks model (by variation of the exponent p). All other laws produce a monotonic function of the total fracture energy. Another aspect that has been addressed by researchers from a continuum mechanical point of view [39, 40] is whether the direction of the traction vector and the direction of the separation should be collinear according to the balance of angular momentum. The calculation with a mixed-mode ratio rmix = 0.5 has been examined in this respect. The results are shown in Fig. 11 in terms of the angle of the traction vector over the loading history. While the separation is applied with a constant angle of 45 with respect to the normal interface plane, the angle of the traction vector is not 45 , even in the beginning. In addition, the angle depends strongly on the TSL used and (for some laws) also on separation itself. If the cohesive strength in normal and tangential directions are equal (which is rather unusual), traction and separation are collinear in the beginning for all TSLs, but with increasing separation, the angle varies in any case for the Xu and Needleman model, with Γn0 6¼ Γt0 also for the Park et al. model. From a material scientist’s point of view, the discussion of collinear traction and separation vector depends on the material considered. If material separation occurs, it depends on the damage mechanism as exemplified in Fig. 12. The damage representation of a cohesive interface (a) can be fiber bridging (b), where indeed

1736

I. Scheider

Fig. 11 Angle of the traction vector (90 means pure normal traction, 0 is pure tangential traction) with a prescribed separation angle of 45 (equal normal and tangential separation) for the two fracture energy scenarios given in Table 3: (a) Γt0 < Γn0 , (b) Γt0 > Γn0

Fig. 12 Cohesive zone and different damage mechanisms it may represent

the separation direction is similar to the traction direction. However, if the cohesive zone represents a ductile damage mechanism (c) or even interlocking (d), the separation is in general not in the same direction as the traction.

3.6

Constitutive Laws Adopted from Plasticity Theory

During the past decade, a new class of constitutive models for the cohesive zone came up, namely, the plasticity-based cohesive zone models. The main assumption within these models is that the material separation can be divided into an elastic part and a plastic part: δ ¼ δel þ δpl

(14)

The reasons for such models are manifold: – If the cohesive interface is used to model a structural interface such as an adhesive, it covers the complete material behavior of this thin layer. This could contain plasticity as well.

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1737

– The damage mechanism itself can lead to a permanent irreversible strain, which can be described by the cohesive zone model with a plastic material separation, for example, the void growth mechanism in ductile metals or crazing in polymers. – A physical interface may have an influence on the stress and/or strain field of the adjacent material. This change in the neighboring layer can be included in the cohesive zone model. An example for this behavior is the microstructural debonding in a metal–ceramic composite, where some localized plastic strain may occur in the metal close to the interface. As long as purely monotonic loading with a constant load direction exists, the use of plastic separation can be avoided. However, if unloading occurs in the aforementioned situations, the plastic unloading type described in Fig. 6 can easily be described by the introduction of an elastic-plastic separation. While such an unloading behavior has been modeled without plastic separation in [25], its definition leads to a thermodynamically consistent framework. An early work using the split between elastic and plastic separation and a plastic yield function was conducted by Corigliano and Ricci in 2001 [41]. They even included a viscoplastic theory for their application of rate-dependent crack propagation behavior. A purely elastic-plastic cohesive law for material separation has been published by Su et al. in 2004 [42]. They introduced a yield function Φ:



Φ t, δpl ¼ f ðtÞ  ty δpl  0

(15)

with f being a function of the traction vector, ty being the resistance function (i.e., Ð initial yield and hardening/softening) with its dependent variable δpl ¼ j δ_pl j dt, an accumulated plastic separation. In case of an associated flow rule, the yield function is equal to the yield potential, and the flow direction and the plastic separation rate, δ_ pl , are both calculated by the derivative of Eq. (15) for the case Φ = 0: @Φ @f ðtÞ ¼ @t @t _δpl ¼ kn _ δ_ pl ¼ k_jnj n¼

(16)

_ ¼ 0, which holds for active yielding, leads to a The consistency condition, Φ further equation for the identification of the plastic multiplier:

@t 

t_ @ty _ δ @δpl pl

@f ¼ 0 , @f @t 

t_ ¼@ty @δpl

k_jnj

(17)

A very recent version of an elastic-plastic interface proposed in [43] uses a free energy function of the form:

1738

I. Scheider

1 Ψðδel , kÞ ¼ cδ2el þ Ψpl ðkÞ 2    kQ0 Ψpl ðkÞ ¼ Γ0 1  exp   Q0 k Γ0 The resistance function, which can be derived by ty ðκÞ ¼ softening in accordance with Eq. (15), defined by:

@ψ pl ðκ Þ @κ ,

(18)

is an exponential

   kQ ty ðkÞ ¼ Q0 1 þ exp  0 Γ0

(19)

The stress function f(t) in Eq. (15) is usually taken as a weighted norm of the traction vector, sometimes denoted as effective traction: f ðtÞ ¼ teff ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi htn i2 þ β½ðtt1 Þ2 þ ðtt2 Þ2 ,

(20)

The interface models presented in [41–43] are elastic-plastic interfaces without an additional damage term, and the plastic separation can, in general, be of hardening or softening type, that is, the change of the traction vector is calculated by an incremental equation of the linear form:  t_ ¼ K  δ_el ¼ K  δ_  δ_pl ,

(21)

which can be solved using the Eqs. (16) to (20). For their specific problem, the authors of [42] introduced the resistance function stepwise, starting with a hardening behavior up to a given accumulated plastic separation, δpl, c, and a softening behavior afterwards until the resistance function and thus the traction reaches zero. The model was successfully used to simulate a peel test of an aluminum strip bonded to an aluminum block by a ductile polymeric adhesive. The parameters were identified by two experiments, a tension test with a butt joint and a lap shear test, respectively. A similar approach with emphasis on thermodynamic consistency was derived in [40] using the reference configuration of the interface.6 A more prominent version of this type of cohesive laws has been promoted by several research groups, which couple the plasticity with a reduction of the free energy by a damage term already discussed in Eq. (8). The free energy proposed by the author in 2011 [44], more recently also by Xu and Lu [45], is written as: Ψðδel , k, DÞ ¼ ½1  DΨel ðδel Þ þ Ψpl ðkÞ ¼ 1=2½1  Dδel  K  δel þ Hk2

6

(22)

Since the surface deformation gradient necessary to define a separation in the reference (material) configuration has not been defined here and belongs to the field of continuum mechanics rather than material mechanics, details are skipped here.

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1739

where H is the hardening modulus and K the elastic stiffness tensor. The traction is then calculated by the derivative with respect to the elastic traction: t¼

@Ψ ¼ ½1  DK  δel @δel

(23)

Similar to Eq. (22), but without the hardening in the free energy function, is the damage-plasticity model introduced by Grassl and Rempling for concrete [46]. An equivalent approach was also introduced based on the material configuration of the interface, see for example, [47]. In any of the publications, an additional evolution equation for the damage is needed for the identification of κ. For this evolution, however, no general recommendation can be given, since it is defined completely differently by the references given above. While a very simple formulation depending on the total separation is used in [45], the damage depends on the plastic deformation only in [46, 47]. A second potential g depending on the damage driving @ψ force Y ¼ @D ¼ 1=2 δel  K  δel is used in [44]: gðk, D, Y Þ ¼ f ðtÞ  ty ðkÞ 

SY 1D

@g k_ ¼ SðY ÞM1 D_ ¼ k_ @Y 1  D

(24)

The main difference between a model including a yield function only for the softening (Eq. 15) and a model including an additional damage variable, Eqs. (22)–(24), is that the elastic compliance depends on elasticity and damage in the latter. Consequently, the unloading stiffness is constant throughout the whole separation process for an elastic-plastic cohesive law, while the stiffness is reduced by [1  D] as soon as damage sets in a damage-plasticity law. Note also that the damage in the plasticity-type cohesive zone models has a completely different evolution than the ones shown in Eq. (9) and Fig. 7b, it may increase smoothly similar to continuum damage models. However, since significant material separation may occur before damage evolves due to plastic separation, the accumulated plastic separation is a better damage indicator. A similar combined damage-plasticity approach has been introduced by Kolluri et al. [101]. They included a potential based plasticity formulation and a damage evolution based on maximum separation in each direction, and they varied the unloading/reloading behavior between plastic and elastic unloading by a so-called plastic limit of the separation. In addition, they emphasized that the yield traction can be 0 in the beginning in order to model a nonlinear shape of the TSL. The proposed model was adapted to the shape in [35], and a procedure was presented to identify the plastic limit for normal and tangential direction, respectively.

3.7

Additional Dependencies of the Constitutive Law

Quite often, the traction at the cohesive interface does not only depend on the separation, but on other quantities as well:

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I. Scheider

– Time. As in other constitutive laws, the time can be involved by rate-dependent parameters, which are important for dynamic effects, but also by viscous effects, usually found in crack propagation of polymers and high-temperature applications. – Additional field variables. The most common is the temperature field, but other quantities may be involved as well, for example, stress triaxiality or a hydrogen embrittlement.

3.7.1 Cohesive Laws with Time Involved There are two possibilities how time can affect the material separation: viscosity and loading rate. High-speed impact shock and similar applications usually ask for strain rate dependent models in the bulk material. Viscous effects are of minor importance in those short loading durations. On the other hand, if the applied loading speed is very low (maybe even kept constant for some time) viscous effects can play a major role. In order to deal with high crack speeds, the easiest way is to include a dependence of the cohesive parameters on a separation rate. However, for a constant load, ratesensitive parameters have no effect. This difference has been depicted in [48] by a two-step process: (1) increasing separation and (2) constant separation, see Fig. 13. The rate effects can be captured by a rate-dependent model as long as the load is monotonously increasing but fall back to the static solution as soon as the load is kept constant.

Rate Dependence The theory behind rate-dependent models is much simpler than that of viscoelastic or viscoplastic models; therefore, these types are described first. It is known from fracture mechanics that the dynamic fracture toughness is higher than the quasistatic one. Based on this finding, it can be assumed that the cohesive energy is a function of the separation rate, Γ0 ¼ f δ_ . However, this does not imply any information about the rate dependence of the cohesive strength. Experimentally, the effect of the applied strain rate on fracture strain has been measured by dynamic tensile tests. The results, for example, shown in [49] for aluminum alloys are a strong indicator that the cohesive strength increases with strain rate as well. One can find various ways to consider the effect of loading rate on cohesive parameters in publications: In [48], it is assumed that the rate-dependent cohesive strength, T rd 0 , increases the quasi-static value by the same factor ß as the energy, such that the critical separation remains constant:

_ T rd 0 ¼ β δ T0,



_ Γrd 0 ¼ β δ Γ0 ,

δrd 0 ¼ δ0

(25)

Other researchers assume that the energy increases by an increase of both the strength and the separation, as for example in [50, 51], while others increase only separation, see for example [52].

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1741

Fig. 13 Effect of different types of cohesive zone models for a given loading function (a) with prescribed separation; (b) rate-independent model; (c) rate-dependent model; (d) viscoelastic model [48]

Viscous Models If viscous effects have to be taken into account, a cohesive zone model usually contains a rheological element of viscosity either parallel and/or in series to the TSL mentioned above. The simplest version of a viscoelastic model of a dashpot in parallel to the law was already proposed by Costanzo and Walton in 1998 [53]. Such a simple so-called Kelvin–Voigt type model is also not able to represent relaxation shown in Fig. 13d, but it is able to describe a creep separation behavior. For these models, the relaxation is then usually considered in the bulk material only. Bazant and Li, however, considered relaxation and creep in cohesive laws in 1997 already [54], Xu and Siegmund introduced a cohesive zone model based on the standard linear material model in viscoelasticy [55] for crazing in polymers, and Geißler and Kaliske [48] considered a generalized Maxwell model (an arbitrary

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I. Scheider

Fig. 14 Viscoelastic cohesive behavior. (a) Principle of a simple viscoelastic material with one Maxwell series parallel to a cohesive spring; (b) traction-separation behavior at various separation rates, here denoted by u n, taken from [55]

series of Maxwell elements) for the simulation of adhesive peel tests. However, the latter used a single Maxwell element for their peel-off example, such that this one [48] and the one by Xu and Siegmund [55] share the same basis. The principle of the standard linear solid is shown in Fig. 14a, consisting of a spring with a nonlinear TSL Ⓐ and a Maxwell element with a linear spring Ⓑ and a linear dashpot Ⓒ. A strong assumption of Xu and Siegmund was that the material fails at the critical separation δc regardless of the separation rate and the stress carried by the linear elastic spring Ⓑ. Therefore, the failure of the cohesive element may occur in a sudden manner at high separation rates as shown in Fig. 14b. The TSL used in [55] is an exponential one similar to the Xu and Needleman law presented in Table 1. The one-dimensional differential equation to be solved for the stress in the cohesive element is written as:   dtCZ _ ηt þ kt ¼ η þ k δ_ þ ktCZ dδ

(26)

with tCZ being the traction in the spring Ⓐ (the index CZ is used to emphasize that this spring has a cohesive behavior, while the other one is just a linear elastic spring). The variables κ and η are the additional parameters for the viscoelastic behavior. A viscoplastic model has also been used particularly for time-dependent behavior of polymers. Those models follow the same principle as the constitutive laws adapted from plasticity discussed in the previous section. Here, the material separation is split into an elastic and a viscoplastic part: T = f(δ  δvp). The foundations that are used also in current publications are set by Corigliano and Ricci [41] or Tijssens et al. [56]. The latter were aiming at the modeling of craze widening, for which they distinguish between elastic behavior, craze widening (after craze initiation), and craze breakdown. Interestingly, the craze initiation is assumed to be dependent on

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1743

the hydrostatic stress (it will be discussed later how a stress triaxiality can be incorporated into cohesive elements), while craze breakdown is a constant quantity independent of time, temperature, or constraint. The evolution of viscoplastic normal and tangential separations is written in the following form: n δ_vp

   A σ c T n _t δ 1 ¼ δ_0 exp  σ c  vp   θ      Tt A τ c Tt A τ c _ 1þ ¼ Γ0 exp 1  exp  θ τc θ τc

(27)

with the parametersδ0 , A , σ c , Γ_ c , τc 7 The temperature is denoted by θ. This viscoplastic behavior starts when the yield potential reaches zero: f ðtn , σ h Þ ¼ 3=2σ h  1=2A þ

B  tn ¼ 0 6σ h

(28)

Note that this function only contains the normal traction and the hydrostatic stress, σ h, but not the tangential traction.

3.7.2 Temperature Dependence The temperature imposed to a modeled structure may imply several effects to the cohesive zone: 1. An elevated or low temperature may affect the strength or fracture energy of the cohesive zone, that is, the cohesive zone parameters may change compared to the room temperature values. 2. If temperature gradients exist in the structure, the temperature may be transferred from one crack face to the other, and thus laws of heat conduction, convection, and radiation may apply, depending on the status of the cohesive zone, whether the crack has already formed or damage is just evolving. 3. The dissipated energy due to localized straining and damage may lead to a significant temperature increase, so-called adiabatic heating, which can be considered by the cohesive interface as a heat source, when the crack propagation rate is very high. Temperature-dependent parameters are simple to implement and often used in the literature whenever needed without placing special emphasis on the implementation. This effect is important in general for hot cracking. If cracks occur in structures loaded at high temperatures, the heat can be transferred across the crack flanks via radiation and convection. In the damaged region ahead of the crack tip, heat conduction of the finite volume of the cohesive interface plays a significant role as well. Hattiangadi and Siegmund [57] developed a Note that Γ_ 0 is the parameter used in [55], which in contrast to what one might expect has the dimension of a separation rate.

7

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I. Scheider

thermomechanical cohesive zone model for fiber reinforced ceramics, which includes radiation and heat conduction of the bridging fiber and the gas in the micro-cracked region. The heat flux within the cohesive zone is then calculated by qCZ ¼ hCZ ½½θ ¼ ½hf þ hg þ hr ½½θ

(29)

with hf, hg being the thermal conductivity in the fiber and the gaseous regions, respectively, and hr the radiation. While the radiation was supposed in [57] to depend only on the temperature difference [[θ]], the conductivity depended also on the separation. In [58], it was assumed that the direction of the fibers bridging the crack, m, has to be taken into account for the heat transfer, and the fiber conductivity depends on the separation via a damage variable d = δ/δ0: qcz ¼ ½½1  dhf þ hr ½½θ ¼ ½½1  dhf þ hr ½½θm qnCZ ¼ ½½1  dhf þ hr ½½θm  n qtCZ ¼ ½½1  dhf þ hr ½½θm  t

(30)

While heat conduction is a major issue in high-temperature applications, the third point, adiabatic heating, considered here, is also relevant for crack propagation without an external heat source: The energy dissipation by generation of new crack surfaces may lead to a significant temperature increase, such that temperature-dependent material parameters may affect the results. This issue has been discussed for various material systems, for example, by Fagerström and Larsson [47] for high-speed cutting of ductile metals, or by Bjerke and Lambros [59] for dynamic fracture of PMMA. The latter used the following equation for the temperature increase: 1 ΔθðtÞ ¼ ρcp hdet

ðt

tðτÞ  δ_ðτÞdτ

(31)

0

Of course, the consideration of an increasing temperature is only useful, if cohesive zone parameters or material parameters in the adjacent bulk material (e.g., yield strength) are temperature dependent.

3.7.3 Dependence on Further Field Variables Until date, the TSL was always dependent on quantities, which could be retrieved from the interface itself (displacement/separation, temperature). However, several investigations can be found in the literature, where the constitutive behavior depends on additional quantities associated with the adjacent bulk material, only. The reason can be (a) So-called multi-physics problems, for example, hydrogen diffusion leading to embrittlement of the structure or (b) Dependence of the TSL on tensor quantities like strains (and not only separations), the stress tensor (not only the stress vector at the interface), or a strain rate (instead of the separation rate)

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

1745

For multi-physics problems, it is usually a matter of the finite element system to provide the respective field quantities, which the cohesive parameters depend on. In the following, the problem of hydrogen embrittlement will be touched exemplarily due to its practical relevance. Like temperature, which was already mentioned above, the hydrogen content can be given at the nodes. The physical basis for hydrogen diffusion (Fick’s law of diffusion) is equivalent to heat transfer and can be modeled by substituting the hydrogen content with temperature, if a solver for temperature-dependent problems exists [60]. Extended laws, which include, for example, stress dependence of the hydrogen diffusion, have been used in [61], and an even more sophisticated law can distinguish between a hydrogen concentration in the metal lattice and a second concentration of trapped hydrogen, naturally depending on plastic deformation. This has been employed for crack propagation analyses, for example, in [62, 63]. In these publications, it is assumed that the cohesive strength and energy decrease with increasing hydrogen concentration, while the critical separation remains constant. The second point in the regime of additional field quantities is the dependence on stress, strains, and associated tensors. While a dependence on strain has been reported in the literature [64] and the strain rate has been used for dynamic simulations instead of the separation rate [51], the most common application is the dependence on stress quantities, particularly the hydrostatic stress, σ h ¼ 1=3 σ ii , or the stress triaxiality h = σ h/σ eff (σ eff is the von Mises stress). The reason is that it is well known that the failure strain of real materials is highly dependent on stress triaxiality (often denoted by constraint dependence). It will be shown in the following section that at least for ductile metals the cohesive parameters should be constraint dependent; namely, the cohesive strength is increasing with triaxiality while the cohesive energy is decreasing [51, 65–67]. It is commonly agreed that both quantities saturate at high stress triaxialities, let say, above h = 3 [68]. Reasons for constant parameters in a simulation are either that the constant values are valid within a certain range for which they have been identified, or that the effect is negligible. The latter is particularly effective for ductile materials where the fracture energy contains a large portion of plastic energy, such that the sensitivity of Γ0 is small. The problem with triaxiality-dependent parameters is that h is neither available within the cohesive element nor at the nodes, it has to be retrieved from the adjacent continuum element in a finite element context, that is, the finite element problem becomes nonlocal, which involves additional numerical difficulties.

4

Considerations for Cyclic Loading

Whenever fatigue, that is, damage due to cyclic loading, is involved, the TSLs used for monotonous loading cannot be used anymore, since the unloading-reloading path is the same, see Fig. 6, and a repeated load does not change the material status. Therefore, specific laws for cyclic loading have been introduced with the beginning

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I. Scheider

of the twenty-first century. There are several different routes for the fatigue simulations using the cohesive zone model: 1. Each cycle is simulated explicitly, that is, the complete load cycles are considered. Since fatigue may evolve after thousands or even millions of cycles, this technique can only be used for low cycle fatigue (depending on the size of the model, for at most hundreds of cycles). 2. For high cycle fatigue, it is possible to introduce a so-called pseudo-time, which represents the cycles. For this case, one has to consider the loading amplitude and (Fmin/Fmax) ratio, which then triggers damage and thus material degeneration. 3. Another possibility for a larger number of cycles is the so-called cycle skipping technique. First, a number of cycles is modeled according to item 1 or 2, and the damage evolution per cycle is observed. When a trend is recognized, the damage evolution can be extrapolated for a larger number of cycles. These two steps can be repeated to simulate a considerable number of cycles. Two features are common to all fatigue approaches: Fatigue occurs due to repeated loading even if the physical material separation, δ, does not increase; and there is an endurance limit below which the fatigue damage does not evolve. However, beside these similarities, principle differences exist in the constitutive laws between the models described in items 1 and 2 above. In the review article by Kuna and Roth [69], these models, which “are formulated in terms of damage rate per load cycle [item 2 in the enumeration above] or which include external load parameters (e.g. load ratio) explicitly in the damage evolution equation, are assessed as not constitutive.” As an extension of the TSL for monotonic loading, several different laws for damage accumulation for cyclic loading have been introduced in the literature, starting in 2001: – A reduction of the cohesive strength during repeated loading – An unloading/reloading hysteresis – A damage term similar to Eq. (8), which includes an evolution based on repeated loading The reduction of cohesive strength was promoted first by Roe and Siegmund [28, 70]. They introduced a damage variable, D, which reduces the cohesive strength: n=t

n=t

T 0, cyc ¼ ½1  DT 0

(32)

The damage evolution has a stress threshold, σ f, and a separation threshold δf; If either the effective separation (Eq. 5) or the effective traction (Eq. 20) is below this threshold, damage does not increase. The law for the damage rate under cyclic loading is given in a very simple manner by:

52

Numerical Simulation of Material Separation Using Cohesive Zone Models

 

σf δeff T eff _ Dcyc ¼ P  H δacc  δf δ T 0, cyc T 0

1747

(33)

The Heaviside function, H, and the Macaulay brackets h. . .i are responsible for avoiding damage accumulation below the thresholds. The accumulated separation, δacc, is calculated by. δacc ¼

ðt

j δ jdτ

(34)

0

While this value is always increasing and thus the Heaviside function will be greater than zero after a number of cycles, the stress level provides a strict barrier, such that damage does not evolve for Teff < σ f . In order to increase damage also under monotonic loading, a second damage variable, Dmon, is used which is simply the increase of separation (also after a threshold), divided by a final separation, at which the cohesive zone fails: δeff  δf Dmon ¼ P δ  δf

(35)

The total damage in this model is then calculated based on Dmon and Dcyc by: D¼

ðt

max D_ cyc , D_ mon Þdτ

(36)

0

This law has been used, modified, and extended by several researchers, for example, Siegmund’s own group [71], who, opposite to their first investigations, did not use a plastic unloading mechanism but an elastic unloading. Also Xu and Yuan [72] and Jha and Banerjee [73] developed the model further. The latter used a different monotonic TSL, which is even triaxiality dependent. While features are added to the original model, other characteristics are left behind (e.g., only one separation component is considered in [73]). Li and Yuan [74] have proposed a different damage evolution without any separation threshold, but a nonlinear stress component with an additional exponent, m, and a stress ratio, n:  m j δ_eff j tn _ Dcyc ¼ P  n Hðσ eq  Sf Þ δ T n0

(37)

The second approach models material degradation by different slopes of the unloading and reloading curves, which may change by evolution of damage. Also, the slope may change over the load cycles. This approach was developed in 2001 by Nguyen [75], parallel to the first models reducing the cohesive strength, see above. The unloading/reloading stiffness is actually independent of the TSL under monotonic loading, which can be understood as an envelope to which the model

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I. Scheider

always returns, if the separation is large enough. However, damage occurs also below this envelope, under any reloading, which can be expressed by the unloading/ reloading relation:

T0 þ K  δ_ δ_ < 0 _ f _ T¼ (38) , with K   ¼ , K_ ¼ K þ δ=δ þ_ _ K δ δ>0 δ0 This approach was adopted by other groups as well, see e.g [76]. While the unloading stiffness does not change in (38), later publications by other researchers have combined the unloading/reloading stiffness with a damage evolution law, for example, [77]: T n0 ½1  Dδn δn0 D_ ¼ α½T n β htn  T f iγ

tn ¼

(39)

A version of unloading stiffness was introduced by Maiti and Geubelle [78] in order to extend the range of applications to the high cycle fatigue regime with the following stiffness change depending on the number of cycles: K_ ¼

8
KIC) can be measured. The tensile sample geometry and surface finish conditions are illustrated in Fig. 9a. Five specimens were prepared and tested for each α-ratio. The notch a was introduced through wire cutting with the notch width less than 0.2 mm. Fatigue precracking from the narrow notch is omitted in order to simplify the multiple tests. This is probably justified because of the relatively large crack-tip plastic zone Δap in those small samples with width W of only 44 mm. Under tensile loading, the plastic deformation at the crack tip and associated crack opening will induce crack blunting even at a sharp fatigue crack tip. If a blunt crack/notch tip does have a strong influence on KC measurements, it will overestimate the plane stress fracture toughness, which will be checked and discussed in a later section. A group of load and displacement curves of Q345B specimens are shown in Fig. 9b, where the structural yield load PY is estimated by the point where the load and displacement curve is deviated from the structural linear elastic behavior [23]. The definition of the structural yielding at PY has been illustrated earlier in Fig. 3. The yield strength σ Y is a material constant, but the structural yield load PY varies, as shown in Fig. 9b, depending on the structure/specimen size and geometry, a/W or α-ratio, and loading condition. With the PY measurements from Fig. 9, the nominal stress σ n can be evaluated for a specified restriction λ and crack-tip plastic zone Δap. The extreme case of λ = 100% and Δap = 0 (unrealistic, only used as a reference) is shown in Fig. 10a, and the case of λ = 90% and Δap = 0 is shown in Fig. 10b. The more realistic cases are shown in Fig. 10c, b, with 90% specimen end restriction, as

54

Elastic-Plastic and Quasi-Brittle Fracture

Fig. 9 (a) Tensile specimen size W = 40 mm, α-ratio (= a/W ) = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, (b) load and displacement curves of Q345B specimens with different α-ratios

1803

a

L = 60 mm W = 40 mm

L

a

W

Wire-cutting As-received W

α = a/W = 0.0, 0.1, 0.2, 0.3, … , 0.7

BT

b 210 PUT or Pax

180 PY

150

P (kN)

a = 0.1 a = 0.2 a = 0.3 a = 0.4 a = 0.5 a = 0.6 a = 0.7

120 90 60 30 0 0

3

6

9

12

15

Displacement (mm)

illustrated in Fig. 4, and assumed crack-tip plastic zone Δap = 1 and 2 mm. After λ and Δap in Eq. 16 are specified, the nominal stress σ n can be determined for every PY measurement. The equivalent crack ae is fully determined by Eqs. 12 and 13 for every given initial crack/notch and α-ratio. The linear relation between 1/σ n2 and ae, shown in Eq. 15, can then be used to determine the yield strength σ Y and fracture toughness KIC through curve fitting. The curve-fitted straight lines from Eq. 15 together with the experimental results and estimated σ Y and KC (for thin plates) values are shown in Fig. 10. It should be mentioned that curve fitting by Eq. 15 as shown in Fig. 10 is not required if both σ Y and KC (or KIC) are known for a given metal structure. For instance, Eq. 14 together with the results σ Y = 518 MPa and KC = 66 MPa√m in Fig. 10c (estimations in (d) can also be used) can be used to predict the elastic and plastic fracture of 9.7 mm thick plate structures of Q345B steel. Based on the results

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X. Hu and L. Liang

Fig. 10 (a) 100% restriction or λ = 1.0 and no crack-tip plastic zone or Δap = 0, (b) 90% restriction or λ = 0.9 and Δap = 0, (c) λ = 0.9 and Δap = 1 mm, (d) λ = 0.9 and Δap = 2 mm

in Fig. 10c, the complete failure curve, or safe design diagram, together with the elastic and plastic fracture results is shown in Fig. 11. As shown in Fig. 11, the yield strengths of AISI 1040 (σ Y = 515 MPa) and Q345B (σ Y = 515–520 MPa) listed in Table 1 have been accurately matched by the extrapolated yield strength σ Y = 518 MPa. In comparison to KIC = 54 MPa√m of AISI 1040 steel, KC = 66 MPa√m also appears to be reasonable (KC > KIC). If KC = 66 MPa√m is indeed correct, the minimum specimen size a and (W  a)  40 mm or 10a*1, i.e., W  80 mm would be required. However, the ASTM standard, specified by Eq. 3, only considers the separate influence from the front and back specimen boundaries. Instead of using a and (W  a), Eq. 14 relies on the equivalent crack ratio ae/a*1, which considers the combined influence from both the front and back boundaries. Considering the condition, ae/a*1 > 10, based on the combined boundary influence, the minimum crack a needs to be at least 140 mm (= 700 0.2) to ensure LEFM and KC are valid (the crack-tip plastic zone Δap  1 mm). It can be seen that the test results shown in Fig. 11 are well within the EPFM region (0.1 < ae/a*1 < 10), but closer to the yield strength σ Y controlled fracture region. Yet the EPFM results are still able to give a reasonable estimation on KC and LEFM region (ae/a*1 > 10), which is further away from the test range. The LEFM

54

Elastic-Plastic and Quasi-Brittle Fracture

1805

Fig. 11 Yield strength σ Y controlled region with ae/a*1 < 0.1, EPFM region with 0.1 < ae/ a*1 < 10, and LEFM KC region with ae/a*1 > 10, as determined by results in Fig. 10c

curves with KC = 60 and 70 MPa√m are also shown in Fig. 11, together with KIC of AISI 1040 carbon steel (= 54 MPa√m). Considering the scatters in experiments, KC may still be used in design even if W is less than 700 mm. However, W = 80 mm does appear to be too small to use KC according to the results in Fig. 11 if the combined influence from both the front and back specimen/structure boundaries are considered together. The test results of Aluminum Alloy 6061 of the test geometry illustrated in Fig. 9 (BT = 6 mm) are shown in Fig. 12. Four specimens were tested for each α-ratio. The specimen end restriction is somewhat smaller, λ  75%, for the softer aluminum in comparison with the harder Q345B carbon steel. Again, after the end restriction parameter λ and crack-tip plastic zone Δap are estimated, the nominal stress σ n can be evaluated from every PY measurement by Eq. 16. The equivalent crack ae is a pure crack geometry measurement, so ae values determined previously for Q345B specimens can also be used for Aluminum Alloy 6061 specimens with the same α-ratio (size W is the same for both metals). Then the linear relation specified by Eq. 15 can be used to determine both the yield strength σ Y and plane stress fracture toughness KC as shown in Fig. 12. In some cases, an accurate crack-tip plastic zone measurement may not be possible, so reasonable estimations for Δap are required. To show the sensitivity of σ Y and KC with the variation in Δap, Δap = 2 and 4 mm have been selected and shown in Fig. 12. The relatively flat stress distribution at the crack plane as illustrated in Fig. 4 explains why the final results are not too sensitive to Δap. Therefore, although the specimen end restriction parameter λ adds some degree of difficulty and uncertainty in application, the test condition also makes the final results be less dependent on the accuracy of Δap measurements.

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X. Hu and L. Liang

The results in Fig. 12a have been rearranged in the form of Eq. 14 and shown in Fig. 13. The strength of Aluminum Alloy 6061 listed in Table 1 (σ Y = 515 MPa) is matched very well by the extrapolated value of 279 MPa from the EPFM test results. Two extra LEFM lines with KC of 40 and 55 MPa√m are also given in Fig. 13 to compare with the LEFM curve with KC of 48 MPa√m determined from Fig. 12a. The KC value of 51 MPa√m for 6 mm thick Aluminum Alloy 6061 has been found in the literature [28, 29] and the complete KC – BT curve is shown in Fig. 14. Considering the KC scatters for BT = 25 mm obtained from other methods [28], the extrapolation of KC = 48 MPa√m for BT = 6 mm in Fig. 13 is satisfactory. The averaged KC value from Fig. 12a, b, i.e., KC = 50 MPa√m for BT = 6 mm, is shown in Fig. 14. It is clear from the comparison shown in Fig. 14 that the test results from the plate specimens

Fig. 12 (a) Crack-tip plastic zone Δap = 2 mm, λ  75%, (b) Δap = 4 mm, λ  75%

Fig. 13 The complete failure or design curve for Aluminum Alloy 6061 with Δap = 2 mm

54

Elastic-Plastic and Quasi-Brittle Fracture

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Fig. 14 The plane stress fracture toughness KC and specimen thickness BT [28, 29], the plane strain fracture toughness KIC = 29 MPa√m, as listed in Table 1 for Aluminum Alloy 6061

with 0.2 mm wide notches did not overestimate the plane stress fracture toughness KC of Aluminum Alloy 6061. It is a big bonus that fracture toughness measurements can be performed with notched specimens without the need of fatigue precracking. Since all tests are well within the EPFM region, as shown in Fig. 13, and fairly close to the yielding strength controlled fracture zone (ae/a*1 < 0.1), crack-tip yielding and crack blunting are expected. Probably, the final crack tip condition of a narrowly machined notch is similar to that of a sharp fatigue crack after crack blunting. However, to ensure valid EPFM measurements, an initial notch should be as narrow as possible, e.g., 0.2 mm. Using σ Y and KIC of Aluminum alloy 6061 listed in Table 1, the ASTM standard or Eq. 3 indicates the minimum specimen thickness BT should be at least 28 mm to have valid KIC measurements. The KC curve in Fig. 14 suggests that the required thickness should be around 50–60 mm, which is twice as much as the ASTM requirement. If KC = 50 MPa√m for BT = 6 mm is used in the estimation of crack length and (W  a), the ASTM requirement is 82 mm. But in Fig. 13, the true LEFM condition should be 1260 0.2 = 250 mm so that the crack-tip plastic zone, Δap = 2 mm, can be ignored. However, 160 mm (82 2) is probably already satisfactory in practice. It is also worthwhile checking the tensile test results from unnotched Q345B specimens (α = 0), listed in Table 2. Note that the specimens were not polished, having the as-received and wire-cutting surface conditions, as illustrated in Fig. 9a, and five tests were performed in total. The average value of 354.9 MPa is only about 70% of 515 MPa listed in Table 1, showing that the surface finish of a specimen or structure can have a strong influence on yielding. If a notch is introduced in an unpolished specimen, the stress concentration from the notch is much greater than that of any microdefects on the specimen surface. That is the reason why the EPFM model, Eq. 14 or Eq. 15, can determine the yield strength σ Y from notched specimens without polishing. Therefore, the EPFM results of Aluminum Alloy 6061 and Q345B carbon steel (or AISI 1040) shown here seem to suggest that the standard tests using polished specimens for measurements of the

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Table 2 Apparent yield strength from unpolished tensile specimens of Q345B Specimen number 1 2 3 4 5 Average value

Apparent yield load PY (kN) 140.6 143.7 132.1 132.5 137.3 137.2

Apparent yield strength σ Y (MPa) 363.9 371.2 342.5 342.1 354.7 354.9

yield strength σ Y and large specimens for the fracture toughness KIC or KC can be replaced by small notched metal specimens without polishing.

4.2

Three-Point-Bending (3-p-b) Tests of Concrete

Concrete fracture is unique since experiments such as measurements of the maximum loads Pmax of small notched 3-p-b specimens are very easy to perform, but interpretations are very difficult because of the coarse concrete structures and associated discontinuous crack growth. It can be understood that quasi-brittle fracture of concrete has been studied extensively in the past 40 years [e.g., 24, 30–42] in search of an adequate fracture model and suitable material parameters to describe the complicated fracture phenomena in such coarse-structured composites as illustrated in Fig. 5. Since nearly all the current fracture models are based on continuum mechanics analyses, unique fracture behaviors associated with the coarse concrete structures as illustrated in Fig. 5 cannot be satisfactorily explained. It has been noticed that even the J-integral approach [10, 30] requiring a welldefined constitutive cohesive stress relation and fracture energy is no longer valid because the specific fracture energy is size dependent or influenced by the specimen boundaries [34, 37–41]. Besides, different size effect models exist due to the lack of clear physical definitions of those empirical parameters used in curve fitting of the experimental results [e.g., 36, 43]. It is not too surprising to see that quasi-brittle fracture of concrete still remains challenging up to now if most of the current fracture models are still based on continuum mechanics and the coarse structure characteristics, such as the maximum aggregate dmax, have not been clearly specified. In this chapter, the approach or concrete fracture model developed in Sect. 3 will be adopted. It has two major advantages: (1) it has the modification to consider the discrete nature of concrete fracture; and (2) it is almost identical to the EPFM model developed in Sect. 2 for metals. Above all, the new approach for quasi-brittle fracture of concrete and elastic-plastic fracture of metals is simple enough mathematically for graduate students and design engineers to use. Comprehensive 3-p-b tests of concrete with the maximum aggregate dmax = 10 mm have recently been reported [43], which provides a good example for detailed data analyses using the new quasi-brittle fracture model. Those 3-p-b specimens had four different sizes, W = 40, 93, 215, 500 mm, and five different

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notch/size ratios, α = a/W = 0.0, 0.02, 0.075, 0.15, and 0.30. All measurements of the maximum load Pmax were listed in the study so that the quasi-brittle fracture analyses using Eqs. 18, 19, 20 and 21 can be performed relatively straight forward. The experimental results together with the linear relations specified by Eq. 18 are shown in Fig. 15 for three different stepwise cracking assumptions, Δafic = 0, 10, and 20 mm or the discrete number β = 0, 1, and 2. The tensile strength ff and fracture toughness KIC have also been calculated with a smaller β increment of 0.2 over the range from 0 to 3 and listed in Table 3 so that the influence of the averaged discrete number can be clearly seen.

Fig. 15 (a) and (b) for the case that the discrete number β = 0 so no fictitious crack growth, or Δafic = 0, (c) and (d) β = 1, or Δafic = 10 mm as dmax = 10 mm, (e) and (f) β = 2, or Δafic = 20 mm [44].

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Table 3 Influence of the average discrete number b on ft and KIC Δafic (mm) 0.0 3.8 7.6 11.4 15.2 19.0 22.8 26.6 30.4 34.2 36 41.8 45.6 49.4 53.2 57.0

ft (MPa) 5.50 5.25 5.01 4.78 4.57 4.38 4.20 4.03 3.88 3.73 3.60 3.47 3.35 3.24 3.13 3.03

KIC (MPam1/2) 1.22 1.24 1.24 1.26 1.29 1.32 1.35 1.38 1.41 1.49 1.53 1.63 1.75 1.83 2.00 2.23

a* 1 (mm) 12.22 13.96 15.35 17.48 19.92 22.65 25.77 29.29 33.25 39.89 45.47 55.40 68.62 79.58 102.1 135.4

76"mm"

W = 305 mm

β 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

dmax = 19 mm

Assumed straight-line fictitious crack Δafic is an average measurement of a real crack

To clearly show the difference between a straight TTT fictitious crack growth Δafic and an irregular crack growth formed by stepwise cracking, the real crack front measurement [25, 26] previously shown in Fig. 5 is also included in Table 3. The scatters in Δafic (and Pmax), due to the difference in stepwise irregular crack growth in the “identical” concrete specimens, can be matched by the discrete number β-range, e.g., between 1 and 2, as the corresponding ft and KIC values are fairly close to the commonly reported results for concrete. The “discrete particle” analysis can be further simplified by selecting the mid-point of the β-distribution as illustrated in Fig. 6. This average β value is fairly close to 1 [14, 44] if W/dmax < 20. Figure 15a, b is the same, and so are (c) and (d), and (e) and (f) as well. The difference is that Fig. 15a, c, e is for size W and Fig. 15b, d, f is for the α-ratio. Estimations of the tensile strength ft and fracture toughness KIC from Fig. 15 are hardly changed if measurements from one particular size W, e.g., 40, 93, 215, or 500 mm, or from one particular α-ratio are removed from the whole group [44]. It shows that the extrapolations of ft and KIC are fairly reliable although the scatters in experimental results shown in Fig. 15 are quite large. Figure 15c, e is presented again in Fig. 16a, b for the complete fracture range (or design diagram) from the tensile strength ft-controlled fracture to quasi-brittle fracture and then finally to fracture toughness KIC-controlled fracture (LEFM). It means that quasi-brittle fracture of an un-reinforced concrete structure can be predicted if both ft and KIC are known.

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a

b

Fig. 16 (a) The average discrete number β = 1, the fictitious crack Δafic = dmax = 10 mm, and (b) β = 2, and Δafic = 20 mm [44].

It should be mentioned that if the yield strength σ Y of a metal has been proven to be sensitive to the surface finish condition, as shown by the results in Table 2, the tensile strength ft of brittle concrete can only be even more sensitive to preexisting defects (which are plenty in coarse-structured concrete). While polishing can remove microdefects on a metal surface, it is not helpful to a concrete specimen because preexisting defects, in the same scale of maximum aggregate dmax, exist throughout the entire concrete structure. Similar to the metal cases discussed in this chapter, notched concrete specimens are less influenced by those preexisting defects. Therefore, more reliable tensile strength ft can be extrapolated from notched concrete specimens by the quasi-brittle fracture model introduced in this chapter. Generally speaking, this extrapolated tensile strength ft in the region of the discrete particle

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model, as illustrated in Fig. 7, is higher than that measured from direct tensile tests because the influence of preexisting defects cannot be ignored without a dominant notch. The results in Table 2 tell the same story. It is not unrealistic that the true tensile strength ft of concrete can be over 4 MPa if direct tensile tests show the average strength is around 3 MPa [45].

5

Summary of Major Equations of the Elastic/Plastic and Quasi-Brittle Fracture Model

The well-known stress intensity factor K equation, commonly seen in textbooks, has been reformulated in this chapter and used to derive the simple elastic and plastic fracture model in presented in Sect. 2. The quasi-brittle fracture model presented in Sect. 3 is virtually the same as that in Sect. 2 if the crack-tip plastic zone Δap is replaced by the fictitious crack growth Δafic. In other words, the elastic and plastic fracture of metal structures and quasi-brittle fracture of concrete and rock can be analyzed and predicted by the non-LEFM model presented in Sects. 2 and 3 of this chapter. Based on the analysis in Sect. 2, the stress intensity factor K for LEFM can be expressed in two different forms: pffiffiffiffiffiffiffiffiffi K ¼ Y  σN π  a σY σ N ¼ rffiffiffiffiffiffi a a1 under the condition a/a*1  1, and the specimen/structure back boundary is far away (or the large plate condition). For a finite structure, the nominal stress σ n at the crack plane should be used and LEFM relation becomes: σY σ n ¼ rffiffiffiffiffiffi ae a1 The equivalent crack ae is introduced to consider the combined influence from the front and back specimen/structure boundaries. The general elastic-plastic and quasibrittle fracture relation is given by: σY σ n ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ae 1þ  a1 which is practically as simple as the above LEFM equation containing √(ae/a*1). The influence of the crack-tip plastic zone Δap at the structural yielding load PY in a metal structure/specimen and the fictitious crack growth Δafic at the maximum load

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Pmax in a concrete structure/specimen is considered by a bi-linear stress distribution for the nominal stress σ n. That is a constant stress distribution over Δap or Δafic, so both metals and concrete are modeled by exactly the same bi-linear stress distribution. Based on the properties of cold-drawn AISI-1040, σ Y = 515 MPa and KIC = 54 MPa√m, it can be estimated from the ASTM standard for plane strain fracture toughness that the specimen thickness B has to be thicker than 27.5 mm to have valid KIC measurements. To use this KIC value of 54 MPa√m in a structural analysis, crack a and un-cracked ligament (W  a) have to be longer than 27.5 mm as well. So the application of LEFM KIC is rather limited even for this relatively brittle steel. Q345B tested in this study is almost identical to AISI-1040 in terms of yield strength (520 vs. 515 MPa) and ultimate tensile strength (598 vs. 585 MPa). Since the crack-tip yielding zone determines the energy dissipation and crack growth resistance, it is expected that the fracture toughness of Q345B would be similar to that of AISI-1040, which has KIC value of 54 MPa√m. The plane stress fracture toughness KC of Q345B determined in this study is around 66 MPa√m. It should be mentioned that in general, a blunt crack/notch has the tendency of overestimating the fracture toughness of a metal. However, we deal with elastic and plastic fracture in this study, and the size of the crack-tip plastic zone is not small in comparison with the sample width W. Crack tip plastic deformation and crack blunting are expected even from a sharp fatigue crack. It seems from the plane stress fracture toughness values determined in this study that any over-estimation effect could be ignored for the metals tested.

6

Conclusions

The concept of fracture toughness KIC and LEFM methodology as explained in textbooks is very important to structural integrity analyses, which is best explained by the inclusion of LEFM in university education around the world. However, direct applications of LEFM and fracture toughness KIC as outlined in many textbooks and handbooks in design are very limited. Safety designs of engineering structures containing (fatigue) cracks urgently require a simple and useful non-LEFM method, which can be handled easily by graduate students and design engineers. Taking the above consideration into account, a simple EPFM approach based on the stress intensity factor K commonly used for LEFM analysis as described in textbooks has been presented in this chapter. To verify the new EPFM approach and illustrate its usefulness in safety design, the model has been used to analyze elastic and plastic fracture of AISI 1040 like Q345B carbon steel and Aluminum Alloy 6061, and quasi-brittle fracture of concrete. Graduate students and design engineers, who can understand the concepts of the stress intensity factor K and fracture toughness KIC, should be able to use the new EPFM model without much difficulty. The main purpose of this chapter is to introduce the simple non-LEFM model to graduate students and design engineers so that they can use it in safe design and

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fracture analysis when LEFM and traditional strength criterion cannot be used. Hopefully, the objective has been achieved although the simplified approach presented in this chapter is only a simple approximate method for modeling of elastic and plastic fracture of metals and quasi-brittle fracture of brittle coarse-structured composites such as concrete and rock. The major conclusions of this chapter are: 1. Elastic and plastic fracture of a metal structure is fully predictable if both yield strength σ Y and LEFM fracture toughness KIC (or KC) are given. 2. LEFM fracture toughness KIC (or KC) and yield strength σ Y can be determined from small metal samples with different initial cracks/notches. 3. There is no material constant within the elastic and plastic fracture region. Hopefully, the simple EPFM model presented in this chapter can provide a useful example that a simple solution does exist even for a complicated engineering and mechanics problem. In the case discussed in this chapter, the complicated elastic and plastic fracture of metals and quasi-brittle fracture of concrete and similar brittle composites has been formulated by the common stress intensity factor of LEFM. From the viewpoint of structural integrity analysis, a simple and reasonably accurate engineering solution or analytical model, which can be used easily by graduate students and design engineers to predict structure safety or failure, is always desirable.

References 1. Atkins AG, Mai YW. Elastic and plastic fracture: metals, polymers, ceramics, composites, biological materials. Chichester: Ellis Horwood Limited; 1988. 2. Anderson TL. Fracture mechanics fundamentals and applications. Boca Raton: CRC Press; 1995. 3. Hertzberg RW. Deformation and fracture mechanics of engineering materials. 4th ed. New York: Wiley; 1996. 4. Callister WD Jr, Rethwisch DG. Materials science and engineering an introduction. 8th ed. Hoboken: Wiley; 2010. 5. Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids. 1960;8:110–04. 6. Well AA. Unstable crack propagation in metals: cleavage and fast fracture. In: Proceedings of the Crack Propagation Symposium, vol. 1, Paper 84, Cranfield; 1961. 7. Irwin GR. Plastic zone near a crack and fracture toughness. Sagamore Research Conference Proceedings, vol. 4; 1961. 8. Barenblatt GI. Mathematical theory of equilibrium cracks in brittle fracture. In: Advances in applied mechanics, vol. VII. New York: Academic; 1962. 9. Burdekin FM, Sone DEW. The crack opening displacement approach to fracture mechanics in yielding materials. J Strain Anal. 1966;1:145–53. 10. Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech. 1968;35:379–86. 11. Cotterell B, Reddel JK. The essential work of plane stress ductile fracture. Int J Fract. 1977;13:267–77. 12. Cotterell B. The past, present and future of fracture mechanics. Int J Fract. 2002;69:533–53.

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13. Zhu XK, Joyce JA. Review of fracture toughness (G, K, J, CTOD, CTOA) testing and standardization. Eng Fract Mech. 2012;85:1–46. 14. Wang YS, Hu XZ, Liang L, Zhu WC. Determination of tensile strength and fracture toughness of concrete using notched 3-p-b specimens. Eng Fract Mech. 2016;160:67–77. 15. http://www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwi 74pPTs8fLAhXLq5QKHf0BC0EQFggdMAA&url=http%3A%2F%2Fhighered.mheducation. com%2Fsites%2Fdl%2Ffree%2F0072953586%2F240416%2FMaterials_Properties_Database. xls&usg=AFQjCNFZCsXRbEJ3VN3QvgXHXSS_qcNvBQ&bvm=bv.117218890,d.dGo. 16. ASTM E1820-15a. Standard test method for measurement of fracture toughness. West Conshohocken: ASTM International; 2015. 17. Hu XZ, Wittmann F. Size effect on toughness induced by crack close to free surface. Eng Fract Mech. 2000;65:209–21. 18. Duan K, Hu XZ. Specimen boundary induced size effect on quasi-brittle fracture. Strength Fract Complex. 2004;2:47–68. 19. Duan K, Hu XZ, Wittmann F. Scaling of quasi-brittle fracture: boundary and size effect. Mech Mater. 2006;38:128–41. 20. Hu XZ, Duan K. Size effect: influence of proximity of fracture process zone to specimen boundary. Eng Fract Mech. 2007;74:1093–100. 21. Hu XZ, Duan K. Size effect and quasi-brittle fracture: the role of FPZ. Int J Fract. 2008;154:3–14. 22. Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. 3rd ed. New York: ASME Press; 2000. 23. Hu XZ, Guan JF, Geng L. A simple elastic and plastic fracture model. Inter J Fract. 2017. 24. Cotterell B, Mai YW. Fracture mechanics of cementitious materials. London: Blackie Academic & Professional, An Imprint of Chapman & Hall; 1996. 25. Swartz SE, Yap ST. Evaluation of recently proposed recommendations for the determination of fracture parameters for concrete in bending. In: Proceedings of 8th International Conference on Experimental Stress Analysis; 1986. p. 233–44. 26. Swartz SE, Refai TME. Influence of size effects on opening mode fracture parameters for precracked concrete beams in bending. In: Shah SP, Swartz SE, editors. Fracture of concrete and rock; SEM-RILEM international conference, Houston. 17–19 June; 1987. p. 242–54. 27. Quality Certificate Y25-05-05. Anyang Iron and Steel Company Limited (http://www.aysteel. com.cn). 28. Amiri S, Lecis N, Manes A, Giglio M. A study of a micro-indentation technique for estimating the fracture toughness of Al6061-T6. Mech Res Commun. 2014;58:10–6. 29. NASGRO 4.11. Reference Manual, NASA Johnson Space Center; 2004. 30. Hillerborg A, Modeer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res. 1976;6 (6):773–82. 31. Bažant ZP, Oh BH. Crack band theory for fracture concrete. Mater Struct. 1983;16(3):155–77. 32. Jenq YS, Shah SP. Two parameter fracture model for concrete. J Eng Mech. 1985;111(10): 1227–41. 33. Karihaloo BL, Nallathambi P. An improved effective crack model for the determination of fracture toughness of concrete. Cem Concr Res. 1989;19(4):603–10. 34. Hu XZ, Wittmann FH. Fracture energy and fracture process zone. Mater Struct. 1992;25:319–26. 35. Carpinteri A. Fractal nature of material microstructure and size effects on apparent mechanical properties. Mech Mater. 1994;18(2):89–101. 36. Karihaloo BL, Abdalla HM, Xiao QZ. Size effect in concrete beams. Eng Fract Mech. 2003;70:979–93. 37. Abdalla HM, Karihaloo BL. Determination of size-independent specific fracture energy of concrete from three-point bend and wedge splitting tests. Mag Concr Res. 2003;55(2):133–42.

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38. Duan K, Hu XZ, Wittmann FH. Boundary effect on concrete fracture and non-constant fracture energy distribution. Eng Fract Mech. 2006;70(16):2257–68. 39. Yang ST, Hu XZ, Wu ZM. Influence of local fracture energy distribution on maximum fracture load of three-point-bending notched concrete beams. Eng Fract Mech. 2011;78:3289–99. 40. Karihaloo BL, Murthy AM, Iyer NR. Determination of size-independent specific fracture energy of concrete mixes by the tri-linear model. Cem Concr Res. 2013;49:82–8. 41. Cifuentes H, Alcalde M, Medina F. Measuring the size-independent fracture energy of concrete. Strain. 2013;49:54–9. 42. Hoover CG, Bazant ZP. Comparison of the Hu-Duan boundary effect model with the size-shape effect law for quasi-brittle fracture based on new comprehensive fracture tests. J Eng Mech ASCE. 2014;140:480–6. 43. Caglar Y, Sener S. Size effect tests of different notch depth specimens with support rotation measurements. Eng Fract Mech. 2016;157:43–55. 44. Guan JF, Hu XZ, Li QB. In-depth analysis of notched 3-p-b concrete fracture. Eng Fract Mech. 2016;165:57. 45. Hu XZ. Size effect on tensile softening relation. Mater Struct. 2010;44:129–38.

Coupling Models of New Material Synthesis in Modern Technologies

55

Anna Knyazeva, Olga Kryukova, Svetlana Sorokova, and Sergey Shanin

Contents 1 2 3 4 5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Reactions in Technological Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Multiphase and Multicomponent Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Coupling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Surface Electron-Beam Treatment with Modified Particles . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Coating Composition Formation during Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Controlling Synthesis of Materials on a Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Today, additive manufacturing (AM) technologies attract large attention. One can define these technologies as step-by-step construction or synthesis of parts from identical or different materials. Stereolithography, selective laser melting, selective laser sintering, hot isostatic pressing, and combined technologies belong to AM technologies. Electron-beam (EB) technologies are also popular. Coating synthesis and surface treatment using EB, composite material synthesis, and A. Knyazeva (*) Tomsk Polytechnic University, Tomsk, Russia Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia O. Kryukova Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia e-mail: [email protected] S. Sorokova · S. Shanin Tomsk Polytechnic University, Tomsk, Russia # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_39

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various technologies of material joining could be also added to additive manufacturing. Since the experimental investigation of technological processes (dynamics of chemical composition, evolution of structure and properties) is very complicated, mathematical modeling can help in this field. This section presents the approach to predictive model construction. Together with chemical reactions accompanying the change of properties, the technological conditions are analyzed. General equations include the energy equation, balance equations for species, equilibrium equation, and governing equation containing terms describing numerous cross effects. Examples of particular model are presented for accepted technologies of surface treatment. The first model describes the surface modification using electron beam and particles that dissolve in a melting pool and change the composition. The second model describes the composition change during the coating deposition and includes coupling effects between transfer processes and mechanical ones. Multilayered coating forms on the metal surface during ion deposition from gas, solution, or plasma. The third model gives the modeling concept for choosing the technological conditions for homogeneous coating creation on the substrate using chemical reactions and external heating. These models relate immediately to additive manufacturing where metals are used.

1

Introduction

Technologies of material treatment and synthesis are quite diverse. New complex methods have been developed to refine the properties of materials and expand the area of their application [1–3]. This inevitably complicates the technologies and increases the variety of factors, which hinder the control. Modern science-intensive technologies demand the engagement of mathematical modeling methods substituting full-scale experiment. More sophisticated models bring additional problems. On the one hand, the development of new technology is impossible without comprehensive theoretical foundation; on the other hand, this necessitates the improvement of old and devising of new mathematical modeling methods. This relates to the models of material synthesis and surface treatment. In modern surface engineering, hybrid technologies have been developed when the interrelation between various physical and chemical factors generates unique properties of multicomponent, multilayer, and gradient coatings [4–6]. Mathematical models of surface treatment and coating synthesis taking into account the interrelation between processes of various physical natures are related to the category of coupling models. Similar models are usual in theoretical chemical engineering, in combustion science, and in biophysics and have many common elements with them as they are based on thermodynamics of irreversible processes where numerous cross effects have been studied. The formation of the surface and coating properties occurs immediately during treatment and is connected with physicochemical conversions. Hence, the

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description of staging of chemical reactions in accordance with the laws of thermodynamics and kinetics is the cornerstone of the coupling model. The change in composition and effective properties is the consequence of physicochemical conversions. To enhance the prognostic properties of the models, it is necessary to include the calculation of the properties using various known approaches. The concomitant phenomena of thermal and nonthermal nature change the staging of chemical processes and affect the result ambiguously. And, finally, technological conditions should be chosen that will ensure the anticipated result. These conditions include not only the type and character of external action (basic and additional ones) but also the compositions of green mixtures, geometrical parameters, the rates the conditions change with, etc. All these factors should be taken into account when physicochemical phenomena are modeled for modern additive manufacturing technologies involving laser or electron beam [7, 8].

2

Chemical Reactions in Technological Conditions

Synthetic generation of chemical substances and new materials in chemical engineering is performed with the help of special apparatuses and equipment – chemical reactors [9, 10]. As for chemical reactor, working regime of which should be chosen, one can analyze the surface layer modified due to the activation of physical and chemical processes by external action, growing coating, whose phase composition changes during deposition, etc. At the conditions of surface treatment and coating deposition, the final material or article (desired product in the theory of chemical reactors) is obtained immediately in the form suitable for practical application. Chemical reactor “coincides” with area where desired product forms. In modern technologies of surface treatment and synthesis of new materials, chemical reactions can proceed in different aggregate state (solid, liquid, and gaseous), and as any reactions they are divided into quick and slow, exothermic and endothermic, simple (elementary) and complex, etc. Additionally, it is very problematic to identify the detailed reaction mechanisms leading to certain material structure. To construct the models for technological processes as well as to model the chemical reactors, initial data are necessary: the information on physicochemical processes forming expected composition and structure and also the information on processes that can prevent the necessary result. Further, the data are required on heat release or absorption during the reaction, on the possible conversion level of initial substances into final product, and on the stability of intermediate and final phases. Partially, the chemical thermodynamics answers these questions. However, it is not enough. Because the reagents should meet with each other for the reaction to start, and the diffusion in condensed phase is a slow process, it is necessary to crush or stir the reagents. The effectiveness of stirring depends on the viscosity of species, on the mutual solubility of reagents and products, on the velocities of reagent and product fluxes, on the reaction rates and locations, on the arrangement of reagent input, etc. It is necessary to support specific temperature regime to optimize the reaction rates. The acceleration or retardation of reaction can be connected with the action of external fields of various

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physical natures, with external mechanical loading and internal mechanical stresses (when reactions proceed with the participation of solid substances). As in the theory of chemical reactor [9, 10] and macro kinetics [11], when the model of technological process of surface treatment and/or coating synthesis is built, it is allowed to restrict the manifold of reactions by some summary reaction scheme reflecting the transition of the reagents into final reaction product. The rate ϕ of chemical reaction ν1 R1 þ ν2 R2 þ . . . ! products

(1)

depends on the reagent concentrations Ri (i = 1, 2, . . .) raised to some powers and on the temperature T corresponding to the Arrhenius equation:
 E ϕ ¼ k½R1 ν1 ½R2 ν2 . . . exp  , RT

(2)

where k is the pre-exponential factor, [Ri] are the concentrations of the reagents, νi are the corresponding stoichiometric coefficients in Eq. (1), E is the activation energy, and R is the universal gas constant. It should be noticed that mass action law allowing relation (1) was formulated historically for reactions proceeding in reversible conditions. However, the formal application of this law to irreversible processes makes possible the construction of verisimilar models with prognostic properties. For elementary reactions, the powers in (2) are integer numbers. For reactions reflecting the course of events only summarily, the powers can be fractional and not equal to stoichiometric coefficients. Additionally, when solid phases participate in the reactions, the function Φ(product) appears in the kinetical laws and reflects the reaction retardation by the product layer:
 E ϕ ¼ k½R1 ν1 ½R2 ν2 . . . exp  ΦðproductÞ: (3) RT The formal kinetic parameters and heat effects of reactions Qi are determined experimentally or calculated with the help of classic and modern thermodynamical theories. An important part of macro kinetics of chemical reactions lies in the establishment of the dependence of the reaction rate on the pressure and/or stresses. According to Evans–Polanyi equation, connecting the rate constant of the process in solid phase with the pressure and deduced on the base of the transition state theory @ln k ΔV 6¼ ¼ , @p RT the reaction rate decreases with the pressure р (because ΔV6¼ > 0). Here, ΔV6¼ is the difference between the volumes of activated complex and substances entering into a reaction.

55

Coupling Models of New Material Synthesis in Modern Technologies

1821

When kinetical theory of the strength for solid substances was constructed in [12], major attention was devoted to the decomposition reactions of elastically strained chemical bonds. In this case, the Eyring and Cozman equation
 Ea  σV 6¼ k ¼ k0 exp  RT

(4)

describes the stress action on the reaction rate, where σV6¼ is the work of elastic stresses in this reaction and V6¼ is the activation volume. Here it is implied that acting stresses lighten the transition through energy barrier but do not affect its height. In displacement reactions, the reactive capacity of deformable fragments of organic molecules is characterized by the sign and value of the coefficient of sensitivity to stresses [13]. Literature also discusses the quantum mechanical approach to the problem of disintegration of elastically stressed chemical bonds. Stress field appearing as the result of mechanical action in certain areas of solid substance can relax in various ways: heat release, new surface formation, generation of various defects in crystals, and initiation of chemical reactions in solid phase. Dominating way of relaxation depends on material properties, loading conditions, and particle characteristics (size, shape, etc.). In general case of condensed matter, each particle is exposed to permanent and changing in time action of surrounded molecules. Hence, the local properties of media are significant for the reaction kinetics additionally to the properties of reacting particles. To take into account the features of the reagent structure, it is possible to use the concept of micro stresses and other internal parameters or the transition to the tensor description of the concentrations and chemical potentials of reacting substances. The influence of stresses and strains at macro level is stipulated by the dependence of reaction rate in a solid body on pressure (stresses) [14]: ln

k ΔEðσ Þ Π ¼γ : ¼ k0 RT RT

The evaluations have shown that the potential energy density of deformations Π can be compared with the activation energy of a chemical reaction [15]. In general, the chemical reaction rate can be controlled in two ways (as it takes a place in any thermodynamical system): one can change the internal energy (that happen during heating-cooling) and do the work. The work is performed due to free energy F diminution. In classical thermodynamics, dF ¼ pdV: In thermodynamics of deformable body, the change of local free energy is df = σ ijeij; so, for isotropic body [16], one can write dF ¼ κ

σ kk mdekk , 3 ρ0

(5)

1822

A. Knyazeva et al.

where m is average molar mass of substances entering into a reaction, ρ0 is the density of reacting mixture (or solid phase reagent) in non-deformed state, κ is the coefficient of reaction rate sensitivity to the stress work, and mdekk/ρ0 is molar activation volume, σ kk/3 =  p. Hence, the Eqs. (4) and (5) are equivalent. When the detailed reaction scheme in a multicomponent and multiphase system is written, the mass action law, Arrhenius law, and the relations of types (4) and (5) are assumed correct for each of reactions. Additionally, the conditions of interphase heat and mass transfer determine the time of occurrence for the reagents in the conditions required for reaction implementation.

3

Properties of Multiphase and Multicomponent Materials

The majority of materials obtained by modern technologies of treatment and synthesis and used in various industrial branches are multiphase and multicomponent. Especially this relates to composites. Multiphase and multicomponent coatings are deposited from gaseous phase and plasma or synthesized on the substrate immediately with the employment of the heat of chemical reactions and external heating; composites on metallic base with ceramic inclusions can be synthesized using chemical heat release as well. Physical and chemical properties of such materials are not known a priori and demand special experimental determination or the calculation with independent well-founded models. The methods of micromechanics [17] unfortunately could not be used here because the material structure is not known a priori and forms during synthesis or treatment. At present, phase field method [18] allows describing the structure of such materials with known structure and properties. Besides, the realization of this method demands the determination of large number of parameters that is a very sophisticated stand-alone work. However, based on thermodynamics [19, 20], one can add the calculation of equivalent properties that alter with the composition formation into the models of technological processes. Therefore, according to the definition, heat capacity is determined as partial temperature derivative of entropy S under fixed conditions CV ¼


 1 @S , T @T V

and because the entropy is additive value, S = SA[A] þ SB[AB] þ . . ., the heat capacity can be calculated by “mixture law”: CV ¼ CVA ½A þ CVB ½AB  þ . . . This result is correct for either multiphase material (each phase occupies a certain area in a thermodynamical system) or multicomponent material (each species potentially presents in each point of the system). The choice of measurement units depends on the problem under study.

55

Coupling Models of New Material Synthesis in Modern Technologies

1823

Thermal conductivity coefficient, as any other property characterizing the transfer process, is a more complex value. The assignment of the following relation: λ ¼ f ðλk , ξk Þ, where ξk are the concentrations or volume fractions of phases and λk are their thermal conductivity coefficients, is the basic task of different theories. To examine this task, some detailed information is necessary on geometry, orientation, and location of all phases of heterogeneous material. In this connection, different investigators model the structure of heterogeneous system and use various mathematical approximations [21]. So, in the case of the simplest structure represented by the system of two unrestricted different plates paralleled to the heat flux, the following estimation takes place: λ ¼ λ1 ξ1 þ λ2 ξ2 :

(6)

If the layers are perpendicular to the heat flux, the identity
 ξ1 ξ2 1 λ¼ þ λ1 λ2

(7)

is correct for equivalent thermal conductivity coefficient. If the structure is previously unknown, the thermal conductivity coefficient is some average value, for example, "
 # 1 ξ1 ξ2 1 λ¼ λ1 ξ1 þ λ2 ξ2 þ þ : 2 λ1 λ2 Averaging can be carried out by various ways. If each phase is multicomponent, it is not possible to validate one or another formula to calculate the dependence of properties on composition. The properties of each multicomponent phase are experimentally measured values. Two-way estimations of equivalent thermal conductivity coefficient (6) and (7) are similar to the investigations of equivalent (or effective) elastic properties. Formula (6) corresponds to Voigt approximation, and formula (7) is Reuss approximation. For equivalent (effective) elastic bulk modulus K and shear modulus μ of two-phase composite, these formulae are as follows: K V ¼ K 1 ξ1 þ K 2 ξ2 ; μV ¼ μ1 ξ1 þ μ2 ξ2 and
KR ¼

ξ1 ξ þ 2 K1 K2

1


; μR ¼

ξ1 ξ2 þ μ1 μ2

1

:

(8) (9)

For any averaging way [22], allowing finding the properties K and μ, the inequalities

1824

A. Knyazeva et al.

K R  K  K V , μR  μ  μV

(10)

take place. These simple estimations are sufficient to create the models with prognostic properties. The formulae are easily generalized for multiphase materials. Of course, one can use the latest theories.

4

General Equations

To formulate the models of technological processes of material synthesis and their surface treatment taking into account physicochemical conversions in different phases, the models of multicomponent deformable media are used. They include continuity equation: dρ þ ρ∇  v ¼ 0, dt

(11)

balance equation for internal energy u: ρ

du ¼ A þ σ  ∇v, dt

(12)

ρ

dv ¼ ∇  σ þ F, dt

(13)

dCk ¼ ∇  Jk þ ωk , k ¼ 1, 2, . . . , n, dt

(14)

motion equation:

and equations for species: ρ

where ρ is medium density, v is velocity vector for mass center, σ is total stress tensor, F is mass forces vector, Ck are mass concentrations for species, Ck = ρk/ρ, ρk n n n P P P are their partial densities, ρ ¼ ρk , so Ck ¼ 1; A ¼ ∇  JT þ Fk Jk , ρv ¼

n P k¼1

k¼1

k¼1

k¼1

ρk vk, and Jk  ρkwk = ρk(vk  v) are diffusion fluxes, JT is the heat flux, and ωk

is the sum of sources and sinks of k species in chemical reactions: ωk ¼

r X i¼1

νki mk ϕi ,

55

Coupling Models of New Material Synthesis in Modern Technologies

1825

where νki are stoichiometric coefficients of k-species in reaction i, mk is molar mass of k-species, r is the reaction quantity, and ϕi is the rate of the ith reaction in mol/(m3s). When models of technology processes with physical and chemical conversions are constructed, it is convenient to go from (12) to thermal conductivity equation in the form depending on the medium under study. So, for isotropic solid deformable (elastic) body, this equation takes the form of ρCe

dT dekk ¼ ∇  Jq þ W ch  3KαT T , dt dt

(15)

where Jq is summary heat flux including heat flux due to species diffusion, W ch ¼ r P Qi ϕi , Qi is the heat effect of reaction i, K is the isothermal bulk modulus, αT is

i¼1

the linear coefficient of thermal expansion, and Ce is the heat capacity at constant strains. In viscoelastic materials, the term connected with viscous dissipation appears in the right part of Eq. (15). The terms connected with external heating or with the action of other physical forces can also emerge. In the case of incompressible media, the total derivatives can be replaced by partial ones. The approximation ρ = ρ(T, Ck) is relevant in numerous applications. In particular case of immobile (solid) reaction product according to Eq. (14) ρ

dCk ¼ ωk : dt

(16)

These equations are completed by governing equations including the equations for the fluxes of heat and masses, source terms from these equations, and equation system of state. Governing equations contain various variants of cross effects between different physical phenomena. The construction of governing equations is grounded on irreversible thermodynamics and experiment. In chemical technology and chemistry, molar concentrations yk = ρk/mk (in mol/ m3) or relative molar concentrations are often used nk Yk ¼ X ðiÞ

ni

y ¼ Xk , yi ð iÞ

where nk is the number of moles of k-species. Relative molar concentrations Yk connect with the mass concentrations by relations Ck =mk Y k mk y mk Yk ¼ X and Ck ¼ X  Xk Ci =mi Y i mi yi m i ðiÞ

ðiÞ

The examples of coupling models are presented below.

ðiÞ

(17)

1826

A. Knyazeva et al.

5

Examples of Coupling Models

5.1

Surface Electron-Beam Treatment with Modified Particles

The mathematical model of electron-beam surface treatment using modified particles is intended to interpret the phenomena observed in the technology described in [23]. Particles can be soluble and insoluble; the substances entering the solution can react with each other and with basic material, which leads to the formation of new composition. Therefore, the model should be a more complex than known thermal models calculating only the temperature [7, 8]. This comes from the necessity of determined phase and chemical composition of the surface layer after the treatment and predicted possible range of properties after the treatment depending on technological parameters. Mathematical model [24] supplemented by chemical reactions in the melting pool includes thermal conductivity equation averaged along the thickness Hof treated layer (Fig. 1), ceq ρeq



 @T @ @T @ @T ¼ λeq λeq þ þ W ch þ Qd ϕ @t @x @x @y @y     e0 σ 0 T 4  T 40  qe ,  H

(18)

kinetical equation for volume fraction ηp of particles entering the melt from special device,   @ηp ¼ qm  ϕ T, ηp , . . . , @t

(19)

and equations of chemical kinetics for phases forming in the treated layer. The thermal conductivity equation, in general case, includes volume source due to chemical reactions Wch and heat effect of dissolution Qd. The heat source in (18), connected with the action of electron beam, is written as

Fig. 1 The scheme of electron-beam building up 1 – specimen (detail), 2 – workaccelerated gun, 3 – powder portioner

2

V

3

particles

Electron beam molten pool L1

1 L2

H

55

Coupling Models of New Material Synthesis in Modern Technologies

( qe ¼

1827

0   , jyj > y0 =2; , 2 2 q0 exp ðx  VtÞ =at , jyj  y0 =2,

where y0 is the scanning thickness, V is the velocity of scanning electron-beam motion along axis Ox, and at is the effective radius of the beam. At some distance xa from the external energy release (or in the same place), particles enter the melting pool. Their properties and composition are different from those of treated material. The density of particle beam is distributed according to Gauss’ law h   i qm ¼ qm0 exp  ðx  xa  VtÞ2 þ y2 =a2p , where qm0 is maximal density of particle beam and the value of ap is determined by the radius of the pipe ejecting the particles. The area particle source can be similar to the area energy source. In terms of thermo-kinetic model [24], the fraction of particles in melt ηp (or undissolved inclusions in forming solid solution) follows from Eq. (19), where function ϕ, characterizing the dissolution rate, in general case, depends on the temperature, particle dispersity, solubility of elements in each other, and local parameters of hydrodynamic flow in the melt. Corresponding to dissolution theory [25], function ϕ(T, ηp) can be presented as     ϕ T, ηp ¼ ϕ0 ηp k0 expðEa =RT Þ and reflects the basic regularities of dissolution process. If dissolution rate in the melt is limited by diffusion, activation energy Ea is close to the activation energy for diffusion. If the transfer of elements from solid into liquid phase is accompanied by the chemical interrelation in the interphase, then Ea is some effective value depending on the dominance of that or other processes. The form of kinetic function ϕ0(ηp) depends on the processes determining the dissolution rate at micro level. Constant k0, according to known dissolution models [9, 25], is determined by local characteristics of hydrodynamic flow. In general case, dissolution process can be endothermic Qd < 0 and exothermic Qd > 0, which depends on various physicochemical factors determining the dissolution mechanism. The heat capacity in (7) is calculated by different formulae depending on the system under study and temperature [24, 26]. 1. In the case of melting/crystallization of the base (without particles), the heat capacity change close to the melting temperature is described as   ðcρÞeq ¼ ρb, s Lph δ T  T ph þ where δ is Dirac delta function:



ðcρÞb, s , T < T ph , ðcρÞb, l , T  T ph

(20)

1828

A. Knyazeva et al.

δ ! 1, T ¼ T ph ; δ ¼ 0, T 6¼ T ph , Lph is latent heat of melting, Tph is the melting temperature of pure basic material, index “b” corresponds to the material of the base, index “s” corresponds to solid phase, and index “l,” to the liquid phase. 2. In the case of the system with insoluble and nonfusible particles, their heat capacity was included in the calculation of equivalent heat capacity:     ðcρÞeq ¼ 1  ηp Lph ρb, s δ T  T ph (   ðcρÞb, s 1  ηp þ cp ρp ηp , T < T ph   þ ðcρÞb, l 1  ηp þ cp ρp ηp , T  T ph ,

(21)

where index “p” corresponds to particles. 3. In the case of the system with dissolving particles without chemical interaction, the theory of two-phase zone has been used for forming solution; then ðcρÞeq ¼ cρ þ Lph ρb, s

@ηl , @T

cρ ¼ ðcρÞs ð1  ηl Þ þ ðcρÞl ηl ,

(22)

where ðcρÞs ¼ ðcρÞp, s ηp þ ðcρÞp, s ξ þ ð1  ξÞðcρÞb, s ðcρÞl ¼ ðcρÞp, l ξ þ ð1  ξÞðcρÞb, l ηl þ ηs þ ηp ¼ 1: Here, expression 1  ηl = ηs þ ηp is the volume fraction of total solid phase (the base with undissolved particles); ηp is the volume fraction of undissolved particles; (1  ηp) is the volume fraction of the solution (solid or liquid); ηl is the volume fraction of the liquid phase calculated from relation
 T liq  T n ; (23) ηl ¼ 1  T liq  T sol (1  ξ) is the volume fraction of the basic material in the solution; ξ is the volume fraction of the materials of the solution particles that complies with equation   @ξ ¼ ϕ T, ηp , . . . ; @t

(24)

ηs = 1  ηp  ηl is the fraction of solid solution; Tliq is liquidus temperature; Tsol is the solidus temperature on the phase diagram of the studied system; parameter n

55

Coupling Models of New Material Synthesis in Modern Technologies

1829

varies in the theory for different alloys. When T  Tliq the fraction of liquid phase is ηl = 1  ηp; when T  Tsol liquid phase is absent ηl = 0. The temperatures of liquidus and solidus, Tliq and Tsol, are approximated by suitable curves. 4. In the case of the system with chemically reacting species, the problem is supplemented by kinetic equations for volume concentrations. Equivalent heat capacity depends on the capacities of all species and has been calculated by formula (22), where n X   ðcρÞs ¼ ðcρÞp, s ηp þ ξ1 þ ξ2 ðcρÞb, s þ ðcρÞk ξk and

(25)

k¼3

ðcρÞl ¼ ðcρÞp, l ξ1 þ ξ2 ðcρÞb, l þ

n P k¼3

ðcρÞk ξk .

Here, ξ1 is the volume fraction of the materials of the solution particles, ξ2 ¼ 1 ξ1 

n P k¼3

ξk is the volume fraction of the base material in the solution, and ξk are the

fractions (relative molar or volume) of forming chemical compounds. The boundary conditions do not take into account the heat losses from the plate ends: x ¼ 0, 1 :

@T ¼ 0 and y ¼ 0, 1 : @x

@T ¼0 @y

At initial time moment, the temperature is equal to initial one T0, liquid phase and particles are absent: t ¼ 0 : T ¼ T 0 , ηl ¼ 0, ηp ¼ 0, ηs ¼ 1; y2 ¼ 0:502: The system Cu-Al (Cu is the particles, Al is the base) is a suitable model. Dissolution of these metals in each other is accompanied by physicochemical conversions. In the area of copper mass concentration from 0 to 0.7% on the state diagram of this system [27], there are two chemical compounds: CuAl and CuAl2 that can appear through the following reactions: Cu þ Al ! CuAl, ðIÞ CuAl þ Al ! CuAl2 , ðIIÞ Cu þ 2Al ! CuAl2 : ðIIIÞ

(26)

The designations of molar concentrations of reagents and products as y1 = [Cu], y2 = [Al], y3 = [CuAl], and y4 = [CuAl2] and the approximation of equality of individual velocities of the species lead to the following system of balance equations:

1830

A. Knyazeva et al.

  dy1 ρ ¼ Cu ϕ T, ηp  ðϕ1 þ ϕ3 Þ, dt MCu dy2 ¼ ðϕ1 þ ϕ2 þ 2ϕ3 Þ, dt dy3 ¼ ϕ1  ϕ2 , dt dy4 ¼ ϕ2 þ ϕ 3 : dt The rate of each reaction depends both on the concentrations correspondingly to the law of mass action and on the temperature corresponding to Arrhenius law (2). However, for described reactions, the data on formally kinetic parameters are absent or data of various authors are quite different. A good way for evaluating formally kinetic parameters is based on thermodynamics and described in [28]. Calculated values of pre-exponential factors k0i, activation energies Eai, and reaction heats Qi are presented in Table 1. Second reaction goes practically without heat effect. Mass concentrations Ck are calculated by formulae (17), if necessary. In principle, under irreversible conditions, the chemical compounds corresponding to another part of the state diagram can form as well. The model can be supplemented by other compounds similarly to previous method. For this system, the temperatures of liquidus Tliq and solidus Tsol in (23) are approximated as T sol ¼ as þ bs  Cm þ cs  C2m and T liq ¼ al þ bl  Cm þ cl  C2m , where Cm is mass fraction of Cu (in free and bounded state) following from Cm ¼

X1 : X1 þ X 2

    Here, X1 ¼ y1 þ y23 þ 23 y4 m1 corresponds to copper; X2 ¼ y2 þ y23 þ 13 y4 m2 corresponds to aluminum.

Table 1 Formally kinetic parameters of the reactions for the system Cu-Al Reaction I II III

k0i 1.64  108 1.517  108 0.478  107

Eai [J/mol] 33406 32234 57030

Qi [J/cm3] 2779 3.676 1460

55

Coupling Models of New Material Synthesis in Modern Technologies

1831

To calculate the thermal conductivity coefficient, in the first place, the thermal conductivity of solution has been calculated according to the mixture rule: λb ¼ λ1 C1 þ λ2 C2 þ λ3 C3 þ λ4 C4 , and then, the effective properties of composite have been found by formulae (6) and (7): λef f ¼ λb ð1  ηp Þ þ λp ηp

1 1 1 ¼ ð1  ηp Þ þ ηp λef f λb λp

The elasticity modulus has been calculated in a similar way. The problem was solved numerically with implicit difference scheme of second approximation order with spatial step and first approximation order with time step using coordinate splitting and linear double sweep method. The calculation has been carried until a quasi-stationary regime was established with given accuracy. As in pure thermal-physical problems corresponding to surface thermal treatment, welding, and buildup, the process quickly reaches the quasi-stationary regime, when the shapes of melting pool and heat-affected zone and the value of maximal temperature in the treated zone do not change. Liquidus temperature corresponds to the first value; the temperature before the structure change in the substance corresponds to second value. The example of temperature field evolution is shown in Fig. 2. The maximal temperature is observed evidently in the area of maxima of energy release from external source (Fig. 2a). In Fig. 2b, dark gray color corresponds to the melting pool. Maximal size of lightly gray color corresponds to the thickness of the heat-affected zone along axis Oy and is determined for T = 500 K. In contrast to purely thermal models [9, 10], the model presented gives additional information. For example, there is a two-phase zone in addition to the melt and solid phases. Therefore, the shape of the area where liquid phase presents (Fig. 3a) does not replicate that of the melt area. As a whole, two types of critical conditions exist in this model [24]. The first of them determines the start of the dissolution process or the initiation of chemical reactions. The second one determines the conditions for the formation of homogeneous coating (without embedded particles) and composite coating (with prior or with the matrix changing due to the partial dissolution of particles and chemical reactions with remaining particles). Various variants of the coatings are observed in the technology [23]. The dependence of the particle fraction on technology parameters is the most interesting for the evaluation of the properties of forming coating. The distribution of particle fraction ηp(x, 0) along the axis of energy source motion, shown in Fig. 4 for different time moments t and various values of source flux density q0, argues that if the particles dissolve only partially, the composite obtained is inhomogeneous: the left part of treated surface contains more undissolved particles than the right one. Obviously, the higher the power density of the source, the larger part of particles dissolves in the melting pool.

1832

A. Knyazeva et al.

Fig. 2 Evolution of temperature field during establishment of the quasi-stationary regime, q0 = 4000 W/cm2, qm0 = 0.9 1/sec, V = 1 cm/sec

Phase composition of the matrix depends on all parameters: power density q0, mass flow qm0, and motion velocity V. Increasing mass particle flow (Fig. 5) allows both evaluating their fraction in the coating and CuAl2 fraction. As a result, the matrix consists of copper dissolved in aluminum, CuAl2 phase, and small quantity of CuAl phase. Surface treatment with modifying particles enables obtaining coatings with different properties. Therefore, the thermal conductivity coefficient (Fig. 6a) should be both higher than the initial coefficient in the left part of the plate and lower in the right part of that; however, the elasticity modulus (Fig. 6b) increases everywhere. The dotted lines in Fig. 6 correspond to the calculation by formulae (6) and (8); the solid lines are obtained from formulae (7) and (9). Real welding of Fe-B-Ti [29] is a more complex process. The results of experimental investigation of this together with corresponding mathematical model are presented in [30]. The buildup of the coating was carried out on low-carbon steel. Modifying particles included boron and titanium. Dissolution process of these

55

Coupling Models of New Material Synthesis in Modern Technologies

1833

Fig. 3 The fraction of liquid phase (а) and solid phase (without particles, b) at time moment t = 9.5 sec; q0 = 4000 W/cm2; qm0 = 0.9 1/sec

b 0,32 0,24 0,16 4 0,08 2 3

7

6

5

0,00 1

0

2

4

6

x,cm

8

10

Volume fraction of particles,hp

Volume fraction of particles,hp

a

0,28 0,21 0,14 4

0,07 2 3 0,00 1 0

2

5

4

7

6

6

x,cm

8

10

Fig. 4 Distribution of particle fraction along the energy source motion for different time moments: qm0 = 0.9 1/sec; t (sec) = (1) 0.025, (2) 0.25, (3) 1, (4) 2.1, (5) 3, (6) 4.5, (7) 6.5. (a) q0 = 2500 W/cm2; (b) q0 = 4000 W/cm2

metals is accompanied by various conversions. Correspondingly to the state diagrams for Fe-B-Ti system, one can write down six following reactions with nine participating species (Fe, B, Ti, FeB, FeTi, TiB2, TiB, Fe2B, TiFe2): 1: Fe þ B ! FeB 2: Fe þ Ti ! FeTi 3: Ti þ 2B ! TiB2

4: Ti þ B ! TiB 5: 2Fe þ B ! Fe2 B 6: 2Fe þ B ! TiFe2

A. Knyazeva et al.

Relative mass concentration, Al

a 1,00

1

2

3

4

5

0,95 0,90 0,85 0

2

x, cm

4

6

Relative mass concentration, CuAl2

1834

b 0,16 0,12 0,08 0,04

0

2

5

4

3

2

1

0,00

x, cm

4

6

Fig. 5 Mass concentrations of phases Al (a) and CuAl2 (b) in the coating for different time moments. q0 = 4000 W/cm2; for dotted lines qm0 = 0.5 1/sec, for solid lines qm0 = 0.9 1/sec; t(sec) = (1) 0.7, (2) 1.7, (3) 2.4, (4) 3.5, (5) 5

b 2,75

85

2,50

2,25

5

3

2

6

4

1 0

Eeff, GPa

leff, Wt/(K*cm)

a

80 75

6

5

4

3

1 2 70

2

4

6

x,cm

8

10

0

2

4

x,cm

6

8

10

Fig. 6 Thermal conductivity coefficient (а) and elasticity modulus (b), calculated for different time moments; q0 = 4000 W/cm2; qm0 = 0.9 1/sec; t(sec) = (1) 0.2, (2) 0.9, (3) 2.75, (4) 4.75, (5) 7, (6) 9.5; for dotted lines, formulae (6) and (8); for solid lines, formulae (7) and (9)

There is a specific feature in this variant of the model. When the temperature exceeds the melting temperature of the base, turbulent intermixing could become significant for particle redistribution in the melting pool under pressure action of the moving source. Thus, the process can be taken into account as an additional term in the following equation for particles: ! @ηp, j @ 2 ηp, j @ 2 ηp, j ¼ qm, j  Φj ðT, ηp, j Þ þ Def f þ , j ¼ B, Ti, @t @x2 @y2 where Deff is effective intermixing coefficient of particles in the melt,

(27)

Concentrations

a

Coupling Models of New Material Synthesis in Modern Technologies

0.75

b

1

0.50

2

0.25 0.00 0.0

3 4

0.5

1.0

1.5

5, 6 2.0

t, sec

Concentrations

55

0.75

1835

1

0.50

2

0.25

3

0.00 0.0

0.5

1.0

5, 6

1.5

4 2.0

t, sec

Fig. 7 Phase concentration versus time in the point with coordinates x = 1 cm and y = 0 cm; (1) FeTi, (2) FeB, (3) TiB, (4) TiFe2, (5) Fe2B, (6) TiB2. q0 = 7000 W/cm2, qmo,B = 1.3; (а) qmo, Ti = 1.7; (b) qmo,Ti = 2.1

Def f ¼ DT , T  T ph ;

Def f ¼ DS , T < T ph :

In this case, the quasi-stationary regime establishes together with advancing of the source along the treated surface. But the obtained composition is more uniform. Different composition of modified layer corresponds to different technological conditions (Fig. 7a and b) which naturally leads to different effective properties. Thus, the suggested model of technology process of electron-beam surfacing with modifying particles predicts the composition of the surface layer and allows estimating the possible scatter of some effective properties when technology parameters are varied. It was shown that the phase composition of forming coating depends essentially both on initial species fraction (particles fraction) and on the parameters characterizing the heating rate (source motion velocity, energy beam density, and mass flow). Similar regularities are observed in real systems. The model refers specifically to additive technologies. Additive technologies can be optimized on the base of similar models by studying physical and chemical phenomena during the buildup of layers with step-by-step change of composition or layers from materials with different properties.

5.2

Coating Composition Formation during Deposition

The operation life of parts can be sufficiently enhanced by surface modification due to deposition of coatings from compounds of refractory metals. Such coatings are widely applicable in modern engineering. Multilayer coatings hold a special place in surface modification. Implementation of multilayer coatings containing films from wear-resistant and antifriction materials enables formation of composite structures with unique properties [31, 32]. The method of ion-plasma coating deposition, involving the sputtering of various targets, is one of the perspective methods for multilayer coating deposition [2, 3]. During the deposition, the distribution of elements and chemical compounds depends on various physical factors: thermal

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A. Knyazeva et al.

conductivity, diffusion, internal mechanical stresses due to temperature and composition gradients, and numerous cross effects, including thermal diffusion and diffusion thermal conductivity. The role of various factors was discussed in many papers. Experimental investigation of different processes interrelating with each other is very difficult. Mathematical modeling is a good alternative for similar experiments. It allows investigating each factor separately and predicting the composition and macroscopic properties for forming coating under variable technological conditions. Coupling models could be used for numerical experiment and can allow detecting new phenomena. The authors of [33] analyze the role of cross effects for different metals deposited from plasma together with nitrogen or carbon. The generalized variant of the models deals with a specimen in the form of a hollow cylinder with a coating growing on the external surface. The cylinder turns around the axis. This provides uniform coating formation (Fig. 8). The rate of the coating growth is given from experiment and is determined by the velocities of ions coming to the surface. Multilayer coatings are obtained when ions of refractory metals alternately enter the plasma. For example, in a coating growing from plasma containing carbon and alternating titanium and chromium ions, the carbides of metals are formed corresponding to three summary reactions: Fe þ C ! FeC; ðIÞ 3Cr þ 2C ! Cr3 C2 ; ðIIÞ Ti þ C ! TiC:ðIIIÞ

(28)

When a coating on the base of aluminum Al, nitrogen N2, and titanium Ti (metals enter the plasma also alternately) grows, the following chemical reactions are possible: 2Al þ N2 ! 2AlN; ðIÞ 2Ti þ N2 ! 2TiN; ðIIÞ 4Fe þ N2 ! 2Fe2 N; ðIIIÞ 8Fe þ N2 ! 2Fe4 N:ðIVÞ

(29)

Fig. 8 Illustration to the problem formulation on the coating deposition

Z R1 m1y1 q0 m2y2

R2

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Coupling Models of New Material Synthesis in Modern Technologies

1837

There are other compounds on the state diagrams. They could be taken into account, if necessary. Principally, one can write more detailed reaction schemes. Mathematical model in [33] consists of two coupling parts. The equation system necessary to determine the composition of a growing coating at any time moment includes energy equation in form of thermal conductivity Eq. (15) and balance equation of diffusion type (14) and kinetic type (16). The equations for the heat and mass fluxes follow from [34, 35]: Jq ¼ λT ∇T 

n1 X

Aj ∇Cj þ Aq ∇σ eii ,

(30)

j¼1

Jk ¼ ρ

n1 X

Dkj ∇Cj  ρCk STk Dkk ∇T þ Bk ∇σ eii , k ¼ 1, 2, . . . , n,

(31)

j¼1

where λT is thermal conductivity coefficient, Dkj are partial diffusion coefficients, STk are Soret coefficients, Bk are the coefficients of mass transfer under stress gradient action, the coefficients Aj describe the heat transfer due to the presence of concentration gradients (these coefficients are calculated through heats of transfer and Dkj), Aq is the coefficient of heat transfer under stress gradient (this coefficient is the derivative of other physical parameters and is not independent), and n is the general number of species. The detailed description of the coefficients for different systems is contained in [33, 36, 37]. These works demonstrate that coefficients Aj are zero when and only when the heats of transfer or Soret coefficients are zero. This indicates that cross effects cannot be omitted alternately. If there is a direct effect, the opposite effect also appears. The condition of infinitesimality of coefficients Ak and Aq in comparison to other coefficients is not obtained explicitly from some verbal proofs and cannot be used here. In whole, transfer coefficients are not independent. This is connected with the symmetry of the Onsager coefficient matrix [19, 20] and irreversible state eqs [34, 35]. The boundary conditions are native. Mass fluxes are absent in the internal surface of the cylinder. This surface is insulated. In the interface (between the coating and base material), the heat fluxes and temperatures are equal; the mass fluxes and chemical potentials of diffusing species are equal. This is the ideal contact. The conditions on the growing surface take into account the heating due to losses of kinetic energy by particles, heat losses due to thermal radiation, and mass entrance from plasma: Jq ¼ q0

  dξ  σ 0 e0 T 4  T 4W , dt

Jk ¼ mk yk

dξ , dt

where TW is the temperature of vacuum chamber walls,

(32)

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A. Knyazeva et al.

q0 ¼ k

X V2 i mi yi , 2 ð iÞ

(33)

and Vi are the velocities of the particles (ions) near the surface. The law of the coating growth is the implication of momentum conservation law: X

yi V i dξ ð iÞ ¼ X : dt yi

(34)

ðiÞ

In these equations, the summation goes with the species in plasma. Ions lose the charge when they reach the surface. At the initial time moment, substrate temperature is given and equal to T0; the coating is absent. Because the properties change due to phase and chemical conversions, all transfer coefficients are the functions of composition and temperature. Naturally, the velocities of elements and their concentrations near the growing coating can vary due to numerous random factors: nonuniform structure of the sputtered cathodes, presence of impurities, voltage oscillations, etc. To take into account the probability of random oscillations, pseudorandom coefficients αi were introduced into Eq. (34): X dξ ¼ dt

ðiÞ

α i yi V i

X ðiÞ

yi

:

(35)

These coefficients are random quantities in some given interval from m1 to m2 changing each tk seconds: αi ¼ rand ðm1 . . . m2 Þ, t ¼ tk : The second part of the problem is the problem of mechanical equilibrium of a hollow cylinder with changing thickness due to the growing coating. Because the melting temperature is not achieved at experimental conditions, one can use an analog of the quasi-static theory of thermal elasticity [38, 39], the quasi-static theory of thermal elastic diffusion [40] supplemented by physicochemical conversions. If strains are small, chemical compounds are immobile, and the observation time is high in comparison with the time of mechanical perturbation attenuation (which, as a rule, is true for real experimental conditions), one can come from motion Eqs. (13) to equilibrium equations. Stress and strains components are connected with temperature and composition as σ ij ¼ 2μeij þ δij ½λekk  Kw,

55

Coupling Models of New Material Synthesis in Modern Technologies

1839

where " ω ¼ 3 α T ðT  T 0 Þ þ

n X

# αk ðCk  Ck0 Þ ,

k¼1

αT are linear thermal expansion coefficient, αk are concentration expansion coefficients (they are calculated from molar volumes of species), and λ, μ are Lame coefficients. The cylinder is free from the action of external forces; its ends are not fixed. Taking into account the symmetry of the problem, only three components of strain tensor components will differ from zero, err ¼

du u , eɸɸ ¼ , and ezz ¼ const, dr r

and unknown constant can be derived during equilibrium problem solution. Analytical solutions of similar problems for constant properties are known very well in the thermal elasticity theory and in thermal elastic diffusion theory [38–40]. Time is a parameter in quasi-static problems. In contrast to known problems, the stresses in the treated zone appear not only due to nonuniform heating along the thickness and the difference in mechanical properties and thermal expansion coefficients of layers but due to the composition change corresponding to irreversible processes, diffusion, and chemical reactions [35]. When analytical methods are used, the properties are taken constant. In general case, the problem is solved numerically. In the coupling models based on Eqs. (13)–(15), kinetic-diffusion part is formulated in mass concentrations. Detailed form of source terms and coefficients included in the model depends on the system under study (on species quantity, number or reactions, individual properties of substances). As it was in the previous problem, the rates of chemical reactions depend on concentrations according to mass action law and on the temperature according to Arrhenius law (2). The reaction retardation by the reaction product is not taken into account because the transport of species is explicitly included in the model. Therefore, for the system with carbon, the reaction rates are ɸ1 ¼ k1 ðTÞC1 C4 , ɸ2 ¼ k2 ðTÞC21 C32 , and ɸ3 ¼ k3 ðTÞC1 C3 : Because the mass concentrations are dimensionless variables and ki(T) = ki0 exp (Ei/RT), so all dimension are hidden in pre-exponential factors. Hence, source terms ω1 ¼ m1 ðϕ1 þ 2ϕ2 þ ϕ3 Þ and ω2 ¼ 3m2 ϕ2 ; . . . ω7 ¼ m7 ϕ3

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are measured in kg/(m3sec), while the corresponding term in thermal conductivity equation is measured in J/(m3sec). From seven concentrations (four pure substances – C, Cr, Ti, and Fe – and three chemical compounds – FeC, Cr3C2, and TiC), only six are independent, because 7 X

Ck ¼ 1:

k¼1

The problem was solved numerically. A specially developed numerical algorithm was described in [41]. The concentrations of the species (elements and chemical compounds) for consecutive time moments were determined; the temperature, stresses, and strains were calculated for various variants of the model. The conditions in plasma were varied. The physical parameters used for the calculations are presented in Table 2. The data from [42] were used. The parameters of chemical reactions were calculated based on thermodynamics and are given in Table 3. Figures 9, 10, 11, and 12 illustrate the possibilities of the model. The distributions of mass concentrations in the coating and in the adjacent part of the substrate are shown in Fig. 9а–f for different time moments for conditions (34). Because the coating and substrate consist of the materials with high thermal conductivity, and the thickness of the substrate is small, the temperature in the system “substrate – coating” becomes uniform very quickly and changes only in time. The end of curves in the figures corresponds to the coating thickness for the corresponding time. The vertical dotted lines conditionally divide the substrate and coating changing from initial position of interface. At the initial stage, (t < 1200 sec), Table 2 Properties of substances Parameter λ ρ Cp m E ν αT

C 2 3.51 20.8 12.01 76 0.27 2

Cr 93.9 7.19 23.3 51.9 185 0.3 3.6

Ti 21.9 4.54 25.1 47.8 232 0.26 2.8

Fe 80.1 7.87 25.14 55.84 190 0.28 4

TiC 36.2 4.93 34.23 59.89 296 0.29 3.2

Cr3C2 67.3 6.68 98.449 180.009 217 0.3 3.8

FeC 60.1 7.6 29.5 68 200 0.29 4.7

Dimension W/(mК) 103 kg/m3 J/(molК) 103 kg/mol GPa 106 К1

Table 3 Kinetic parameters of the reactions Forming chemical compound FeC Cr3C2 TiC

Activation energy [J/ mol] 2∙105 1.7∙105 21,406

Pre-exponential factor 1.2∙1011 2∙104 2.46∙1014

Reaction heat [kJ/ mol] 51.8 85.43 11.64

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Coupling Models of New Material Synthesis in Modern Technologies

1841

chromium carbide forms during coating deposition. When iron penetrates into the coating, or carbon penetrates into the substrate, ferric carbide appears, which is accompanied by the decrease of chromium carbide. While the coating is thin, the formation of ferric carbide accelerates due to temperature growth. Further, the radiation hinders the temperature growth, atoms of carbon have to cover larger distance to iron, and ferric carbide formation is essentially retarded, i.e., the product layer itself retards this reaction. During t  [1200, 2400] the plasma composition changes, titanium comes to the surface instead of chromium, which is accompanied by titanium carbide formation (Fig. 9b). Because chromium is absent in plasma, its concentration in the coating is practically zero (Fig. 9c and d); chromium can appear

Fig. 9 The distributions of concentrations of carbon (a), titanium carbide (b), chromium (c), chromium carbide (d), ferric carbide (e), and titanium (f) in growing coating along the radius at different time moments (in sec): t = (1) 2800, (2) 3200, (3) 3600; y1 = 0.5, y2 = 0.45; the velocities of ions of carbon, chromium, and titanium (in km/sec): V1 = 12, V2 = 18, V3 = 3

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here only due to diffusion. Secondary change of the plasma composition occurs when t = 2400 sec. Carbon is permanently present in the plasma and its concentration rises all time (Fig. 9a). The interface shifts into the depth of the substrate due to the formation of the transient layer included to the coating. Ferric carbide does not appear in consequent layers. Similarly, when the coating from a metal and carbon deposits on the substrate from permanent plasma [33], the iron carbides form only near the interface. The coating composition depends on the velocities of ions, duration of the stages (changing in the plasma composition), relative fractions of ions in plasma, etc. Evidently, the increasing concentrations of Cr and Ti lead to the increasing of both these elements in the coating and corresponding carbides. The ion velocity increase leads to temperature growth and to the acceleration of diffusion and chemical reactions. The composition of the coating is not uniform along the thickness. One can distinguish the area occupied primarily by single phases. Numerical investigation has shown that numerical values of concentrations and the size of the transient zone depend on the phenomena taken into account in the calculation, similarly to [33]. For example, the activation of cross diffusion fluxes leads to the constriction of the area where ferric carbides form in comparison with the calculation where cross diffusion fluxes are not considered. On the one hand, cross diffusion fluxes prevent the iron penetrating into the coating; on the other hand, they enable the increase of carbon, chromium, and titanium in the coating. Because of the slight temperature gradient for the system under study, the thermal diffusion and diffusion thermal conductivity affect the concentration distributions and temperature value only during short initial stage. Temperature gradient practically equals to zero starting from 10 to 30 s. Stronger influence of these factors can be anticipated for the case of carbide and nitride coating deposition on the materials with low thermal conductivity. Stresses and strains in the coating can be quite high [40]. It was detected that stresses connected with the composition change are higher the thermal stresses. This agrees with experimental data and calculation. Oscillations in the plasma composition those included into the model with special coefficients give more nonuniformity of the coating composition and lead to the formation of wider diffusion zones between the coating and substrate (Fig. 10). The features appear also in stress and strain distributions (Fig. 11). Practically linear character of radial stresses in the coating area [33] changes into wavelike one. Obviously, in the external surface of the growing coating, σ rr = 0. However, the tangential stresses in the coating grow. The axis stresses in the coating are lower than in the substrate. The values of stresses depend both on ion velocities and on composition oscillations in plasma. In the system with nitrogen, four chemical reactions take participation with the following rates: ɸ1 ¼ k1 ðTÞC23 C1 ; ɸ2 ¼ k2 ðTÞC22 C1 ;

55

Coupling Models of New Material Synthesis in Modern Technologies

1843

Fig. 10 The distribution of substance concentrations along the radius at t = 3600 sec for different velocities of ions of carbon, chromium, and titanium (in m/sec): (1) V1 = 12, V2 = 18, V3 = 3; (2) V1 = 16, V2 = 22, V3 = 7; αi = 0.8

ɸ3 ¼ k3 ðTÞC44 C1 ; ɸ4 ¼ k4 ðTÞC84 C1 : Hence, for eight species, one can write ω1 ¼ m1 ðϕ1 þ ϕ2 þ ϕ3 þ ϕ4 Þ; ω2 ¼ 2m2 ϕ2 ;  ω8 ¼ 2m8 ϕ4 : The properties of individual substances are presented in Table 4, and formal kinetic parameters are given in Table 5. In this case, the transition zones between the layers of growing coating are also wide, and the composition changes contiguously during the layer buildup. First, at the initial stage (t < 900 sec), the titanium carbide forms in the coating. The formation of iron nitride is observed near the interface due to diffusion. As it was noted above, the acceleration of similar reactions (formation of nitrides Fe2N и Fe4N) in thin coating happens due to temperature growth with the evaluation of ion

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Fig. 11 Distributions of tangential, axis, and radial stresses along the radius at t = 3600 sec for different ion velocities (а) and for different values of coefficient αi (b). (a) αi = 0.6; (1) V1 = 12, V2 = 18, V3 = 3 m/sec; (2) V1 = 16, V2 = 22, V3 = 7 m/sec; (b) V1 = 16, V2 = 22, V3 = 7 m/sec; (1) αi = 0.6; (2) αi = 0.8 Table 4 Properties of individual substances Parameter λ ρ Cp m E ν αT

N2 0.026 0.8 29.1 14 0.14 0.24 2

Ti 21.9 4.54 25.1 47.8 112 0.26 2.8

Al 203.5 2.7 24.35 26.98 70 0.28 3.1

AlN 30.14 3.12 30.1 40.98 310 0.3 3.4

TiN 36.2 4.93 34.23 61.8 261 0.29 3.2

Fe2N 67.3 6.68 98.44 125.6 217 0.3 3.8

Fe4N 60.1 7.6 29.5 237.2 200 0.29 4.7

Dimensions W/(mK) 103, kg/m3 J/(molК) 103, kg/mol GPa 106, К1

55

Coupling Models of New Material Synthesis in Modern Technologies

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Table 5 Kinetic parameters of the reactions Forming chemical compound Fe2N Fe4N TiN AlN

Activation energy 80,508 79,225 200,204 319,000

Pre-exponential factor 2.9108 3.67108 2.461014 3.691018

Reaction heat [кJ/ mol] 7.54 21.78 11.64 9.63

velocities. Further, the diffusion way rises, and nitride formation retards. During time interval t  [900, 1800] seconds, the plasma composition changes: aluminum ions enter the coating surface which yields aluminum nitride in the coating. During this time interval, new layer forms, and numerical algorithm allows titanium to appear in corresponding area of the coating only by diffusion. Secondary change in the plasma composition occurs at t = 1800 seconds. In general, the picture is similar to the previous one. Figure 12 illustrates that nonuniform properties of the coating are connected with nonuniform composition. All properties presented in the equations are calculated explicitly, but only heat capacity, thermal conductivity coefficient, and elastic modulus are shown here. Numerical analysis has shown that the heat capacity of carbide coating is close to the heat capacity of the substrate; however, nitride coating has much higher heat capacity. The thermal conductivity coefficient and elastic modulus were calculated using mixture rules (6) and (8). Since the structure of forming coating is unknown, the implementation of more sophisticated approaches is not reasonable. In terms of the discussed model, the thermal conductivity of carbide coating is smaller than that of the substrate. Nitride coating has practically the same thermal conductivity coefficient on average; however, it changes in the layers threefold. The model does not give the increase of the elasticity modulus of the carbide coating in comparison with the substrate. However, it predicts the increase of elastic modulus of nitride coating in some points twice. Taking into account the nonuniformity of the properties along the radius, it is necessary to carry out simple averaging. In so doing, for curves 1 and 2, the averaging with five marked points gives for nitride coating E 138 and E 169 GPa; the averaging for carbide coating leads to E 214 and E 272 GPa, correspondingly. Hence, it is arguable that ion velocity rise leads to the increase of elastic modulus. Because the explicit correlation between elastic modulus and strength properties is absent, and there are no fail-safe methods to calculate the strength properties, this model is useless. Thus, the models of technology processes of coating formation taking into account multiscale phenomena could be the instrument for numerical experiments. Using initial information on technological conditions, geometry, and technological regimes, users would calculate the composition of material or coating obtained. This allows reducing the number of real experiments and obtaining information that is hard to get experimentally. The developed numerical algorithm can be used for

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Fig. 12 Distribution of physical properties along the radius at t = 3600 sec at different ion velocities (m/sec): (1) V1 = 12, V2 = 18, V3 = 3; (2) V1 = 16, V2 = 22, V3 = 7; (a) Al-Ti-N-Fe system; (b) Ti-Cr-C-Fe system

different coupling problems. The models give new information on the interrelation between processes and allow new phenomena to be studied. Such models are useful for the formulation of the problem of technology optimization, when the difference between calculated and given values must be minimized. The models could also be used to find unknown parameters when the theoretical result is equaled to experimental one. Finally, this model could be useful for students and investigators who wish to understand the phenomena. The model could be modified for more complex technology situations, combined technologies, more complex plasma compositions, more complex geometry, and different substrate materials.

55

Coupling Models of New Material Synthesis in Modern Technologies

1847

The process of multilayered coating deposition can be also called “additive technology,” when a product has been obtained by step-by-step material addition.

5.3

Controlling Synthesis of Materials on a Substrate

The third example demonstrates the role of conjugate heat exchange in the formation of phase composition of a new material. Currently, the technologies are popular where the coating synthesis and surface layers formation are controlled by external high-energy action. Electron or laser beams are used as the heat source. During coating synthesis, the transitional zone between condensed materials scarcely forms; the formation of transitional zone requires additional thermal treatment. However, the substrate receiving the heat from the heated zone can essentially affect the reaction initiation conditions in the forming coating and reaction propagation along the surface, similarly to known processes. Conjugate heat exchange plays a significant role in heating and conversion regimes in systems with inert inclusions [43], inert rod [44], or joined materials [45]. Similar effects take place in modern technologies of material creation, under the conditions of laser or electron-beam synthesis of articles from intermetallides [46]. By analogy to the previous one, the mathematical model could be developed to carry out a numerical experiment and to optimize new technologies. The examples of similar models were presented in [47, 48]. Two-dimensional model of multiphase coating synthesis for the Ti  Ni system is presented below. When a new material synthesis is carried out on the surface of a steel cylindrical workpiece (Fig. 13), preliminarily heated to temperature Tw, and the energy distribution in the external immobile source is symmetric relatively to the axis of the cylinder, the problem can be formulated for cylindrical coordinate system. In this case, the temperature and distributions of the concentrations of species follow from the combined solution of thermal conductivity and equations of chemical kinetics. The coupling between mechanical and heat processes is not taken into account. Thermal conductivity Eq. (15) takes the form of cρ



 @T 1 @ @T @ @T ¼ rλ λ þ þ W, @t r @r @r @z @z

Fig. 13 Illustration to the model of intermetallic coating synthesis on the substrate

(36)

z

z = hb + hc

Rc

z = hc zz=
00

Rbb

r

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A. Knyazeva et al.

where W = Wch in above layer (coating made from pressed metallic powders) with radius equal or smaller the steel workpiece (substrate) radius and W = 0 in the workpiece (substrate);   c c ρc , h b  z  h c ; λc , for hb  z  hc ; cρ ¼ λ¼ c b ρb , 0  z  h b , λb , for 0  z  hb , index ‘c’ corresponds to the pressed powders; index ‘b’ corresponds to the substrate. Boundary conditions correspond to Fig. 13:   @T ¼ eb σ 0 T 4  T 4W ; @z 8 @T < Tjhb ¼ Tjþhb , λb @T @z hb ¼ λc @z þhb , for 0  r  Rc ; z ¼ hb :   @T : λb ¼ eb σ 0 T 4  T 4W þ qðt, r Þ; for Rc  r  Rb @z z ¼ 0 : λb

z ¼ hb þ hc : λc r¼0:

  @T ¼ ec σ 0 T 4  T 4W þ qðt, r Þ; @z

  @T @T ¼ 0; r ¼ Rc : c  λc ¼ ec σ 0 T 4  T 4W at hc þ hb  z  hc ; @r @r and r ¼ Rb : λb

@T ¼ eb σ 0 ðT 4  T 4W Þ, at hc  z  0: @r

Here σ 0 is Stephen-Boltzmann constant, ec, eb are blackness levels for the coating and substrate. The external heat source q(t,r) connects with electron beam or laser:  qðt, r Þ ¼

q0 f ðr Þ, for T  T S 0, for T > T S

where TS is the average temperature of the pressed powders, when the external heating is interrupted (the averaging is carried out along the coating volume); q0 is maximal power density of the source (flux density) known from the experiment. The calculation way for q0 depends on the source type; in this model, this value is given. Correspondingly to state diagram of Ti-Ni system [27], in the pressed powders, the reactions Ti þ Ni ! TiNi, 2Ti þ Ni ! Ti2 Ni, and Ti þ 3Ni ! TiNi3 can be expected during the heating. Then for five species with molar concentrations y1 ¼ ½Ti, y2 ¼ ½Ni, y3 ¼ ½TiNi, y4 ¼ ½Ti2 Ni, and y5 ¼ ½TiNi3 , the kinetic equation system will take the form of Eq. (16):

(37)

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Coupling Models of New Material Synthesis in Modern Technologies

1849

dyk ¼ ωk , k ¼ 1, 2, 3, 4, 5, dt where source terms correspond to Scheme (37): ω1 ¼ y1 y2 ɸ1  y21 y2 ɸ2  y1 y32 ɸ3 ; ω2 ¼ y1 y2 ɸ1  y21 y2 ɸ2  y1 y32 ɸ3 ; ω3 ¼ y1 y2 ɸ1 ; ω4 ¼ y21 y2 ɸ2 ; ω5 ¼ y1 y32 ɸ3 : The functions for the reaction rates take into account the reaction retardations by the reaction product (3): ϕi ¼ ki expðEai =RT Þexpðmi yÞyni ; i ¼ 1, 2, 3, where ki, Eai are pre-exponential factors and activation energies for the reactions, y = y4 + y5 are the concentration of summary reaction product, and mi, ni are the retardation parameters. The summary heat source due to the chemical reactions in thermal conductivity Eq. (36) takes the form of W ch ¼ Q1 y1 y2 ϕ1 þ Q2 y21 y2 ϕ2 þ Q3 y32 y1 ϕ3 : Initial conditions: t ¼ 0 : T ðx, 0Þ ¼ T 0 at hc þ hb  z  hc and T ðx, 0Þ ¼ T w at hc  z  0; y1 ¼ y10 , y2 ¼ y20 , y3 ¼ y4 ¼ y5 ¼ 0: This problem allows investigating the dynamics of the synthesis process for various heating conditions and various relations between concentrations of initial substances. Algorithm of numerical solution is similar to [47, 48]. The value of power density was calculated as q0 ¼

IU E ¼ , S S

where I is current intensity, U is voltage, and S is the area of action. If the area of 2 actionis uniform,  then S = πR , where R is the area radius. If the area is a ring, then 2 2 S ¼ π R1  R2 , R1 > R2, where R1and R2 are external and internal radiuses of the ring, correspondingly. Thermodynamical and physical parameters are known from the experiment or were calculated on the base of thermodynamical data. All data used for the calculations are presented in Tables 6 and 7. It was assumed that all retardation parameters are the same, that is, n = n1 = n2 = n3 and m = m1 = m2 = m3. Their values were chosen from the condition that the reactions must start when the specimen has temperature T 600К: n = 0 and m = 5 (this corresponds to a strong retardation by the reaction product [11]). Further, these parameters could be varied to coordinate the theory and experiment.

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Table 6 Formal kinetic parameters of reactions Ti þ Ni ! TiNi 2.8  1018 2.11  105 3.765  103

ki Ea, i[J/mol] Qi [J/cm3]

2Ti þ Ni ! Ti2Ni 1.1  1018 2.3  105 3.178  103

Ti þ 3Ni ! TiNi3 2.53  1018 2.7  105 3.732  103

Table 7 Thermal-physical properties used for calculations Steel Ti + Ni

ρ [g/cm3] 7.9 2.3

с[J/(g К)] 0.502 0.623

λ[J/cm sec К)] 0.163 0.08

c z

b z

a z

d z

q0

q0

r

d [cm] 2.5 1 2.5

q0

r

r

h [cm] 0.5 0.5

q0

r

Fig. 14 Possible variants of process arrangement

Additionally to concentrations and temperature distribution, the integral curves can characterize the dynamics of the synthesis. Integral concentrations are calculated as 1 hY ii ¼ h c Rc

hc ð þhb R ðc

yi drdz: hc

(38)

0

The values hYii determine the average composition of synthesized material to the end of the deposition process. The synthesis process could be arranged by various ways (Fig. 14). Additionally to the preliminary heating of the substrate to some given temperature TW or the change of the electron-beam energy E, one can control the process dynamics by varying the relations between the geometrical parameters of the substrate and pressed powder and by changing the heating area. Detailed investigations have shown that the most uniform composition has been obtained when the diameter of detail from pressed powders (coating) is less than that of the substrate and external source heats only the surface of the pressed powders (Fig. 14a). Other variants (Fig. 14b, c, and d) lead to nonuniform composition. The dynamics of the synthesis for initial substrate temperature TW = 1073K. External heating stops when the average temperature of pressed powders reaches Ts = 1073 K.

55

Coupling Models of New Material Synthesis in Modern Technologies

a

b

c

d

e

f

1851

Fig. 15 Spatial distribution of titanium y1(Ti) in the coating for different time moments (in sec): (a) t = 0.3, (b) t = 0.4, (c) t = 0.5, (d) t = 0.6, (e) t = 0.9, (f) t = 1.6

The spatial distributions of Ti and TiNi are shown in Figs. 15 and 16 for different time moments. The concentration of nickel changes similarly to the concentration of titanium. Ti2Ni and TiNi3 phases appear in very small quantities. Their spatial distribution qualitatively repeats the distribution of TiNi. Because initially the substrate has a high temperature, open surface cools only due to thermal radiation; in the interface, the heat exchange with powder layer takes place through thermal conductivity; the reactions start from a side of the substrate practically uniformly excluding the small area near the open surface (Figs. 15 and 16a). Further, the uniformity of the process is violated (Figs. 15 and 16b–f), which is due to heat losses. The reaction is initiated from the free surface up to t = 0.4 second, and only a small part of initial substances stays until t = 1.6 seconds near the substrate close to open surface (Figs. 15 and 16f). TiNi phase is distributed not uniformly as well (Fig. 16) both during the process (Figs. 16b–e) and up to the final time (Fig. 16f). However, the nonuniformity level is low. The nonuniformity of the temperature field is shown in Fig. 17a–f. The synthesis can be considered as completed up to t = 1.6, because the composition does not

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a

b

c

d

e

f

Fig. 16 Spatial distribution of phase y3(TiNi) in the coating for different time moments (in sec): (a) t = 0.3, (b) t = 0.4, (c) t = 0.5, (d) t = 0.6, (e) t = 0.9, (f) t = 1.6

further change. The external heating, the heating of the reagent from the substrate, chemical heat release, and heat losses to the environment act together. Additionally the dynamics of the synthesis is presented in Figs. 18a–e. Here, three stages are discernible. Average concentrations change at each stage by various ways. I. The first stage is characterized by slow increase of product concentrations and slow decrease of reagent concentrations. This is induction period. II. Reaction acceleration and quick product accumulation are typical for the second stage. It is accompanied by the confluence of initial reaction areas. III. Retardation and termination of the reactions due to heat losses and reagent exhausting correspond to the third stage. Sharp ruptures on the curves hYi(t)i at given scale are actually flex points. Their position does not depend on the parameters of different schemes and is determined by physical factors. For example, the change of the temperature of preliminary heating leads to the process rate rise and to the change in the dynamics of the

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Fig. 17 Spatial distribution of temperature in the system at different time moments (in sec): (a) t = 0.3; (b) t = 0.4; (c) t = 0.5; (d) t = 0.6; (e) t = 0.9; (f) t = 1.6

reaction product formation. In any case, for the chosen initial composition, the product consists predominately of TiNi phase. The average integral concentration for different values of initial substrate temperature is presented in Fig. 18. When temperature TW rises, the synthesis starts earlier, which does not contradict to common sense and experimental data. A more complete reaction is observed. With the temperature TW increase, the concentrations of Ti2Ni and TiNi3 phases grow, while the concentration of y3(TiNi) decreases. Different synthesis regime is observed when the surface of the pressed powders is heated by the source with constant intensity and with the radius less than powder layer (Fig. 14b). In this case, the temperature field is quite nonuniform, and the synthesis goes nonuniformly. In spite of preliminary heating, the reactions start in the area of electron-beam action. Only by the moment t = 0.08 second, the reactions are initiated in the coating near the substrate. Further, the reactions propagate from the external surface and from the substrate, and the reaction zones coalesce; however, they retard due to the heat losses and the synthesis stops. All phases are distributed in the coating not uniformly. Integral curves for different

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Fig. 18 Integral concentrations in time for different values of the substrate temperature: (1) Ts = 1073K; (2) Ts = 1273K; (3) Ts = 1473K. (a–e) synthesis under the conditions of 14а; (f) synthesis under the conditions of 14b

phases mirror this, for example, the dynamics of TiNi phase accumulation change (Fig. 18f). Different stages are marked on the curves Yi(t): I. Product appearance (or reagent concentration decrease) in the area of external heating II. Reaction retardation due to partial exhaustion in the area of external heating III. Reaction acceleration due to their involvement in the area close to the substrate

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IV. Reaction retardation accompanied by the coalescence of reaction zones V. Retardation or termination of the reactions due to heat loss and reagent exhaustion The influence of the substrate initial temperature is more evident than in the previous case. Here, there is a large area in the pressed powders where the reactions are not complete. Such process engineering does not give desired result and is not suitable for practice. Thus, for any type of heating used in experiment, the coating has nonuniform composition. However, there is a possibility to arrange the process in a way when the nonuniformity will be inessential. It is obvious that the result depends on initial composition of the powders, on geometrical parameters, etc. The model could be modified for other chemical systems.

6

Conclusions

The work has shown the possibility of the models of technology processes with predictive properties. Similar models contain many parameters. Therefore, the necessity arises to calculate or discover parameters in the literature. Known data, as a rule, are different. Independent theories give approximate information. Experimental data could be used for parameter identification, and thermodynamics calculations can be the initial point to calculate kinetic parameters. The thermodynamical approach allows reducing the number of independent values and suggesting the effective formulae to calculate some coefficients for scarce information on simple properties and thermodynamical data for individual substances. As a result, one can obtain a closed theory that can be used in practice and modified for various technological and experimental situations. The key strength is in the way of model construction and the way of coefficient calculation. Unfortunately, the questions of effective property calculation for numerically obtained materials (and coatings) are not discussed in detail here because they are a special problem. Numerous questions arise when the rheology of complex materials is described. The investigation of this problem is pending. New materials are under study, and new technological conditions lead to new mathematical problems that will need further investigation.

References 1. Poate JM, Foti G, Jacobson DC. Surface modification and alloying: by laser, ion, and electron beams. New York: Springer; 1983. 2. Tracton AA, editor. Coating technology. Fundamentals, testing, and processing. Boca Raton: CRC Press, Taylor & Francis Group; 2007. 3. Bullinger H-J, editor. Technology guide. Principles – applications – trends. Berlin/Heidelberg: Springer; 2009.

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4. Mazurkiewicz A, Smolik J. The innovative directions in development and implementations of hybrid technologies in surface engineering. Archives of metallurgy and material. 2015; 60(3):2161–72. 5. Zenker R, Spies H-J, Buchwalder A, Sacher G. Combination of high energy beam processing with thermochemical treatment and hard protective coating: state of the art. International Heat Treatment and Surface Engineering. 2007;1(4):152–5. 6. Kostyuk GI. The perspective of development of combined technologies with help of ion, electron, light-beam and plasma fluxes for the receipt of the improved surface properties. 19th International Symposium on Discharges and Electrical Insulation in Vacuum, Xi'an 2000;2:663–6. 7. Murr LE, Gaytan SM, Ramirez DA, Martinez E, Hernandez J, Amato KN, Shindo PW, Medina FR, Wicker RB. Metal fabrication by additive manufacturing using laser and electron beam melting technologies. J Mater Sci Technology. 2012;28(1):1–14. 8. Travitzky N, Bonet A, Dermeik B, Fey T, Filbert-Demut I, Schlier L, Schlordt T, Greil P. Additive manufacturing of ceramic-based materials. Adv Eng Mater. 2014;16(6):729–54. 9. Levenspiel O. Chemical reaction engineering. 3rd ed. New York/Chichester/Weinheim/ Brisbane/Singapore/Toronto: Wiley; 1999. 10. Davis ME. Davis RJ. Fundamentals of Chemical Reaction Engineering: Published by McGrawHili; 2003. 11. Merzhanov AG, Mukasyan AS. Solid-flame combustion (Tverdoplamennoe gorenie – in Russian). Moscow: Torus Press; 2007. 12. Regel VR, Slutsker AI, Tomashevskii EE. Kinetical nature of the strength of solid bodies (Kineticheskaya priroda prochnosti tverdykh tel – in Russian). Moscow: Nauka; 1974. 13. Casale A, Porter RS. Polymer stress reactions. (Reakzii polimerov pod deistviem napryazhenii – in Russian). Khimiya: Leningrad; 1983. 14. Knyazeva AG. Ignition of a condensed substance by a hot plate with consideration of thermal stresses. Combustion, Explosion, and Shock Waves. 1992;28(1):10–5. 15. Chupakhin AP, Sidelnikov AA, Boldyrev VV. Influence of mechanical stresses appearing during solid-phase chemical conversions on their kinetics. General approach (Vliyanie voznikayushchikh pri tverdofaznykh khimicheskich prevrashcheniykh mechanicheskikh napryazheniy na ikh kinetiku. Obshchiy podkhod – in Russian). Izv SB AN SSSR., ser Chimich, nauki 1985;6:31–8. 16. Knyazeva AG. Hot-spot thermal explosion in deformed solids. Combustion, Explosion, and Shock Waves. 1993;29(4):419–28. 17. Nemat-Nasser S, Hori M. Micromechanics: overall properties of heterogeneous materials. 2nd ed. Amsterdam: North-Holland; 1999. 18. Krivilyov MD, Mesarovic SD, Sekulic DP. Phase-field model of interface migration and powder consolidation in additive manufacturing of metals. J Mater Sci. 2017;52(8):4155–63. 19. De Groot SR, Mazur P. Non-equilibrium thermodynamics. Amsterdam: North-Holland Publishing Company; 1962. 20. Kondepudi D, Prigogine I. Modern thermodynamics: from heat engines to dissipative structures. Chichester: Wiley; 2014. 21. Kachanov M, Sevostianov I. Effective properties of heterogeneous materials. Dordrecht: Springer; 2013. 22. Hill R. Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids. 1963;11:357–72. 23. Belyuk SI, Panin VE. Vacuum electron-beam powder technology: equipment, technology, and application. Fizicheskaya Mesomechanica. 2002;5(1):99–104. (in Russian) 24. Kryukova ON, Knyazeva AG. Critical phenomena in particle dissolution in the melt during electron-beam surfacing. J Appl Mech Tech Phys. 2007;48(1):109–18. 25. Romankov PG, Rashkovskaya NB, Frolov VF. Mass exchange processes in the chemical engineering (systems with solid phase) (Massoobmennye processy v chimicheskoi technologii. Systemy s tverdoi fasoi. – in Russian). Khimiya: Leningrad; 1975.

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26. Knyazeva AG, Pobol IL, Gordienko AI, Demidov VN, Kryukova ON, Oleschuk IG. Simulation of thermophysical and physico-chemical processes occurring at coating formation in electronbeam technologies of surface modification of metallic materials. Physical Mesomechamics. 2007;10(3–4):207–20. 27. Hansen M, Anderko K. Constitution of binary alloys. New York: McGraw-Hill; 1958. 28. Karapetiync MK. Chemical thermodynamics. Khimiya: Moscow; 1975. (in Russian) 29. Galchenko NK, Belyuk SI, Panin VE, Samartsev VP, Shilenko AV, Lepakova OK. Electronbeam facing of the composite coatings on the base of titanium Diboride. Fizika i Khimiya Obraotki Materialov. 2002;4:68–72. (in Russian) 30. Kryukova O, Kolesnikova K, Gal'chenko N. Numerical and experimental study of electronbeam coatings with modifying particles FeB and FeTi. IOP Conference Series-Materials Science and Engineering. 2016;140:012011. 31. Sarakinos K, Alami J, Konstantinidis S. High power pulsed magnetron sputtering: a review on scientific and engineering state of the art. J Surface & Coatings Technology. 2010;204:1661–84. 32. Ali R, Sebastiani M, Bemporad E. Influence of Ti–TiN multilayer PVD-coatings design on residual stresses and adhesion. J Materials & Design. 2015;75:47–56. 33. Knyazeva AG, Shanin SA. Modeling of evolution of growing coating composition. Acta Mech. 2016;227:75–104. 34. Knyazeva AG. Connected equations of heat and mass transfer in a chemically reacting solid mixture with allowance for deformation and damage. J Appl Mech Tech Phys. 1996;37(3): 381–90. 35. Knyazeva AG, Demidov VN. Transfer coefficients for three component deformable alloy. Vestnik PermGTU. Mechanika. 2011;3:84–99. (in Russian) 36. Shanin SA, Knyazeva AG. Multilayer coating formation at the deposition from plasma. IOP Conf. Series. Mater Sci Eng. 2016;116:012003. 37. Shanin S, Knyazeva A, Kryukova O. Coating composition evolution during the deposition of al and N from plasma on a cylindrical substrate. Key Eng Mater. 2015;685:690–4. 38. Timoshenko SP, Goodier JN. Theory of elasticity. New York: McCraw-Hill; 1970. 39. Boly BA, Weiner JH. Theory of thermal stresses. Chichester: Wiley; 1960. 40. Barvinok VA. Controlling of stressed state and properties of plasmatic coatings. (Upravlenie napryazhennym sostoyaniem i svoistva plazmennykh pokrytiy. – in Russian). Moscow: MIR; 1964. 41. Shanin SA, Knyazeva AG. On the numerical solution of non-isothermal multicomponent diffusion with variable coefficients. Computational technologies. 2016;21(2):88–97. 42. Grigoriev IC, Meylikhov EZ. Physical values. (Fizicheskie velichiny. Spravochnik - in Russian). Moscow: Energoatomizdat; 1991. 43. Prokof’ev VG, Smolyakov VK. Impact of structural factors on unsteady combustion modes of gasless systems. Combustion, Explosion, and Shock Waves. 2003;39(2):167–76. 44. Ivleva TP, Merzhanov AG. Mathematical simulation of three-dimensional spin regimes of gasless combustion. Combustion, Explosion, and Shock Waves. 2002;38(1):41–8. 45. Knyazeva AG, Chashchina AA. Numerical study of the problem of thermal ignition in a thick walled container. Combustion, explosion and. Shock Waves. 2004;40(4):432–7. 46. Shishkovskii IV. Laser synthesis of functional-gradient meso structures and volume articles (Lazernyi sintez functionalno-gradientnykh mesostruktur I ob’emnykh izdelii. – in Russian). FIZMATLIT: Moscow; 2009. 47. Knyazeva АG, Sorokova SN. Simulation of coating phase structure formation in solid phase synthesis assisted by electron-beam treatment. Theor Found Chem Eng. 2008;42(4):457–65. 48. Knyazeva АG, Sorokova SN. Numerical study of the influence of the technological parameters on the composition and stressed-deformed state of a coating synthesized under electron-beam. Theor Found Chem Eng. 2010;44(2):184–97.

Simulation of Fracture Behavior of Weldments

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Haoyun Tu, Siegfried Schmauder, and Yan Li

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Description of Different Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Rousselier Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Gurson-Tvergaard-Needleman (GTN)Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Cohesive Zone Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Crack Propagation in Welded Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In this chapter, the fracture behavior of an S355 electron beam welded joint is simulated with the Rousselier, Gurson-Tvergaard-Needleman (GTN), and cohesive zone models separately. First, each model is discussed and the method identifying the model parameters is given. Second, the simulation results on the crack propagation of compact tension (C(T)) specimens with the initial crack located at different weld regions are given. Finally, the cohesive zone model is compared with the other two models, showing its superiority.

H. Tu (*) · Y. Li School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, People’s Republic of China e-mail: [email protected]; [email protected] S. Schmauder Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_41

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Introduction

Welding, an old fabrication technique which was invented in the late 1800s, is used to melt or join metals in many industry fields. Since its invention, the welding technique or the weldment has been used for manufacturing, such as production of cars, manufacture of railways, or ship building. As the requirement of weight reducing and energy saving from industry, the welding technique has been used to replace riveting, which was used widely to connect different components of structures. According to the information of ISO standard, welding can be roughly divided into several categories that include the arc welding, resistance welding, solid-state welding (e.g., friction stir welding), and other weldings (e.g., electron beam welding) [1]. Among these welding techniques, electron beam welding (EBW) attracts more attention as narrow heat affected zones, little distortion, and less residual stresses are obtained after welding due to the high power density during the EBW. For engineers and scientists, as the fracture behavior of the welded joint influences the serving life of the component, it is worth to investigate the fracture behavior of the weldment experimentally and numerically. In this chapter, simulation results on the fracture behavior of an electron beam welded joint are presented and discussed. Two main types of models are widely used nowadays to describe the ductile damage evolution by void initiation, growth, and coalescence. One is a micromechanically based model (e.g., the Gurson-Tvergaard-Needleman [GTN] model and the Rousselier model) and the other one is a phenomenological model (i.e., the cohesive zone model). In this chapter, the mentioned models will be discussed and the Rousselier, GTN, and cohesive zone models are used simulating the fracture behavior and crack propagation of compact tension (C(T)) specimens extracted from the weldments.

2

Description of Different Models

In this part, the basic equations of the Rousselier, GTN, and cohesive zone models are shown and the influences of model parameters on numerical simulation results are discussed.

2.1

The Rousselier Model

The first model discussed in this chapter is the Rousselier model [2, 3]. In the Rousselier model, damage is defined by a variation of the void volume fraction originating from second-phase particles under tensile loading conditions. The general equation of the Rousselier model is as follows: Φ¼


 σeq σm þ Dfσk exp  R ð pÞ ¼ 0 1f σk ð1  f Þ

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where σeq is the von Mises equivalent stress, σm is the mean stress, f is the void volume fraction (initial value is f0), σk and D are material constants, p is the cumulated plastic strain, and R(p) is the true stress–true plastic strain curve of the material. As we use a finite element model to investigate the ductile failure of different testing structures or specimens (e.g., the precracked compact tension specimen), element size, type, symmetries, etc., has a significant influence on the simulation result (so-called mesh size dependence), so lc (lc stands for the average distance between voids) is assumed as an additional parameter.

2.1.1 Identification of Rousselier Parameters From theoretical considerations [2], D is a material-independent parameter that is always equal to 2. The initial void volume fraction (f0) can be obtained from optical microscopy of the polished material. According to the assumption of Rousselier, when a material is under deformation, the inclusions will nucleate voids. The initial volume fraction of inclusions can be assumed as the initial void volume fraction (f0). However, analysis on 2D images cannot accurately get the real volume fraction of inclusions in the 3D bulk material, instead, the experimental value f0 obtained from the optical microscopy investigation must be calibrated before using. Based on the combination of information from optical microscopy and numerical simulation of tensile test processes of the notched round specimen, the numerical f0-value is fixed. The f0-value influences the drop-down moment of the force versus diameter reduction (F-ΔD) curves of notched round specimens. When the material is under external loading, higher f0-values lead to an earlier void coalescence stage when micro cracks emerge on the surface of the material. Higher f0 results in an earlier fracture stage while the slope of the curves after the final fracture point is not affected by f0-values. The parameter σk is a material parameter as explained by Rousselier [2, 3]. The first suggested value for σk is two thirds that of σm at the rupture moment of the stress–strain curve for the smooth round specimen. However, this is an underestimated value that is lower than the value used for the simulation. According to our previous experiences at IMWF/ MPA [4–6], σk is assumed to be 445 MPa for the current investigated steels (10MnMoNi55, 15 NiCuMoNb5, S355NL). Based on the experimental F-ΔD curves from notched round specimens, σk can be calibrated. The parameter σk has a close relation to the material flow curve. Higher σk leads to later void coalescence and later fracture stage while the slope of the curves after the final fracture point is not affected. The lc value is the average distance between voids. This lc value can also be calibrated with the notched round specimen. The slope of the F-ΔD curve before the fracture position is not affected by the lc-value. Higher lc-values lead to a flatter curve after the breakpoint of the F-ΔD curve because more energy is needed to damage larger elements, which stand for a longer distance between void-forming neighboring particles in the microstructure. In finite element simulations, the material loses its stress-carrying ability completely when the critical fc-value is reached.

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Fig. 1 Schematic influence of Rousselier parameters f0, σk, fc, and lc on the F-ΔD curves of the notched specimens [9]

Experimental investigation and numerical analyses of tensile tests have shown that fc = 0.05 is a reasonable value for most steels and other plastically deforming solids [4, 6–8], which was confirmed by a numerical study of Mohanta [7]. The influence of Rousselier parameters (f0, σk, lc, fc) on the F-ΔD curve of a notched round specimen is shown in Fig. 1.

2.2

The Gurson-Tvergaard-Needleman (GTN)Model

Another porous ductile material model is that developed by Gurson [10], later was modified by Tvergaard [11] and Needleman [12], and the equation of the so-called Gurson-Tvergaard-Needleman model is as follows: Φ¼

σ 2 q σ  eq m þ 2q1 f  cosh 2  1  q3 f  2 ¼ 0 σ 2σ

where σeq is the von Mises equivalent stress, σ is the flow stress for the matrix material of the cell, σm is the mean stress, f is the actual void volume fraction, and q1, q2 and q3 are parameters introduced by Tvergaard which can improve the accuracy of the predictions of the Gurson model [13]. The material is assumed to lose its stress-carrying ability completely when the current void volume fraction reaches 1/q1. Needleman and Tvergaard [12] accelerate the damage evolution when the actual local void volume fraction reaches the void coalescence value (fc ranges from 0.03 to 0.15 when the initial void volume fraction ranges from 0.001 to 0.08.) [14], and the modified damage variable f is a function of f and is given as follows: 8 f > > > > > < f þ kð f  f Þ c c  f ¼ > > > f > > : u

f < fc fc < f < ff , f  ff

k¼

f u  f c ff  fc

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Simulation of Fracture Behavior of Weldments

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When the current f value reaches the final void volume fraction ff, then f reaches (f u ¼ 1=q1 ), and the material loses all its stress-carrying ability. It is assumed that the rate of the void volume fraction consists of the growth of existing voids and the nucleation of new voids [15, 16]: f u

f_ ¼ f_ growth þ f_ nucleation The void growth is given according to the following equation: pl f_ growth ¼ ð1  f Þ_εkk

where ε_ pl kk is the plastic volume dilatation rate and the matrix is assumed behaving plastically incompressible. The contribution from nucleation of new voids is shown as follows:
 1 pl f_ nucleation ¼ A_εkk þ B σ_ m þ σ_ kk 2 The influence of the void nucleation is either strain controlled (B = 0) or stress controlled (A = 0). The void nucleation rate for the strain-controlled nucleation situation, as suggested by Chu [16], is as follows: 2

f_ nucleation

fn 1 εpl pl eq  εn ¼ A_εkk , A ¼ pffiffiffiffiffi exp4 2 sn sn 2π

!2 3 5

where fn is the void volume fraction of the void-nucleating particles, εn is the mean strain when void nucleation occurs, and sn is the corresponding standard deviation of the nucleation strain, which follows a Gaussian distribution [9]. For stress-controlled void nucleation, the void nucleation rate is shown in the equation as follows:

Fig. 2 Influence of GTN parameters f0, fc, fn, ff, En on the force versus cross section reduction curves [9]

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f_ nucleation

"

2 # 1 fn 1 σm þ 13 σkkσn ¼ B σ_ m þ σ_ kk , B ¼ pffiffiffiffiffi exp  3 2 sn sn 2π

where σm þ 13 σkk is an approximate value of the maximum normal stress acting on the particle/matrix interface [17, 18], σm is the mean stress for nucleation.

Fig. 3 Shape of traction separation laws according to (a) Hillerborg [22], (b) Needleman [17], (c) Needleman [23], (d) Tvergaard and Hutchinson [24], and (e) Scheider [25]

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2.2.1 Identification of GTN Parameters In the GTN model, the parameters (f0, fc, ff) control the damage process from void initiation, growth, coalescence until final failure. The parameters (fn, En sn) describe the void nucleation process. As discussed before, the growth of void volume fraction comprises growth of existing voids and the nucleation of new voids; it is suggested to use numerical methods to calibrate the GTN parameters based on metallographic investigations. The parameters q1 and q2 are parameters introduced by Tvergaard to get an accurate prediction of the Gurson model. Normally, the values vary as 1.25 e_I, 2 > e_I, 3 > e_I, 4 , resulting in hardness measures H1 > H2 > H3 > H4. (b) A representative strain rate-stress relations for a power-law creep material, obtained from indentation and conventional uniaxial testing

While indentation is able to measure the stress exponent, its viability to fully characterize the uniaxial creep response will also rely on a quantifiable relation between e_I and e_s . With reference to Fig. 10b, the question at hand is the vertical distance between the two lines. If it is specified for a material (with unknown uniaxial power-law creep properties), then shifting the indentation-measured line over the given distance will produce the uniaxial creep response. This would empower indentation as a robust means to characterize power-law creep behavior without the need to perform conventional creep tests. In Fig. 10b, the e_s =e_I ratio was found to be 0.33. A general relation applicable to a wide range of materials showing power-law creep has not been identified. Another complication is that all the above

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discussion was under the steady-state creep assumption. In reality the effects of transients (primary creep) always exist. For an indentation creep test, the stresses around the indentation site vary with time and position as the indenter moves deeper into the specimen. The transient effects can influence the indentation creep behavior in a different manner from those in uniaxial creep [3].

4.2

Viscoelastic Behavior

Indentation techniques have been developed to characterize creep of polymers which is primarily in the form of viscoelastic deformation. Here the elastic response is represented by the storage modulus, E0 , indicating the material’s capacity to store energy; it is the component in phase with the applied displacement or load. The internal friction and damping are represented by the loss modulus, E00 , indicating the material’s capacity to dissipate energy; it is the component 90 out of phase with the applied 00 displacement or load. The ratio EE0 ¼ tan ϕ is called the loss factor, where ϕ is the phase lag between the load and displacement. These parameters may be best characterized by the dynamic stiffness measurement techniques. The specimen-indenter contact can be modeled, for example, as a spring of stiffness S in parallel with a dashpot with damping Cω, where C is the damping coefficient and ω is the angular frequency of the dynamic oscillation [25]. In the limit of linear viscoelasticity, Sneddon’s stiffness equation may be used to relate the dynamic stiffness of the contact to E0 as pffiffiffi π S E0 pffiffiffi : ¼ 2 2β 1ν A

(26)

By analogy the damping can be related to loss modulus as pffiffiffi π Cω E00 pffiffiffi : ¼ 2β A 1  ν2

(27)

Here it is assumed that the Poisson’s ratio is not time-dependent. Successful application of the methodology requires rigorous characterization of the frequency response of the measurement system itself [25].

5

Residual Stress Measurement

All the indentation theories presented thus far are based on the premise that the test material was stress-free prior to indentation. In many circumstances, significant preexisting (residual) stresses may be present in the specimen. Possible causes include cold working (resulting from surface polishing) and mismatch in thermal expansion between the constituents (such as a thin coating on a substrate) during thermal processing. Nanoindentation may be used as a standalone tool to quantify the near-surface residual stresses.

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Fig. 11 Schematic showing the spherical indentation geometry and definition of parameters

In general, the existence of tensile and compressive residual stresses in an elasticplastic material tends to decrease and increase, respectively, the measured hardness, as was observed using the sharp Berkovich indenter [26]. However, the extent of hardness variation is small and is influenced by the pile-up geometry [27]. A method using the difference in the contact area of stressed and stress-free materials, indented to the same depth, was also proposed [28]. Since the effect of residual stress on the contact area is relatively small, it is not apparent if the method can be practically employed in cases where the residual stress is not too close to the yield strength of the material. A spherical indentation based technique [29], which can be more sensitive to residual stress than measurements with sharp indenters, is described below. Experiments have verified that this method is accurate to within 10–20% of the specimen yield strength and can be useful for making localized measurements. The method is predicated upon the extrapolation of spherical indentation data from the post-yield regime to determine the contact radius at the onset of yielding. It requires the yield strength of the material to be known. The biaxial stress can then be calculated based on a closed-form analytical solution. The geometric parameters involved are shown in the schematic in Fig. 11, where R is the indenter radius, a is the contact radius, h is the total penetration depth when loaded, hc is the depth at which contact occurs during loading, and hf is the final depth of indentation upon unloading. When a is small compared to R during indentation, the classical Hertzian contact theory can be employed up to initial yielding. The relation between the total depth of penetration h and contact radius a follows Eq. 3, h¼

a2 : R

(28)

The applied load P is related to h by Eq. 4, 4 P ¼ Er R1=2 h3=2 , 3

(29)

The mean contact pressure pm can be obtained by combining Eqs. 28 and 29,

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pm ¼

P 4Er a : ¼ πa2 3πR

(30)

The method requires measurement of the following parameters: the peak indentation load Pmax, the maximum indentation depth hmax, the final depth of the contact hf, and the contact radius a. Both Pmax and hmax are directly obtained from the indentation load-displacement data. The determination of hf follows a procedure on the basis that indentation unloading is modeled as elastic unloading of a sphere of radius R1 from a spherical contact impression of another radius R2. The unloading data are then described by  3=2 4 P ¼ Er Re 1=2 h  hf , 3

(31)

which is a modified form of Eq. 29, with Re being an effective radius related to R1

1 and R2 through Re ¼ R11  R12 . The magnitude of hf is obtained by curve fitting (second-order polynomial) the upper 90% of the unloading curve. It is noted that R2 is also unidentified and is treated as an unknown in the regression analysis. To determine the contact radius a, results on the basis of Hertzian contact theory are again used. The contact depth is expressed as [15, 30] hc ¼

 1 hmax þ hf : 2

(32)

The contact radius then follows from the spherical indenter geometry,  1=2 : a ¼ 2Rhc  hc 2

(33)

This method calls for the mean contact pressure, pm, at the onset of yielding. In other words, with reference to Eq. 30, the contact radius at the onset of yielding, a0, needs to be obtained experimentally. This is achieved by first plotting the measured hf Er a hmax against the nondimensional parameter σ y R at various indentation loads, where σ y is h

f ¼ 0 represents fully the yield strength of the test material. Since the condition hmax hf elastic deformation and hmax > 0 when plastic deformation commences, the value of hf Er a σ y R at hmax ¼ 0 signifies the onset of yielding and is used for determining the contact radius a0 at this critical point. Numerical analyses [29, 31] have shown that a hf and log σEyrRa over a wide range of nearly linear relationship exists between hmax

elastic-plastic transition. As a consequence, by extrapolating the discrete data set hf ¼ 0, one obtains the desired a0. Figure 12 shows such a plot from an of log σEyrRa to hmax actual test of an aluminum alloy [32]. The contact pressure pm thus follows according to Eq. 30 with a being a0. The remaining procedure involves correlating pm with the yield condition. In the elastic regime, the maximum shear stress for spherical indentation reaches about

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Fig. 12 Variation of the hf elastic recovery parameter hmax with the nondimensional parameter σEyrRa, as well as the linear fit, obtained from three cases of maximum indentation load

0.47pm underneath the surface directly below the indenter [9]. Taking the maximum shear stress to be 0.5σ y, one obtains pm = 1.06σ y. When a biaxial residual stress σ R exists in the material, the von Mises or Tresca yield criteria will both lead to the following expression of the contact pressure:   pm ¼ 1:06 σ y  σ R

(34)

as long as the subsurface yielding initiates along the axis of symmetry. Combining Eqs. 30 and 34 gives   σR 1:26 Er a0 ¼1 π σyR σy

(35)

at the onset of yielding. If σ y of thehtest imaterial is given, the residual stress can be determined by the measurement of EσryaR0 (using the extrapolation method described above).

6

Fracture Toughness Measurement of Brittle Materials

The indentation technique has been used most extensively for ductile materials which display inelastic behavior. However, it can also be applied to measure fracture toughness of materials of brittle nature. Contrary to conventional large-scale fracture toughness testing where a preexisting crack is needed, the indentation-based technique forces the creation of surface cracks from the corners of sharp indenters. Fracture toughness is then evaluated from the size of induced surface cracks. There exists a large body of literature dealing with indentation cracking. A common approach for the characterization of fracture toughness is summarized here. Figure 13 shows a schematic of cracks caused by a sharp indentation. The

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Fig. 13 Schematic showing the cross section through the center of a Vickers indentation and the semicircular crack with diameter 2c

Cracks

2a 2c cross-section exposes a well developed semicircular crack; the other semicircular crack is perpendicular to the cross-section. The two cracks will leave a total of four traces on the specimen surface, for the case of four-sided pyramidal indentation. From dimensional analysis, it is recognized that fracture toughness is proportional to P , with P being the applied indentation load and c being the crack length from the c3=2 center of the indent to the tip of the crack (Fig. 13). In addition, the hardness H and Young’s modulus E of the material influence the indentation size, which in turn will affect the cracking behavior. The fracture toughness can be expressed as [33] Kc ¼ α


1=2 E P H c3=2

(36)

For the cube corner indenter the constant α = 0.016, while for the Vickers and Berkovich indenters α = 0.032 [34]. Note that the cube corner indenter results in a reduced cracking threshold due to its sharper geometry. When conducting the test, the average value of the crack sizes c from a given indentation is used. It should be noted that cracking is not necessarily induced during the loading phase of the indentation. Local tensile residual stresses upon unloading can lead to crack formation [35]. Slow growth of the cracks may also occur after unloading, so the determination of crack size c may be influenced by the timescale. A general requirement for such measurement is that the crack pattern should be well developed, with c ≳ 2a, with no chipping [33]. The applied load P needs to be adjusted if no well-defined radial cracks are observed. It is also important that the test surface should be under no preexisting stresses prior to indentation. Surface stresses, such as those in tempered glasses, will render Eq. 36 inaccurate.

7

Adhesion Between Coatings and Substrates

Nanoindentation is not limited to the measurement of mechanical properties of materials. It may be used to provide quantitative estimates of the adhesion between a thin film (coating) and its underlying substrate. If the fracture toughness of the filmsubstrate interface is lower than those of the film and substrate materials themselves, decohesion at the interface can be expected if the film is subjected to indentation load

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Fig. 14 Schematic showing the cross section through the Vickers indentation center and the semicircular crack along the film-substrate interface, with diameter 2c

Crack along interface

Film Substrate

Plastic zone 2c

beyond a critical level. Figure 14 shows a schematic of the indented film/substrate specimen, where the cross-section is through the center of the indentation. A pennyshaped crack (shown in Fig. 14 as semicircular) with diameter 2c, originated from the plastic zone of the deformation, was induced along the interface. The crack length can be related to the applied load P and the critical indentation load to form the interface crack Pcr as [36]
 Pcr 1=2 1=4 c¼α 1 P , P

(37)

with 3=2

α2 ¼

α1 tf H 1=2 K Ic

,

(38)

where tf, H, and KIc are film thickness, film hardness, and fracture toughness of the interface, respectively. The parameter α1 is a material independent coefficient, calibrated as 6.4  108 [36]. Note that Pcr itself depends on hardness and adhesion of the film. In experiment, α is obtained by fitting the measured crack lengths at several indentation load levels following Eq. 37. The interface fracture toughness KIc can then be determined by Eq. 38 given the known film thickness and hardness. Subsequent studies have proposed ways to measure the adhesion influenced by residual stress in the film and the degree of buckling during indentation testing etc. [37, 38]. An alternative technique to measure film-substrate adhesion, derived from indentation, is the scratch test. During the test, a scratch is made across the coating surface with a stylus (indenter) while under a progressive normal load. The critical load at which the coating is detached from the substrate provides a measure of adhesion [34, 39]. The method is most commonly employed to quantify adhesion of strongly bonded hard coatings.

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Nanoindentation on Heterogeneous Materials

Although nanoindentation has seen widespread use in all classes of materials, the processes of measurement and data analysis are based on theories which deal with homogeneous materials with a unique form of mechanical response (such as elasticplastic). The focus in this section is on heterogeneous materials, with indentation penetration well into the composite microscopic features. This is an issue of increasing importance, because new composite materials with small-scale heterogeneities are being developed at a rapid pace, and their mechanical characterization has been, and will continue to be, dependent on the technique of nanoindentation. The current status of knowledge about the indentation behavior of heterogeneous materials is quite limited. How material heterogeneity will affect the measurement and interpretation of effective properties extracted from indentation tests are still not well understood. In addition, the complex nature of the microstructure will result in the development of a complex stress state underneath the indentation, which can lead to a nontrivial local damage state. An example is given here, using numerical finite element modeling applied to a multilayered structure consisting of alternating layers of very different mechanical strengths. The model is shown in Fig. 15, with 41 alternating aluminum (Al) and silicon carbide (SiC) layers on a thick silicon (Si) substrate, following an actual material used in the experiment. Both the top layer (in contact with the indenter) and the bottom layer (adjacent to the Si substrate) are Al. A conical diamond indenter with a semiangle of 70.3 is used. The left boundary is the symmetry axis in this axisymmetric model. The thicknesses of the individual Al and SiC layers are 50 nm each. Both the Al and SiC thin films are assumed to be isotropic elastic-plastic materials. The eventual plastic flow strength is 400 MPa for Al and for SiC it is 8770 MPa (estimated from its hardness) [40]. This great disparity in strength renders the structure highly heterogeneous. All the interfaces between different materials in the composite structure are assumed to be perfectly bonded. Fig. 15 Schematic showing the Al/SiC multilayer thin films above a Si substrate and the boundary conditions used in the axisymmetric model. The left boundary is the symmetry axis. The entire specimen is 40 μm in lateral span (radius) and 43 μm in height. The individual Al and SiC layers are 50 nm thick [19]

Indenter

41 layers of Al/SiC

Si substrate 2 1

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Fig. 16 Contour plots of hydrostatic pressure (negative hydrostatic stress) near the indentation site, when the indentation depth is at (a) 500 nm and (b) after unloading [19]

The evolution of stress and deformation fields inside the layered structure during indentation loading and unloading is now presented. Figure 16a, b show the hydrostatic pressure (negative hydrostatic stress) developed in the model near the indentation contact when the indentation depth is equal to 500 nm and after unloading, respectively. The corresponding contour plots of equivalent plastic strain are shown in Fig. 17. The same contour scales are used in each group of plots so the stress and strain magnitudes can be easily compared. The deformed laminate geometry can be discerned in regions where a large shading contrast exists. It can be seen from Fig. 16 that, directly below the indentation contact, very large compressive hydrostatic stresses are generated. The compressive stress magnitudes in the Al layers are generally much greater than those in SiC. This counterintuitive result is due to the different plastic yielding response of the two constituents. Note that the von Mises yield condition, in terms of the principal stress components σ 1, σ 2, and σ 3, is i12 1 h pffiffiffi ðσ 1  σ 2 Þ2 þ ðσ 2  σ 3 Þ2 þ ðσ 3  σ 1 Þ2 ¼ σ y , 2

(39)

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Fig. 17 Contour plots of equivalent plastic strain near the indentation site, when the indentation depth is at (a) 500 nm and (b) after unloading [19]

where σ y is the yield strength of the material under uniaxial loading. The soft Al layers (with a small σ y in Eq. 39) have undergone severe plastic deformation, as observed in Fig. 17, and a large triaxial compressive stress state is developed in these layers. For the SiC layers directly below the indentation, their very high strength (σ y in Eq. 39) requires a much less compressive (or even tensile) stresses of σ 11 and σ 33 than the highly compressive σ 22 (see the general form on the left-hand side of Eq. 39, which is dominated by the differences in the stress components). The combination of these normal stress components thus results in a much reduced hydrostatic compression in SiC because the hydrostatic stress is defined as 13 ðσ 11 þ σ 22 þ σ 33 Þ. Even in the Si substrate, large hydrostatic compression exists below the indentation (Fig. 16). This observation suggests the importance of the substrate material in affecting the indentation response. Upon unloading, the high stresses caused by the indentation are relieved to a great extent. However, considerably large areas, in both the multilayers and substrate, are still under residual compressive stresses well into the GPa range (Fig. 16b). This is a manifestation of the severity of plastic deformation occurred during loading. Another notable attribute about unloading is the evolution of deformation field in

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Fig. 18 Contour plots of σ 22 stress near the indentation site, when the indentation depth is at (a) 500 nm and (b) after unloading. The contour levels are adjusted to highlight the region with tensile stresses [19]

Fig. 17. A comparison between Fig. 17a, b reveals that the equivalent plastic strains in Al have actually increased during the unloading process. This observation is unique because, in a homogeneous material, indentation unloading is a pure elastic recovery process and the accumulated plastic strain remains unchanged during unloading. It is apparent that, in the present case of a heterogeneous structure, the internal constraint induced by the hard and stiff SiC layers has forced the much softer Al to undergo continued deformation even though the composite as a whole is undergoing unloading. It is recognized that, during unloading, if locally two adjacent SiC layers undergo uneven elastic relief, then the Al region sandwiched in between can experience large shear stress which promotes plastic yielding. The present finding does raise the issue of the accuracy of using instrumented indentation, during unloading, to measure the elastic behavior of multilayered materials, because the unloading process is no longer a simple elastic event. Figure 18 shows the contour plots of σ 22 when the indentation depth is at (a) 500 nm and (b) after unloading. The contour levels are adjusted so as to highlight the tensile σ 22 stresses evolved during loading and unloading (otherwise the localized tensile stresses would be largely “hidden” by the predominant compressive field). It

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can be seen that, under indentation, tensile stresses in the vertical direction have developed in certain areas below and slightly outside the edge of indentation. The stress magnitude can be significant, well over 200 MPa in many Al and SiC layers as well as in a small region of the Si substrate (Fig. 18a). Upon unloading, Fig. 18b, the region with high tensile stresses has shifted slightly inward and two distinct highstress areas have developed. It is noticed, by comparing the lighter shades in Fig. 18a, b, that the area with tensile σ 22 stresses has actually expanded during unloading. The expansion is due to the fact that the heavily compressed material directly below the indentation contact has rebounded during unloading. This imposes a vertical pulling action on the region already under tension. In fact, the same qualitative features can be seen even in a homogeneous material, and they are now enhanced by the presence of the laminated structure. Experimental characterization on postindented Al/SiC nanolayers has revealed local cracking and delamination, below and slightly outside the edge of indentation, in mid-layers and along the interface between the multilayers and Si substrate [40]. It is notable that these indentationinduced damages fall into the region where large tensile σ 22 stresses are seen in the modeling. How the indentation-measured quantities will be affected by internal damage remains to be studied.

9

Future Directions and Open Problems

While nanoindentation has seemed to become ubiquitously employed to measure mechanical properties, many challenges still remain. As presented above, microscopic heterogeneity can greatly complicate the measurement and interpretation of data. It should be noted that, in addition to the “man-made” multiphase structures, heterogeneity is also the rule rather than the exception in ordinary materials. Indentation loading itself can induce internal damage, such as delamination in thin film systems. During regular testing, internal damage may not be easily detected, and the influence it has on the measured property is unknown. Even for homogeneous materials, the effects of anisotropy in elastic and inelastic behavior on the indentation data are not well understood. Aside from performing indentation testing on a “big and flat” material surface, it has also been popular to apply the technique to probe discrete surface features (e.g., patterned thin films) and standalone small structures (e.g., nanorods, nanospheres, and biological tissues). It is noticed that the indentation theories are built upon the condition of an indenter brought in contact with a dense “semi-infinite” solid. Errors and implications caused by deviations from this condition are yet to be established. All of these open problems present opportunities for future explorations, especially on emerging material systems.

References 1. Doerner MF, Nix WD. A method for interpreting the data from depth-sensing indentation instruments. J Mater Res. 1986;1:601–9.

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2. Oliver WC, Pharr GM. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res. 1992;7:1564–38. 3. Hay JL, Pharr GM. Instrumented indentation testing. In: Kuhn H, Medlin D, editors. ASM handbook volume 8: mechanical testing and evaluation. Materials Park: ASM International; 2000. 4. Fisher-Cripps A. Nanoindentation. New York: Springer; 2002. 5. Oliver WC, Pharr GM. Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology. J Mater Res. 2004;19:3–20. 6. Gouldstone A, Challacoop N, Dao M, Li J, Minor AM, Shen Y-L. Indentation across size scales and disciplines: recent developments in experimentation and modeling. Acta Mater. 2007;55:4015–39. 7. Lawn BR, Cook RF. Probing material properties with sharp indenters: a retrospective. J Mater Sci. 2012;47:1–22. 8. Hertz H. On the contact of elastic solids. J Reine Angew Math. 1882;92:156–71. 9. Tabor D. The hardness of metals. Oxford: Clarendon Press; 1951. 10. Suresh S. Fatigue of materials. 2nd ed. Cambridge: Cambridge University Press; 1998. 11. Mathews JR. Indentation hardness and hot pressing. Acta Metall. 1980;28:311–8. 12. Sneddon IN. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Eng Sci. 1965;3:47–57. 13. Pharr GM, Oliver WC, Brotzen FR. On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation. J Mater Res. 1992;7:613–7. 14. King RB. Elastic analysis of some punch problems for a layered medium. Int J Solids Struct. 1987;23:1657–64. 15. Field JS, Swain MV. A simple predictive model for spherical indentation. J Mater Res. 1993;8:297–306. 16. Li X, Bhushan B. A review of nanoindentation continuous stiffness measurement technique and its applications. Mater Charact. 2002;48:11–36. 17. Mencik J, Munz D, Quandt E, Weppelmann ER, Swain MV. Determination of elastic modulus of thin layers using nanoindentation. J Mater Res. 1997;12:2475–84. 18. Gao H, Chiu C-H, Lee J. Elastic contact versus indentation modeling of multilayered materials. Int J Solids Struct. 1992;29:2471–92. 19. Shen Y-L. Constrained deformation of materials. New York: Springer; 2010. 20. Yang F, Li JCM. Impression test – a review. Mater Sci Eng R. 2013;74:233–53. 21. Mayo M, Siegel RW, Narayanasamy A, Nix WD. Mechanical properties of nanophase TiO2 as determined by nanoindentation. J Mater Res. 1990;5:1073–82. 22. Lucas BN, Oliver WC. Indentation power-law creep of high-purity indium. Metall Mater Trans A. 1999;30A:601–10. 23. Cheng Y-T, Cheng C-M. What is indentation hardness? Surf Coat Technol. 2000; 133–134:417–24. 24. Martinez NJ, Shen Y-L. Analysis of indentation-derived power-law creep response. J Mater Eng Perform. 2016;25:1109–16. 25. Herbert EG, Oliver WC, Pharr GM. Nanoindentation and the dynamic characterization of viscoelastic solids. J Phys D Appl Phys. 2008;41:074021. 26. Tsui TY, Oliver WC, Pharr GM. Influences of stress on the measurement of mechanical properties using nanoindentation: Part I. Experimental studies in an aluminum alloy. J Mater Res. 1996;11:752–9. 27. Bolshakov A, Oliver WC, Pharr GM. Influences of stress on the measurement of mechanical properties using nanoindentation: Part II. Finite element simulations. J Mater Res. 1996; 11:760–8. 28. Suresh S, Giannalopoulos AE. A new method for estimating residual stresses by instrumented sharp indentation. Acta Mater. 1998;46:5755–67. 29. Swadener JG, Taljat B, Pharr GM. Measurement of residual stress by load and depth sensing indentation with spherical indenters. J Mater Res. 2001;16:2091–102. 30. Francis HA. Phenomenological analysis of plastic spherical indentation. J Eng Mater Technol. 1976;98:272–81.

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31. Taljat B, Pharr GM. Measurement of residual stresses by load and depth sensing spherical indentation. In: Besser P, Shaffer II E, Kraft O, Moody N, Vinci R, editors. Materials research society symposium proceedings volume 594: thin films – stresses & mechanical properties VIII. Cambridge: Cambridge University Press; 2000. p. 519–24. 32. Olivas ER, Swadener JG, Shen Y-L. Nanoindentation measurement of surface residual stresses in particle-reinforced metal matrix composites. Scr Mater. 2006;54:263–8. 33. Anstis GR, Chantikul P, Lawn BR, Marshall DB. A critical evaluation of indentation techniques for measuring fracture toughness: I. Direct crack measurements. J Am Ceram Soc. 1981; 64:533–8. 34. Bhushan B, Li X. Nanomechanical characterization of solid surfaces and thin films. Int Mater Rev. 2003;48:125–64. 35. Meyers MA, Chawla KK. Mechanical behavior of materials. 2nd ed. Cambridge: Cambridge University Press; 2009. 36. Chiang SS, Marshall DB, Evans AG. A simple method for adhesion measurements. In: Pask J, Evans AG, editors. Surfaces and interfaces in ceramics and ceramic–metal systems. New York: Plenum; 1981. p. 603–12. 37. Marshall DB, Evans AG. Measurement of adherence of residually stressed thin films by indentation mechanics of interface delamination. J Appl Phys. 1984;56:2632–8. 38. Vlassak JJ, Drory MD, Nix WD. A simple technique for measuring the adhesion of brittle films to ductile substrates with application to diamond-coated titanium. J Mater Res. 1997; 12:1900–10. 39. Benjamin P, Weaver C. Measurement of adhesion of thin films. Proc R Soc Lond A. 1960; 254:163–76. 40. Tang G, Shen Y-L, Singh DRP, Chawla N. Indentation behavior of metal-ceramic multilayers at the nanoscale: numerical analysis and experimental verification. Acta Mater. 2010;58:2033–44.

3D/4D X-Ray Microtomography: Probing the Mechanical Behavior of Materials

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Sudhanshu S. Singh and Nikhilesh Chawla

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Introduction to X-Ray Microtomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Working Principles of X-Ray Microtomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lab-Scale and Synchrotron Microtomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Modes of Imaging in X-Ray Microtomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Application of X-Ray Microtomography in Materials Science . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3D Microstructural Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 In Situ 4D X-Ray Microtomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 In Situ Mechanical Testing in a Corrosive Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2014 2015 2015 2016 2018 2019 2019 2021 2026 2031 2031

Abstract

A fundamental principle in materials science and engineering is that the microstructure controls material properties. The use of three-dimensional techniques has gained popularity in establishing structure-property relationships in a variety of material systems. In particular, X-ray microtomography is being widely used as it requires minimal sample preparation and is nondestructive in nature. Moreover, being a nondestructive technique, it is very well suited to perform 4D studies (the fourth dimension being time) where the evolution of microstructure can be captured over time. This chapter describes the fundamentals of X-ray S. S. Singh Department of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India e-mail: [email protected] N. Chawla (*) Center for 4D Materials Science, Materials Science and Engineering, Arizona State University, Tempe, AZ, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_47

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microtomography followed by applications of the use of X-ray microtomography to understand the mechanical properties of materials under a variety of loading conditions, such as tensile loading, fatigue loading, corrosion fatigue, and stress corrosion cracking. Keywords

X-Ray microtomography · Microstructure · Mechanical properties · Corrosion

1

Introduction

It is well known that the mechanical properties of materials are dependent on microstructure. Traditionally, the structure of materials has been studied by two-dimensional (2D) techniques. This approach is often inaccurate or inadequate for solving many cutting-edge problems since the properties in the bulk have been observed to be different from those obtained on the surface. Nowadays, several three-dimensional techniques are available (Fig. 1) to visualize and quantify the microstructure in 3D [1]. Depending upon the length scale, serial polishing/optical microscopy (at higher length scale) to atom probe tomography (at smaller length scale) can be used to study the 3D microstructure. The advantages of X-ray microtomography over other techniques are that it is nondestructive in nature and requires minimal sample preparation [2]. X-ray

102

103

104

105

106

107

108

109

1010

Length Scale (m)

Serial Sectioning/Optical Microscopy X-ray Tomography

Focused Ion Beam Tomography

Ag3Sn Intermetallics in Sn

TEM Tomography Al-SiC Composite

Ce-Zr Oxide Particle

Atom Probe Tomography

InGaN

Fig. 1 Three-dimensional techniques with respect to length scale [1]

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3D/4D X-Ray Microtomography: Probing the Mechanical Behavior of Materials

2015

microtomography technique has been used in the field of medicine for decades (CT scan). This technique has gained popularity in materials science. Due to improved optics and computation, the imaging resolution has now reached a level where the structure at the micro- and nanoscale can be studied and correlated to mechanical behavior [2]. Over the years, many ex situ X-ray microtomography experiments have been conducted to study the mechanical behavior of materials. However, these experiments have consisted of postmortem characterization after testing. A complete understanding of the structure-property correlation requires a thorough understanding of the evolution of microstructure as a function of time (also called 4D studies, where fourth dimension is time). The nondestructive nature of X-ray microtomography makes it suitable for in situ experiments such that the evolution of microstructure with time can be quantified. Several in situ experiments have been conducted under tension [3, 4], compression [5, 6], cyclic loading (fatigue) [7–11], stress corrosion cracking (SCC) [12–15], corrosion fatigue [16], and creep [17]. In this chapter we begin with an introduction to X-ray microtomography which includes details about the working principles, mode of imaging, and types of microtomography. The later sections describe the use of X-ray tomography to understand the mechanical properties of materials under tensile loading, fatigue loading, corrosion fatigue (CF), and stress corrosion cracking (SCC).

2

Introduction to X-Ray Microtomography

2.1

Working Principles of X-Ray Microtomography

The setup for X-ray microtomography consists of a source, a rotation stage, and a detector, as shown in Fig. 2. The specimen is mounted on a rotating stage and is placed between the source and detector. X-ray source can be categorized into either a lab-scale or synchrotron-based source [18], details of which are given in the next

Sample

X-ray Source

Detector Rotating Stage

Fig. 2 X-ray microtomography setup consisting of X-ray source, sample mounted on a rotating stage, and a detector to capture the projection

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S. S. Singh and N. Chawla

b

a

0o

45o

180o

135o

90o

2D X-ray Projections

c

At θ = 0o 225o

270o

315o

360o

Fig. 3 (a) Initial orientation of a solder joint sample with respect to the direction of incident X-ray beam, (b) 2D projections of the sample at different angles of rotation of stage (here 45
), and (c) 3D reconstruction of solder joint

section. The detector consists of a camera and a scintillator. The scintillator converts the X-rays into visible light which is captured by a CCD (charge-coupled device) or CMOS (complementary metal-oxide semiconductor) camera. The schematics in Fig. 3 illustrate the working principle of X-ray microtomography. Figure 3a shows the initial orientation of a solder joint sample (the colors of solder and copper are gray and orange, respectively) with respect to the direction of the incident X-ray beam. The stage is rotated by either 180
or 360
in small angular increments (1/8
is typical), and a projection (or radiograph) is captured for each angle. The series of grayscale images in Fig. 3b comprise of the projections of the solder joint sample taken at different angles of rotation. The orientations of the sample with respect to the incident X-ray beam are shown directly above/below the corresponding projections. These projections are then used to obtain the 3D reconstruction of the object (Fig. 3c) by using algorithms such as filtered back projection, Gridrec algorithm, Fourier-transformbased Feldkamp-Davis-Kress (FDK), iterative algebraic reconstruction techniques (ART), etc. [2, 19]. Small values of angular rotation increments, i.e., higher number of projections, result in a higher accuracy of 3D reconstruction of the projections. A typical image set consists of ~1000–1500 projections.

2.2

Lab-Scale and Synchrotron Microtomography

X-ray microtomography can be performed either by using a microfocus X-ray apparatus (lab-scale apparatus) or by synchrotron radiation [20–22]. In a typical

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synchrotron radiation facility, electrons are produced by heating a cathode (~1100–1200
C). These electrons are accelerated to relativistic speeds in a linear accelerator with very high energy (~400–500 MeV) and then injected into the booster synchrotron where the energy of the electrons increases from MeV to GeV (~5–8 GeV). The high-energy electron beam is then introduced into the storage rings where it travels in a circular path with the help of electromagnets. Whenever there is a change in trajectory of these high-energy electrons by the use of bending magnets or insertion device (wiggler or undulators), X-ray beams are produced, which are then used for experiments. Over the years, synchrotron facilities have evolved from first-generation to modern third-generation facilities [23]. Some of the thirdgeneration synchrotron radiation facilities include the Advanced Photon Source (APS) at Argonne National Laboratory in the USA, the 6-GeV European Synchrotron Radiation Facility (ESRF) in France, and the Super Photon Ring 8-GeV (SPring-8) in Japan [20]. Third-generation synchrotron radiation facilities produce high brilliance (flux), highly coherent, and parallel X-ray beams. In the lab-scale microtomography, X-ray tubes are used to produce Bremsstrahlung radiation (or polychromatic radiation) in a cone beam. The tube potential and current can be varied to change the energy and flux, respectively. Table 1 lists the major differences between synchrotron and lab-scale microtomography [21]. The advantages of using synchrotron source over lab-scale microtomography are a higher photon flux, a highly coherent beam, and a parallel beam. Higher flux in synchrotron facility leads to higher temporal resolution (scanning time is very short) than lab-scale microtomography which makes synchrotron radiation to be suitable for fast, dynamic experiments, such as fatigue, stress corrosion cracking, and corrosion fatigue. The parallel X-ray beam in synchrotron source can also be used to perform phase-contrast imaging, which will be discussed in the next section. Moreover, the use of monochromatic and parallel beam in synchrotron source results in fewer artifacts and higher accuracy in reconstructed images. In contrast, polychromatic beam in lab-scale microtomography leads to artefacts in the reconstructed images due to beam hardening [22]. The major disadvantages of synchrotron source include little access for experimentation, significant cost, and space. The advantages of lab-scale microtomography over synchrotron source include higher energies, requirement of a small space, less expensive, and easy accessibility. In many cases, lab-scale and synchrotron microtomography are complementary to each other. As very limited time is usually available to perform experiments in the Table 1 Major differences between synchrotron source and microfocus tube source Synchrotron source Very coherent monochromatic X-ray beam Essentially parallel beam Moderate energy (10–30 keV) High flux Extremely costly, large area, difficult to access

Microfocus tube source Polychromatic beam Conical beam High energy (~10–200 keV) Low flux Significantly less costly, small area, easy to access

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synchrotron beamline, one can use lab-scale microtomography to select suitable samples before using synchrotron source. For example, to study the effect of corrosion pits on fatigue, corrosion fatigue, and stress corrosion cracking behavior of an alloy, lab-scale microtomography can be used to select suitable samples, based on the size and shape of the corrosion pits, before performing experiments in the synchrotron beamline.

2.3

Modes of Imaging in X-Ray Microtomography

X-ray synchrotron microtomography can be conducted using both absorption contrast and phase contrast [20, 24]. The proportion of phase contrast and absorption contrast depends upon the distance between the sample and the X-ray detector (e.g., scintillator), with smaller distances yielding less phase contrast relative to absorption contrast.

2.3.1 Absorption Contrast Imaging This is the common mode of imaging in X-ray tomography and is based on the attenuation of X-rays as they pass through the sample. As X-rays pass though the specimen, the transmitted beam intensity is given by Beer-Lambert’s law as [2] I ¼ I 0 eðρÞρt μ

where Io is the original intensity of the incoming X-rays, t is the thickness of the specimen being imaged, I is the transmitted intensity after the beam has travelled a distance “t,” ρ is the density of the specimen, μ is the linear attenuation coefficient, and (μ/ρ) is the mass attenuation coefficient. Mass attenuation coefficients depend upon the atomic number of the specimen and wavelength or energy of the incident X-rays. The contrast between two phases in the radiography is due to the difference in mass attenuation coefficient and density. Therefore, phases with higher mass attenuation coefficient and density appear darker, and low-density phases appear brighter in the projections. In other words, higher-density phases appear brighter, and lower-density phases appear darker in the reconstructed images as the linear attenuation coefficient (μ) values are used during reconstruction [2]. Figure 4 shows a 2D slice of 3D reconstructed volume of Al 7075 alloy showing different types of inclusions (or constituent particles) and pores [25]. Due to absorption contrast, Fe-bearing inclusions (high mass density) appear bright, and pores (low mass density) appear dark in the reconstructed images.

2.3.2 Phase-Contrast Imaging Phase contrast occurs when X-rays are refracted by specimen interfaces. Phases on both sides of a phase boundary exhibit different refractive indices, which causes a phase shift of the propagating waves, causing interference patterns in the detector [2]. After an absorption-based reconstruction, microstructural features are distinguishable primarily

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Gray boundary around Febearing inclusion

Fe-bearing inclusion Si-bearing inclusion

Pore

S T

White boundary around pores

25 mm

Fig. 4 A 2D X-ray microtomography slice showing both absorption and phase contrast. Absorption contrast leads to brighter Fe-bearing inclusions and black pores in the reconstructed image. Phase contrast leads to the formation of near-field phase-contrast fringes. The periphery of the Si-bearing inclusions and pores is white, and periphery of Fe-bearing inclusions is of gray color [25]

through absorption contrast due to the difference in their mass attenuation coefficients and density. However, near-field phase-contrast fringes (mainly first-order interference Fresnel fringes) are also present at the interfaces due to phase-contrast imaging, as shown in Fig. 4. Due to phase contrast, near-field phase-contrast fringes form along the peripheries of Fe-bearing inclusions (gray boundary), Si-bearing inclusions (white boundary), and the pores (white boundaries). Increasing the distance between the detector and the sample will lead to stronger and broader fringes due to an increase in the proportion of phase contrast compared to absorption contrast. X-ray beams produced at the synchrotron radiation facilities exhibit a high degree of coherence, and therefore phase contrast is primarily used at synchrotron sources. The presence of phase contrast becomes significant in the material systems where two phases are not distinguishable in the reconstructed images due to similar mass attenuation coefficient and density values. One such system is SiC-reinforced Al-matrix composites, where absorption alone could result in very poor contrast difference between SiC particles and the matrix. The presence of phase contrast, however, results in Fresnel fringes around SiC particles, leading to easy detection of particles in the matrix [1, 8]. This system will be described in detail later in this chapter.

3

Application of X-Ray Microtomography in Materials Science

3.1

3D Microstructural Characterization

Over the years, X-ray microtomography has emerged as an excellent technique to characterize the microstructure of materials in three dimensions. Several studies have been performed on the use of X-ray microtomography to visualize and quantify

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(b)

(a)

Pores

Mg2Si Al7Cu2Fe

Al23Fe4Cu

Composite inclusion L

Pores Fe- bearing inclusions

S T

Si- bearing inclusions

100 μm

Fig. 5 (a) 2D reconstructed X-ray microtomography slice showing Si-bearing inclusions (Mg2Si), Fe-bearing inclusions (Al7Cu2Fe, Al23Fe4Cu and composite inclusions), and pores and (b) 3D rendering of inclusions [25]

the microstructure of different material systems, such as Al alloys [25], metal matrix composites [26, 27], Mg alloys [28], etc. In this section, we describe a case study on 3D microstructural characterization of Al 7075 alloy [25]. These materials are used in aerospace industries, as structural materials, due to their high strength-to-weight ratio [29]. This alloy contains Zn, Mg, and Cu as the main alloying elements which form precipitates during aging treatment. They also contain second-phase particles, also called inclusions or constituent particles, which form during casting and affect mechanical and corrosion behavior of the alloys. These particles can be classified as Si-bearing and Fe-bearing particles [30, 31]. The fatigue cracks have been found to initiate and then propagate from these inclusions. X-ray microtomography was performed at the 2-BM beamline of the Advanced Photon Source at Argonne National Laboratory. The details of the microtomography system at 2-BM have been described elsewhere [27]. The energy of X-ray beam was ~24 keV. Figure 5a shows a 2D slice of the reconstructed image, where Fe-bearing inclusions, Si-bearing inclusions, and pores can be seen. Due to the absorption contrast, Fe-bearing inclusions appear white, Si-bearing inclusions appear gray, and the pores appear black. Moreover, based on contrast difference, two different types of Fe-bearing inclusions can also be identified, namely, Al7Cu2Fe and Al23Fe4Cu. The composite inclusions indicate the mixture of both types of Fe-bearing inclusions. Energy dispersive spectroscopy (EDS) in SEM was used to obtain the composition of these inclusions [25]. Figure 5b shows the 3D rendering of inclusions, which was obtained by using Avizo ® Fire (VSG, Burlington, MA). From Fig. 5b it is clear that inclusions having a small volume appear to be spherical, whereas inclusions with high volume are significantly more elongated along rolling direction [25]. Moreover, pores are always associated with inclusions. Although

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the particles appear to be separate from each other in 2D, they are interconnected in three dimensions leading to the accurate quantification in terms of size and aspect ratio. The volume fraction of inclusions and pores was found to be ~1.63% and 0.06%, respectively.

3.2

In Situ 4D X-Ray Microtomography

Over the years, several theories have been developed to understand the mechanisms of crack initiation and propagation by observation of fractured surfaces. Mechanical testing using in situ X-ray microtomography has received attention since it provides a means of direct observation of damage within the material. Such information can also be used to validate analytical or numerical theories. Many in situ studies have been performed using X-ray microtomography [32–36]. The following sections describe the use of X-ray microtomography as a means to understand the mechanical properties of materials.

3.2.1 Loading Stage for In Situ Experiments In situ mechanical testing can be performed in both lab-scale and synchrotron systems. Synchrotron microtomography is preferable in the case of stress-assisted corrosion experiments, such as stress corrosion cracking and corrosion fatigue, where high temporal resolution is required. The in situ experiments require a dedicated loading stage to perform mechanical testing. Singh et al. [37] have listed the challenges in designing in situ mechanical testing stage. The size and weight of the motorized linear actuator are restricted by the weight limits of the motion stages for their precise and reproducible movement. Moreover, the weight of the load frame must be distributed symmetrically about the rotational axis for precise rotational movement during scanning. The stage also requires an X-ray transparent material in the scanning area which should also be able to bear the applied load. The distance between the fixture and the detector should be sufficient enough such that there is no contact between the scintillator and load bearing sleeve while scanning. Singh et al. [37] have described the evolution of three generations of loading stages to perform mechanical testing in synchrotron. Figure 6 shows the thirdgeneration loading stage. The loading stage was designed to be less than 2 kg due to weight limitations on the motion stage at the synchrotron beamline. The loading jig has a load cell (to measure the load), a stepper motor with linear actuator (to impose a displacement on the specimen to achieve the desire load), and two grips (top grip being attached to the stepper motor and bottom grip being attached to the load cell). The load cell had a capacity of 500 N. The load was transmitted from the top to bottom of the specimen through a polymer sleeve [poly(methyl methacrylate) or PMMA]. The stage also had specimen alignment capability. Concentric alignment was done by translating the top grip with respect to the bottom grip using set screws. Axial misalignment was minimized by the use of shims. The top grip was also allowed to rotate freely about its mounting screw to self-align in the third

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Stepper motor with linear actuator

Sample loading is from top

Custom shaft collar (x2)

Load Cell (Can’t be seen in image)

PMMA sleeve Spring -loaded concentric alignment screw (top clamp self aligns in the perpendicular direction)

Fig. 6 Loading stage to perform in situ mechanical testing [37]

rotational direction. The actuator was controlled using the feedback from the load cell (load controlled). The following sections describe the use of these loading stages in understanding the evolution of damage in different materials, such as metal matrix composites (MMC) and Al alloys. The use of X-ray synchrotron microtomography made it possible to visualize and quantify the damage in these materials in 3D/4D. All experiments in the next sections were carried out at the 2-BM beamline of the Advanced Photon Source (APS) at Argonne National Laboratory with the parameters approximately same as mentioned in the Section 3.2.

3.2.2

Damage Evolution in Metal Matrix Composites (MMCs) in Tensile Testing In many structural applications, metals are being replaced by metal matrix composites (MMCs) because MMCs exhibit an optimum combination of high strength, low density, and high stiffness. The damage behavior of MMCs has been studied extensively, and it is often agreed that the damage in particle-reinforced MMCs takes place by a combination of particle fracture and matrix void growth. However, these observations were made either by the surface examination or by postmortem analysis of the fractured surface. In this regard, X-ray synchrotron microtomography is very useful to understand the precise mechanism of the crack initiation and propagation in hot-pressed and extruded SiC particle-reinforced 2080 Al alloy matrix composites [1, 27]. In particular, postmortem analysis was not able to determine whether particles cracked ahead of the crack tip or if particle fracture occurred by crack propagation through the particles. Figure 7a shows the tensile stress-strain curve of the composite indicating the fracture strain to be ~1.6%. To understand the damage evolution, tomography scans of the sample were taken at four strain values, i.e., 0%, 0.4%, 1%, and 1.6%.

0

100

200

300

400

500

0

0.5

1 Strain (%)

0

10

20

30

40

50

1.5

0 500

2

1500

Distance (µm)

1000

2000

0% strain 0.4% strain 1.0% strain 1.6% strain –Fracture

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

Distance (µm)

200 400 600 800 1000 1200

0% 1% 1.6%

3D/4D X-Ray Microtomography: Probing the Mechanical Behavior of Materials

Void volume fraction (%)

Fraction of particles fractured (%)

Fig. 7 (a) Stress-strain curve, (b–d) progression of damage with strain, (e, f) extent of damage near fracture region and away from it [1, 27]

Stress (MPa)

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The evolution of damage in MMC with strain, as shown in Fig. 7b, c, d, shows the occurrence of both particle fracture and debonding at the particle/matrix interface. 3D quantitative analysis showed that particles with larger aspect ratio and larger size were more prone to fracture. It was also concluded that the damage was concentrated in the fracture region and the extent of damage decreased away from fracture region (Fig. 7e, f).

3.2.3 Fatigue Behavior of Al-SiC Composite Most structural materials undergo cyclic loading (fatigue) leading to failure of the component. One of the most studied topics in fatigue is fatigue crack growth. For decades, surface measurement techniques, such as optical microscopy, have been used to measure the fatigue crack growth rates. However, this technique does not provide insights into the bulk crack propagation. Bulk measurement techniques, such as use of clip gauge, provide the bulk behavior. However, it does not give an insight into the interaction of cracks with the microstructure. X-ray microtomography has been used to measure crack growth rates in three dimensions and also to correlate the crack propagation behavior with microstructure [9, 10]. MMCs show higher resistance to fatigue compared to their counterparts, i.e., monolithic alloys. The fatigue resistance of the MMCs depends upon several factors, such as particle size, particle volume fraction, matrix and interfacial microstructure, and testing environment. Using the loading stage described in Fig. 6, Peter et al. [38] studied the damage characteristics of SiC particle-reinforced 2080 Al alloy matrix through in situ X-ray synchrotron microtomography at high and low load ratio (R ratio) [38]. The composition of the MMC and powder metallurgy process for fabrication of MMC were similar to that described in Section 3.2.3. Figure 8 shows the 2D X-ray tomography slices showing fatigue crack growth damage at the crack tip and interaction between the crack and the SiC particles at low (R = 0.1) and high R ratios (R = 0.6). Under the low R ratio, slices at 135 μm from the specimen surface show that very few particles were fractured in front of the crack tip. Figure 8a–c shows that the crack propagated in a linear fashion and not deflected around the particle. In contrast, under the high load ratio, slices at 295 μm from the specimen surface show that several particles were fractured in the area ahead of the crack tip (Fig. 8d–f). The crack path was tortuous and tried to pass through previously fractured particles. Figure 8d shows that the circled particle was not fractured after 7000 fatigue cycles. However, the particle fractured by 8000 cycles (Fig. 8e), and the crack later passed through the same fractured particle (Fig. 8f). 3D renderings of progression of cracks at R = 0.6 are shown in Fig. 8g–i indicating that a large number of particles were fractured and then cracks eventually passed through them. The mechanism of particle fracture in the case of R = 0.6 was even common at the low ΔK due to large Kmax, indicating that the static component of fracture, Kmax, controlled fatigue crack growth and contributed to particle fracture at the crack tip. The quantification of interactions between fatigue cracks and particles was performed in 3D. A statistical summary under both low and high load ratios is given in Table 2. The statistics show that the fatigue crack has a tendency to bypass

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3D/4D X-Ray Microtomography: Probing the Mechanical Behavior of Materials

(a)

5000 cycles

(b)

crack tip

crack tip

(d)

7000 cycles

7400 cycles

(e)

8000 cycles

(c)

2025

7900 cycles

crack tip

(f)

21000 cycles

crack tip

crack tip

crack tip

(g)

(h)

0 Cycles

7000 Cycles

(i) 21000 Cycles

Fig. 8 Fatigue crack growth behavior (a–c) at 135 μm from specimen surface at R = 0.1 for different number of cycles, (d–f) at 295 μm from specimen surface at R = 0.6 for different number of cycles. Locations of crack tip are located. (g–i) Progression of crack at R = 0.6 are shown. Only those particles through which the crack eventually passes are shown. Particles are shown in green if fractured ahead of crack. Particles are shown in blue if crack has passed through. Newly observed instances are displayed as opaque (or solid particles) [38]

Table 2 3D Analysis for Both Low and High R Ratios [38] Total number of observed particles with which crack interacts during fatigue crack growth % particles crack bypasses % particles crack passes through % particles through which crack passes that fractured ahead of crack tip

R = 0.1 98

R = 0.6 29

66.3 33.7 9.1

10.3 89.7 84.6

particles under low load ratio. However, under high R ratio, the large damage zone ahead of the crack tip due to high Kmax contributed to a significant number of particles fracturing. It also shows that cracks passed through ~85% of the fractured particles ahead of the crack tip.

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S. S. Singh and N. Chawla

In Situ Mechanical Testing in a Corrosive Environment

Structural components are not only subjected to stress but are also often exposed to corrosive environment. The design and development of high performance structural materials requires a thorough understanding of the relationship between environment, mechanical stresses, microstructure, and properties. Conducting in situ experiments in a corrosive medium has its own challenges – the most difficult being the design of the loading stage.

3.3.1 Loading Stage Design The biggest challenge in designing the loading stage using a liquid medium is to avoid the reaction of corrosive fluid with any part of the loading stage, especially the grips. Therefore, the material used for the grips must be chemically inert to the solution while also sustaining the applied load. In order to study the effects of corrosion fatigue as well as stress corrosion cracking, Singh et al. [37, 39, 40] incorporated modifications in the loading stage described above. Figure 9a, b shows the modification in the loading stage to perform mechanical testing in liquid and moisture environments, respectively. The PMMA sleeve has not been shown here in order to clearly show the arrangements inside the stage. A polymeric (b)

(a) Motor

Motor Humidity sensor

Plastic wrap

Grip wire

Kapton tube

Corrosive fluid Sample PEEK

Pin Wet sponge Grip

Load cell Base

Base

Fig. 9 Modification in the loading jig to perform in situ mechanical testing in corrosive environment (a) in liquid environment and (b) in moisture [37, 39, 40]

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PEEK (Polyether ether ketone) cylindrical grip was chosen to replace the bottom steel grip since it is chemically resistant to many solutions and has good mechanical strength. To perform the corrosion testing in moisture, an annular wet sponge was placed around the load cell at the base of the loading stage (Fig. 9b). Since SCC experiments require long exposures to moisture, maintaining a constant humidity inside the chamber is a big challenge as fluctuations can lead to variability in results. In order to minimize the loss of moisture and to maintain constant relative humidity, the top part of the loading stage was covered with a plastic wrap. A humidity sensor was placed in front of the notch to measure the relative humidity throughout the test. The sample was clamped using grips attached to the actuator on one side and the load cell on the other.

3.3.2 In Situ Fatigue Corrosion of Al 7075 Alloys Singh et al. [16, 37] performed in situ corrosion-fatigue experiments on AA7075T651 using X-ray synchrotron microtomography in EXCO (exfoliation corrosion) solution (4 M NaCl, 0.5 M KNO3, and 0.1 M HNO3). “Pink beam,” which is a polychromatic beam with low and high energies removed from the white beam spectrum, was used during the corrosion experiments. The higher photon flux in pink beam compared to a monochromatic beam allowed for significantly faster data acquisition. While a typical scan with a double-multilayer monochromatic (DMM) beam takes 20 min to complete, a pink beam scan can take only 0.5–1 s. Although much faster scans are technically possible using a pink beam, significant X-ray attenuation by the corrosive fluid required higher exposure times (2–3 min) to achieve a suitable signal-to-noise ratio. Figure 10a shows the fatigue crack growth rates measurement with and without EXCO solution. It is evident that the presence of corrosive medium accelerated the crack growth rate. Figure 10b shows a 2D X-ray microtomography image of the fatigue crack along with corrosion product and the hydrogen bubbles inside the crack formed due to reaction between AA7075 and the EXCO solution. Figure 10c shows a 3D rendering of the crack, bubbles, corrosive fluid, and corrosion products (segmented on a few slices from the part of crack). It was also shown that the shape of the bubbles changes and new bubbles form in a fatigue cycle. These results provided insights to the fundamentals of evolution of hydrogen bubbles inside a growing fatigue crack which would not have been possible with conventional 2D measurement techniques. 3.3.3 In Situ Stress Corrosion Cracking (SCC) in Al 7075 Alloys Singh et al. [39] performed in situ stress corrosion cracking experiments on Al 7075 alloys in humid environment (~95% RH). The modified arrangement of the loading jig led to fairly constant relative humidity (RH) inside the loading jig, as shown in Fig. 11a. Figure 11b shows the 3D rendering of the SCC crack growth in moisture at a constant load of 110 N. Each color represents cumulative time during which the stress corrosion cracks grew. Two separate cracks were observed to grow independently, where the bottom crack almost covers the entire thickness. Through the

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(a) Corrosion-fatigue

10

–6

10–7 Fatigue

10–8

4

5

(b)

7

8

9 10

Bubble

Corrosive Fluid

Notch

6

Bubble

Corrosion Product

Top View

Corrosive Fluid

(c)

Side View

100 μm

Corrosion product

Fig. 10 (a) Fatigue crack growth rates (da/dN) versus cyclic stress intensity (ΔK) with and without corrosive environment, (b) 2D X-ray microtomography slice showing hydrogen bubbles, corrosive fluid, and corrosion products, and (c) 3D reconstruction of the fatigue crack (bubble + fluid) and corrosion products from selected area of the segmented crack [16]

3D rendering, it was possible to observe a significant variation in the crack length through the thickness and therefore variation in crack growth rate, which would not have been possible by 2D measurements.

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100

Relative Humidity (%)

(a) 98 96 94 92 90

-5

0

5

10

15

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30

Time (Hr)

(b)

Precrack

9.4 h

17.8 h 21.9 h 28.6 h

5.5 h Crack advance

14.8 h

29.9 h

26.6 h 250 mm

Fig. 11 (a) Constant relative humidity inside the loading jig to perform SCC experiments in moisture and (b) 3D SCC crack growth profile in moisture as a function of time. Significant variation of crack length through the thickness was observed [39] Fig. 12 2D slices of the sides of the specimen showing the discontinuous cracks (crack jump) [39]

(a)

Front side

jump

Crack advance

(b)

Back side

jump

2D slices obtained from X-ray microtomography in Fig. 12a, b shows the appearance of discontinuous cracks on both sides of the specimen after about 25 h of SCC. It was postulated in literature that the second crack initiates in front of the first crack and then grows separately. However, 3D renderings of the crack with

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(f)

(a)

No Surface crack

(b)

(g)

(c) Surface crack appears

100 μm Surface crack appears

13.8 h

(d)

17.8 h 19.9 h

100 μm

21.9 h

(e)

24.9 h

Fig. 13 Stepwise growth of crack showing the appearance of the discontinuous crack (c) and then further growth, (f–g); top view of (b) and (c) shows that the surface crack does not appear in (f) but appears at (g) [39] Fig. 14 Comparison of 2D and 3D crack growth rate measurements. The 3D crack growth rate is quite uniform than of the measurement based on 2D [39]

10-7 Average crack length through thickness

da/dt (m/s)

Average of crack lengths at sides

10-8

10-9

0

2

4

6

8

10

12

14

16

K(MPa-m1/2)

time (Fig. 13a–g) show that these discontinuous surface cracks were actually connected in 3D. The time interval is shown in the legend. It can be seen that the main crack grew from inside toward the surface (Fig. 13a, b) and finally appeared as a discontinuous crack on the surface (Fig. 13c) and then continued to grow. The

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appearance of discontinuous surface cracks can clearly be seen in Fig. 13g, which was not in Fig. 3f. Both Fig. 13f, g are the top views of Fig. 13b, c. 2D and 3D crack growth rates were measured and compared. The crack growth rates were measured using the average crack lengths though thickness (3D measurement) and average of the crack lengths at both sides (2D measurement), as shown in Fig. 14. The crack growth rates measured in 2D showed significant amount of variability compared to the growth rate when the average crack length through thickness was used (3D). In addition, a large deviation in the surface crack lengths resulted in significant differences in stress intensity factor, as shown with error bars in Fig. 14. These were attributed to the highly nonuniform growth of the SCC crack at the surface which includes formation of discontinuous cracks and independent growth of the two cracks. The results obtained here show the importance and necessity of 3D techniques, such as X-ray synchrotron microtomography, to study the crack growth behavior during stress corrosion cracking.

4

Future Directions

A great deal of advances has been made in 3D and 4D materials science. The resolution of instruments has improved, as have the capabilities for handling large data sets and processing the data (i.e., image analysis). New techniques based on X-ray tomography are emerging, including diffraction contrast tomography (DCT). This technique has been used at the synchrotron sources but now can be done in the lab as well. Here, 3D orientations of grains, as well as the evolution of these crystalline structures, can be resolved over time and under particular stimuli. Such experiments are very exciting. With the advent of 4D techniques, one of the important problems is how to handle the large amount of data generated. How should one look through all the data to find relevant microstructural features of interest, and can automated scripts and computer codes be of help? In this area, techniques such as machine learning have become quite popular, where algorithms can be “trained” to conduct image segmentation of a larger stack of images, for example. Using two or three segmented images for the “training,” the remaining stack of images can then be characterized automatically. In short, the capabilities for faster, better resolution and larger samples are constantly improving yielding a bright future for X-ray-based techniques and 4D materials science.

References 1. Williams JJ, Chapman NC, Jakkali V, Tanna VA, Chawla N, Xiao X, De Carlo F. Characterization of damage evolution in SiC particle reinforced Al alloy matrix composites by in-situ X-ray synchrotron tomography. Metall Mater Trans A. 2011;42:2999–3005. 2. Stock SR. Microcomputed tomography: methodology and applications. Boca Raton: CRC Press; 2008.

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3. Maire E, Zhou SX, Adrien J, Dimichiel M. Damage quantification in aluminium alloys using in situ tensile tests in X-ray tomography. Eng Fract Mech. 2011;78(15):2679–90. 4. Maire E, Carmona V, Courbon J, Ludwig W. Fast X-ray tomography and acoustic emission study of damage in metals during continuous tensile tests. Acta Mater. 2007;55:6806–15. 5. Patterson BM, Cordes NL, Henderson K, Williams JJ, Stannard T, Singh SS, Ovejero AR, Xiao X, Robinson M, Chawla N. In situ X-ray synchrotron tomographic imaging during the compression of hyper-elastic polymeric materials. J Mater Sci. 2016;51:171–87. 6. Cordes NL, Henderson K, Stannard T, Williams JJ, Xiao X, Robinson MWC, Schaedler TA, Chawla N, Patterson BM. Synchrotron-based X-ray computed tomography during compression loading of cellular materials. Microscopy Today. 2015;23:12–9. 7. Guvenilir A, Breunig TM, Kinney JH, Stock SR. Direct observation of crack opening as a function of applied load in the interior of a notched tensile sample of Al-Li 2090. Acta Mater. 1997;45(5):1977–87. 8. Chapman NC, Silva J, Williams JJ, Chawla N, Xiao X. Characterization of thermal cycling induced cavitation in particle reinforced metal matrix composites by three-dimensional (3D) X-ray synchrotron tomography. Mater Sci Technol. 2014;31(5):573–8. 9. Williams JJ, Yazzie KE, Padilla E, Chawla N, Xiao X, De Carlo F. Understanding fatigue crack growth in aluminum alloys by in situ X-ray synchrotron tomography. Int J Fatigue. 2013;57:79–85. 10. Khor KH, Buffiere JY, Ludwig W, Toda H, Ubhi HS, Gregson PJ, Sinclair I. In situ high resolution synchrotron X-ray tomography of fatigue crack closure micromechanisms. J Phys Condens Matter. 2004;16:S3511–5. 11. Toda H, Sinclair I, Buffiere JY, Maire E, Connolley T, Joyce M, Khor KH, Gregson P. Assessment of the fatigue crack closure phenomenon in damage-tolerant aluminium alloy by in-situ high-resolution synchrotron X-ray microtomography. Philos Mag. 2003;83(21): 2429–48. 12. Babout L, Marrow TJ, Engelberg D, Withers PJ. X-ray microtomographic observation ofintergranular stress corrosion cracking in sensitized austenitic stainless steel. Mater Sci Technol. 2006;22:1068–75. 13. Marrow TJ, Babout L, Jivkov AP, Wood P, Engelberg D, Stevens N, Withers PJ, Newman RC. Three dimensional observations and modelling of intergranular stress corrosion cracking in austenitic stainless steel. J Nucl Mater. 2006;352:62–74. 14. King A, Johnson G, Engelberg D, Ludwig W, Marrow TJ. Observations of intergranular stress corrosion cracking in a grain- mapped polycrystal. Science. 2008;321(5887):382–5. 15. Singh SS, Williams JJ, Stannard TJ, Xiao X, De Carlo F, Chawla N. Measurement of localized corrosion rates at inclusion particles in AA7075 by in situ three dimensional (3D) X-ray synchrotron tomography. Corros Sci. 2016;104:330–5. 16. Singh SS, Williams JJ, Xiao X, De Carlo F, Chawla N. In Situ three dimensional (3D) X-ray synchrotron tomography of corrosion fatigue in Al7075 alloy. In: Srivatsan TS, Imam AM, Srinivasan R, editors. Fatigue of materials II: advances and emergences in understanding. Pittsburgh: Materials Science and Technology; 2012. 17. Isaac A, Sket F, Reimers W, Camin B, Sauthoff G, Pyzalla AR. In situ 3D quantification of the evolution of creep cavity size, shape, and spatial orientation using synchrotron X-ray tomography. Mater Sci Eng A. 2008;478:108–18. 18. Stock SR. Recent advances in X-ray microtomography applied to materials. Int Mater Rev. 2008;53:129–81. 19. Marone F, Stampanoni M. Regridding reconstruction algorithm for real-time tomographic imaging. J Synchrotron Radiat. 2012;19:1029–37. 20. Baruchel J, Buffiere JY, Maire E, Merle P, Peix G. X-ray tomography in material science. Paris: HERMES Science Publications; 2000. 21. Mertens JCE, Williams JJ, Chawla N. Development of a lab-scale, high-resolution, tubegenerated X-ray computed-tomography system for three-dimensional (3D) materials characterization. Mater Charact. 2014;92:36–48.

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22. Mertens JCE, Williams JJ, Chawla N. A study of Pb-rich dendrites in a near-eutectic 63Sn-37Pb solder microstructure via laboratory-scale micro X-ray computed tomography (μXCT). J Electron Mater. 2014;43:4442–56. 23. Stock SR. X-ray computed tomography, X-ray techniques, characterization of materials. Hoboken: Wiley; 2012. 24. Maire E, Withers PJ. Quantitative X-ray tomography. Int Mater Rev. 2014;59:1–43. 25. Singh SS, Schwartzstein C, Williams JJ, Xiao X, De Carlo F, Chawla N. 3D microstructural characterization and mechanical properties of constituent particles in Al 7075 alloys using X-ray synchrotron tomography and nanoindentation. J Alloys Compd. 2014;602:163–74. 26. Silva FA, Williams JJ, Mueller BR, Hentschel MP, Portella PD, Chawla N. 3D microstructure visualization of inclusions and porosity in SiC particle reinforced Al matrix composites by X-ray synchrotron tomography. Metall Mater Trans A. 2010;41:2121–8. 27. Williams JJ, Flom Z, Amell AA, Chawla N, Xiao X, De Carlo F. Damage evolution in SiC particle reinforced Al alloy matrix composites by X-ray synchrotron tomography. Acta Mater. 2010;58:6194–205. 28. Wang M, Xu Y, Zheng Q, Wu S, Jing T, Chawla N. Dendritic growth in Mg-based alloys: phasefield simulations and experimental verification by X-ray synchrotron tomography. Metall Mater Trans A. 2014;45:2562–74. 29. Singh SS, Loza JJ, Merkle AP, Chawla N. Three dimensional microstructural characterization of nanoscale precipitates in AA7075-T651 by focused ion beam (FIB) tomography. Mater Charact. 2016;118:102–11. 30. Singh SS, Guo E, Xie H, Chawla N. Mechanical properties of intermetallic inclusions in Al 7075 alloys by micropillar compression. Intermetallics. 2015;62:69–75. 31. Merkle AP, Lechner L, Steinbach A, Gelb J, Kienle M, Phaneuf MW, Unrau D, Singh SS, Chawla N. Automated correlative tomography using XRM and FIB-SEM to span length scales and modalities in 3D materials. Microsc Microanal. 2014;28:S10–3. 32. Withers PJ, Preuss M. Fatigue and damage in structural materials studied by X-ray tomography. Annu Rev Mater Res. 2012;42:81–103. 33. Buffiere JY, Maire E, Adrien J, Masse JP, Boller E. In situ experiments with X-ray tomography: an attractive tool for experimental mechanics. Exp Mech. 2010;50:289–305. 34. Beckmann F, Grupp R, Haibel A, Huppmann M, Nöthe M, Pyzalla A, Reimers W, Schreyer A, Zettler R. In-situ synchrotron X-ray microtomography studies of microstructure and damage evolution in engineering materials. Adv Eng Mater. 2007;9:939–50. 35. Salvo L, Suery M, Marmottant A, Limodin N, Bernard D. 3D imaging in material science: application of X-ray tomography. C R Phys. 2010;11:641–9. 36. Maire E, Buffiere J-Y, Salvo L, Blandin JJ, Ludwig W, Letang JM. On the application of X-ray microtomography in the field of materials science. Adv Eng Mater. 2001;3:539–46. 37. Singh SS, Williams JJ, Hruby P, Xiao X, De Carlo F, Chawla N. In situ experimental techniques to study the mechanical behavior of materials using X-ray synchrotron tomography. Integr Mater Manuf Innov. 2014;3:1–14. 38. Hruby P, Singh SS, Williams JJ, Xiao X, De Carlo F, Chawla N. Fatigue crack growth in SiC particle reinforced Al alloy matrix composites at high and low R-ratios by in situ X-ray synchrotron tomography. Int J Fatigue. 2014;68:136–43. 39. Singh SS, Williams JJ, Lin MF, Xiao X, De Carlo F, Chawla N. In situ investigation of high humidity stress corrosion cracking of 7075 aluminum alloy by three-dimensional (3D) X-ray synchrotron tomography. Mater Res Lett. 2014;2(4):217–20. 40. Singh SS, Stannard TJ, Xiao X, Chawla N. In situ X-ray microtomography of stress corrosion cracking and corrosion fatigue in aluminum alloys. JOM. 2017;69(8):1404–14.

Mechanical Testing of Single Fibers

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Krishan K. Chawla

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Tensile Testing of a Single Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Recoil Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Loop Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Dynamic Elastic Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Shear Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Transverse Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Micro Raman Strain Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Tensile Testing Stages for Insitu Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Materials in the form of long fibers are used in a variety of structural applications. Examples include cables, ropes, woven or nonwoven fabrics, and as reinforcements in composite materials. Thus, it is important to have knowledge of their mechanical characteristics. Mechanical testing of long, slender individual fibers is a nontrivial matter. Many techniques for testing single fibers are available. We describe some of the important ones in this chapter: tensile testing by using cardboard frame technique, recoil compression test, shear modulus by means of torsional pendulum test, loop test, techniques involving laser interferometry, Raman spectroscopy, and insitu techniques involving the use of scanning electron and transmission electron microscopy.

K. K. Chawla (*) Department of Materials Science and Engineering, University of Alabama at Birmingham, Birmingham, AL, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_48

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Introduction

Materials in a fibrous form are used in used in structural applications. Fibers are used in a variety of forms: unidirectional fibers, two dimensional fabrics (woven or nonwoven), ropes or cables, or by incorporating in a matrix material as a reinforcement to form a composite [1]. In any of these structural applications, the high performance fibers undergo a variety of loading. Hence, it is of great importance to have knowledge of the stress-strain behavior of individual fiber. Ideally, it would be desirable to have knowledge about the stress-strain behavior of individual fibers under tension, compression, torsion, flexure, etc. In this chapter we describe tensile as well as other mechanical testing techniques for single fibers. Many high performance fibers such as aramid, polyethylene, carbon, etc. can have highly anisotropic properties. For example, the elastic modulus, the strength, the coefficient of thermal expansion, thermal conductivity, etc. may be quite different along the fiber axis and transverse to the fiber axis. This anisotropy could have origin in the highly oriented chain structure in the case of organic fibers or from the oriented structure of the precursor fiber. In the case of pure single crystal fibers such as alumina, the anisotropy will have origin in the crystal structure. Generally, these are the longitudinal Young’s modulus of fiber, E11 or E‘, the transverse Young’s modulus E22 or Et, the major and minor Poisson’s ratios, ν12 and ν21 and the principal shear modulus, G12. Fig. 1 shows different moduli (Poisson’s ratios are not shown.) Fig. 1 A schematic showing the longitudinal Young’s modulus of fiber, E11, the transverse Young’s modulus E22, the principal shear modulus, G12. The major and minor Poisson’s ratios, ν12 and ν21 are not shown

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Tensile Testing of a Single Fiber

Tensile test is one of the most simple and important tests that can be done on conventional materials to obtain their mechanical characteristics. The variables involved in a tensile test are stress, strain, and time. Generally, we keep one of these variables constant during the test. In a tensile test, we get a uniform stress field over a relatively large effective volume of test material. A tensile test is important for evaluating inherent flaw distributions and the resultant strength distributions. In a longitudinal tensile test of a fiber, the long, thin fiber is subjected to a tensile force along its length. One end of the fiber is anchored to a fixed crosshead of the machine while the other end is secured to the crosshead that can move at a constant speed. The material, in response to the applied force, undergoes some deformation. As the specimen is elongated, the applied force is measured by means of a load cell which is put in series with the specimen. It is very important to realize that all of the displacement of the crosshead is not transmitted to the gage length section of the specimen. A part of the displacement is transmitted to the load cell, grips, coupling between specimen and the grips, and the machine frame. Ideally, one should measure the fiber elongation directly by means of an extensometer. The output of such a test is thus a force-elongation record which can be transformed into engineering stress-engineering strain or true stress-true strain curves. In the case of conventional materials, the longitudinal deformation over the gage length is measured by means of an extensometer or strain gage. Most of these devices are clip-on type or adhesively stuck on the specimen, which would be fine for conventional tensile specimens but not for thin, single fibers. Most of the high performance fibers have diameters in the micrometer to nanometer range. It is easy to see that such long and thin fibers will be difficult to handle because of a variety of problems: precise measurement of small diameter, measurement of very small loads and deformation, axial alignment, and gripping, etc. An indirect of way of measuring the axial strain in a fine diameter fiber that avoids the use of an extensometer to measure the deformation in the fiber gage length is as follows [1, 2]. We can write for the total cross-head deflection, δt, as follows: δt ¼ δn þ δc þ δm

(1)

where δn is the deflection due to the nominal gage length of the fiber ‘n, δc is the deflection due to the length of the fiber in the jaws, and δm is the deflection due to the load cell and the testing machine. We can reasonably assume that the terms, δc and δm are constants at a given load and independent of the nominal gage length. Also, one can use the relationship δn ¼ e‘n ,

(2)

where e represents strain. Eq. 2 allows one to rewrite Eq. 1 as δt ¼ e‘n þ c,

(3)

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where c is a constant. Thus, at a constant load, if we plot the total deflection, δt against the nominal gage length, ‘n, we should get a straight line whose slope is the strain, e. That is to say, one can measure deflections corresponding to different fiber lengths at a constant load in the elastic regime, then a plot of δt versus ‘n will give us the true elastic strain in the fiber at that load, from which we can obtain the elastic modulus. There is an ASTM standard (ASTM -3379-75) for specimen preparation and testing high modulus (> 20 GPa) single filaments. Other ASTM standards available for measuring the strength of fibers include: D3479-76, D3800-79, D3878-87, D4018-81. There are some very important requirements for tensile testing of a single fiber. Alignment of fiber can be a serious problem with some of the high strength and stiffness fibers because of the likelihood of fiber breakage in the grips. In order to avoid such premature failure, some kind of facing material, polyethylene or neoprene, is generally placed between the fiber and the grips. This technique is fairly commonly used; see refs. [3–5] for works involving the use of this technique to evaluate the properties of carbon fibers. Xu et al. [6] used this technique to study aluminosilicate fibers and Chawla et al. [7] used it to study the stress-strain behavior of alumina fiber, Nextel 610. The general setup is shown in Fig. 2a while a testing machine is shown in Fig. 2b. The single fiber is adhesively bonded to a cardboard frame with a longitudinal slot. The cardboard, with the bonded fiber, is clamped in the grips of a tensile testing machine. After all the adjustments and alignment procedures are done, the cardboard frame is cut off on the two sides, leaving the single fiber ready to be tested. Now when the load is applied, only the fiber experiences the load. The specimen is pulled to failure, and a record of load and apparent deformation is obtained. The tensile strength is the maximum load divided by the cross-sectional area of the fiber. The problem is how to obtain precise strain in the fiber length. One must subtract the machine compliance from the total compliance to get the true fiber deformation.

Fig. 2 (a) The cardboard frame technique for tensile testing of a single fiber. The fiber is adhesively bonded to a cardboard frame with a longitudinal slot. The cardboard, with the bonded fiber, is clamped in the grips of a tensile testing machine. After all the adjustments and alignment procedures are done, the cardboard frame is cut off on the two sides, leaving the single fiber ready to be tested. (b) A testing machine used to test single fiber

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Although direct measurement of strain in the gage length of a fiber is not easy, it can be done. A non-contacting laser correlation sensor can be used obtain the strain in a fine fiber in a single fiber tensile test. The fine fiber diameter makes the application of conventional strain measurement techniques inapplicable. In a non-contacting method, the strain is measured directly and the end-effects from the gripping, etc. are avoided. Extensometer is a device that measures change in length (extension or contraction) of a sample. Commonly, extensometers require mechanical attachment to the sample being measured. Such a procedure will not work with very fine fibers. Laser extensometry provides a noncontact measurement of elongation. A laser diffraction technique can be used for non-contact measurement of strain in the gage length of a fiber. A laser beam will diffract from a carefully mounted fiber sample and produce a series of bright bands. In order to avoid detrimental vibrations, the whole set-up should be mounted on an optical table. The technique measures axial strain by monitoring surface displacement of two points on the specimen. Two laser beams that differ in frequency by a precise amount, say 250 kHz, are brought together at the sample surface at a fixed angle. The location of any two points on the specimen is defined by the point where recombination of the laser beams occurs. When the specimen is strained, relative movement of two points on the specimen surface occurs. This results in a phase change in the recombined beam which can be used as a measure of strain in the specimen. Laser extensometry has been used to measure strain in a tungsten filament at 2700  C and in 8 μm diameter carbon fiber [8]. For textile fibers, there is an ASTM D2101 standard that allows for fibers to be gripped directly because such fibers are generally not damaged by the clamp pressure. In general, the clamps are lined with rubber or elastomeric pads to reduce the friction. So far we have discussed the measurement of mechanical properties at room temperature. Obtaining mechanical property data at high temperature is even more difficult. One needs such data, especially the strength of ceramic fibers at the high temperatures at which they are likely to be used. A major problem in such testing comes from the thermal gradients involved between the fiber specimen gage length and the portion of the fiber in the grips. Unal and Lagerlof [9] tested alumina fibers to 1500  C by putting the entire fiber and grips in the hot zone of the furnace, thus avoiding thermal gradients associated with cold grips, i.e., outside the hot zone. Beads of an alumina adhesive were deposited on the alumina fiber extremities with much skill and patience.

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Recoil Compression Test

Obtaining compressive strength of a slender fiber is even more difficult than measuring its tensile strength. An indirect technique has been devised to obtain compressive strength of fibers. It is well known that recoil of a tensile stress wave generates a compressive wave. This principle is used in the recoil compression test; and it allows one to estimate compressive strength of fibers. The fiber sample is mounted in a paper frame as described above. After mounting the frame in the jaws of the test machine, side arms are cut, the fiber is loaded to a

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predetermined stress level. At this stress level, there is a certain strain energy that is stored in the fiber. The fiber recoil is initiated by cutting the fiber at mid-length. Allen [10] describes the use of mechanical scissors to cut the fiber. The scissors must be carefully aligned with the midpoint or there may occur some lateral displacement. Hayes et al. [11] also used an electrical initiation method. This one involves the use of electrodes at a 10,000 V potential. The high potential creates an ionized path between the electrodes near the fiber that produces an electric current. The fiber is oxidized and/or ionized until the recoil begins. When the fiber is cut, the stress goes to zero at that point and the stress strain energy is converted into kinetic energy and a stress wave travels toward the clamp. We assume that the clamp is rigid. At the clamp, all the strain energy is converted to kinetic energy and a compressive stress wave travels down the fiber length. The magnitude of the compressive stress is assumed to be equal to applied stress. Each half of the cut fiber experiences the same compressive wave, i.e., each mounted fiber results in two test results, Recoil compressive stresses are calculated based on an average fiber diameter. Compressive failure of the fiber occurs if the applied stress exceeds the recoil compressive strength of the fiber.

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Loop Test

A loop test, which is a kind of bend test, can be used to estimate the strength and elastic modulus of fine fibers [12]. It is also referred to as a knot test. There are many variants of this test. Greenwood and Rose [13], among others, used such a test to evaluate Kevlar aramid fiber. Hüttinger [14] used such a loop or knot test to determine the deformation characteristics of carbon fibers. Figure 3 shows the loop or knot test schematically. If d is the fiber diameter and Dε is the loop diameter corresponding to a tensile strain of ε in fiber, then, we can write De  d=ε ¼ Ed=σ where σ is the stress and E is the Young’s modulus. As in any bending beam test, tensile stresses will be highest at the outer surface of the fiber and decrease to zero at the axis of the fiber and then become compressive in nature to the inner surface of the beam or fiber. Clearly, in a fiber having a skin-core structure, such a test will be biased toward the characteristics of the skin region. Hillig [15] used a version of loop test with the fiber lying on a rubber mat and was bent by means of a cylindrical tool. Siemers et al. [16] tested SCS-6 silicon carbide fiber by looping around a series of successively smaller drill bits until the fiber failed. The failure strain ε was computed from the relationship, e ¼ d=D where d is diameter of the monofilament and D is the diameter of the drill bit that led to failure. Since the fiber remains elastic to the point of fracture, one can compute the

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Fig. 3 The loop or knot test, d is the fiber diameter and De is the loop diameter corresponding to a tensile strain of ε in fiber

failure stress from the Hooke’s law, σ = Eε, where E is the Young’s modulus of the fiber. The failure strain was taken as the average of the strain imposed by the drill bit that caused fiber failure and the one immediately before that.

5

Dynamic Elastic Modulus

Speed of sound, c, in a material is related to its Young’s modulus, E, and density, ρ, by the following relationship c2 ¼ E=ρ A transducer is used to send a pulse down the length of the fiber while another transducer is set at a fixed distance from the first one to detect the arriving pulse. Smith [17] used such an ultrasonic technique to measure the Young’s modulus of carbon fibers. The modulus measured by making an acoustic pulse go down a fiber axis is called dynamic elastic modulus to distinguish it from the static elastic modulus measured in a conventional mechanical testing machine.

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Shear Modulus

Fibers are frequently twisted in a variety of operations. Thus, it is important to know the torsional or shear modulus of a fiber. Figure 4 shows a cylindrical fiber of length ‘, being twisted under the action of a torque, resulting in a torsion angle of θ. We apply a torque to the fiber by two opposing but equal forces, F at the two ends. The shear modulus of the fiber is G. The shear strain, γ, is given by γ ¼ θr=‘ while the shear stress, τ, is related to shear strain via the shear modulus, G:

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Fig. 4 A cylindrical fiber of length ‘, being twisted under the action of a torque, results in a torsion angle of θ.

τ ¼ G  θr=‘

(4)

where r is the fiber radius, ‘ is the fiber length, and θ is the angle of twist. Let da be the elemental area over which a small force dF is acting, then τ ¼ dF=da

(5)

This elemental area, da is equal to 2πrdr, where dr is the thickness of the annular area on which the force dF is acting. Thus, τ ¼ dF=2πrdr

(6)

From Eqs. 4 and 6, we can write Gθ  r=‘ ¼ dF=2πrdr Or dF ¼ 2πr 2  Gθ  dr=‘

(7)

The torque at a radius r from the central axis is rdF. The total torque is obtained by integrating from 0 to R, the fiber radius. Thus ð Total torque ¼ rdF ¼

ðR 0

    θ 2πR G dr ¼ πR4 Gθ =2‘ ‘ 3

(8)

The quantity πR4G/2 is called the torsional rigidity which is a combined function of the shear modulus, G and a geometric term containing the fiber radius to the power four! Thus halving the fiber diameter will result in reducing the torsional rigidity by a factor of 16. One can determine the shear modulus of a fiber from a torque per unit area versus twist curve. In practice, a simple apparatus called a torsional pendulum is used more commonly. An experimental set up, due to Mehta [18], to measure shear modulus of

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Fig. 5 A torsional pendulum apparatus used to determine the shear modulus of a fiber [18]

small fibers is shown in Fig. 5. The torsional pendulum, placed in a vacuum oven, allows the measurement of shear modulus as a function of temperature under vacuum. In this particular example, the fiber sample was mounted on a cardboard disk. At the center of the cardboard an aluminum fastener was fixed to hold the paper tab on which the fiber sample was mounted. After mounting the paper tab and sample, the top end was fixed on a clamp stand inside the oven. The two sides of mounted paper tab were then cut and only the fiber was freely attached to the torsional pendulum. A small initial twist is given to the fiber and the amplitude of the successive oscillations is determined. The shear modulus of a fiber of a circular cross-section is given by G ¼ 8πI‘=T 2 R4 where I is the moment of inertia of the torsion pendulum, ‘ is the fiber length, R is the fiber radius, T is the period of oscillations. The period of oscillations is given by T ¼ To=

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 þ ðΔ=2π Þ2

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where To is the experimentally measured period of oscillation, Δ is the logarithmic decrement = ln (A2/A1), where A1 and A2 are the amplitudes of two successive oscillations. Note that the radius term has an exponent of 4, which means that even a very small error in measurement of fiber radius will introduce a significant error in the shear modulus value.

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Transverse Modulus

Many high performance fibers such as aramid, polyethylene, carbon, etc. can be highly anisotropic. More specifically, their elastic modulus, strength, coefficient of thermal expansion, thermal conductivity, etc. are likely to be quite different along the fiber axis and transverse to the fiber axis. This anisotropy stems from the highly oriented chain structure in the case of organic fibers, from the layer structure in the case of carbon and boron carbide, or from the oriented structure obtained by CVD, which is the case for silicon carbide or boron fibers. In the case of pure single crystal fibers such as alumina, the anisotropy will have origin in the crystal structure. Hadley et al. [19] measured transverse modulus of polyethylene and some other monofilaments in the rod form. They compressed the thick rodlike specimen of fibers in the transverse direction between two parallel glass plates and measured, with a microscope, the flattening of the contact area of the fiber. The transverse modulus was obtained from the force applied and the size of the contact area. This method is suitable for large diameter fiber. Jawad and Ward [20] did transverse compression tests on nylon 6.6 and linear polyethylene samples machined from extruded rods (2.5 mm diameter). Thus, most of attempts at measuring the transverse modulus of polyethylene and some other polymeric fibers have involved compressing the monofilament in between two parallel glass plates and measuring the amount of flattening of the contact area of the fiber under compression. Others used the indirect route, viz., testing a fibrous composite and obtaining an estimate of the transverse modulus of the fiber from the transverse modulus of a unidirectionally reinforced fiber/matrix composite. The assumption here is that insitu behavior of the fiber is the same as outside the composite. Kawabata [21] devised a new technique to measure transverse modulus of fibers around 10 μm in diameter by recording compressional force versus diametral change at a constant loading rate. The bottom plane was a mirror-finished, flat steel bed while the top plane was a square plane, mirror-finished. A single fiber was placed horizontally on the bottom plane and compressed by the top plane, which was driven by a rod connected to an electromagnetic driver (load capacity = 50 N). A transducer was used to measure the force and a linear differential variable transformer (LVDT) was used to measure the deformation of the fiber in the diametral direction (resolution ~0.05 μm). An experimental load-deflection curve was obtained and compared with the load deflection curve obtained from a theoretical model, assuming isotropy in the cross-sectional plane. The change in diameter, u, as a function of compressive force, F, can be obtained from the following relationships:

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    0:19 þ sin h1 ðR=bÞ u ¼ ð4F=π Þ ð1=ET Þ  ν2 LT =EL    b2 ¼ ð4FR=π Þ ð1=ET Þ  ν2 LT =EL where u is the change in fiber diameter; b is the half of contact width between the fiber and the punch face; F is the compressive force per unit length of fiber; EL is the longitudinal modulus of fiber; ET is the transverse modulus of fiber; R is the radius of fiber; and νLT is the longitudinal Poisson’s ratio of fiber (defined by the longitudinal strain and the transverse contraction strain when the fiber is uniaxially stretched along its axial direction). As can be seen, the transverse modulus, ET, is an input parameter here. This ET is used as a curve fitting parameter to match the experimental curve. Note also that the Poisson ratio term, ν, enters as a squared term. Poisson’s ratio being very small, the square of Poisson ratio will be an insignificant quantity and can be neglected in most cases. Kawabata’s [21] measurements on carbon and aramid fibers did show them to be highly anisotropic. He observed a correlation between transverse and longitudinal moduli. Carbon and other ceramic fibers failed in a brittle manner and ET decreased sharply with increasing EL. Aramid and polyethylene UHMW PE failed in a ductile manner. Organic fibers showed a small increase in ET with increasing EL. Ward and colleagues [19, 22, 23] measured the transverse modulus of polyethylene and some other polymeric fibers by compressing the monofilament in between two parallel glass plates and measuring the amount of flattening of the contact area of the fiber under compression.

8

Micro Raman Strain Measurement

Raman spectroscopy deals with the interaction between light and matter; in particular the light that is inelastically scattered (the Raman Effect.) Photons of a single wavelength (most commonly from a monochromatic laser source) are focused onto a sample. The photons interact with the molecules and are either reflected, absorbed, transmitted or scattered. With Raman spectroscopy, we study the inelastically scattered photons. Elastic scattering of photons is termed Rayleigh scattering; they have the same wavelength as the incident light. With Raman scattering, the incident photon interacts with matter and its wavelength is either shifted lower or higher. Of course, not every molecule or functional group exhibits Raman scattering. Factors such as the polarization state of the molecule (which determines the Raman scattering intensity) must be considered. The greater the change in polarizability of the functional group, the greater the intensity of the Raman scattering effect. This means that some vibrational or rotational transitions, which exhibit low polarizability, and will not be Raman active. They will not appear in a Raman spectra. In Raman microspectroscopy we use a Raman microspectrometer. A Raman microspectrometer consists of a specially designed Raman spectrometer integrated with an optical microscope. This allows the experimenter to acquire Raman spectra

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of microscopic samples or microscopic areas of larger samples. The advantages are that much smaller sample is required and certain effects may be enhanced over very localized regions. A technique called micro Raman strain measurement has become quite important in measuring strain in non-metallic fibers. We digress a bit and describe the theory behind Raman spectroscopy. Raman spectroscopy deals with the phenomenon of a change in frequency when light is scattered by molecules. An exchange of energy between the scattering molecule and the photon of the incident radiation gives rise to the Raman Effect, named after C.V. Raman who got the Nobel prize for his discovery of this phenomenon. Modern day Raman scattering spectroscopy uses a monochromatic source of radiation, generally obtained from a laser. The Raman frequency (or the wavenumber) depends on the spring constant. The interatomic or intermolecular force being nonlinear, the spring constant will change if we constrain the molecule with a bond length different from that it would have in the unconstrained state. This implies that a shift will occur in the Raman frequency if we stress the solid. This phenomenon has been successfully employed to study the deformation behavior of organic as well as inorganic materials. Raman spectroscopy is a very powerful technique that can provide a significant structural information. Newer equipment involving use of charge coupled device (CCD) detector and more computer control have extended the Raman microprobe into Raman microscope in which one can obtain images of the Raman scattered light and record the spectra under confocal conditions [24–26]. An experimental set up to obtain the characteristic Raman spectra from fibers is shown in Fig. 6 [27]. Under tension, the peaks of Raman bands shift to lower frequencies. Figure 7 shows such a frequency shift for a polyethylene fiber. The 1127 cm1 Raman band in polyethylene fiber corresponds to symmetric stretching of the C─C

Fig. 6 An experimental set up to obtain the characteristic Raman spectra from fibers (Adapted from Ref. [27])

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Fig. 7 Frequency shift for a polyethylene fiber. The 1127 cm1 Raman band in polyethylene fiber corresponds to symmetric stretching of the C-C bond. (Adapted from Ref. [27])

bond. When a polyethylene fiber is strained, this peak would shift to lower frequencies. Similar results have been obtained with aramid fibers. There is an approximately linear shift in the peak position, Δv with strain, e. The slope of the line, dΔv/de, is proportional to the fiber modulus, Ef (Young, 1996). This means that one can use the Raman spectroscopy to investigate molecular stretching, since elastic modulus is a measure of molecular bonding. In reality, the magnitude of frequency shift is a function of the material, Raman band under consideration, and the Young’s modulus of the material. The shift in Raman bands results from changes in force constants due to changes in molecular or atomic bond lengths, and bond angles. We can write

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hνi þ E1 ¼ hνs þ E2 where h is Planck’s constant, νi is the frequency of incident radiation, νs is the frequency of scattered radiation, and E1 and E2 are the initial and final energies of the molecule, respectively. The frequency shift (+ or –) or Raman frequency is given by Δν ¼ νs  νi A set of Raman frequencies of the scattering species forms the Raman spectrum. Raman frequency shift is commonly expressed in term of a wave number (cm1). Thus ν

¼ ðνi  νs Þ=c ¼ ðE2  E1 Þ=h

where c is the velocity of light. The necessary condition for Raman scattering is that the incident photon energy hvi must be greater than the energy difference between the final and initial states of actual transition. Characteristic Raman spectra can be obtained from these fibers and that under tension, the peaks of the Raman bands shift to lower frequencies. The magnitude of frequency shift is a function of the material, Raman band under consideration, and the Young’s modulus of the material. The shift in Raman bands results from changes in force constants due to changes in molecular or atomic bond lengths and bond angles.

9

Tensile Testing Stages for Insitu Measurements

Tensile testing stages are available that can be mounted inside a scanning electron microscope or optical microscope to test simple fibers. These stages allow insitu testing fibers at very low forces (but high stresses because of the extremely small cross-section of fibers). One such stage with a heating unit by Kammarath and Weiss is shown in Fig. 8. It allows fine fibers (2.5 μm in diameter), several mm long to be tested. Example of an annealed copper filament undergoing tensile test in such an insitu stage is shown in Fig. 9. Mention should be made of in situ transmission electron microscope tensile testing of nanofibers. Beese et al. [28] performed tensile testing on carbon nanofibers in a transmission electron microscope (TEM) using a microelectromechanical system (MEMS) tensile testing device. Specifically, they tested single carbon nanofiber by mounting the sample onto a MEMS device using nanomanipulation inside the chamber of a scanning electron microscope, and followed that by insitu tensile testing inside a transmission electron microscope. As expected, they observed a dependence of strength and modulus carbon nanofibers on diameter; both decreasing with increasing fiber diameter.

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Fig. 8 An SEM in situ testing stage with a heating unit. (Courtesy of Kammarath and Weiss). It allows fine fibers (2.5 μm in diameter), several mm long to be tested

Fig. 9 An annealed copper filament undergoing tensile test in an in situ stage. (Courtesy of Kammarath and Weiss)

10

Summary

Fibers are used in many structural applications. For example cables, ropes, woven or nonwoven fabrics, and, of course, as reinforcements in composite materials. This makes it imperative that we know their mechanical characteristics. Mechanical testing of long, slender fibers is a very important topic. We have described some important techniques of testing single fibers. These include tensile testing by using cardboard frame technique, recoil compression test, shear modulus by means of torsional pendulum test, loop test, techniques involving laser interferometry, Raman spectroscopy, and insitu techniques involving the use of scanning electron and transmission electron microscopy.

References 1. Nunes J, Klein W. Trans ASM. 1967;60:726. 2. Chawla KK. Fibrous materials. 2nd ed. Cambridge: Cambridge University Press; 2016. 3. McMahon PE. ASTM Special technical publication 521. Philadelphia: ASTM; 1973. p. 357. 4. Jain MK, Abhiraman AS. J Mater Sci. 1987;22:278. 5. Jain MK, Balasubramanian M, Desai P, Abhiraman AS. J Mater Sci. 1987;22:301. 6. Xu ZR, Chawla KK, Li X. Mater Sci Eng A. 1993;A171:249. 7. Chawla N, Kerr M, Chawla KK. J Am Ceram Soc. 2005;88:101.

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8. Reder C, Loidl D, Puchegger S, Gitschthaler D, Peterlik H, Kromp K, Khatibi G, BetzwarKotas A, Zimprich P, Weiss B. Compos Part A. 2003;34:1029–33. 9. Unal D, Lagerlof KP. J Am Ceram Soc. 1993;76:3167. 10. Allen SR. J Mater Sci. 1988;22:853. 11. Hayes GJ, Edie DD, Kennedy JM. J Mater Sci. 1993;28:3247. 12. Sinclair D. J Appl Phys. 1950;21:380. 13. Greenwood JH, Rose PG. J Mater Sci. 1974;9:1809. 14. Hüttinger KJ. Adv Mater. 1990;2:349. 15. Hillig WB. Am Ceram Soc Bull. 1981;66:373. 16. Siemers PA, Mehan RL, Moran H. J Mater Sci. 1988;23:329. 17. Smith RE. J Appl Phys. 1972;43:255. 18. Mehta VR. PhD. Thesis, Georgia Tech, Atlanta; 1996. 19. Hadley DW, Pinnock PR, Ward IM. J Mater Sci. 1969;4:152. 20. Jawad SA, Ward IM. J Mater Sci. 1978;13:1381. 21. Kawabata S. J Text Inst. 1990;81:432. 22. Hadley DW, Ward IM, Ward J. Proc R Soc. 1965;A285:275. 23. Pinnock PR, Ward IM, Wolfe JM. Proc R Soc. 1966;A291:267. 24. Batchelder DN, Bloor D. J Polym Sci Polym Phys Ed. 1979;17:569. 25. Sudiwala RV, Cheng C, Wilson EG, Batchelder DN. Thin Solid Films. 1992;210:452. 26. Williams KPJ, Pitt GD, Batchelder DN, Kip BJ. Appl Spectrosc. 1994;48:232. 27. Berger LP, Kausch HH, Plummer CG. Polymer. 2003;19:5877. 28. Beese AM, Pankov D, Li S, Dzenis Y, Espinosa HD. Carbon. 2013;60:246.

Stress Measurement in Thin Films Using Wafer Curvature: Principles and Applications

63

Eric Chason

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Principles of Wafer Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Stoney’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Wafer Curvature Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Application of Wafer Curvature to Thin Film Growth Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Stress Versus Thickness for Low-/High-Mobility Systems . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dependence of the Stress on Temperature and Growth Rate . . . . . . . . . . . . . . . . . . . . . . . 3.3 Mechanisms for Thin Film Stress Generation Suggested by Curvature Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Kinetic Model for Thin Film Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Effect of Grain Growth on Stress Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Stress Modification with Energetic Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Stress in Sn Leading to Whisker Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Stress, Intermetallic Compound Formation, and Whisker Growth . . . . . . . . . . . . . . . . . . 4.2 Mechanical Properties of Sn and Sn Alloys: Stress Relaxation Measured Using Thermally Induced Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dependence of Nucleation Kinetics and Whisker Growth Rate on Thermally Induced Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Future Directions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2052 2054 2054 2058 2061 2061 2063 2064 2066 2068 2069 2070 2070 2073 2075 2078 2079

Abstract

Wafer curvature is a useful and sensitive technique for measuring stress in thin films. This work describes the basis for wafer curvature measurements and how they can be used to understand stress evolution. Various approaches to the measurement of curvature are discussed with their strengths and drawbacks. The information that can be obtained from wafer curvature is illustrated by E. Chason (*) School of Engineering, Brown University, Providence, RI, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_49

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examples from different studies. Measurements during thin film growth (evaporation, electrodeposition, and sputter deposition) enable the relationship between residual stress and many different processing parameters (temperature, growth rate, microstructure, material types, deposition process, etc.) to be quantified. Studies of the real-time evolution have played a large part in identifying the physical mechanisms that are responsible for the resulting stress. A kinetic model is described that incorporates these mechanisms into equations that predict the stress under different conditions. Another set of examples describes the stress that leads to the formation of whiskers in Pb-free Sn coatings on Cu. Simultaneous measurement of the Sn-Cu intermetallic volume, stress, and whisker density show how these processes are correlated. The controlled application of stress via thermal expansion mismatch is used to study the mechanical properties and plasticity in Sn layers. It also allows the nucleation kinetics as a function of applied stress to be measured.

1

Introduction

It is hard to overstate the importance of thin films in advanced technology. For instance, they are at the heart of semiconductor manufacturing. From the front-end to the back-end of the fab line, thin film processing makes the gates and contacts for individual transistors, the metal interconnects that allow communication between them, and the coating on the packaging that attaches the device to a circuit board. In other applications, they provide hard coatings for tool bits and thermal barrier coatings for jet engine turbine blades. Optical components use thin film coatings for antireflection and filtering. Thin film fabrication methods enable the manufacture of ultra-small MEMS (micro-electromechanical systems) devices that could not be produced by other means. The list could go on to include many different applications and corresponding deposition techniques designed to optimize the film properties. An important property that affects many applications is stress in the film. This has been a persistent issue for many years. As-grown films very often have large residual stress that can significantly affect their performance or lead to failure. Some illustrative examples are shown in Fig. 1. In the first case (Fig. 1a), residual stress has caused a Ni film to delaminate from the Si substrate onto which it has been evaporated [1]. Compressive stress in a diamond film (Fig. 1b) has caused it to buckle [2] while tensile stress in a NiW coating has caused it to crack (Fig. 1c [3]). The MEMS device in Fig. 1d [4] is supposed to be flat, but stress gradients in the material have caused it to bend. In other cases, external factors (such as thermal cycling or phase transformations) that are not directly related to the process of film growth can have a large effect on the stress. The mechanical response of the film determines whether the film can relax this stress gracefully or if leads to catastrophic failure. Figure 1e

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Stress Measurement in Thin Films Using Wafer Curvature: Principles and. . .

2053

Fig. 1 Examples of failures due to stress in thin films. (a) delamination of Ni film evaporated onto Si, (b) buckling in diamond film (c) cracking in NiW coating (d) elastic deformation in MEMS device due to internal stress, (e) spalling of Al2O3 on Ni, (f) whisker growing out of Sn layer (Figures a–e are adapted from Refs. [1–5], with permission)

shows an example in which stress has led to buckling and spalling of an Al2O3 layer on Ni [5]. The whiskers shown in Fig. 1f grow out of Sn coatings and have caused numerous system failures including the crash of the Galaxy IV satellite. The source of the whiskers is believed to be stress induced by the reaction between the Sn and the underlying Cu layer and is discussed further in Sect. 4. Stress is not always a bad thing for thin films. The bandgap of III-V semiconductors can be modified [6] and the mobility of Si transistors can be enhanced by using strained layers [7]. Strain can also be applied to a thin film to measure its mechanical properties, for example by heating a film on a substrate with a different thermal expansion coefficient [8–11]. An example is shown in Fig. 2 for an Al film with an anodic oxide deposited on oxidized Si [8]. The resulting stress versus temperature is equivalent to a stress-strain curve obtained by mechanical loading. Because of the difficulty of mechanically straining a thin film, there can be advantages to using this method, in particular for studying the effects of compressive stress. Measuring the stress evolution when a strain is applied can be used to determine what the strain relaxation mechanisms are. This is an important application of the curvature technique. For instance, it has been used to study time-dependent stress relaxation in Sn films and relate it to power-law creep [12].

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E. Chason 400 Biaxial Film Stress(MPa)

Fig. 2 Film stress due to thermal expansion mismatch induced by heating in Al films with oxide coating (Figure from Ref. [8] used with permission)

200 100 0 –100 –200

2

Anodized Film Thickness ∼ 500Å

300

100

0

200 300 Temperature (°C)

400

500

Principles of Wafer Curvature

In order to understand or control thin film stress, it is necessary to be able to measure it. Wafer curvature is a sensitive technique for measuring thin film stress that can be used in real-time while the film is being grown or post-processed. As discussed below, measuring the stress while the film is growing has the added advantage of being able to estimate the distribution of stress within the film. The following section focuses on the principles underlying wafer curvature and then describes examples from several systems in which wafer curvature has been used to study the stress.

2.1

Stoney’s Formula

The physical basis of the wafer curvature technique is that a stressed film will induce curvature in the underlying substrate or wafer. The relationship between curvature and stress was first calculated in 1909 by Stoney, who was interested in understanding delamination of copper layers from searchlight reflectors [13]. Correcting Stoney’s original calculation to consider biaxial rather than uniaxial stress in the film [14], the curvature (κ) is κ¼

6σhf Ms h2s

(1)

where hf is the film thickness, hs is the substrate thickness, Ms is the biaxial modulus of the substrate, and σ is the average in-plane stress in the film. The wafer curvature depends on the product of the average stress and thickness (sometimes referred to as the stress-thickness or the force per unit width). For application of the Stoney formula, the stress is assumed to be uniform in the plane of the film but does not have to be uniform throughout the thickness of the film.

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Stress Measurement in Thin Films Using Wafer Curvature: Principles and. . .

s (z)

a

Film with stress distribution Substrate b 2.5

Ag on SiO2 23oC

2.0 Stress-thickness (N/m)

Fig. 3 (a) Schematic of a uniform thin film on a substrate showing stress distribution through the thickness of the film. (b) Stress-thickness measured during deposition of Ag on SiO2. The slope of the dotted line on the figure represents the average stress (σ) and the dashed line represents the incremental stress (σ(hf))

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1.5 1.0 0.5

s

0.0 –0.5

s (hf)

hf 0

50 Thickness (nm)

100

A schematic of the geometry corresponding to this situation is shown in Fig. 3a. Defining σ(z) as the in-plane stress at height z from the substrate, σ is given by 1 σ¼ hf

hðf

σ ðzÞdz

(2)

0

where the integral is over the thickness of the film. Conceptually, the effect of the thin film stress is equivalent to stretching an elastic membrane isotropically with a force on the edge equal to σhf . When this membrane is attached to the substrate and released, the substrate responds by developing a curvature. The magnitude of the curvature is determined by setting the bending moment and the sum of the forces acting on a cross-section of the system equal to zero. This calculation assumes that the curvature is sufficiently small so that the stress in the film does not change due to the substrate deformation. Extensions of the Stoney formula for large curvatures can be found in the literature [15]. The approach can be extended to multiple layers so that the total curvature is equal to the curvature that would be induced by each layer:

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κ¼

6 X σ i hi Ms h2s i

(3)

In this expression, σ i and hi are the average stress and film thickness for each layer in the film. There can be an additional contribution from interfacial stress [16] that is not included here. Since the curvature is proportional to the average stress, the stress profile as a function of depth (σ(z)) cannot be determined from a single curvature measurement. However, it can be estimated from the evolution of the stress as the film grows. Consider a growing film in which the thickness h(t) changes with time. Then 2 dκ ðtÞ 6 6 dhðtÞ ¼ þ 4σ ðhðtÞÞ dt dt Ms h2s

hð ðtÞ

0

3 @σ ðz, tÞ 7 dz5: @t

(4)

The quantity on the left is the derivative of the curvature measured as a function of time. It changes in different ways as indicated by the 2 terms on the right side. The first term corresponds to adding a new layer to the top of the film (at z = h(t)) with a value of the in-plane stress equal to σ(h(t)). This is called the instantaneous or incremental stress, because it corresponds to changing the stress by adding a new layer to the film. The second term on the right corresponds to changing the curvature by changing the stress in layers that have already been deposited. This expression can be used to estimate the stress distribution in the film in the following way. For plots of the curvature versus thickness, the slope is equal to the time derivative divided by the growth rate. Assuming that the stress in the existing film does not change (i.e., @σ(z, t)/@t = 0), the integral on the right-hand side of Eq. 4 is equal to zero and the slope of the curvature versus thickness (dκ/dh) is proportional to the stress in the layer deposited at the surface σ(hf). The incremental stress (i.e., in the new layer being deposited) can therefore be determined from the slope of the curvature versus thickness measurement if it can be assumed that the stress in the underlying film is not changing. Note that this is a strong assumption and may not be true. The stress in the existing layers of the film may change due to a number of effects (thermal expansion, grain growth, etc.) that will be discussed further below. However, if it can be assumed that the stress distribution is locked into the film after it is grown, the stress in each layer can be determined from the slope of the curvature measurement with thickness. To show how the average and incremental stress can be determined from curvature measurements, an example is shown in Fig. 3b for Ag deposited on SiO2 [17]. The measured curvature has been multiplied by Ms h2s =6 so that it is proportional to the stress-thickness. For a given thickness (designated as hf on the figure), the average stress in the film can be determined by dividing the stressthickness by hf. This average value is equal to the slope of the dotted line in the figure.

Stress Measurement in Thin Films Using Wafer Curvature: Principles and. . .

Fig. 4 Stress in sputtered Cu film during growth and after growth is terminated (at time indicated by vertical line). The large change in stress is due primarily to sample cooling

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50 Stress-thickness (N/m)

63

0

-50

growth

cooling

100 0

1000

2000 Time (s)

3000

4000

It is clear from the figure that the average stress does not change linearly with time. This indicates that the stress in the layers that are being deposited is changing with the thickness (again, this is assuming that the stress in the existing layers does not change during the deposition). The incremental stress in the new layer (σ(hf)) can be estimated from the slope of the curve in Fig. 3b (shown by the dashed line drawn tangent to the curve). In the case shown, the incremental stress has a different sign than the average stress. This indicates that the average stress in the existing film is tensile but that the new layers being deposited are compressive. From these kinds of measurements, the stress profile within the layer can be determined. The evolution of the slope in Fig. 3 with thickness indicates that the film initially grows with layers that have tensile stress but then the stress in new layers becomes compressive. This would create a significant stress gradient within the film. Understanding why the stress may change so dramatically during thin film growth is discussed in the section on thin film growth stress. As an alternative example, consider the case where the film is not growing so that any change in curvature with time must be due to changes in the stress in the existing layers in the film. Figure 4 shows measurements of stress-thickness during sputter deposition of a Cu film (growth conditions described in Ref. [18]). When the growth is stopped at the point indicated by the vertical line, the stress changes significantly, going from compressive to tensile. Since no layers are being added, the stress change is occurring in the layers of the film that have already been deposited. In this case the temperature of the sample increased during deposition and the stress change is due to thermal expansion mismatch between the film and the substrate as the sample cools. This also shows that the stress that develops during growth is not necessarily the same stress that is measured after growth if there is significant change in temperature of the sample. Additional factors besides temperature change can lead to changes in the stress in the existing layers. Grain growth in the film can lead to additional stress that is not due to the growth process [19–25]. Also, in films that have relatively high atomic mobility, there is often a change in the stress when the growth is terminated [20, 23, 25–28].

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Wafer Curvature Measurement Techniques

With the use of Stoney’s equation, the average stress in a thin film can be determined by measuring the curvature of the substrate. There have been multiple experimental techniques developed to do this. Each has some benefits and drawbacks in terms of factors such as sensitivity, ease of use, and speed. Some of the approaches are described here. Capacitive: In this approach, the curvature is monitored by measuring the motion of the end of a cantilever sample. This is done by measuring the capacitance change induced by changing the spacing between the sample and a fixed reference electrode [29, 30]. This approach can have high sensitivity. However, it assumes that there are no other sources of motion of the sample (e.g., drift of the sample position or vibration) that can lead to a capacitance change. This can be especially important in deposition systems where there may be a lot of mechanical noise and the experiments can last a long time (several hours) at elevated temperature. In one variation of this technique, the samples are created by microfabrication directly out of the substrate wafer, which reduces some of the mechanical noise [26]. This variation of the technique can also be very fast so that data can be acquired at the rate of 1000 measurements/s. The drawback is the need to fabricate special sample structures. Interferometric: Optical interferometric methods can also be used to measure the deformation of a sample due to stress-induced curvature. One benefit of this approach is that it does not require special fabrication of the sample and the optical detection makes it compatible with many deposition methods. Depending on the approach it may be more or less sensitive to sample motion or vibration. An optical dilatometer can be used to measure the displacement of the center of a sample as it bows due to stress. The setup is simple and easy to use but the analysis assumes that the displacement is due to uniform curvature. In another variation, the deformation of an entire region of the sample is measured simultaneously to determine the shape change due to stress in the film [31]. Deflection of reflected beams: This technique uses the fact that curvature in the sample will cause rotations or displacements of the substrate that will deflect the orientation of light beams reflected from it. It requires optical access to the sample for shining an incident beam(s) on the sample and measuring the reflected beam(s). It is compatible with in situ monitoring in many deposition environments as long as the light is not too strongly absorbed. The surface must also remain reflective enough to be able to monitor the reflected beams; this can be done in some situations by monitoring the side of the substrate that is not being deposited on. As long as these conditions can be achieved, it can be used in many deposition environments (evaporation, chemical vapor deposition, sputtering, electrodeposition, pulsed laser deposition, etc.) The principle of the beam deflection method is that curvature makes the orientation of the surface normal rotate relative to a flat sample. The geometry (shown schematically in Fig. 5a) compares the orientation of a sample that is flat (dashed line) with a sample with uniform curvature (solid lines). The diagram is not to scale;

63

Stress Measurement in Thin Films Using Wafer Curvature: Principles and. . .

a)

2059

b)

x substrate film

dq

a

Ldetector etalon

d

dd CCD

laser

detector Fig. 5 (a) Schematic showing geometry of film on substrate for flat sample (dashed lines) and curved sample (solid lines). The surface of the curved sample at position x is rotated by an angle δθ. The parallel red lines represent two beams impinging on the sample with a separation of d and being reflected into a detector at a distance of Ldetector. The beam from the curved sample is deflected from the flat sample by δd. (b) Schematic of multi-beam optical stress system implemented on a vacuum deposition chamber with dual evaporation sources

the gap in the diagram is meant to show that the distance x is much larger relative to the surface displacement than shown in the diagram. The distance from the sample to the detector (Ldetector) is also much larger than the spacing between beams. For the curved sample, the angle of rotation (δθ) changes with the position (x) from a fixed reference point: δθ ¼ xk

(5)

where the curvature κ is equal to the reciprocal of the radius of curvature. The red lines in the diagram correspond to parallel light beams that illuminate the surface at different positions separated by a distance x. Rotation of the substrate surface induces corresponding rotation in the direction of beams reflected from different positions, which causes the measured beam position to change by an amount δd on a detector. Several variations of this technique use different configurations for both the incident light and the reflected light, some of which are described below. Single fixed beam: A single beam can be used to illuminate the tip of a sample that is in the form of a cantilever, which is held at the other end in a fixed position [32, 33]. The position of the reflected beam can be monitored with a position-sensitive detector or a CCD camera. The corresponding rotation of the beam by curvature of the sample can be calculated and used to relate the displacement of the reflected

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beam to the stress. The technique is sensitive to small angles of deflection, but the reflected beam can also be influenced by sample motion or vibration. This can be reduced by implementing the system on an optical bench so that high sensitivity can still be achieved. Scanning beam: The sensitivity to sample position can be reduced by measuring at several positions on the sample. The curvature can then be calculated from the difference in the orientations of the beams reflected from different positions on the sample. This can be done by translating the sample under a stationary beam and detector; this approach is used in some commercial bench-top systems for measuring wafer curvature. However, this is not easily compatible with in situ monitoring during deposition. Alternatively, the measurement can be made at different positions by using a mirror to scan the incident beam across the sample [19, 27, 34]. The use of a scanning beam in principle reduces the sensitivity to overall displacement of the sample. However, because the beam is scanned the measurements at different positions are made at different times. Vibrational motion of the sample that is rapid relative to the time required to scan the beams can decrease the sensitivity of the measurement. Multiple beams: A third method is based on the use of multiple parallel beams that illuminate different positions on the sample simultaneously. As shown schematically in Fig. 5a, two incident beams are spaced by an amount d. They are incident at an angle a relative to the normal so that they are spaced by an amount x = d/ cos (α) on the surface. The detector is placed at a position Ldetector from the sample. If the sample is flat, then the two parallel beams striking the surface are reflected at the same angle and the spacing between them measured on the detector is d. However, if the sample has curvature, then the spacing between the beams changes. The curvature can be determined from the deflection of the beam positions by κ¼

δd cos α d 2Ldetector

(6)

The factor of 2 in the equation is because the reflected beam rotates by twice the angle of rotation of the surface. The initial spacing between the parallel beams can be determined by the use of a flat mirror as a standard. However, for many in situ measurements, it is the change in the curvature as a film is growing that is of interest. In this case, the change in d from the beginning of the measurements to the end is more important than the absolute magnitude of d for determining the change in the film stress. The multi-beam method requires parallel incident beams, and these can be produced by a beam-splitter [35–37] or a specialized optic element (referred to as an etalon) in the multi-beam optical stress (MOSS) technique [38]. A benefit of the multi-beam technique is that it has much less sensitivity to vibration of the sample than some of the other techniques. Because reflections from different positions on the sample are obtained simultaneously, even rapid motion of the sample does not affect the sensitivity. In studies of the system sensitivity, the RMS variation in the position of one of the beams was measured to be 4.3 pixels (over a time of 100 s). In

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comparison, the RMS variation in the spacing between two neighboring beams was 0.9 pixels over the same period, showing that the use of multiple beams reduces the noise due to vibration by a factor of almost 50. Because of its high sensitivity, the multi-beam technique has even been used to measure the surface stress from single layers of As on Si [35]. A schematic of a MOSS system is shown in Fig. 5b implemented on a vacuum chamber. Multiple parallel beams are produced from a single laser beam by the etalon (i.e., a piece of glass that has been coated with reflective layers on both sides). Reflections of the incident beam at these two surfaces produce an array of parallel beams with relatively uniform intensity and spacing. A second etalon oriented orthogonal to the first can be used to produce a two-dimensional array of parallel beams for measurement of the curvature in the orthogonal direction. The optics are all mounted outside of the chamber and the only requirement is that there be viewports for the incident beams and detection of the specularly reflected beams. When this is not possible, mirrors have been used to bring in beams through available viewports.

3

Application of Wafer Curvature to Thin Film Growth Stress

One of the most important issues in thin film stress is understanding the amount of stress that develops from the process of thin film growth (sometimes referred to as intrinsic stress). This is a longstanding issue and there is a large literature of stress for many systems and deposition conditions [19, 20, 28, 38, 39]. The following section reviews some of the trends that have been studied and describes how wafer curvature measurements have been used to identify some of the mechanisms creating the stress.

3.1

Stress Versus Thickness for Low-/High-Mobility Systems

Measurements of the curvature evolution as a function of film thickness have shown that there are different types of behavior depending on the material system. This was identified by Abermann [25] who divided them into type I and type II behavior. Type I behavior is found in materials with low atomic mobility (or alternatively high melting point). An example is shown in Fig. 6a for the evaporation of Cr at room temperature [40] (the growth temperatures are indicated in the figure). The positive slope of the stress-thickness indicates that the incremental stress (i.e., the stress in new layers) is tensile. The near linear behavior indicates that the incremental stress is similar to the average stress in the film. Measurements of the corresponding microstructure of the film indicate that the grain size does not change significantly during the growth, consistent with the low atomic mobility. Therefore, subsequent layers of the film are grown on top of a film that is not changing significantly with thickness. This is consistent with the fact that the stress in new layers is the same as in the layers that have already been deposited. Another feature of stress evolution in low-mobility

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b

a 27

Cr

Force per unit width (N/m)

Force per unit width (N/m)

150 oC

100

200 oC

50

0 -20

300 oC 0

20

40

60

Thickness (nm)

80

2 0 -2 -4 -6

Cu

-8 0

20

Time (min)

0

25

Thickness (nm)

50 0

500

Time (min)

Fig. 6 Measurements of stress-thickness (force/width) for (a) Cr evaporated on MgF2 at temperatures indicated in figure and (b) Cu evaporated onto oxidized Si at nominal room temperature. The vertical line in the figures corresponds to the point where the growth flux is terminated (Figures in (a) adapted from Ref. [40] and in (b) from Ref. [27] with permission)

materials is that when film growth is stopped, the stress in the film does not change. This further suggests that the stress and microstructure in the layers that have already been deposited do not change after the layer is deposited for this material. This is in contrast with the behavior seen in high-mobility metals, such as the example shown in Fig. 6b for the case of Cu grown at room temperature by e-beam evaporation [27]. In this case (referred to as type II behavior), the incremental stress changes significantly as the film thickness changes. In the early stages, the incremental stress is low or slightly compressive (for thicknesses less than 5 nm). As the film gets thicker, the slope becomes increasingly positive and the incremental stress becomes tensile (for thickness between approximately 5–12 nm). The average stress reaches a maximum after which the incremental stress becomes negative (for h > 12 nm). At even greater thickness, the slope approaches a near constant value and the corresponding incremental stress is referred to as the steadystate stress. The steady-state stress depends on the growth rate and temperature as discussed below. The different stages of stress seen in type II films are mirrored by an evolution in the microstructure of the film [41, 42]. In the early stages, the film consists of discrete islands (pre-coalescence regime) when the corresponding stress in the film is small. As the film gets thicker, the individual islands impinge and start to form a continuous network (coalescence regime) during which the stress is tensile. At even larger thicknesses, the film becomes continuous with nearly uniform thickness (steady-state regime) and the system reaches a steady-state morphology. For the example shown in Fig. 6b, the incremental stress in the steady-state regime also reaches a nearly constant value. However, for many high-mobility materials, the steady-state stress does not reach a constant value (e.g., [24, 43]) but becomes less compressive/more tensile as the thickness increases. One explanation for this is

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that the microstructure changes in the steady-state regime due to grain growth. As discussed below, grain growth can lead to additional stress evolution growth that can cause the incremental stress to change with thickness in the steady-state regime. In addition to the different evolution of stress with thickness, high-mobility materials also exhibit different behavior when the growth is terminated. If the steady-state stress is compressive, there is typically a relaxation of the stress in the tensile direction (as seen in Fig. 6b where the vertical line delineates the thickness where the growth is terminated). Because this behavior correlates with the atomic mobility, it suggests that the relaxation process is related to atomic diffusion.

3.2

Dependence of the Stress on Temperature and Growth Rate

The discussion of stress evolution above divides materials into low and high mobility. However, the mobility (and hence the type of stress evolution behavior) can be changed by changing the temperature of the material. This can be seen by looking at the evolution of the same material at different growth temperatures. In Fig. 6a, results are shown for Cr films evaporated at temperatures of 27,200, and 300  C. The stress evolution changes from highly tensile type I behavior (at room temperature) to a series of stress stages associated with type II behavior (at 300  C). Although the stress does not change when the growth is terminated at low temperature, there is significant tensile relaxation when the growth is terminated at the higher growth temperature. The behavior for Cr at 300  C is similar to that for the Cu at room temperature, indicating that raising the temperature can induce a change from type I to type II stress evolution. Similar results can be seen for Ag films grown at different temperatures from Ref. [44]. In addition, the distinction of high and low mobility is relative to other kinetic processes occurring on the surface. This can be seen by changing the growth rate for the same material at the same temperature. An example of stress-thickness evolution in electrodeposited Ni films [45] is shown in Fig. 7 for different growth rates. The films were grown over Au underlayers so that the initial stress (below 100 nm) is influenced by the lattice mismatch between the Au and Ni layers. As the film gets thicker, different behavior is observed for the different growth rates. For high growth rates, the behavior is type I, similar to low-mobility materials. For the low growth rates, the incremental stress becomes compressive at larger thickness as in type II behavior. Further insight can be obtained by looking at the dependence of the steady-state stress on the growth rate (Fig. 7b). As the growth rate is increased (while the atomic mobility is held constant), the stress becomes more tensile. These results suggest that the dependence on the atomic mobility is relative to the deposition rate (R). For the Ni electrodeposition results shown in Fig. 7, the grain size (L ) was measured and determined to remain constant during the growth. Other work has been performed that studied the dependence of the steady-state stress on L [24, 46]. This work shows that the grain size can also have an influence on the stress evolution.

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b

250

14 nm/s 200 150

7 nm/s

100

2.8 nm/s 50 0

0

1.4 nm/s 200 400 600 800 Thickness (nm)

1000

Steady-state stress (Mpa)

Stress-Thickness (N/m)

a

E. Chason

400 200 0 -200 -400 -600 0

4

8 12 16 Growth rate (nm/s)

20

Fig. 7 (a) Measurements of stress-thickness in electrodeposited Ni for different growth rates indicated in figure. (b) Steady-state stress versus growth rate determined from stress-thickness measurements. Solid line in figure is fit to stress model described in text (Figures in (a) adapted from Ref. [45] and in (b) from Ref. [44] with permission)

3.3

Mechanisms for Thin Film Stress Generation Suggested by Curvature Measurements

The curvature measurements described above illustrate some general trends that are seen in multiple material systems. In summary, low-mobility materials develop tensile stress, and the stress remains tensile as the thickness increases. High-mobility materials go through a series of stress states that change with the microstructure of the film with a steady-state stress that may be compressive. The stress in lowmobility materials remains relatively unchanged when the growth is terminated, while the stress in high-mobility materials relaxes. The stress evolution during growth may be changed through changing the temperature, growth rate, or grain size. The variation of the stress with the kinetics and microstructure suggests that there are multiple mechanisms of stress generation and relaxation occurring during the growth of a film. The observations suggest that the resulting stress is due to a balance among the different mechanisms, determined by the film kinetics and growth conditions. This section describes some of the mechanisms that have been suggested to explain the curvature measurements. A model is also described that has been developed to explain this balance that depends on the ratio of the diffusivity, the growth rate, and the grain size (D/RL). The first mechanism is the development of tensile stress due to island coalescence, originally proposed by Hoffman [47]. The mechanism is based on considering the energetics associated with the formation of a grain boundary as two adjacent islands impinge upon each other. As new grain boundary forms, the total interfacial energy changes by Δγ = (γ gb  2γ s) where γ gb is the grain boundary energy and γ s is the surface energy. Assuming that there is initially a small gap between the adjacent islands, the islands may elastically deform to facilitate the creation of the grain

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boundary from the free surfaces. This will occur as long as the increase in strain energy in the islands is less than the corresponding decrease in the energy due to grain boundary formation. This leads to a formula for the critical tensile stress due to  1= island coalescence that depends on Mf Δγ=L 2 . The physical origin of the stress in this picture is similar to the growth of a crack in the Griffith’s model. Instead of a crack opening in order to relieve the applied elastic stress (Griffith’s model), the gap between adjacent islands closes at the expense of creating stress in the grain. This picture has been extended by others [48–51] but the driving force and results are similar. It has also been recognized that new grain boundaries form by adjacent islands growing toward each other (i.e., accumulating additional atoms) rather than the islands elastically deforming [44]. So therefore this mechanism of grain boundary closure should be envisioned to occur on a layer-by-layer basis rather than all at once. The coalescence mechanism is consistent with the development of tensile stress observed in low-mobility materials. However, there must be another mechanism operating to explain why the stress changes from tensile to compressive stress in high-mobility materials. The change in stress is not just a relaxation of the tensile stress because the average stress in the film becomes compressive as the thickness increases. The increase in the strain energy density with thickness suggests that there must be an energetic reason (i.e., driving force) for the compressive stress to develop. The fact that the stress can be changed from tensile to compressive by increasing the atomic mobility (temperature) further suggests that it is related to a diffusive process. A mechanism has been proposed to explain the compressive stress that addresses the issues described above. It is based on the insertion of atoms into the top of the grain boundary from the surface [17, 44] while the film is growing (other mechanisms have been proposed by others that will not be discussed further here [20, 52–54]). The process is shown schematically by the arrow in Fig. 8 that represents atom diffusion into layer i; the layers in the adjacent grains are assumed to have already grown together to form a new section of grain boundary. We assume that the layer deforms elastically to accommodate this atom so that its insertion creates a strain approximated as a/L where a is the nominal diameter of the atom. The resulting stress is compressive and proportional to the number of atoms inserted into the layer. An important feature of the adatom insertion mechanism is that there is a driving force for the compressive stress due to the fact that the surface is not at equilibrium when the film is growing. This leads to a surface supersaturation that raises the chemical potential of atoms on the surface and promotes their incorporation into the growing grain boundary. The flow of atoms into the grain boundary creates stress that raises the chemical potential of the atoms that are incorporated there. The system ultimately reaches a steady-state where the chemical potential in the grain boundary with stress balances the elevated chemical potential on the surface. This mechanism also explains the reversible changes in the stress that are observed when the growth is interrupted. Decreasing the chemical potential of atoms on the surface (by

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stopping the growth) leads to the out-diffusion of atoms from the grain boundary to the surface. This mechanism has been supported by recent experiments [22, 23, 55, 56] and modeling [57, 58]. The origin of the compressive stress is still controversial and the grain boundary insertion mechanism is not universally accepted. However, the grain boundary insertion mechanism has enabled the development of a kinetic model for stress (described in the following section) that provides some insight into the processes controlling film growth.

3.4

Kinetic Model for Thin Film Stress

The mechanisms described above have been incorporated into a set of rate equations that describe the evolution of the stress under different conditions. Details of how the equations are derived can be found in Ref. [38], so only the results are summarized here. The model focuses on the stress that develops at the top of the grain boundary as two neighboring islands grow together. A schematic of the corresponding geometry is shown in Fig. 8; the top of the grain boundary is referred to as the triple junction. The stress that develops in each layer is assumed to be independent of the other layers (Guduru has termed this the linear spring model [59]. The total stress in each layer is taken to be a combination of tensile stress due to grain boundary coalescence and compressive stress due to atoms inserted into the layer. If we assume that the mobility is not high enough for atoms to diffuse deeper into the grain boundary than the triple junction, then we can derive analytical expressions that predict the stress under different conditions. Grain boundary diffusion could be included by numerical calculations, but it is not considered here. An alternative formulation for very high grain boundary mobility has also been derived [17]. For films that are in the steady-state regime, the magnitude of the stress predicted by the model is given by Fig. 8 Schematic of the geometry around the top of a grain boundary (triple junction) showing the stressgenerating processes included in the model. The formation of new grain boundary between adjacent grains initially leads to tensile stress in the layer. Subsequent insertion of atoms from the surface into the grain boundary leads to compressive stress.

Deposition flux

ith layer

hgb=ai a

triple junction grain boundary

63

Stress Measurement in Thin Films Using Wafer Curvature: Principles and. . .

  βD σ ss ¼ σ C þ ðσ T  σ C Þ  exp  LR

2067

(7)

In this expression, σ T is the tensile stress developed due to the coalescence of neighboring islands as described by the Hoffman mechanism; it depends inversely on the square root of the grain size. σ c is the compressive stress due to the nonequilibrium chemical potential on the surface caused by the growth flux. β is a kinetic parameter related to the surface and mechanical properties of the layer. D is an effective diffusivity based on the transition rate of atoms from the surface into the grain boundary. The model predicts that the stress depends exponentially on the parameter βD/RL. This is consistent with the experimental results from wafer curvature described above. For films where the mobility is low or the growth rate is high, the steadystate stress is predicted to be more tensile. Alternatively, for high mobility or low growth rates, the stress becomes more compressive. This is in agreement with the experiments. A nonlinear least squares fit of the model to the data for electrodeposited Ni at different growth rates is shown as the solid line in Fig. 7b (using a constant grain size as confirmed experimentally). The dependence of the stress on the grain size for the same temperature and growth rate is also consistent with measurements in Fe made by Koch et al. [46]. The predicted dependence of the stress on the ratio of D/RL suggests that the changes in stress due to the atomic mobility can be compensated by changing the growth rate or microstructure. For films in the early stages of growth, the rate at which the grain boundary height is changing (dhgb/dt) is not the same as the average growth rate (R). In this regime, the rate equations indicate that the average growth rate R in Eq. 7 should be replaced by the rate at which the grain boundary height changes (dhgb/dt). The grain boundary growth rate can change dramatically in the early stages of film growth. Considering islands with hemispherical geometry, the high contact angle between the islands at the point of coalescence means the grain boundary height increases much faster than the average growth rate. Since the model predicts that faster growth rates lead to more tensile stress, the rapid increase in the grain boundary height at the beginning of coalescence will make the incremental stress more tensile. As the film gets thicker, the grain boundary growth rate decreases to the average growth rate and the stress becomes less tensile/more compressive. The effect of the changing grain boundary velocity on the stress is consistent with the microstructural evolution observed in type II films. To further investigate the coupling between microstructure and stress, recent measurements have been made of stress evolution during the growth of patterned arrays of islands with the shape of hemispheres [60] or half-cylinders [61]. Because the geometry is known, the grain boundary growth rate at different film thickness is well known. Calculations of the stress evolution using the grain boundary growth rate determined by the pattern geometry are in good agreement with the measurements providing support for the use of the model to understand the stress as a function of thickness.

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3.5

E. Chason

Effect of Grain Growth on Stress Evolution

The model described above is able to account for many observed features of stress induced by thin film growth. One effect that has not been discussed is grain growth that occurs during film growth. As described by Chaudhari [21] and others, an increase in the grain size leads to a decrease in the number of grain boundaries. Since grain boundaries are low-density regions in the film, eliminating them tends to densify the film. Since the film is constrained to remain fixed to the substrate, this leads to the development of tensile stress in the film. The effect of grain growth can have a large impact on the stress. However, since it is difficult to measure the grain size evolution, it is not always easy to include this effect. In the experiments analyzed above, the grain size was measured to remain fixed during the stress measurements so that grain growth effects did not need to be considered for comparison with the model. Recent experiments by Yu and Thompson [24] studied evaporated Ni films in which there was substantial grain growth. Measurements indicated that the grain size was proportional to the film thickness. Importantly the grains retained a columnar morphology, i.e., they had the same thickness throughout the film with essentially vertical grain boundaries. In other words, the grain size increased with the film thickness, and this was accompanied by grain growth throughout the film that made the grain size the same for all depths in the film. This grain growth had a substantial effect on the stress evolution and the corresponding curvature measurements. As shown in Fig. 9, the slope of the stress-thickness does not reach a constant value. Because the grain size is changing in the layers that have already been deposited, the slope of the stress-thickness cannot be interpreted as the incremental stress (i.e., the integral throughout the thickness of the film in Eq. 4 changes with time). However, since the microstructural Fig. 9 Stress-thickness measured in evaporated Ni films at different growth rates indicated in figure. Measurements of the grain size (not shown) indicate that the microstructure remains columnar as the grain size increases with thickness (Figures adapted from Ref. [24] with permission)

30

0.25 nm/s 20

0.13 nm/s

10

0.08 nm/s 0

0.05 nm/s 0.03 nm/s

-10 0

50 Thickness (nm)

100

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evolution is known, the effect of grain growth on the stress-thickness can be calculated for comparison with the experiments. Grain growth can affect the film stress in two ways, corresponding to the two terms describing the curvature evolution in Eq. 4, i.e., adding new stressed layers to the surface or by changing the stress in the layers that have already been deposited. Equation 7 describes how changing the grain size L changes the stress in new layers being deposited at thesurface. L affects the amount of tensile stress that is generated  1 by grain coalescence σ T  1=L =2 and it also modifies the amount of stress that is generated by insertion of atoms into the grain boundary (in the term βD/RL). In addition, grain growth can create additional tensile stress in the layers that have already been deposited by densifying the film. Preliminary analysis suggests that the measurements in Fig. 9 can be understood as a superposition of grain-size dependent incremental stress (Eq. 7) and stress from grain growth. However, further work remains to be done to confirm this.

3.6

Stress Modification with Energetic Beams

The stress in thin films can be modified significantly by the use of different deposition processes. One of these is sputter deposition where energetic inert gas atoms are used to knock atoms out of the source material. This creates an impinging flux on the substrate that consists of deposited species with hyperthermal energy and energetic inert gas ions or atoms. The energy of the incoming particle flux can be affected by changing the gas pressure. Decreasing the pressure results in fewer atomic collisions in the gas so that the species have a higher energy. For charged species, the incident energy can also be modified by applying a bias to the substrate. Wafer curvature can play a useful role in determining how the energetic processes modify the stress as well as quantifying how much the stress is modified. Many studies have used wafer curvature to determine that the stress can be made less tensile/more compressive by the use of lower gas pressure [62–64]. The effect can be large, for example, the stress in W films can be changed from 2.1 GPa (highly tensile) to 2.3 GPa (highly compressive) by changing the pressure. In situ stress measurements by Fillon et al. [65] (Fig. 10) show how the stress evolution in Mo films sputter-deposited at room temperature is changed for different growth pressures. The grain size of these samples remained constant (appr. 50 nm) so grain growth did not play a role. At high pressure (low particle energy), the behavior is typical of a low-mobility material with a highly tensile steady-state stress of 3.3 GPa. At lower pressures, the behavior becomes more like a type II material, with a steady-state compressive stress. The application of a bias voltage (60 V) makes the stress even more compressive. These results show how the energetic component of the deposition beam plays an additional role in the stress beyond the nonenergetic growth processes. It is beyond the scope of this chapter to go into detail with respect to the different mechanisms that have been proposed. One factor is through the creation of energetic defects in the bulk that create stress in the film.

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80 0.43 Pa

60 40

F/w (N/m)

Fig. 10 Measurements of stress-thickness versus thickness during sputter deposition of Mo films at a growth rate of 0.06 nm/s and different conditions of pressure and bias indicated in figure (Adapted from Ref. [65] with permission)

20

0.24 Pa

0 -20

0.11 Pa R = 0.06 nm/s

-40 -60

0.11 Pa/-60 V 0

20

40

60

h (nm)

Other studies indicate that some of the stress generation occurs at grain boundaries [66], which was shown by comparing stress evolution in films with different grain sizes.

4

Stress in Sn Leading to Whisker Formation

Another example where wafer curvature has been used to characterize thin film stress is in understanding the formation of Sn whiskers. This topic is discussed in this chapter because their formation is believed to be caused by stress in the Sn layer. Whiskers grow in the form of thin filaments that develop spontaneously out of Sn coatings over Cu conductors (an example is shown in Fig. 1f). Growing to lengths of mm’s, they have been responsible for many system failures [67] and are a critical reliability issue in electronic components. The problem has become more acute since the removal of Pb from manufacturing for environmental reasons [68]. Before this, coatings made of Pb-Sn alloys were used that suppressed whisker formation.

4.1

Stress, Intermetallic Compound Formation, and Whisker Growth

It has been suggested that the source of the stress is the formation of an intermetallic compound (IMC) between the Sn and Cu layers. Tu [69] explained this in terms of the different diffusivity of each species so that the IMC grows preferentially on the Sn side of the interface. Adding Cu into the constrained volume of the Sn film creates

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Stress Measurement in Thin Films Using Wafer Curvature: Principles and. . .

Fig. 11 Cross-section of region around a Sn whisker showing the microstructure of the layers

2071

whisker

oxide Sn IMC Cu Si

stress in the layer (the volume is constrained by the presence of an oxide that prevents atoms from diffusing to the Sn surface). The corresponding microstructure of a Sn coating over Cu on a Si substrate is shown in the FIB (focused ion beam) cross-section of a region that formed a whisker in Fig. 11. The layers are identified on the figure; the Sn has a columnar microstructure with large grains while the Cu layer has a much finer grain structure. A thin oxide covers the Sn layer. The IMC is visible at the base of the Sn layer, next to the Cu layer. Although it cannot be seen in this image, the IMC grows preferentially into the Sn-Sn grain boundaries at the interface between the Cu and Sn layers. More details showing the morphology of the IMC can be seen in Ref. [70]. The use of wafer curvature to measure the stress in the Sn layer is complicated by the fact that there are multiple layers (Cu, IMC, and Sn) on the Si substrate. As shown by Eq. 3, the measured curvature is due to the stress in all of these layers: κ¼

6 ðσ Cu hCu þ σ IMC hIMC þ σ Sn hSn Þ Ms h2s

(8)

The stress in the Sn layer therefore cannot be determined by a single curvature measurement, but it can be determined from the following series of measurements. First, the curvature is measured in the full multilayer system. Then, the Sn layer is selectively removed by a chemical etch that does not affect the other layers and the curvature is measured again. The curvature is now due to the two remaining layers (Cu and IMC). Assuming that the etching does not change the stress in these layers, the difference in the curvature between the two measurements can be attributed to the stress in the Sn layer. Note that this requires knowing the thickness of the Sn layer that has not reacted with Cu to form IMC. This can be determined by comparing the change in the weight when the Sn layer is removed with the amount of weight in the original layer (this can be done with sufficient accuracy using a microbalance). This also enables the volume of intermetallic to be determined at different times. An example of the results from such curvature measurement made at different times after deposition is shown in Fig. 12. The blue diamonds correspond to

2072

Total

200 stress-thickness (N/m)

Fig. 12 Measurements of stress-thickness in Sn layers grown over Cu on Si substrates for different times after deposition. Diamonds (◊) represent total stressthickness; squares (□) represent measured stressthickness after Sn layer is removed by chemical etching; triangles (Δ) are the difference corresponding to stressthickness of Sn layer

E. Chason

After Sn stripping

160

Difference (Sn only)

120 80 40 0 -40

0

5

10

15

Time (days)

measurements on the full Cu-Sn-Si sample. The red squares represent the curvature measurements after the Sn layer was stripped away. Because the measurement is destructive, it was performed on different samples prepared identically and held for different amounts of time. The difference between the measurements, shown by the triangles, indicates that the stress in the Sn is compressive even though the total curvature is tensile. The co-evolution of the IMC volume and the average stress in a pure Sn film obtained from a series of samples is shown on the left side of Fig. 13a, b) [70]. The whisker density versus time was measured over the same time period on an identical sample using optical microscopy (Fig. 13c). The results illustrate how the IMC, stress, and whisker density are related. The IMC is observed to grow continuously over the 5 day period of the measurement. The corresponding stress in the Sn layer initially becomes compressive and then saturates, indicating the onset of strain relaxation processes in the Sn layer. Strain relaxation can occur by many different processes including creep, dislocation-mediated plasticity, or whisker formation. Importantly, the whisker density does not start to increase until the stress has reached its maximum compressive value. As new whiskers nucleate, the stress in the Sn layer remains essentially constant. This same approach can also be used to investigate Pb-Sn layers to understand why they are less prone to form whiskers than pure Sn. Figure 13d–f shows measurements of the IMC volume, stress, and whisker density in Pb-Sn samples of the same thickness as for the Sn layers described above. Comparison of the measurements of the IMC volume in the Pb-Sn with the pure Sn samples shows that the presence of Pb in the layer does not dramatically change the growth kinetics of the intermetallic layer. However, the evolution of the stress in the Pb-Sn and Sn layer is very different. For roughly the same amount of IMC formation, there is almost no stress developed in the Pb-Sn layer. Because the driving force (IMC formation) is nearly the same for both the Sn and the Pb-Sn, the lower stress suggests that there is more stress relaxation occurring in the Sn layer. Importantly, the lack of

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Fig. 13 Measurements in a pure Sn film (left side) and a Pb-Sn alloy film (right side) over a Cu layer of (a, b) IMC volume; (c, d) stress; and (e, f) whisker density (Reprinted from [70] with permission)

stress in the Pb-Sn is correlated with a corresponding decrease in the density of whiskers observed.

4.2

Mechanical Properties of Sn and Sn Alloys: Stress Relaxation Measured Using Thermally Induced Strain

In order to understand why there is more stress relaxation in Pb-Sn layers than in pure Sn, a comparison of the microstructure of the different layers is shown in Fig. 14a and d) [71]. The grains in the pure Sn layer are columnar, with a single grain spanning the full thickness of the Sn layer so that the grain boundaries are primarily vertical in orientation. In contrast, the Pb-Sn layer has a more equi-axed structure with a mixture of grain boundaries, some of which are horizontal. Boettinger et al. [72] have proposed that the presence of these horizontal grain boundaries makes it possible for atoms to diffuse to them and relax the volumetric strain by pushing the layer outward, without inducing stress in the layer. Because the Sn layer has a tenacious oxide, it prevents atoms from diffusing to the surface in order to relieve the stress in the absence of internal boundaries. This proposed mechanism for stress relaxation can be tested by measuring the evolution of stress imposed by thermal expansion mismatch as was done for experiments like those shown in Fig. 2 to study plastic relaxation of thin films. To

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do this, Sn and Sn-alloy layers were deposited directly onto Si substrates over a thin Ti interlayer to enhance adhesion. Because there are no other layers, the measured curvature upon heating can be attributed to the stress in the Sn-based layer. The difference in thermal expansion coefficients between Sn and Si creates a strain of 1.88  105 per degree of heating (in  C). The results of thermal cycling are shown for 7.5 μm of pure Sn and 10% Pb-Sn in Fig. 14b, and e, respectively [71]. The measurements were obtained during thermal cycles in which the sample was heated by different amounts (ΔT = 10, 20 and 40  C) as shown in the figure. The different stages of the thermal cycle for ΔT = 40  C are delineated by the vertical dashed lines in the figure: (1) the sample was heated at a constant rate of 0.5  C/s, (2) sample was held at the maximum temperature for 5 h, (3) the sample was cooled to room temperature, (4) sample was monitored at room temperature for 6 h. Considering the Sn sample first, the stress initially increases linearly with time as the temperature is raised. For the higher temperatures (ΔT = 20 and 40  C), the stress starts to relax before the maximum temperature is reached, as shown by the deviation from linearity. When the temperature is held at the maximum, the stress decays at different rates and by different amounts. The time evolution of the stress at constant temperature has been explained by a power-law creep mechanism [12], which has also been found for bulk Sn [73]. The stress as a function of temperature is shown in Fig. 14c. For low temperatures the stress is proportional to the temperature with a ratio that is consistent with the

Fig. 14 Comparison of microstructure and stress relaxation in (a–c) pure Sn films and (d–f) Pb-Sn alloy films. (a, d) SEM images of FIB cross-sections; (b, e) stress versus time during thermal cycling; (c, f) stress versus temperature. The numbers (1–4) in (b, e) indicate the four stages of the heating cycle (stage 1, heating; stage 2, holding; stage 3, air-cooling; stage 4, room temperature monitoring). The values of ΔT indicate the amount the sample was heated for each cycle (Figure adapted from Ref. [71] with permission)

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thermal expansion coefficients. This corresponds to elastic behavior in the sample. At higher temperature (i.e., strain), the stress does not increase linearly with temperature, which corresponds to the onset of plastic deformation via power law creep. The stress in the 10% Pb-Sn sample for the same thermal cycle is shown in Fig. 14e) and f). The degree of stress relaxation is much higher than for Sn and the maximum stress obtained in the layer is much lower. In order to confirm that the enhanced relaxation is due to the grain structure of PbSn, different sample configurations were studied. To confirm that the effect was not due to an effect at the surface (e.g., a weaker oxide), different layered structures were measured in which thin layers of Sn were deposited over Pb-Sn in order to change the surface termination without changing the bulk of the film. These structures still exhibited enhanced relaxation even though the surface consisted of Sn and not PbSn. Thin layers of Pb-Sn (1 μm thick) were also studied, which had columnar grains, rather than the equi-axed structure found in the thicker Pb-Sn layers. These layers had similar behavior to Sn layers of the same thickness, again indicating that the grain structure of the Pb-Sn is responsible for the enhanced stress relaxation. In another study, 10% Bi-Sn alloys were grown that had equi-axed grain structures similar to Pb-Sn [74]. These also exhibited enhanced stress relaxation relative to Sn layers of the same thickness.

4.3

Dependence of Nucleation Kinetics and Whisker Growth Rate on Thermally Induced Stress

In the final sets of Sn whisker examples to be considered, thermally induced strain was used to simultaneously measure the stress and the whisker growth kinetics. To perform these measurements, a wafer curvature setup was coupled with an optical microscope [75]. The samples consisted of Sn layer deposited directly on Si substrates as in the measurements of mechanical properties discussed in the previous section. The surface facing the optical microscope was illuminated at an oblique angle, which highlighted the development of surface features. Comparison of the optical measurements with SEM measurements of the same region indicated that features with a volume greater than 0.5 μm3 could be identified reliably. This enabled the density of new surface features (whiskers and hillocks) to be quantified as the sample was heated. Measurements of the temperature, the resulting stress, and the corresponding whisker/hillock density are shown in Fig. 15 [76, 77]. The temperature evolution is shown in the upper part of the figure as the sample is heated to 65  C at a rate of 2  C /min and then held. The stress (middle part of figure) initially changes linearly in proportion to the thermal strain. As the temperature approaches the maximum value, the stress deviates from linearity, corresponding to the onset of plastic deformation. The corresponding whisker density (lower part of figure) starts to increase around the same time. As the sample is held at the maximum temperature, the stress decreases but the number of whiskers continues to grow. Ultimately, the stress saturates at a value of approximately 12 MPa and the number of whiskers saturates also. The

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Fig. 15 Measurements of (a) temperature, (b) stress, and (c) whisker density and nucleation rate during thermal cycle of pure Sn film on Si (Figure adapted from Ref. [77] with permission)

nucleation rate is shown superimposed on the measurement of the whisker density. The peak of the nucleation rate is seen to correspond with the maximum in the compressive stress. Similar measurements of the stress and whisker density were performed at the same heating rate for maximum temperatures of 45, 55, and 75  C [76]. Because the stress was measured at the same time as the nucleation rate, the data can be used to plot the nucleation rate versus the measured stress in the film for the different temperatures (shown in Fig. 16a)). These data have been corrected for the small amount of time that it took for the whiskers/hillocks to become large enough to be observed by the optical microscope (details can be found in Ref. [76]). The results show that there is a correlation between the measured nucleation rate and the stress in the film. In addition, for the same amount of stress the nucleation rate is greater for higher temperatures. The dependence on the stress is significant, with the nucleation rate increasing by a factor of more than 50 as the compressive stress increases from 18 MPa to 30 MPa at 75  C.

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To understand the processes controlling nucleation, these data were analyzed in terms of classical nucleation theory. The nucleation rate is assumed to be dN ΔG 1βc ðT Þe kT dt

(9)

where ΔG is the nucleation barrier and βc(T ) is the rate at which a cluster grows beyond its critical size. ΔG is assumed to depend on the stress since nucleation of a whisker can lower the strain energy in the system. However, unlike phase transformations, it is not expected to depend on the undercooling since the phase of the Sn does not change when a whisker nucleates, only the grain structure. βc(T) is expected to depend on the temperature; any possible dependence on the stress is not considered here. By inverting this equation, the dependence of ΔG on stress can be obtained from the measured nucleation rate. Assuming that this barrier depends on the stress but not the temperature, then dividing dN/dt by the appropriate value of βc(T ) should make all the data collapse to the same curve for G* versus stress. This is shown in Fig. 16b) for the different measured temperatures. The values of βc(T ) that produced this curve Fig. 16 (a) Nucleation rate versus stress from heating to temperatures indicated on figure. (b) Change in activation barrier for nucleation (Adapted from Ref. [76] with permission)

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have an Arrhenius temperature dependence with an activation energy of 0.77 eV. The results in Fig. 16 indicate that the nucleation barrier decreases by approximately 0.08 eV as the compressive stress increases from 18 GPa to 30 GPa. By using a controlled stress via thermal expansion to nucleate the whiskers, it was possible to determine this stress dependence quantitatively. It would be difficult to obtain these results by other means.

5

Future Directions and Open Problems

Wafer curvature is a straightforward and versatile technique that enables the stress in thin films to be measured as a film grows, as its microstructure evolves, or as it responds to external sources of strain (e.g., thermal expansion, phase changes, or ion irradiation). Despite its relative simplicity, it can be used to understand a wide range of phenomena by systematically changing the film structure, composition, or the processing conditions. Case studies based on wafer curvature in thin film growth and Sn whisker formation illustrate how it can be used to elucidate the fundamental mechanisms controlling stress and surface evolution. Real-time curvature studies have given tremendous insight into the processes that control residual stress as thin films are growing. Based on many studies of thin film stress a comprehensive picture is beginning to emerge. Some of this insight is captured in a kinetic model of the stress based on processes that occur at the growing boundary between adjacent grains as the film is deposited. Although this model has many appealing features, it requires comparison with systematic experiments in order to either validate it or refine the mechanisms included in it. Because of the large number of parameters that play a role in stress evolution, these studies require that multiple features be controlled and investigated (e.g., grain size, temperature, growth rate). Applying the model to analyze measurements in multiple materials will contribute to creating a database of parameters that can be used to predict stress in different systems under different conditions. These parameters can be measured across different deposition methods, so that for instance the effective diffusivity during electrodeposition can be compared with the effective diffusivity during evaporation to determine how the kinetic parameters vary in each case. In addition, the parameters determined from the model need to be compared with more fundamental calculation of these parameters (e.g., surface/interface energy difference for comparison with the tensile stress parameter). If this modeling approach is found to be successful then it can be extended to other systems. Preliminary work has already been done to extend the model to include energetic particles in sputter deposition to analyze results such as those shown in Fig. 10. This is done by adding the effects of ballistic collisions and ioninduced defect production to the other processes already in the growth model. Another area in which further study is needed is the deposition of alloy films that are commonly used in many applications. Although some excellent experimental studies have been done [78, 79], the range of phenomena that has been studied is not nearly as extensive as for single elements. A wider range of systematic studies is

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needed to quantify the dependence on growth rate, grain size, etc. for different alloy compositions to understand how the parameters compare with the elemental systems. In addition, the different thermodynamics of solid solutions versus immiscible systems must be considered. Based on the experimental results, the model can be extended to alloy systems to understand how multiple components modify the stress evolution in the film. In the area of Sn whiskers, a large amount of experimental study is being done to understand the underlying causes and potential solutions. Novel alloy compositions are being investigated to understand how they modify the film properties in ways that can suppress whisker growth. Based on these results, a more fundamental understanding of the mechanisms controlling whiskers is being developed [80]. The work presented here represents just a few examples of how curvature measurements can have an impact on thin film technologies. The technology landscape suggests that the importance of thin film materials will only continue to increase. Stress in new two-dimensional materials such as graphene can induce defects or buckling of the layers. Increased performance in hard coatings and thermal barriers are needed for even more extreme environments. Pushing the film growth conditions farther from equilibrium is likely to lead to even more stress in the film. This provides an opportunity for wafer curvature to study the relationship of stress to the processing conditions and determine the underlying mechanisms controlling it. The work on thin film stress was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award # DE-SC0008799. The work on Sn whisker formation was supported by the National Science Foundation under contract DMR1206138 and DMR1501411. The author acknowledges helpful contributions from Alison Engwall, Hang Yu, Carl Thompson, Ben Freund, Allan Bower, Jae Wook Shin, Sean Hearne, Greg Abadias, Kostas Sarakinos, and Gery Stafford.

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58. Buehler MJ, Hartmaier A, Gao H. Atomistic and continuum studies of crack-like diffusion wedges and associated dislocation mechanisms in thin films on substrates. J Mech Phys Solids. 2003;51(11–12):2105–25. 59. Guduru PR, Chason E, Freund LB. Mechanics of compressive stress evolution during thin film growth. J Mech Phys Solids. 2003;51(11–12):2127–48. 60. Chason E, Shin JW, Chen CH, Engwall AM, Miller CM, Hearne SJ, et al. Growth of patterned island arrays to identify origins of thin film stress. J Appl Phys. 2014;115(12):123519. 61. Chason E, Engwall AM, Miller CM, Chen CH, Bhandari A, Soni SK, et al. Stress evolution during growth of 1-D island arrays: kinetics and length scaling. Scr Mater. 2015;97:33–6. 62. Thornton JA, Hoffman DW. Internal-stresses in titanium, nickel, molybdenum, and tantalum films deposited by cylindrical magnetron sputtering. J Vac Sci Technol. 1977;14(1):164–8. 63. Hoffman DW, Thornton JA. Compressive stress transition in AL, V, ZR, NB and W metal-films sputtered at low working pressures. Thin Solid Films. 1977;45(2):387–96. 64. Thornton JA, Hoffman DW. Stress-related effects in thin-films. Thin Solid Films. 1989;171 (1):5–31. 65. Fillon A, Abadias G, Michel A, Jaouen C. Stress and microstructure evolution during growth of magnetron-sputtered low-mobility metal films: influence of the nucleation conditions. Thin Solid Films. 2010;519(5):1655–61. 66. Magnfält D, Abadias G, Sarakinos K. Atom insertion into grain boundaries and stress generation in physically vapor deposited films. Appl Phys Lett. 2013;103(5):051910. 67. NASA. http://nepp.nasa.gov/whisker/. 68. Puttlitz KJ, Galyon GT. Impact of the ROHS directive on high-performance electronic systems part II: key reliability issues preventing the implementation of lead-free solders. Journal of materials science-materials in. Electronics. 2007;18(1–3):347–65. 69. Tu KN. Irreversible-processes of spontaneous whisker growth in bimetallic cu-sn thin-film reactions. Phys Rev B. 1994;49(3):2030–4. 70. Chason E, Jadhav N, Pei F, Buchovecky E, Bower A. Growth of whiskers from Sn surfaces: driving forces and growth mechanisms. Prog Surf Sci. 2013;88(2):103–31. 71. Jadhav N, Wasserman J, Pei F, Chason E. Stress relaxation in Sn-based films: effects of Pb alloying, grain size, and microstructure. J Electron Mater. 2011;41(3):588–95. 72. Boettinger WJ, Johnson CE, Bendersky LA, Moon KW, Williams ME, Stafford GR. Whisker and hillock formation on Sn, Sn-Cu and Sn-Pb electrodeposits. Acta Mater. 2005;53(19): 5033–50. 73. Weertman J, Breen JE. Creep of tin single crystals. J Appl Phys. 1956;27(10):1189–93. 74. Jadhav N, Williams M, Pei F, Stafford G, Chason E. Altering the mechanical properties of Sn films by alloying with bi: mimicking the effect of Pb to suppress whiskers. J Electron Mater. 2012;42(2):312–8. 75. Pei F, Chason E. In situ measurement of stress and whisker/hillock density during thermal cycling of Sn layers. J Electron Mater. 2013;43(1):80–7. 76. Pei F, Briant CL, Kesari H, Bower AF, Chason E. Kinetics of Sn whisker nucleation using thermally induced stress. Scr Mater. 2014;93:16–9. 77. Chason E, Pei F. Measuring the stress dependence of nucleation and growth processes in Sn whisker formation. JOM. 2015;67(10):2416–24. 78. Fu B, Thompson GB. Compositional dependent thin film stress states. J Appl Phys. 2010; 108(4):043506. 79. Kapoor M, Thompson GB. Role of atomic migration in nanocrystalline stability: grain size and thin film stress states. Curr Opinion Solid State Mater Sci. 2015;19(2):138–46. 80. Chason E, Pei F, Briant CL, Kesari H, Bower AF. Significance of nucleation kinetics in Sn whisker formation. J Electron Mater. 2014;43(12):4435–41.

Testing of Foams

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Nikhil Gupta, Steven Eric Zeltmann, Dung D. Luong, and Mrityunjay Doddamani

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Classification and Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Open-Cell Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Closed-Cell Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Composite Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Selection of Test Methods for Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Applications of Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Selection of Test Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Compression Test Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Compression Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Impact Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dynamic Mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Vibration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Foams are lightweight cellular materials that are widely used in applications such as packaging, thermal insulation, sound absorption, underwater vehicle structures, and as the core in sandwich structures used in aircraft. Testing of foams to obtain reliable properties that are relevant to a given application is a significant N. Gupta (*) · S. E. Zeltmann · D. D. Luong Composite Materials and Mechanics Laboratory, Department of Mechanical and Aerospace Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA e-mail: [email protected] M. Doddamani Department of Mechanical Engineering, National Institute of Technology Karanataka, Surathkal, Mangalore, Karanataka, India # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_50

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challenge. High damping, high compressive or tensile strain, and high volume of air in the structure are among the challenges that make it difficult to apply the common test methods to these materials. For example, use of strain gauges for tensile or compression testing is usually not possible because bonding the strain gauges to the surface of a cellular material may not be possible, the small measurement range of a strain gauge may not be enough to capture the strain in the entire loading range, and microscopic material structure may dominate the measurement. This chapter discusses test techniques that include quasi-static compression, high strain rate compression, impact, dynamic mechanical analysis, vibration methods, and imaging techniques that are relevant to testing of foams. The imaging methods include ultrasonic imaging and microCT-scanning. Test techniques are described and results on representative foam materials are presented to understand the test outcomes. Keywords

Foam · Cellular material · Compression · Impact · Vibration · Dynamic mechanical properties · MicroCT-scan · Imaging

1

Introduction

Foams are cellular materials having gas porosity incorporated in a matrix material. Two important characteristics, namely, light weight and high damage tolerance, result in numerous applications of these materials. In many cases, foams are used for dual purposes. For example, automobile seat cushions are designed to provide comfort in daily driving, while these foams are also optimized to protect the passenger during a crash. Almost any material can be converted to a foam structure by incorporating gas porosity for catering to specific requirements of an application. For example, polymer foams are extensively used in packaging and seat cushion applications, whereas metal foams are used in automobiles and trains for energy absorption during impact, and ceramic foams are used in biomedical scaffold structures. Characterization of foams for various properties is more challenging compared to a solid specimen of the same material. The complications of testing foams include determining the size and shape of a representative specimen of the foam material to obtain properties that are representative of the bulk material. Localized variations of the cellular structure can result in vastly different measured properties. In such cases, the standard deviation in the properties measured on a batch of specimens obtained from the same foam slab may be large and statistical analysis methods need to be employed to interpret such results. This chapter is focused on describing the test techniques and presenting results for a variety of classes of foams. Standard test methods, such as those described by ASTM standards, are not always directly applicable to foams, unless standards specifically developed for foams are available. Often a plurality of methods may be partially applicable to such materials. Usually, a conservative approach is

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beneficial in such cases to select test conditions that can provide a representative set of properties from the selected samples. This chapter first discusses foam microstructure and classification and then discusses methods to test foams under the most relevant loading conditions. A majority of applications of foams are related to damage tolerance, which requires understanding their properties under dynamic loading conditions. Therefore, several dynamic test methods are described in this chapter.

2

Classification and Microstructure

A variety of classifications for foams are available in the published literature. In one possible classification, foams can be divided into the categories of nanoporous, mesoporous, and microporous foams, which are based on the size scale of porosity present in the foam structure. However, these classifications do not transmit any information about the microstructure and morphology of the foam, which are major contributors to the foam’s performance in a given application. A more illustrative classification of foams is presented in Fig. 1. In general, two different classes of foams can be proposed, which include: • Two-phase foams: Containing gas porosity in a matrix material • Composite foams: Containing three or more phases (matrix, gas porosity, and the additional reinforcing phases) In the case of composite foams, numerous possibilities exist to incorporate reinforcement in the foam. For example, micro- or nanoscale fibers or particles can be incorporated in the cell walls to make them stronger, a coating of a reinforcing Fig. 1 Classification of foams

Open Cell Foams Two Phase Foams Closed Cell Foams

Foams Composite Foams

Syntactic Foams

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material can be provided on the cell walls to strengthen them, or a grid-like structure can be generated in the pores to reinforce them. Several subclasses can be proposed in these two categories based on the type of materials used and the morphology of porosity [1]. These classes include open-cell foams, where the gas pores are interconnected; closed-cell foams, where the gas pores are separate and not connected to one another; and syntactic foams, a composite foam in which the porosity is added by dispersing gas-filled hollow particles in the matrix medium. Examples of foams that cross these boundaries are also available, such as open-cell foams that contain hollow particles in the matrix phase [2]. Examples of these types of foams are presented in the following sections but manufacturing methods of foams are not discussed here as significant literature is available in published books [3, 4], and the discussion in this chapter is focused on the test methods for foams.

2.1

Open-Cell Foams

Open-cell foams comprise of interconnected porosity, just like a sponge. Air can pass freely between the cells of such foams. Figure 2a shows an example of open-cell aluminum foam. The cell walls of open-cell foams can vary from having only small openings to connect the cells to being only struts creating a large open volume in the foam. An example of a polymer foam containing very high level of porosity is shown in Fig. 2b. Air or working fluid circulation in foams made of conducting materials Fig. 2 (a) An open-cell aluminum foam (Specimens provided for imaging by Dr. Dirk Lehmhus, IFAM, Bremen, Germany). (b) Micrograph of an opencell polyurethane foam [9]

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Fig. 3 Aluminum foam core sandwich composite having aluminum face-sheets (Specimen provided for imaging by Dr. Dirk Lehmhus, IFAM, Bremen, Germany)

such as aluminum alloys makes them effective as heat exchangers. Due to high volume fraction of porosity (as high as 99%), these foams have low strength and stiffness under compression and impact. Important material parameters in open-cell foams are cell size, cell shape, gas permeability, and density. These parameters define the mechanical properties of such foams. Open-cell foams are often used in a sandwich configuration as shown in Fig. 3 for aluminum foam core sandwich having aluminum face-sheets. The sandwich configuration provides them with high bending stiffness and strength.

2.2

Closed-Cell Foams

In closed-cell foams, each gas filled cell is surrounded by solid cell walls and the cells do not have interconnected porosity. Figure 4 presents an example of closedcell polyvinyl chloride (PVC) foam. The completely enclosed gas porosity provides closed-cell foams with higher density and mechanical properties than open-cell foams. The closed-cell foams find applications in packaging or impact protection. Apart from foams of polymers such as polyethylene, polystyrene, and polypropylene [5–7], closed-cell metal foams have also been studied for energy absorption capabilities under impact and high strain rate loading conditions [8–10]. Back pressure applied by the gas enclosed in closed cells provides higher properties to these foams compared to similar foams having open-cell porosity. Closed-cell foams can be tailored to have different wall thickness, cell size, and cell shape. These parameters contribute to the mechanical properties of such foams [11].

2.3

Composite Foams

2.3.1 Reinforced Open or Closed-Cell Foams Due to thin walls or ligaments, open- and closed-cell foams are not strong enough for structural applications. Many studies have relied on reinforcing the walls of such foams with micro- or nanoscale reinforcements. For example, carbon nanotubes, carbon nanofibers, and graphene have been used as reinforcements in polymer foams

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Fig. 4 As-cut surface of a closed-cell PVC foam specimen

[12, 13]. Complete dispersion of nanoscale reinforcements is a challenge in such foams because the cell walls or ligaments are usually very thin. The nanoscale reinforcement is useful in improving the tensile and shear properties of foams. However, the benefit of such reinforcement in compressive properties is limited because of reasons such as folding and buckling of very-long-aspect-ratio nanoscale reinforcements during mixing operations. Metal foams have also been reinforced with carbon nanotubes and ceramic particles [14–16]. Several forms of reinforcements are incorporated in metal foams, including nanotubes, whiskers, and particles [17–19]. In metal foams, additional effects such as grain refinement and change in precipitate size and location may occur due to the presence of an additional phase. Such effects also need to be studied while characterizing these materials.

2.3.2 Syntactic Foams Reinforcing each gas pore of a foam is an attractive possibility for improving its mechanical properties. Hollow particles dispersed in a matrix material, called syntactic foams, are such foams where shell of hollow particles provides reinforcing effect to each cell [10]. Syntactic foams can be fabricated over a large density range by varying wall thickness and/or volume fraction of hollow particles. These foams possess high strength and modulus that can be tailored over a wide range for specific applications [11]. In addition to buoyancy effect, high specific strength and low moisture absorption make them useful in the present applications in marine vessel and underwater vehicle structures [20, 21]. The matrices used in syntactic foams can be polymers, metals, or ceramics. An example of polymer matrix syntactic foam microstructure is presented in Fig.5. Selection of hollow particles of a given size and wall thickness can help in tailoring the porosity characteristics in syntactic foams and obtaining properties in the desired narrow range [22, 23]. The main benefit of syntactic foams is that the volume fraction and wall thickness of hollow particles can be selected as two independent parameters. A combination of these two parameters can provide a range of novel properties that cannot be achieved in two-phase composites [24, 25].

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Fig. 5 Glass hollow particle reinforced vinyl ester matrix syntactic foam

Polymer matrices have been most extensively used in syntactic foams and polymer matrix syntactic foams have found a wide range of applications in underwater vehicles, submarines, and aircraft structures [22, 23]. Thermoplastics and thermosets are the two basic categories in the polymer regime; both have been used in syntactic foams. Thermoplastics are polymeric materials that soften upon heating and harden upon cooling, making it possible to fabricate syntactic foams by injection molding and compression molding processes [26, 27]. Thermosets are currently the predominant matrix resins for syntactic foam that are used in industrial applications due to the well-established processing parameters, processing methods, availability of material properties, and understanding of failure mechanisms [28, 29]. Considerable focus has been observed in metal matrix syntactic foams with magnesium and aluminum alloy matrices [30–32]. These syntactic foams can be useful for higher-temperature applications than polymer matrix syntactic foams. Syntactic foams of a wide variety of other matrix materials including iron, invar, titanium, and zinc have also been studied [33–36]. Syntactic foams are designed for use as matrix material in sandwich composites. There are many studies available on fabrication and characterization of polymer and metal matrix syntactic foams for compressive, bending, and dynamic properties [37–40]. Syntactic foams have high compressive strength and stiffness but their bending stiffness and tensile properties are relatively poor. Sandwich configuration provides improvement in bending properties and makes them suitable for structural applications.

3

Selection of Test Methods for Foams

3.1

Applications of Foams

Testing and analysis techniques for materials and structures depend on their intended use. A number of applications of foams are based on their ability to absorb energy and damage under quasi-static and dynamic loading condition, ability to dampen

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vibrations, and ability to provide thermal insulation. Examples of some of the typical applications of foams are • Energy absorption in structures • Packaging • Insulation – Thermal – Acoustic – Mechanical vibration • Sports and personal protective gear • Radiation shielding Details of many of these applications and case studies for designing of materials for these applications are available in the published literature [41]. Many applications combine different foam types in the same structure to achieve properties that are not possible in just one foam type. For example, the Brazuca soccer ball used at the 2014 World Cup contains layers of both open-cell foams and syntactic foams, as shown in Fig. 6. Beneath the outer skin is a layer of syntactic foam created using expanded polymeric hollow particles. This layer provides stiffness and prevents moisture uptake that could make the ball heavier and off balance during wet pitch conditions. For the purpose of efficient energy absorption in structures, functionally graded foams have been widely studied and optimized for different loading conditions [42–46]. In all cases, the selection of test methods needs to make sure that the characteristics and properties most appropriate to the application are measured. The possibility of tailoring materials for different required properties is presented in Fig. 7 [23]. The Porfiri–Gupta model [47] is used to develop elastic modulus estimates with respect to density (ρc) for syntactic foams containing hollow glass

Fig. 6 MicroCT-scan of construction of Brazuca soccer ball, showing layers of various foam types

Testing of Foams

Fig. 7 Variation of the elastic modulus of vinyl ester/glass hollow particle syntactic foams with respect to the foam density. The numbers associated with each line refer to the true particle density of the glass filler in kg/m3. The resin density is 1160 kg/m3 [23]. The line of 0 represents closed-cell foam of vinyl ester and 2540 kg/m3 represents solid glass particles

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Modulus (GPa)

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0 400 2540

200 600 Matrix resin

6

3

0 500

750

1000 1250 ρc (kg/m3)

1500

1750

particles. In this figure, two extreme cases are also plotted for comparison, where “0” represents foam that has no particle shell around the gas porosity and is only a closed-cell polymer foam and 2540 kg/m3 represents solid glass particles embedded in the composite. It can be observed in Fig. 7 that the modulus of syntactic foams can be significantly higher than that of closed-cell foams at the same density level and the use of syntactic foams can lead to significant weight saving in structural applications. Such figures can be used for material selection. Availability of such figures depends on theoretical models that are validated with experimental data for accuracy of prediction.

3.2

Selection of Test Standard

Selection of a test standard for foams can be a challenge. A number of different standards can be applicable to the same testing procedure at the first glance. An example of possible ASTM test methods for compression testing is presented in Table 1. In different situations, any of these standards may be applicable for testing a specific type of foam. In addition to the standards available from ASTM, numerous agencies develop standards for materials and applications. For international use, there are also equivalent methods published by ISO. In some instances, for example, ASTM D1621 for rigid cellular plastics, the ASTM standard is mirrored by ISO 844. However, standards developed by different agencies may differ from each other even for exactly the same purpose. Many countries have national standard agencies that independently develop standards for that country. In such cases, the standards developed by different countries may not have close correspondence with each other even if developed for the same purpose. This has emerged as an important challenge in the present era of global supply chains and is a constant consideration for engineers and product designers.

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Table 1 ASTM Standards that may be relevant to compression testing of foams for a particular application Standard # ASTM D695–10 ASTM D6641/ D6641M-14 ASTM D1621–16 ASTM C365/ C365M-16 ASTM F1551–09

Title of the Standard Standard test method for compressive properties of rigid plastics Standard test method for compressive properties of polymer matrix composite materials using a combined loading compression (CLC) test fixture Standard test method for compressive properties of rigid cellular plastics Standard test method for flatwise compressive properties of sandwich cores Standard test methods for comprehensive characterization of synthetic turf playing surfaces and materials

Selection of the appropriate test method should be conducted keeping in mind the actual application conditions for the material. Only then the testing will ensure that the appropriate material behavior is captured. Among the possible differences in the test methods may be the difference in the specimen size, aspect ratio, loading rate, and specimen end or gripping conditions. The properties of a specimen of a cellular material depend on the cell size and shape distribution present in the specimen. Smaller-size specimens would show more prominent effect of localized microstructural features, while larger-size specimens would be more representative of bulk properties of the material. Therefore, it should be ensured that the number of cells in the specimen in any dimension is enough to provide a representative sample of material properties.

4

Compression Test Methods

4.1

Compression Testing

Dynamic loading conditions such as high strain rate compression or impact are important for many applications of foams, especially in protective applications such as helmets, packaging foams, and armors. Apart from the microstructural effects due to the presence of cellular structure and porosity, some of the materials such as thermoplastic polymers have high viscoelasticity. Therefore, the properties measured for these materials are strongly dependent on the rate of deformation applied during the test. This rate of deformation is quantified by the strain rate (e). An overview of the classification of strain rate regimes given by Meyers [48] is shown in Fig. 8. Note that the exact boundaries of each regime are vague and tend to depend on the author. The equipment used to achieve these strain rates are also given for each regime. The discussion in this chapter will focus primarily on quasi-static and high strain rate dynamic loading, because of their relevance to the protective applications of foams. The quasi-static and low strain rates can be achieved by the electric screw-driven and servo-hydraulic universal test machines that are widely used for general materials characterization. When no strain rate is specified for a test

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Inertial Forces Negligible

Inertial Forces Important Dynamic: Low Modified Hydraulic & Pneumatic

Quasi-Static Servo-Hydraulic, Screw-Driven Machines

Dynamic: High Taylor Impact, Hopkinson bar, Expanding Ring Creep and Stress Relaxation Conventional Machines & Creep Testers

High Velocity Impact Explosives, Plate Impact

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107 (s-1)

Fig. 8 Strain rate regimes in mechanical testing and typical test methods, following the definitions given by Meyers [48]

one generally would assume it to be in this regime, and the most common ASTM test methods for materials characterization specify rates that fall close to the middle of this regime. Strain rates in the high-dynamic range are most commonly achieved by the split-Hopkinson pressure bar (SHPB) technique and Taylor plate impact method.

4.1.1 Quasi-Static Compression Quasi-static compression testing is used to determine the mechanical properties of foams, such as the elastic modulus, strength, and energy absorption. Although the standards should be used as per the applications of the foams, several precautions are necessary. For example, the absolute specimen size given in some standards may not be the best choice due to the cellular nature of these materials. The test specimens may not be truly representative of the properties of the entire material if the foam cell size is large and only a few cells are present in any direction of the specimens. In such cases, localized microstructural effects and defects may dominate the observed behavior of the specimens. At least 10 cells (50 or more is preferred) should exist along any dimension of the specimen. Lubricating the specimens between the test platens can also be a challenge because of porosity in the foam. In some cases, solidstate lubricants or high-viscosity lubricants may be preferred as they may not flow inside the cells and stay at the interface between the specimen and the compression test platens. A typical stress–strain graph for a closed-cell PVC foam is presented in Fig. 9a. It has an initial linear elastic region (marked with A), followed by a small peak, and then a long stress-plateau region (marked with B). The modulus of the foam is calculated from the initial linear region. The peak represents initial collapse of some of the weak cells. Cell size and cell wall thickness vary over a wide range in any given sample of foam. Some of these cells are weaker than others, which trigger the initial failure. The peak represents the first activity of cell collapse. Once a cell

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a 16

Stress (MPa)

12 C

B

8 4 A 0 0

b

0.2

0.4 Strain

0.6

0.8

c

Fig. 9 (a) A representative stress–strain graph for a closed-cell PVC foam having 200 kg/m3 density. (b) Failed specimen showing densification and (c) close-up of the cells shows cell wall buckling and folding due to high compressive strain

collapses, the stress is redistributed in the nearby region, causing some of the nearby cells also to collapse due to the increased stress level. The height of this peak is defined by the number of cells that collapse due to this initial cascading effect. This activity continues until the stress reduces below the failure stress level for cells in that region and the stress attains a plateau, which represents stable compression of the foam. The long stress-plateau is responsible for the high energy absorption characteristics of foams. Compression reduces the porosity and increases the density of the foam. Eventually, the entire foam densifies and the material behaves similar to a dense polymer and the stress level starts to increase again. This part of the stress–stain graph is marked with C in Fig. 9a. Figure 9b and c shows the microstructure of closed-cell PVC foam at two different magnifications after compression to about 70% strain [49]. There is some elastic recovery after the stress is removed from the foam. The foam cell wall buckling can be observed in these figures due to high compressive strain. It is also observed that continued compression leads to folding of the cell walls and plastic deformation leading to compaction, which results in the stress rise after the densification stage marked with C in Fig. 9a. Similar characteristics are observed in open-cell foams as well, where buckling of struts and eventual compaction leads to densification.

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The height of the peak between linear elastic and stress-plateau regions depends on the inhomogeneity in the foam cell morphology. If the cell wall thickness, cell size, and cell shape vary over a large range, the strength of various cells is likely to be vastly different and the collapse of one cell may trigger a longer chain reaction leading to a higher peak. An inhomogeneous cell morphology also results in a positive slope in the stress-plateau region. A homogeneous cell morphology results in a smaller peak and a more stable stress-plateau. Higher-density foams usually have higher modulus, higher plateau stress, and lower densification strain. Shearing of cell walls becomes prominent for high-density foams due to lower aspect ratio and reduced buckling effects. Using the factors such as cell size and cell wall thickness, the foam microstructure can be tailored to obtain the desired characteristics in the compressive properties. The characteristics of stress–strain graphs for syntactic foams are similar to those observed for other open- and closed-cell foams [50–52]. The stress-plateau region in syntactic foam compression is associated with crushing of hollow particles and their densification. The failure pattern in a metal matrix syntactic foam is shown in Fig. 10a [52]. At low strains, the brittle particle walls crack but the overall shape of the particle is maintained. As the deformation increases, shear bands form by plastic flow of the matrix, which causes shear fracture of particles and also displacement of the particle fragments. The broken particles densify and reduce the porosity of the material. In regions outside of shear zone, particles may undergo crushing but their fracture mechanism is not shear. The particle to matrix stiffness ratio is expected to play an important role in defining the onset of shear band and particle crushing mechanisms. The final failure is a combination of the two mechanisms as shown in Fig. 10b. In the present case, SiC is used as the particle material, which has a modulus of about 420 GPa, compared to about 70 GPa for the aluminum alloy. Failure observations are available in the literature for a number of syntactic foam systems including polymer (epoxy, vinyl ester, high-density polyethylene) and metal (aluminum, magnesium, titanium, iron, and Invar) matrix syntactic foams reinforced with glass, fly ash, alumina, silica, and silicon carbide hollow particles [22, 33, 53].

4.1.2 High Strain Rate Compression Testing High strain rate compressive characterization of foams is important because of their applications requiring damage tolerance and energy absorption under dynamic loading conditions. Highly time-dependent mechanical response of polymers makes such testing relevant for polymeric foams. Metal foams have also been tested at high strain rate loading to quantify the structural effects due to the presence of porosity and to understand the strain rate sensitivity of metals such as iron. One of the most common techniques for mechanical tests at high strain rates is SHPB [54]. An example of the apparatus for SHPB experiments is shown in Fig. 11, and a schematic of the setup is shown in Fig. 12 [55]. The principle of SHPB testing is based on wave propagation in materials, where a striker bar is launched at a high velocity at the incident bar. The impact between the striker and incident bars generates an elastic impulse in the incident bar that travels through the bar length. A small specimen is sandwiched between the incident and transmitter bars. The

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Fig. 10 Failure mechanisms in Al alloy matrix syntactic foam reinforced with SiC hollow particles. (a) Specimen failure under quasi-static compression in the stress-plateau region. Crushing of brittle SiC particles is marked with 1 and is illustrated in (a1), shear band formation and shearing of SiC particles in that region is marked with 2 and is illustrated in (a2). Failure of the specimen at higher strain has remarkable effect of shear band formation and shearing of particles

Fig. 11 Picture of a split-Hopkinson pressure bar

traveling elastic pulse interacts with the specimen. The impedance mismatch between the specimen and the bar material causes a part of the wave to be transmitted through the specimen, and a part to be reflected back in the incident bar as a tensile wave. The portion of the wave in the specimen is then reflected/transmitted again at the specimen–transmitter bar interface. Strain gages on each bar measure the strain

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Pressure Chamber

Incident bar

Specimen

Strain gage

Oscilloscope

Data acquisition

Transmitter bar

Time (t)

Striker bar Pulse shaper

Distance (x)

Fig. 12 Schematic illustration and working principle of the split-Hopkinson pressure bar technique

pulses over time, which are amplified by a Wheatstone bridge and recorded on an oscilloscope. The stress and strain in the specimen are then calculated by [56] σ ðtÞ ¼

A0 Eet ðtÞ A

(1)

e_ðtÞ ¼

2c0 er ðtÞ l

(2)

ðt eðtÞ ¼ e_ðτÞdτ

(3)

0

where A and l are the cross-sectional area and thickness of the specimen; A0, c0, and E are the cross-sectional area, sound wave velocity, and Young’s modulus of the bars; er and et are the reflected and transmitted strain pulses; and τ is a time variable used for integration. Detailed derivation of these equations is given in [48]. The equation set (1, 2, and 3) is obtained from the one-dimensional linear elastic wave propagation assumption, which ignores any transverse effects in the bars. Long incident and transmitter bars are used to obtain a close approximation with this onedimensional wave propagation assumption. A more thorough analysis would account for the fact that the wave is dispersive, and include reflections off the sides and losses to the surrounding air. The simplified solution also ignores the different velocities of waves of different frequencies [57]. The solution to the wave equation becomes far more complex when accounting for these effects, so different approaches are used to compensate for the dispersion of the wave. For smalldiameter (typically 13 mm or less) bars, these effects are mostly ignored. Above this size, numerical solutions are used to compute correction factors that are applied to the measured data [57]. When this approach is not sufficient, as in the case of bars

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that are themselves time- and frequency-dependent (i.e., viscoelastic, made of materials such as nylon), a more thorough approach is conducted that includes the dissipation effects [58]. In such case, calibration experiments are performed on the bar with no specimen in order to measure the attenuation and frequency shifting of the pulse. The results are then used to develop the correction factors for the experimental data acquired with the specimen in place. Theoretical models can also be formulated to use the viscoelastic data for the bar obtained from separate dynamic mechanical analysis experiments. Significant difference in the length of the specimen and the bars ensures that a large number of internal reflections are obtained within the specimen before the secondary wave from either of the bars interacts with the specimen again. This condition is necessary to obtain stress equilibrium in the specimen and uniform compression in order for the test to be valid. Specific challenges related to testing soft materials using SHPB techniques have been discussed in the published literature [59]. Large impedance mismatch between the bar material and the foam material, difficulty in reaching stress equilibrium in the specimen, and high damping effects resulting in very little transmitted wave are among challenges that provide low-quality output from foams [60]. However, advancements have resulted in testing very soft materials using the SHPB technique [61–63]. A typical set of incident and reflected strain pulses captured during an SHPB experiment on closed-cell PVC foams is shown in Fig. 13a. The reflected and transmitted pulses are combined and then compared with the incident pulse in Fig. 13b to determine the validity of the test. If this test is failed and the excessive losses are not accounted for, then corrections need to be applied in order to obtain valid results from this test. The strain rate sensitivity observed in closed-cell PVC foams and in iron/glass hollow particle syntactic foams is shown in Fig. 14. The strain rate sensitivity may be observed in modulus, yield strength, plateau stress, or in failure mode of the material.

a

b

5

Strain (× 10-3 )

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Ref lected pulse

1 -1800

-3

1200

1600

2000

Transmitted pulse Incident pulse

-5

Time (μs)

Fig. 13 A representative set of (a) strain signals from incident and transmitter bars and (b) stress equilibrium obtained in a HP100 PVC foam specimen at strain rate of 2176 s1

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0.0001 /s 0.001 /s 0.01 /s 1542 /s 1923 /s 2290 /s

20 15 10 5 0

0

0.2

0.4 Strain

0.6

0.8

Fig. 14 A representative set of compressive stress–strain graphs for (a) HP200 closed-cell PVC foam and (b) iron matrix syntactic foam contain 10 wt.% glass hollow particles

Not all the properties may have the same level of strain rate sensitivity, and the sensitivity level may be different in different strain rate ranges. It should be noted that foams are capable of withstanding a large compressive strain, which can be a concern in SHPB-based testing. The end of the test (see Fig. 14a) may not be the densification limit of the specimen or the fracture point. Since it is a wave propagation method and the pulse has a certain width as observed in Fig. 13a, a long specimen subjected to a short pulse may only be partially compressed before the pulse reflects. In such a case, the total strain obtained in the specimen may be a characteristic of the incident pulse, rather than being indicative of specimen densification limit or fracture point. A variety of methods including selection of appropriate material and length of the striker bar and pulse shaper are important for obtaining the desired characteristics in the pulse. The rise time of the pulse can be effectively controlled by means of a pulse shaper used between striker bar and incident bar. Figure 13a shows that the total pulse duration is about 400 μs for this test setup. This is a very short time frame for observing transient effects such as crack initiation and propagation. High-speed image acquisition systems are often used with the SHPB method to observe the specimen deformation and fracture pattern. Figure 15 shows a set of images obtained at 7000 frames/second acquisition rate for a magnesium matrix syntactic foam filled with SiC hollow particles. Crack initiation, progression, and fragment formation are observed in this set of images for a specimen that is tested at 1480 s1 strain rate. Such imaging techniques can be combined with digital image correlation (DIC) methods to conduct quantitative strain measurements in the specimen. The SHPB method has been extended to testing materials at low and high temperatures [64–66]. New challenges are involved in such testing because the wave propagation speed in the bars is dependent on the temperature and may change in the zone that is maintained at low or high temperature. Correction factors and careful calibration studies are required for such testing before the results can be interpreted with confidence.

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Fig. 15 High-speed images at 70,000 frame/second of a magnesium alloy matrix syntactic foam filled with SiC hollow spheres tested at strain rate of 1480 s1. Image frames are sequentially numbered as per increasing time

4.2

Impact Testing

Impact testing is widely used for foams because of their use in energy absorbing protective structures. Two types of impact testers are used, which include pendulum impact and drop-weight impact testers. The pendulum and the drop-weight impact test equipment are shown in Fig. 16. In both cases, a dead weight is raised to certain height and then released on a specimen that is fixed using certain boundary conditions. The potential energy of the dead weight is converted to kinetic energy as the weight is released and allowed to impact the specimen. Pneumatic attachments can be used to accelerate the weight for increasing the impact energy. A number of standards are available for impact testing, some of which are included in Table 2. Some of these standards are intended for very specific forms of materials or applications. An appropriate standard needs to be selected for conducting the test.

4.2.1 Pendulum Impact A typical setup for pendulum impact testing is shown in the left side of Fig. 16. The impactor tip is equipped with an accelerometer, which is used to measure the

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Weight Impact Tup

Velocity Sensor Specimen Clamp Pendulum impact tester

Drop-weight impact tester

Fig. 16 Pendulum (left) and drop tower (right) impact test equipment

Table 2 Examples of impact test method standards Standard # ASTM D6110–10 ASTM D256–10 ASTM D4812–11 ASTM E2316b ASTM D3420–14 ASTM E2298–15

Title of the Standard Standard test method for determining the Charpy impact resistance of notched specimens of plastics Standard test methods for determining the Izod pendulum impact resistance of plastics Standard test method for unnotched cantilever beam impact resistance of plastics Standard test methods for notched bar impact testing of metallic materials Standard test method for pendulum impact resistance of plastic film Standard test method for instrumented impact testing of metallic materials

accelerations and thus impact forces during the experiment. These measurements are used to calculate the impact resistance of the material, with units of energy per area. The tests can be conducted as per either Izod or Charpy configurations in accordance with the methods listed in Table 2. Izod test conditions use a cantilever, and Charpy tests used a simply supported configuration for the specimen. Both types of tests can be conducted on notched or unnotched specimens. A notched Izod impact test specimen fixed in a standard test fixture is shown in Fig. 17. The impact failure can be divided into three stages, namely, crack initiation, propagation, and final failure. Comparing the tests

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Fig. 17 A syntactic foam specimen gripped in the Izod test fixture for pendulum impact testing

VE460-30

VE460-40

VE460-50

VE460-60

b

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Energy (J)

Load (kN)

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VE460-40

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1

Deflection (mm)

1.25

0.00 0

0.25

0.5

0.75

1

1.25

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Fig. 18 (a) Load-deflection and (b) cumulative energy absorbed-deflection curves for vinyl ester matrix syntactic foams containing glass hollow particles of 460 kg/m3 true particle density in 30–60 vol.%. Material types are named according to VE-[true particle density]-[volume fraction]

results of notched and unnotched specimen conditions can provide information on the energy required for initiating a crack in the specimen. Pendulum impact tests are generally unsuitable for flexible foams and are mostly conducted for rigid foams. An example of results for syntactic foams is presented in Fig. 18, where load and cumulative energy absorption are plotted with respect to the deflection for four different types of syntactic foams. The specimen strength can be calculated from the load-deflection graphs, and the energy absorption profile can provide insight into the energy absorbed in various stages of deformation and failure. The impact strength is presented with respect to syntactic foam density in Fig. 19 for a number of syntactic foam compositions. These results can be useful in tailoring the foams for energy absorption application. The studies on impact testing can be combined with microstructural analysis to understand the failure mechanism in the foams [67]. Such information can be used to improve the structure and properties of foams for impact applications.

4.2.2 Drop-Weight Impact Drop-weight impact testing is similar to pendulum impact testing in principle, except that a vertically falling weight is used instead of a swinging pendulum. Izod and

Testing of Foams

Fig. 19 Experimentally measured impact strength with respect to density for the various syntactic foams. Vinyl ester matrix syntactic foams containing glass hollow particles of 220, 320, and 460 kg/m3 true particle density in various volume fractions are tested

Fig. 20 Three types of tups commonly used in dropweight impact test: A – flat, B – conical and C – hemispherical

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8 6 4

VE220 VE320

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VE460 0 500 600 700 800 900 1000 Syntactic foam density (kg/m3)

A

B

C

Charpy testing can be conducted on drop-weight tower, but this setup is not commonly used for those tests. In drop-weight tower usually, a long tup is used as the loading head. The tup can be of various designs as per the requirements. Three commonly used designs are presented in Fig. 20, which are flat, conical, and hemispherical. Depending on the type of expected loading in the actual application, the tup design can be selected. In drop-weight tower, the specimen is mounted below the tup. A beam-like specimen can be used in this method. However, plate-like specimens are most commonly used in this testing. The plate can be either simply supported or rigidly clamped on all sides. Testing can also be conducted in other configurations such as on back supported specimens. This kind of impact testing is more common for flexible foams because the clamped-on-all-sides boundary condition allows testing of soft foam specimens to failure even after large deformation. In case of rigid foams,

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Fig. 21 A hemispherical tup of a drop-weight impact tester (a) just before impact with a closed-cell PVC foam, (b) deforming the foam specimen, and (c) fracture of the foam. The grid spacing is 5 mm

sometimes the specimens may not fracture under the applied load and the tup may rebound. The test equipment can be equipped with air brakes to prevent rebound and multiple impacts if the specimen does not fracture. This mechanism provides added flexibility in the test method to generate more test conditions. An example of testing closed-cell PVC foams using the drop-weight test method is presented in Fig. 21, where a closed-cell PVC foam specimen is tested using a hemispherical tup. The image sequence shows the specimen just before the impact, deformation of foam due to the tup, and fracture of the foam. After the fracture, the fractured portion moves down with the tup, while rest of the foam specimen recovers its elastic deformation [49]. This type of testing also provides load, energy, and deflection data that can be used to understand the behavior of the foam and to tailor its properties.

4.3

Dynamic Mechanical Analysis

Dynamic mechanical analysis is a widely-used technique for characterizing viscoelastic properties of materials and has been used for foams [68–70]. In this technique, a sinusoidal oscillating load is applied to the specimen and the resulting deformation is recorded as schematically represented in Fig. 22. This figure shows the typical dual-cantilever bending configuration used with soft solids, and the waveforms measured during the test. Most DMA machines have a

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a

b e(t) s(t)

w

t

Fig. 22 (a) Loading conditions in typical DMA experiment, where the specimen is clamped at two ends and a central clamp is oscillated vertically at a prescribed frequency. (b) Representation of the measurands in DMA. A sinusoidal displacement is applied and the scaling and shifting in the stress sinusoid is measured

relatively limited range of force and displacement, so different clamping configurations are used to adjust the specimen stiffness into the acceptable range. Other possible specimen configurations include the single cantilever, compression [71], tension, and 3-point bending. DMA is considered the most sensitive method to locate thermal transitions in polymers [72–76], including those in crystallization and resin curing. When combined with other spectroscopy methods, information from DMA can reveal the activation of different modes of motion of polymer chains [77–79]. DMA is also used to gain information on the behavior of polymer blends, such as to determine the miscibility of different polymers [80, 81]. DMA testing can be conducted at a wide range of temperatures and loading frequencies. In most available studies, the testing is conducted at 1 Hz frequency over a range of temperatures. In most cases, the interest is in determining the temperature-dependent behavior of the material, such as change in stiffness with respect to temperature, glass transition temperature, and maximum use temperature for the material. The direct usefulness of frequency-domain data is limited, which results in the use of a constant frequency of 1 Hz in most studies. A transform procedure is described in this section that can provide significant value to frequency-domain data because it can be converted to time domain in order to extract stiffness over a range of strain rates [82]. The transform technique has been validated on syntactic foams to ensure its applicability to porous materials [83].

4.3.1 Storage and Loss Moduli A typical set of results obtained from DMA testing is presented in Fig. 23. Storage (E0 ) and loss (E00 ) moduli represent the energy stored and energy lost during one loading cycle in the material, respectively. The loss tangent, also referred to as damping parameter, is calculated as tan δ ¼

E} 0 E

(4)

The results in Fig. 23 refer to a magnesium alloy matrix syntactic foam containing silicon carbide hollow particles tested at 1 Hz frequency. The storage modulus results can also be used to find the maximum temperature until which the material

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E' (GPa)

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T emperature (˚C)

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tan δ

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0.30 0.20 0.10

0.00 100

200

300

400

500

T emperature (˚C)

Fig. 23 Result for the DMA analysis (temperature sweep at constant frequency of 1 Hz): (a) storage modulus, (b) loss modulus, and (c) damping parameter

can be used with acceptable level of loss in the stiffness. The loss modulus curve shows a peak, which is associated with a thermal transition that depends on material type. This peak is very clear in polymers and corresponds to glass transition temperature. In magnesium foams, this peak appears very close to the melting temperature, where material starts to flow. The results on most syntactic foams show a similar trend. The materials can also be tested over a range of frequencies (1–100 Hz for commercially available equipment) as shown in Fig. 24a. Metals do not have very strong strain rate sensitivity, which results in relatively small slope of this graph. The figure shows testing conducted over a range of temperatures for the selected frequency range. The time-temperature superposition (TTS) principle can be used to develop master curves using any of these cures as the pivot. TTS allows extrapolating the material behavior over a wider range of frequencies than those used in the actual test as shown in Fig. 24b [84]. The useful range of frequencies that are relevant to an application can be selected from such graphs.

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335°C 260°C 185°C 110°C

310°C 235°C 160°C

E' (MPa)

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Testing of Foams

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10 1

10 2

Frequency (Hz)

Fig. 24 (a) Storage modulus over a frequency range and (b) master curve obtained by TTS for a magnesium alloy matrix syntactic foam reinforced with silicon carbide hollow particles, showing the low sensitivity of metallic syntactic foams to frequency variation

4.3.2

Obtaining Elastic Modulus at Various Strain Rates from DMA Data A number of viscoelastic and thermal properties are obtained from DMA testing. Methods are available that use DMA results to obtain the elastic modulus of the material at various strain rates. The value of such a transform is immense because elastic modulus at various strain rates is required for designing a number of applications that are subjected to dynamic loading conditions. In this way, this method allows using only one specimen on DMA to obtain all the viscoelastic properties, many thermal properties, and also elastic modulus at a wide range of strain rates. In this technique, the frequency-domain storage modulus function obtained from DMA is inverted to obtain the time-domain relaxation function, which can then be used to obtain the linear viscoelastic response of the materials at a given strain rate [82]. Various material properties can be found from this transformed data, such as secant modulus at a given strain and energy absorption at a given elastic strain. First, the frequency response curves are obtained for the specimen from a set of frequency sweeps from 1 to 100 Hz at temperatures from 10  C to 120  C in 5  C increments. These sweeps are shifted along the frequency axis using TTS to yield the master curves shown in Fig. 25. These results are obtained on HDPE matrix syntactic foams containing fly ash cenospheres [85]. The number in the material type indicates the weight fraction of hollow particles in the foam. The shift factors for most polymers are found to obey the Williams-Landel-Ferry (WLF) equation [86]: log10 aT ¼

c1 ðT  T 0 Þ c2 þ T  T 0

(5)

where aT is the frequency shift factor, c1 and c2 are the WLF coefficients, T is the temperature each data set is acquired at, and T0 is the reference or pivot temperature. Once the shift factors for a given material are determined, the corresponding WLF coefficients are independent of the choice of reference temperature.

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Storage Modulus (MPa)

Fig. 25 Master curves for HDPE/fly ash syntactic foams. HDPE20 and HDPE40 refer to syntactic foams with 20 and 40% fly ash by weight

2500 2000 1500 HDPE20

1000

HDPE40

500 0

10-15

10-9

103 10-3 -1 w (s )

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1015

Once the master curve E0 (ω) is known, it can be transformed into the time-domain relaxation function E(t) by the relationship using [87] 2 EðtÞ ¼ π

1 ð

0

E 0 ð ωÞ sin ðωtÞdω ω

(6)

where ω is the frequency and t is the time. In order to perform the inversion of this frequency-domain material function to the time domain, it needs to be extrapolated to zero and infinite frequencies. This is done by fitting a mixture of sigmoidal functions to the master curve data. Since a sigmoidal storage modulus curve yields a bell-shaped loss modulus curve, this choice of fitting functions can be interpreted as using one sigmoid for each observed transition in the master curve. So, in the case when one transition is observed, the fit is of the form E0 ðωÞ ¼ atanhðbðlogðωÞ þ cÞÞ þ d

(7)

where a, b, c, and d are the fit coefficients and log(ω) is the natural logarithm of frequency. If instead two step transitions are observed in the storage modulus master curve, corresponding to two peaks in the loss modulus master curve, the fit would take the form E0 ðωÞ ¼ atanhðbðlogðωÞ þ cÞÞ þ dtanhðeðlogðωÞ þ f ÞÞ þ g

(8)

where f and g are additional fit parameters. The fitting function is then transformed by numerically evaluating the integral transform (Eq. 6) using an accelerated convergence method to approximate the improper integral. According to viscoelasticity theory, the stress at a given point in time is a function of all the entire strain history of the material and therefore can be written as a functional of the strain

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history. Using only the assumptions that the strain history is continuous and the function is linear, this relationship can be written as [87] ðt σ ðtÞ ¼

Eð t  τ Þ 1

deðτÞ dτ dτ

(9)

where σ, e, and τ represent stress, strain, and time variables used for integration, respectively. This type of integral is known as a Stieltjes convolution. This representation shows that the stress at time t is the product of the relaxation function E(τ) with the instantaneous strain rate, integrated over all previous times. For constant strain rate deformation with a strain rate of e_ beginning at t = 0, which is the idealized deformation state in a standard compression or tension test, the convolution integral simplifies to ðt σ ðtÞ ¼ e_ EðτÞdτ

(10)

0

where e_ represents strain rate. Using this procedure, the linear viscoelastic stress–strain response of the materials can be predicted for any strain rate. This technique has been validated on HDPE over a wide range of strain rates [82]. Fig. 26 compares the predictions of this model with the experimentally measured secant modulus of HDPE matrix syntactic foams filled with 20 and 40 wt.% fly ash cenospheres. A close matching of the results is observed between the theoretical and experimental results. Although this technique is new, it has been successful in providing results for various polymers including thermosets and thermoplastics.

b

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Fig. 26 Comparison of tensile testing results with predictions from transform of DMA data for two HDPE/fly ash syntactic foams containing (a) 20% and (b) 40% hollow particles by weight [85]

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Vibration Methods

Vibration methods are employed for measuring the dynamic properties of relatively stiff foams, especially thermoplastic polymer foams and metallic foams. These techniques are also advantageous when examining sandwich structures, because there are fewer constraints on the specimen dimensions than in DMA. Due to their ease of implementation, vibration-based methods are advantageous over wave propagation methods for dynamic characterization [88, 89]. In addition to material properties such as modulus and damping coefficient, this type of test can also provide information such as the natural frequencies and vibration modes of beams. In vibration methods, the specimen is clamped at one end in a single cantilever configuration. A laser source is focused on the free end of the beam, which reflects off the specimen surface and onto a position -sensitive detector (PSD). A photograph of this setup is shown in Fig. 27. The free end of the specimen is excited using an instrumented hammer that captures the loading history. To cover a wide range of frequencies, the length of the beam can be varied. The scope of these experiments is to characterize the material by performing a dynamic modal analysis. The focus is on Young’s modulus and the damping parameter tan δ. The storage modulus E0 and the loss modulus E00 are also determined from the analysis. The storage and loss moduli are related to Young’s modulus by E¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 E0 þ E00

(11)

In addition, the damping parameter tan δ represents the lag between the response of the stress and the strain due to the applied load. The PSD measurement provides the beam deflection y(τ) by yð τ Þ ¼ hð x 1 , τ Þ

Fig. 27 Experimental setup used for vibration test. A mirror located on the surface of a metal matrix syntactic foam is reflecting the incident laser to a PSD

(12)

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where τ is the time. The deflection is given by hðx, τÞ ¼

N X

Xn ðτÞϕn ðxÞ

(13)

n¼1

where Xn(τ) is the nth modal coefficient and ϕn(x) is the nth mode function for the cantilever computed according to [90] 

   λn x λn x  sinh e e      sin ðλn Þ þ sinhðλn Þ λn x λn x  cos  cosh cos ðλn Þ þ cos ðλn Þ e e

ϕn ðxÞ ¼ sin

(14)

where e is the beam free length and λ are the solutions of the transcendental eigenvalue equation cos ðλeÞcoshðλeÞ ¼ 1

(15)

The fast Fourier transform of Xn = a1 is then performed. The magnitude of the first modal coefficient allows computing the phase of the signal by acquiring the relative bin number. The governing equation of motion for the specimen vibration in the Fourier domain, neglecting air damping, can be written according to [88] E  ð ωÞ

I@ 4 h~ðx, ωÞ  ρAω2 h~ðx, ωÞ ¼ Pψ ðx  x2 Þ @x4

(16)

where E(ω) is the complex Young’s modulus, I is the cross-section moment of inertia with respect to the centroidal axis orthogonal to the vibration plane, ρ is the mass density, A is the area of the cross section, ψ is the Dirac delta generalized function, and Pψ(x  x2) is the load applied by the impact hammer. Projecting Eq. 16 on the mode shapes set ðl E



0

¼

ðωÞβ4n Xn ðxÞXn ðxÞe φ n ðωÞdx ðl 0



ðl 0

e n ðωÞXn ðxÞXn ðxÞdx ρAω2 φ

Pψ ðx  x2 ÞXn ðxÞdx

(17)

Taking Xn(x)Xn(x) = 1 and ψ(x  x2)Xn(x) = PXn(x2) provides e n ðωÞ  ρAω2 φ e n ðωÞ ¼ PXn ðx2 Þ E ðωÞIβ4n φ

(18)

For moderately low values of structural damping, in the proximity of the mth resonance frequency, the system can be modeled with a single degree of freedom and Young’s modulus can be assumed to be constant over the frequency range. The resonance frequency is given by

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ω2m ¼

E0 ðωm ÞIβ4n ρA

(19)

Since Young’s modulus can be assumed constant, especially for narrow frequency ranges, it can be computed by [91–93] E¼

    2 2 M 12 l 2 ð Þ 2πf n 3 L bt βn 2 l2

(20)

where M is the total mass of the beam, L is the total length of the beam, b is the width of the beam, t is the thickness of the beam, fn is the nth mode of the natural frequency, l is the free length of the beam, and βnl is the value corresponding to the nth mode of the natural frequency. During the study of a three-layered skin sandwich beam, Barbieri et al. [94] used the damping ratio ζ to approximate the loss factor. They consider tan (δ) = ζ, where ζ is computed by   x1 log xn tan ðδÞ ¼ ζ ¼ 2nπ

(21)

00

Since tan ðδÞ ¼ EE0 , the loss modulus E00 is E00 ¼ tan ðδÞE0

(22)

For small damping ratios, the damping in the material can be incorporated into a complex modulus [92]: E ¼ E0 ð1 þ 2iζ Þ

(23)

Using this scheme, free vibration method can be used to determine various properties. Equipment with software is available to conduct this test. However, the test is relatively simple and based on fundamental vibration principles and can be recreated in the laboratory with relative ease. Calibration studies can be conducted using standard specimens before testing complex materials such as foams and composite materials.

4.5

Imaging

A number of imaging techniques including photography, optical microscopy, scanning electron microscopy, and transmission electron microscopy have been used for various purposes including measurement of cell size and shape, visualization of foam ligament size and shape, and distribution of porosity in the foam structure. Availability of high-quality lenses has made it possible to photograph minute details

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using digital cameras. These imaging methods are used for initial foam structure characterization or for postmortem of the specimens after testing. All these techniques can provide information only about the surface of observation. If the damage is not on the surface or if the information is required along the specimen thickness, then the usefulness of such imaging technique is limited. Two nondestructive imaging techniques that are used for damage evaluation in the entire specimen volume of the foams are discussed here. These techniques are ultrasonic c-scan imaging (UI) and micro CT-scan.

4.5.1 Ultrasonic Imaging UI has been used in a variety of fields for damage detection inside specimens and image acquisition of internal features [95]. This technique is well developed and is available in numerous different configurations. In most cases, a coupling agent is applied between an ultrasonic transducer and a specimen to facilitate wave propagation through the interface. A part of the ultrasonic waves reflects back from any feature present inside the specimen along the wave path. Rastering the transducer over the specimen provides a three-dimensional profile of the specimen with an image of internal features and defects. In principle, when an ultrasound plane wave proceeds in a material, part of it reflects and part of it propagates through the surface to further depth. The percentage of energy reflected is given by  R¼

Z1  Z2 Z1 þ Z2

2 (24)

while the percentage of the energy transmitted through the material is given by D¼

4Z 1 Z2 ðZ1 þ Z2 Þ2

(25)

where R is the reflection coefficient, D is the transmission coefficient, Z1 is the acoustic impedance for the upper medium (usually the coupling medium), and Z2 is the acoustic impedance for the lower medium (the specimen). The acoustic impedances can be formulated as Z1 = v1ρ1 and Z2 = v2ρ2, where v1, v2 are the ultrasonic velocities and ρ1, ρ2 are the relative density for medium 1 (coupling medium) and medium 2 (specimen), respectively. However, when the acoustic wave travels through the material, it undergoes attenuation. The attenuation is characterized as p

 p0

¼ eαd

(26)

where po and p are the sound pressures at the start and the end of a traveling distance d, respectively, and α is the attenuation coefficient of the material. Application of UI to foams is challenging because air causes an attenuation effect on the ultrasonic waves and makes foams highly attenuating materials. Even thin specimens of foams may have very high damping, making it difficult to scan them. The attenuation

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coefficient is inversely proportional to the ultrasonic wave frequency. Higher frequencies can provide higher scan resolution but result in poor penetration depth due to high attenuation. Obtaining information for specimens of certain thickness requires lower ultrasonic frequencies, which can be as low as 500 kHz. The UI technique has been used extensively on laminated composites to detect internal damage such as interfacial separation, delamination, and single ply failure [96–98]. It is known that the damage to materials usually spreads beyond the visible damage region. Foams have an excellent capability to keep the damage localized. Figure 28 shows high-speed camera images of an impactor interacting with a closedcell PVC foam in a drop-weight impact test. The grid on the specimens surface is manually drawn for visualization and calculation of displacement. The impactor penetrates the foam and causes complete fracture in the localized area under the tup. Figure 29a shows the damaged specimen. The same specimen is scanned using UI in Fig. 29b and c. The damage zone detected by UI is larger than the visible damage in the specimen. The information on the damage zone can be used to determine residual properties of the foam and make decisions about repair of the structure. UI has been used with DIC to obtain quantitative strain measurements during deformation [99]. The scan resolution is often poor in foams compared to fully dense materials due to the porosity present in their structure. This makes signal and image processing algorithms especially valuable for foam materials.

Fig. 28 Sequence of images (left to right) showing the fracture pattern in a HP60 tested at an impact velocity of 3.48 m/s. The total time period for the entire sequence is 250 μs. The grid spacing is 5 mm

Fig. 29 Ultrasonic C-scan images of HP60 foam, showing (a) a photograph of the specimen, (b) a C-scan image of amplitude, and (c) a C-scan image of time of flight (ToF)

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4.5.2 MicroCT-Scan Computerized tomography (CT scan) is a powerful technique in which X-ray (or other types of radiation) is used to obtain projection of the specimen structure through the thickness. The specimen is rotated to obtain projections from different sections of the specimens, and then the images are reconstructed to obtain a three-dimensional model of the material. With the current capabilities in the field, it is now possible to obtain submicrometer resolution in the images. The technique works for a wide range of materials. High-resolution imaging of internal microstructure, defects, and damage of foams is highly desired. MicroCT-scan is a technique that is now widely adopted for this purpose. One of the main limitations of this technique is the large size of the dataset that is generated. The total dataset size for high-resolution scanning of a small specimen of 10  10  10 mm3 volume can be 1 TB, which requires high computational power to open and manipulate the image files. A representative two-dimensional (2D) image set of a high-density polyethylene matrix syntactic foam containing fly ash cenospheres is shown in Fig. 30. Such 2D images are combined to obtain the 3D model shown in Fig. 31. Syntactic foams comprise of a matrix material with embedded particles of a different material. The absorption coefficient of both materials is different and it is possible to resolve them separately in image processing. Figure 31a shows only the hollow particles, Fig. 31b shows only the matrix, and Fig. 31c shows the color-coded image of both hollow particles and matrix together. Such capabilties are useful for a variety of reasons including the possibilities such as

Fig. 30 Orthogonal micro-CT-scan slices of a HDPE matrix syntactic foam

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• Quantitative analysis can be conducted about the volume fraction of each constituent in the composite foam. • The entire 3D model can be imported into finite element analysis programs to preserve the structural details down to micron levels. • Each material can be imported separately into a finite element analysis program or image analysis program to conduct separate analysis on each phase. • Various phases can be identified individually based on their absorption coefficients. Figure 32a shows an example of a cement-based syntactic foam containing glass hollow particles. A portion of this image is enlarged in Fig. 32b, where individual hollow particles can be observed in the form of pores surrounded by thin shell. Such high-resolution images are useful in capturing all the microstructural details of the material system. Similar studies in open- and closed-cell foams have focused on capturing the cell size, shape, and wall thickness to conduct theoretical analysis with more realistic input parameters [100, 101].

Fig. 31 MicroCT-scan images of HDPE matrix syntactic foams containing 20 wt.% of fly ash particles. (a) Distribution of particles in the composite, (b) visualization of only matrix, and (c) matrix and particles shown together in different colors

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Fig. 32 (a) A high-resolution microCT-scan of a cementitious syntactic foam containing 40 vol.% of glass hollow spheres. Some gas pores are also (larger-size pores) seen in this image. (b) A small part of the image is enlarged to observe individual hollow particles

5

Summary

Foams are of interest in a wide range of applications. Characterization of mechanical properties of foams is very challenging due to factors such as microstructural variations within the same material and presence of porosity in the structure. This chapter is focused on discussing techniques for characterizing mechanical properties of foams under quasi-static and dynamic loading conditions. Challenges specific to testing various foams for such properties are discussed with a representative set of results on various kinds of foams. High strain rate characterization conducted using SHPB and dynamic characterization conducted using vibration or cyclic loading methods are very challenging due to viscoelastic nature of the material in polymer foams and high damping effects. Imaging methods such as high-speed imaging, ultrasonic imaging, and microCT-scan are also discussed in context of their strengths and limitations for foam materials. High-performance foams with tailored properties are in great demand for weight-sensitive applications. It is expected that the capabilities of test and imaging methods will continue to improve as the demand for these materials is expected to remain high. Acknowledgments Support provided by Office of Naval Research grant N00014-10-1-0988 is acknowledged. The authors thank the NYU-AD Dean of Engineering for providing partial funding for visit of MRD to NYU-AD to work on the manuscript. Dr. Dirk Lehmhus, Fraunhofer IFAM, Bremen, Germany, is acknowledged for providing several specimens for imaging and for useful discussions. Funding from DAAD (grant # P91583818) and NYU for visit of Dr. Lehmhus to NYU is acknowledged to facilitate discussions. The views expressed in this article are those of the authors, not of funding agencies.

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Crack-Dislocation Interactions Ahead of a Crack Tip

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 BCS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Modification of BCS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Majumdar and Burns’ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lin and Thomson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Pande, Masamura, and Chou: Mode-III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 TEM Studies on Crack Tip Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Application to Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

An overview of interactions of crack and dislocations emitted from the crack is provided with particular emphasis on theoretical approaches. Several existing models, such as due to Bilby, Cottrell and Swinden, Burns and Majumdar, Lin and Thomson, and Pande and Masamura, are presented. These models provide a formalism that gives the equilibrium positions of dislocations in an array ahead of crack tip, from which important parameters such as plastic zone size, dislocationfree zone, and dislocation stress intensity factor can be determined, which are useful in discussing the phenomenon of fatigue. Following these overviews, the experimental results on the dislocation-free zone and plastic zone observed ahead of the cracks in aluminum using transmission electron microscopy are presented and compared with the existing models. The experimentally measured values of these zones are shown to be in reasonably good agreement with theoretical models of crack-dislocation configuration based on a continuum distribution of R. Goswami (*) · C. S. Pande Materials Science and Technology Division, Naval Research Laboratory, Washington, DC, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2019 C.-H. Hsueh et al. (eds.), Handbook of Mechanics of Materials, https://doi.org/10.1007/978-981-10-6884-3_52

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dislocations ahead of the crack. However, these models fail to predict the total number of emitted dislocations, underlying the need for better analytical models. We also provide a brief overview on crack tip dislocation behavior under fatigue loading. Keywords

Crack tip dislocations · TEM · Modeling · Fatigue · Dislocation configurations

1

Introduction

Designing materials for high damage tolerance requires a basic understanding of interaction of crack and dislocations. Significant efforts have been made in attempts to understand the crack-tip deformation behavior in metals and alloys during tensile deformation in an electron microscope [1–6]. It is found that deformation occurs mostly by the emission of dislocations from the crack tip, which acts as a possible dislocation source because it is associated with the highest stress. The behavior of dislocations ahead of the crack tip has been correlated with the propagation of cracks, and the local crack tip stress intensity factor has been determined. Theoretically [7–10] several authors have examined the process of dislocation emission from the crack tip by considering the elastic interaction between a crack and a dislocation. For an excellent summary of the theoretical work, see Thomson [11]. In addition, considerable attention has been paid recently in understanding the dislocation-crack tip interactions that influences the kinetics of fatigue crack growth [12], and the factors that determine fatigue thresholds using dislocation concept. It is now well known that dislocations are involved both in the initiation and propagation of cracks in metallic materials. In order to understand and establish a physical basis of crack growth and propagation, several dislocation models have [7–11, 13] been proposed. Most of them deal with the behavior of monotonically loaded cracks, but some also deal with the cyclically loaded cracks. We will present here brief overviews of the models, particularly those of Bilby, Cottrell, and Swinden (BCS) [13], Burns and Majumdar [7, 8], Lin and Thomson [14], and Pande et al. [15]. Following these overviews, we present recent results on dislocation configurations ahead of crack tip in Al alloys.

2

BCS Model

A detailed model for the monotonically loaded cracks was first given by Bilby et al. [13]. The BCS model considers a finite crack of the length 2a in an infinite isotropic elastic medium (Fig. 1). In their theory, the displacement in the crack as well as the plastic zone near crack tip was modeled by the continuous distribution of dislocations along the crack plane. Thus, they considered crack formally as a group of

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Fig. 1 A schematic diagram showing the configuration of a crack with the plastic zone and the dislocation distribution

infinitesimal dislocations pressed up against its sharp ends by applied shear stress. Some of these dislocations, “leaking through,” form the plastic zone at both ends of the crack. So, the resistance to dislocation slip is zero in the crack (freely slipping crack) and equal to the friction stress in the plastic zone. In this picture, the crack obviously deforms in the anti-plane mode III. The advantage of using continuum of infinitesimal dislocations is that the mathematical apparatus of singular integral equations can be used to obtain an analytical solution for dislocation distribution function in the crack and in the plastic zone satisfying all the boundary conditions. This function can then be used to determine various crack parameters such as crack opening displacement and length of the plastic zone. The BCS model, thus, introduced the dislocation modeling of crack and its interactions with other dislocations around the crack, and by 1990 this model was cited in over 460 papers. In particular, the BCS model has been applied by Weertman [16] to model fatigue crack growth. The main weakness of BCS model is that it does not account for the existence of a dislocation-free zone (DFZ) between the crack tip and the plastic zone. A series of in-situ transmission electron microscopic (TEM) observations by Ohr and his group on Al, Ni, Cu, Nb, W, and Mo established that the dislocation configuration ahead of the crack tip is in fact in the form of inverse pile-ups in the plastic zone, and they showed the existence of a DFZ at the tip of the crack [1–6]. Several authors have investigated the condition and formation mechanism of the dislocation-free zone. Ohr and coworkers made an observation of the distribution of dislocations in the plastic zone during in situ tensile deformation of thin films obtained from the bulk in a transmission electron microscope and the DFZ. Park and coworkers [17] and subsequently few others have made a similar observation. Ding and coworkers [18] showed that by in-situ straining of pure tin solder foils in a transmission electron microscope the dislocations were emanated from the blunted main crack tip, and dislocation-free zones were formed between the crack tip and the emitted dislocations. Pande [15] in his detailed study obtained direct evidence of (i) dislocations

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being emitted at the crack tip, (ii) internal stresses due to a crack, and (iii) dislocation-free zone, by straining a (111) oriented copper specimen thinned from the bulk using an in-situ transmission electron microscope. A mechanism for DFZ formation and its relation to fracture criteria of metallic systems have been extensively studied by many researchers [9–11, 14]. From these experimental works it is clear that the crack propagation involves both plastic and elastic processes. The plastic portion is created by the emitting dislocation from the crack tip. The elastic process seems to be associated with a brittle fracture of the DFZ. The BCS theory, which is mostly a plastic theory of crack propagation, has been extended by Chang and Ohr [9], Majumdar and Burns [7, 8], and Weertman et al. [19–23] to take this fact into account. We will discuss some of these theories in the subsequent sections.

3

Modification of BCS Model

The shear stress (σ) required to emit a dislocation form the crack can be written as [6, 9] σ ¼ kIII =ð2πxÞ0:5  μb=4πx

(1)

The first term is due to elastic crack stress and the second term is due to image stress. Here kIII is the local stress intensity factor of mode III crack, μ is the shear modulus, and b is the Burgers vector. In order to glide, the dislocation must overcome the frictional stress of the lattice, τfric. Once a dislocation is emitted, it will glide away from the tip of the crack, and will come to a rest at a distance, Xl, where the forces on dislocation are balanced by the frictional force. The region between the crack tip and XI is free of dislocations, and is termed as dislocation-free zone as shown in Fig. 2. It should be noted that these analyses use the concept of dislocation continuum so that the discrete dislocations are smeared into a continuously distributed continuum.

3.1

Majumdar and Burns’ Analysis

Dislocations are generated at the crack tip by the applied stress, when a metallic solid with a crack is under load. These dislocations generate a stress intensity factor on the sharp crack, which opposes the stress intensity factor from the applied stress. The sharp crack’s stress intensity factor is thus “shielded” from the external stress intensity factor by crack tip dislocations. If the crack tip remains sharp and each crack tip dislocation contributes to a reduction of the applied stress intensity factor, the sharp crack satisfies Griffith’s fracture criteria based on the true surface energy of the solid. The surface energy, thus, has a significant role even in the presence of substantial plasticity near the crack.

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Crack-Dislocation Interactions Ahead of a Crack Tip

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Fig. 2 A schematic diagram showing the crack, dislocation free zone and dislocations ahead of crack tip

An exact analysis of a Griffith-type crack shielded by a dislocation pile-up has been provided Majumdar and Burns [7, 8]. It is a two-dimensional analysis and includes a dislocation-free zone at the tip of a mode-III shear crack with a pile-up of screw dislocations parallel to the crack front, in anti-plane shear. As in BCS analysis, it is a continuous distribution of dislocations to represent discrete dislocations. The equilibrium of various forces due to crack and dislocations can be calculated and an equilibrium distribution of dislocations is possible in such a way that the dislocation distribution gives stress on each dislocation equal to a friction stress. The crack tip is assumed to remain sharp and it is also assumed to satisfy Griffith’s fracture criteria using the local crack tip stress intensity factor. The dislocation pile-up should shield the sharp crack tip from the applied stress intensity factor, which is calculated by a simple addition of each dislocation’s negative contribution to the applied stress intensity value. The analysis differs substantially from the well-known BCS theory because of the inclusion of a dislocation-free zone and also because in the analysis the local crack tip fracture criteria enters into the dislocation distributions. It should be noted that their analysis does not consider crack tip blunting. Coming to the detailed calculations, Majumdar and Burns consider an array of continuously distributed infinitesimal screw dislocations lying on the symmetry plane of a semiinfinite, anti-plane, shear crack, and parallel to the crack plane. The plastic zone extends from in front of the crack with a DFZ in between along the x-axis. Then if dislocation density is represented by f(x), so that f(x)dx is the number of dislocations between x and (x + dx), in the interval x to (x + dx) the Burgers’ vector is db = bf(x) dx. The solutions for f(x) in explicit form is given as f ðtÞ ¼ ½K ðmÞEðφ, mÞ  EðmÞK ðφ, mÞ ð4σ 0 =πμbÞ

(2)

where t = c (1  m2sin2φ)1, m = [1  (c/d)]0.5, α = m [t/(t  c)]0.5, sin φ = 1/α, K (m) is the complete elliptical integral of the first kind, E(m) is the complete elliptical integral of the second kind, c is the distance from the crack tip to first dislocation, and d is the distance from the crack tip to last dislocation. There are basic differences between the BCS theory and the analysis of Majumdar and Burns. The BCS theory assumes a stress level at the leading edge of the plastic zone. The crack tip in the BCS case has no stress intensity factor. The final results of the two analyses are also significantly different. In Majumdar and Burns’ analysis, an increase in the friction stress changes the number of dislocations and decreases the fracture toughness. In

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the BCS theory, on the other hand, an increase in friction stress increases the fracture toughness. It should be noted that both models do not independently predict the number of dislocations in the dislocation pile-up. Once the distribution function is known (see Eq. 2), several parameters, such as crack intensity factor and number of dislocations, are obtained. (See the Ref. [11] for details).

3.2

Lin and Thomson Model

Lin and Thomson [14] have considered the conditions for the existence of a cleavage crack in a lattice and its response to all types of external loads when shielded by neighboring dislocations. They examined the situation where two symmetrically inclined slip planes radiate from a semi-infinite crack tip in modeI. They considered the local stress field of the crack tip, the shielding contributions from the dislocations, and the complete specification of the forces on the two types of defects. One major approximation was the replacement of the dislocation arrays by a superdislocation. With this approximation they were able to determine many parameters of dislocation-crack behavior analytically such as the criteria for crack cleavage and emission under static conditions. They also found that the condition for equilibrium of a cleavage crack involves primarily only the mode-I part of the local stress field. On the other hand, dislocation emission depends on the full specification of all three components of the local stress field. Their primary result is that the combined cleavage and emission states are possible under mixed loading wherein a cleavage crack can generate significant plasticity in mode-II and III dislocation configurations. They also derived fracture toughness relations. Their work thus established the framework for dislocation-crack interaction for understanding the overall stability of a shielded cleavage crack under general loading. In the next section, similar analysis by Pande et al. [15] but without the use of superdislocation approximation will be described. Most of the results obtained by Lin and Thomson are shown to be approximately true. In addition, they obtain an expression for the length of DFZ, which is not possible in Lin-Thomson model due to superdislocation approximation.

3.2.1 Pande, Masamura, and Chou: Mode-I As mentioned before this model is similar to Lin and Thomson (LT) model except that the superdislocation approximation is not used. This gives an expression for the size of DFZ not possible in LT model. We consider this model in detail as it incorporates LT model, and examine here two symmetrically inclined slip planes radiate from a semi-infinite crack tip in mode-I. The geometry and parameters that are used in this analysis are shown in Fig. 3. Edge dislocations on just two inclined slip planes were considered. The forces on the i-th dislocation on the slip plane are due to (1) the crack tip stress field, (2) stress field due to other dislocations, and the (3) image forces due to the crack surface. The image contribution of the dislocation is only radial for an edge dislocation and is inversely proportional to ri, the distance from the crack tip to the dislocation [24]. The image force, Fimage, can be written as

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Crack-Dislocation Interactions Ahead of a Crack Tip

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Fig. 3 A schematic diagram showing symmetric dislocation arrays at crack tip in mode-I

Fimage ¼ 

μb2 4π ð1  υÞr i

(3)

where ν is Poisson’s ratio. The force due to the crack, Fcr, is     bK A θ θ Fcr ¼ pffiffiffiffiffiffiffiffiffi sin cos2 2 2 2πr i

(4)

where KA is the stress intensity factor due to the externally applied stress. The contribution Fd due to the other dislocations is obtained by superposition of forces of the elastic field obtained by the complex potential method. At equilibrium, the sum of the three forces is equal to the lattice friction force, Ffric. Thus Ffric ¼ b τfric ¼ Fimage þ Fcr þ Fd ,

(5)

where Fd can be written as Fd ¼

n X

Fkd

(6)

k¼1k6¼i

Fdk can be solved from the complex potential developed by Muskhelishvili [25] and formulated by Zhang et al. [26] for an infinite slit crack and a single edge dislocation on an inclined slip plane. Zhang et al. have developed the complex potentials, φ(z) and ψ(z), appropriate for this case from which the shear stress σ rθ can be obtained:

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R. Goswami and C. S. Pande

h i h 0 i 0 σ θθ þ iσ rθ ¼ e2iθ zϕ} ðzÞ þ ψ ðzÞ þ 2 Re ϕ ðzÞ

(7a)

h 0 i σ θθ þ σ rr ¼ 4 Re ϕ ðzÞ

(7b)

where Re denotes the real part of the complex function in the brackets, z = x + iy, z ¼ x  iy and i = (1)1/2. Equation (7a) is convenient as it can be used to check the boundary conditions along the crack. For the exact form of φ(z) and ψ(z), see the work of Zhang et al. [26]. These expressions are used to determine the stresses on the i-th dislocation from other ones in the array. From the Peach-Koehler equation, the force acting on the i-th dislocation due to the k-th dislocation is FKd ¼ b σ krθ

(8)

Thus, for the i-th dislocation, the scaled force equilibrium condition is given as τi

    n 1 K θ 1 X 2 θ ¼ þ pffiffiffiffi sin fk cos þ 2ρi 2 2 2 k¼1 d ρi

(9)

k6¼i

where τi ¼

τfric σk ri μ KA and K  ¼ pffiffiffiffiffiffiffiffi , f kd ¼ rθ , ρi ¼ , A ¼ 2π ð1  υÞ A A b A 2πb

For n dislocations, Eq. (9) becomes a system of n equations whose solution will yield the equilibrium positions, ri, for a given K* and τi*. Using the expressions given by Zhang et al. [26], the set of n by n nonlinear simultaneous equations was numerically solved. The result that the interaction between opposing dislocations in each array is zero [15] was used to solve the equations. DFZ size is one of the few parameters that are not easy to obtain. Figure 4 shows the first and last equilibrium positions as a function of the number of dislocations for K* = 10.0 and τi* = 0.1, parameters which will be subsequently used unless specified otherwise. Here the slip plane inclination is θ = 70.54 , the maximum effect of the total applied stress field in mode-I. The plastic zone size is taken to be distance from the crack tip to the last dislocation in the array. In a similar manner, the distance from the tip to the first dislocation is considered to be the DFZ. As can be seen, the plastic zone size increases in almost linear fashion as n increases while the DFZ decreases as expected with a larger array. The greatest extent of the plastic zone size occurs when θ = 70.54, the maximum of the total applied stress field. Other equilibrium properties of the array can also be seen in Fig. 4. The centroid of the super-dislocation is calculated as a simple mean and is fairly linear over a large range of dislocations. The length of each array reflects the dominance of the plastic zone size over the DFZ. Also, we see in Fig. 5 that the various zones decrease with decreasing K*, e.g., externally applied load. The elastic field at a crack tip is shielded

Crack-Dislocation Interactions Ahead of a Crack Tip

Fig. 4 Theoretical results using Pande, Masamura and Chou model showing equilibrium properties of the dislocation array

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6 Equilibrium Positions (×/1000b)

65

Length of array

4 3

Centroid of superdislocation

2

Initial equilibrium position

1 0

Fig. 5 Effect of applied and friction stress on KD

Final equilibrium position

5

0

20 40 60 Number of dislocations

100 (1) (2) (3)

K D /[A (2pb)1/2]

80

K+

t1*

10.0 15.0 10.0

0.10 0.10 0.05

80

(1)

(2) 60 (3) 40

20 Computed Super dislocation 0

0

20

40

60

80

Number of dislocations

(b > 0) or enhanced (b < 0) depending on the sign of the Burgers vector. This can be characterized by the stress intensity factor, KD, due to the dislocations and have been given by Lin and Thomson [14] and Zhang et al. [26]. For the i-th dislocation, its contribution is

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R. Goswami and C. S. Pande

(

#) rffiffiffi" π 1 1 z  z d d kiD ¼ Re iAb pffiffiffiffi þ pffiffiffiffi þ 2 zd zd ðzd Þ3=2

(10)

where zd is the position of the dislocation and the bar denotes complex conjugation. Equation (10) is summed over all of the dislocations: KD ¼

n X i¼1

K iD ðupperÞ þ

n X

K iD ðlowerÞ

(11)

i¼1

A discussion about shielding/anti-shielding is given by Lin and Thomson [14]. To obviate the problem of solving large set of nonlinear equations for the equilibrium positions, Lin and Thomson utilized the “super” dislocation concept. Here, the second to the n-th dislocations are replaced with one at a mean position with the strength (n-1) b. The relation for KD for such a case is given by Lin and Thomson [14]:  K D ¼ 6A

   π θ ð n  1Þ 1 sin ðθÞ cos þ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 2 2 ρmean ρ1

(12)

where n is number of dislocations. Equations (10) and (11) are plotted in Fig. 5. For purposes of comparison, ρmean was taken as the arithmetic average of the dislocations positions. Figure 5 shows that increasing K*, the externally applied tensile stress, reduces the effect of shielding while decreasing the friction stress can also lower the KD. These effects can influence the crack behavior and response to fatigue loads. The super dislocation computation can deviate significantly from the actual KD. Briefly, Pande et al. [15] show that the DFZ size is proportional to log n, where n is the number of dislocations in the array, and n is also shown to be proportional to the load. A minimum load is necessary for the crack to emit a dislocation. The complex potential method developed by Zhang et al. [26] for the crackdislocation field can be utilized to study more complex geometries of dislocation configurations. This technique can be used to examine dislocation shielding in symmetrically inclined arrays ahead of a crack. This formulation leads to equilibrium position of dislocations in each array from which important parameters such as plastic zone size, dislocation-free zone, and dislocation stress intensity factor can be determined. Both the applied K field and the intrinsic friction stress affect the dislocation shielding. The interaction between these two parameters is inhomogeneous and nonlinear and such scaling becomes an important issue. The concept of a super dislocation is useful to obtain a measure of KD, provided that a realistic measure of the mean position of dislocations, ρmean, can be determined. The use of an inverted pile-up may not be a good approximation to these arrays where crackdislocation interaction plays an important role. A certain minimum value of K* is necessary for dislocation emission and its movement away from the crack. It is not yet clear if this value of K* can be associated with one of the two fatigue thresholds. In summary, a formalism has been discussed that deals with the equilibrium positions of dislocations in an array ahead of the dislocation-free zone. The shielding effects are

65

Crack-Dislocation Interactions Ahead of a Crack Tip

2133

determined. These should prove useful in discussing the phenomenon of fatigue. Of particular interest is the ability to determine directly the displacements at the crack surfaces, and, therefore, it should be useful in any discussion on crack closure.

3.3

Pande, Masamura, and Chou: Mode-III

A detailed analysis similar to the case of mode-I has been done by Pande and coworkers [27]. A brief summary is presented here as the analysis is very similar to mode-I, and the conclusions are also not very different. They considered two symmetrical screw dislocation arrays similar to mode-I, where dislocations were of the edge character. The results were obtained as a function angle of inclination, θ. The simplest case will be when θ = 0, which reduces to Dai and Li analysis [10]. The stress intensity factor is similar to that of Dai and Li. It increases with increasing angle by a significant amount. Similar results are also obtained for other parameters such as length of pile up and DFZ. So if the arrays are inclined, they may significantly affect dislocation nucleation and crack propagation. This may be true if the loading mode is not exactly mode-III.

4

TEM Studies on Crack Tip Dislocations

Although considerable theoretical efforts have been made to study the crack tip deformation behavior by invoking the concept of dislocation continuum, a few experimental investigations have focused on verifying the concept of dislocation continuum and the shape of the plastic zone ahead of a crack tip. TEM has led to direct observation of the dislocation-crack interactions, in particular to the discovery of DFZ. This in turn has led to improved theoretical predictions of the crack tip behavior, which is critical in the study of fracture behavior of materials. However, theory often needs to assume drastic simplifying assumptions such as continuous distribution of dislocations. This has often necessitated the need for computation simulations. Experimental results on crack tip dislocation in Al [28] are presented here and compared with three methods of analysis. It was observed that the agreement between these three methods of analysis is only qualitative and further development both in analytical modeling and computation simulations are necessary before reliable quantitative predictions can be made of mechanical behavior due to crackdislocation interactions. To study the crack tip dislocations, a thin slice of an Al 1100 alloy was initially annealed at 550  C for 2 h under inert atmosphere and then quenched in water kept at room temperature. In-situ tensile stage in the TEM was then employed to monitor crack tip deformation. TEM specimens were mounted on 1 mm thick Cu tape containing a hole in the middle. One end of the Cu was attached to a fixed point and the other end was attached to a movable crosshead of the tensile stage. The specimens were elongated via a micrometer screw driven by a DC motor, Gatan Accutroller Model 902. A constant elongation rate was applied to the TEM

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specimen using a Gatan single-tilt straining holder (Model 672). To examine the underlying mechanisms of crack tip deformation behavior, TEM observations of dislocations near several crack tips were recorded. TEM samples were prepared by initially polishing mechanically arc-cut disk samples of 3 mm in diameter, and finally by thinning in an ion mill with a gun voltage of 4 kV and a sputtering angle of 10 . The thin section close to a central hole contains a number of sharp cracks. Tecnai G2 analytical transmission electron microscopes operating at 300 keV were used to characterize the crack tip dislocations. A bright-field TEM image close to the [110] zone exhibiting the dislocations emitted from a crack tip for Al is shown in Fig. 6. A dislocation-free zone (DFZ) of approximately 1.35 μm in length just ahead of the crack tip was observed. To determine the length of DFZ clearly, the specimen was tilted to minimize the contrast due to presence of the bend contour close to the crack. Approximately 90 dislocations were emitted from the crack tip and they were pushed away up to 3.85 μm from the crack tip. As the stacking fault energy of Al is high, the dislocation can easily cross slip, which results in considerable spread of 55 of dislocations around the crack tip. In the plastic zone, dislocations were well separated, and no intersection or entanglement could be detected. The plastic zone shape was observed to be approximately elliptical, which is somewhat in agreement with the simulation results using the discrete

Fig. 6 A bright-field TEM image close to the [110] zone showing dislocations ahead of a sharp crack and a dislocation free zone. The shape of the plastic zone is approximately elliptical

Crack-Dislocation Interactions Ahead of a Crack Tip

Fig. 7 The distribution of dislocations in the plastic zone as a function of distance from the crack tip showing that most dislocations are located between 1.35 and 3.85 μm

2135

30 25 20

counts

65

15 10 5 0 1.35

1.85

2.35

2.85

3.35

3.85

Distance from crack tip, µm

Fig. 8 (a) Two beam TEM image showing of the crack tip dislocations are extinct with most

g ¼ 11 1 (b) Two beam image with g ¼ 111 showing most of crack tip dislocations are visible

dislocation model [29]. The distribution of dislocation as a function of distance from crack tip is given in Fig. 7, showing that the mean position of dislocation is at 2.6 μm from the crack tip. The dislocations are approximately in the form of inverse pile-up, which is consistent with the previous observations of crack tip dislocation distribution in the plastic zone for stainless steel, Cu, Ni, Al, and Mo [6]. The Burgers’ vector of the dislocations were determined by the invisibility criterion of g.b = 0. Figure 8a is the two beam image of dislocations with g ¼ 111, showing most dislocations were extinct. However, most dislocations were visible with g ¼ 111 and g = 002 as shown in Fig. 8b, suggesting that the dislocations with Burgers’ vector b = a/2[011] would be the most dominant. Those dislocations were emitted in order to accommodate mode-III stress intensity around the crack tip. In mode-III loading, most dislocations emitted at the crack tip are of screw type, and the displacement provide by the dislocations is parallel to the crack front.

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To understand the dislocation distribution in the plastic zone, Majumdar and Burns’ model [7, 8] is used. This model uses continuum distribution concept because exact solution of the discrete dislocation-crack interaction problem is fairly complex and can be done only numerically. The assumption of dislocation continuum suggests that the discrete dislocations are smeared into a continuously distributed continuum so that the Muskhelishvili’s singular integral technique [25] can be incorporated. The advantage of this approximation is that problem of crack-dislocation interaction has an exact analytic solution in terms of elliptic integrals. According to Majumdar and Burns [7, 8], the distance of the peak in dislocation distribution Xm in front of the crack as measured from the crack tip is given by Xm ¼ K ð1  c=d Þ0:5 E ð1  c=dÞ0:5 c

(13)

The parameters in Eq. 13 have already been defined in section 3.1. Neglecting the higher order terms and using a continuum approximation, the elliptic integrals, K and E, can be written as K ¼ π=2 þ mπ=8 þ 9πm2 =128þ

(14)

E ¼ π=2  πm=8  3πm2 =128

(15)

The Xm can be calculated using the experimentally measured parameters, c and d. For Al, the experimentally observed values of c and d are 1.35 μm and 3.85 μm, respectively, and the Xm turned out to be 2.15 μm, which is close to the experimentally measured mean distance of 2.6 μm. This suggests that the continuum approximation can adequately describe the plastic zone. Now other parameters, such as number of emitted dislocations, can be estimated and compared with the theory. For this purpose, following Weertman et al. [21] and Thomson [11], further simplification by using the limit where c 1000
C

Forms borides and nitrides Reduces oxide, e.g., Al ~400
C

Forms Al4C3 with Al Inert

Al, Si, Ge Forms Al4C3 > 700
C with Al Forms SiC > 1200
C with Si Forms borides

Metalloids

2224 M. A. Monclús and J. M. Molina-Aldareguia

68

High Temperature Nanomechanical Testing

a

2225

b

2.0 1.8

1.8

1.5

1.5

1.3 Y[µm]

Z[nm] 0 –200 2.0

1.3

1.0 Y[µm] 1.0 0.8 0.8 0.5 0.5 0.3 Z[nm]

0.0 0.0

0.3 3000

0.3

0.5

0.8 1.0 1.3 1.5 2500 1.8 2.0 X[µm] Z[nm] c 1400 1200 1000 2.0

0.0 0.0

0.3

0.5 0.8 1.0 1.3 X[µm]

1.5 1.8 2.0

1.8 1.5 1.3 Y[µm] 1.0 0.8 0.5 0.3 0.0 0.0

0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0 X[µm]

Fig. 2 AFM 3D topography images (2 μm scan size) of (a) a new diamond Berkovich tip; (b) after nanoindentations in SiC coating at 300
C and (c) after nanoindentation in Al coating at 300
C (From [13]). These observations emphasize the importance of carrying out indenter tip shape calibrations on a regular basis, especially when indenting and scanning hard samples and metals that readily oxidize at T > 200
C, so that the tip gets easily contaminated

2.4

Ambient Atmosphere

The ambient atmosphere during high temperature testing is an important consideration because many metals oxidized readily in air. Oxidation rates increase with increasing temperature, and the evolution of an oxide on a sample will greatly affect the measured properties. When testing metals, it is therefore necessary to reduce or eliminate oxidation in order to achieve accurate mechanical property measurements. Gas lines are usually used to purge the air around the testing area and slow down the oxidation process during testing. Argon gas is typically used to purge the chamber and establish positive pressure achieving oxygen levels of 0.01%. In other configurations, compressed dry air and inert gases (Ar or N2 with a few percentage of H2)

2226

M. A. Monclús and J. M. Molina-Aldareguia

are used to purge the testing area to prevent heated gases reaching the transducer and reduce oxidation. Another possibility is to deposit a few nm of a passivating layer (e.g., Ti or Cr) in order to protect the surface from further oxidation. Tests should be performed at sufficient penetration depths in order to minimize the effect of the passivating layer on the measured properties. There is also the option to perform tests in an almost oxygen-free environment by housing the instrument within a high vacuum chamber which can achieve ultimate vacuum levels of 105 Pa [10]. In this case, thermal management is more difficult as heat convection is eliminated as a means of equilibrating temperatures, but at the same time, temperature stabilization is faster (particularly above 600
C) as radiation heating assists equalization of tip and sample temperatures. Appropriate design of cooling systems and thermal drift correction methods make high temperature vacuum indentation the preferred method for testing metals that readily oxidize, where high temperature measurements up to 950
C have been reported [10].

3

Nanomechanical Testing Studies Revealing TemperatureDependent Strength

Up to the beginning of the twenty-first century, the study of mechanical properties at small scales was almost exclusively limited to the application of instrumented nanoindentation. However, with the availability and application of focused ion beam (FIB) milling to micromachine micrometer-scale 3D geometries with high precision, a novel nanomechanical testing discipline has emerged, by which conventional mechanical tests, like tensile, compression, or bending tests, are reproduced at the micro- and/or nanoscale. Among them, micropillar compression has become widely popular to assess mechanical properties at small scales, particularly at high temperatures. Therefore, in the following, the progress on elevated temperature nanomechanical testing has been divided into two main topics: instrumented nanoindentation and micropillar compression. Nanoindentation involves pressing a hard tip with a well-defined geometry into the sample surface, while force and depth are measured continuously during loading and unloading. Commonly used indenter tips have self-similar geometries with depth (i.e., the ratio between contact area and contact depth is constant), like conical, pyramidal Berkovich and cube-corner indenters. For self-similar indenters, the elastoplastic deformation induced by the indenter is, in principle and if not affected by indentation size effects, independent of indentation depth and representative of an equivalent plastic strain eeq, which depends on the included angle of the indenter (eeq  0.08 for Berkovich and  0.2 for cube-corner geometries). Despite the complex strain and stress fields underneath the indenter, hardness and modulus are commonly obtained from such nanoindentation tests, using contact mechanics and the well-known Oliver and Pharr method [14]. The reduced modulus, Er, is calculated from the stiffness at the onset of unloading (S) and the projected area of contact between the indenter and the material (Ac), as:

68

High Temperature Nanomechanical Testing

2227

Fig. 3 (a) Cross section of nanoindentation profile at peak load and (b) loadingunloading sequence for an indentation in Cu at room temperature (blue curve) and at 200
C, indicating the parameters used to estimate hardness and modulus. Arrows point at the increased creep taking place at high temperature

pffiffiffi π S pffiffiffiffiffi Er ¼ 2 Ac

(1)

The unloading slope (S) is obtained from the elastic unloading, and the hardness, H, is calculated as: H¼

Pmax Ac

(2)

where Pmax is the maximum applied load (see Fig. 3a). By calibration into a reference material of known elastic properties (e.g., fused silica), the indenter area function can be determined relating the contact depth hc to the projected contact area Ac. Guidelines for accurate room temperature measurements of hardness and modulus are given in ISO 14577(1–4) [15]; however, high temperature nanomechanical testing requires adapting the measurement protocols described in the ISO approach so that the factors described in the previous section are taken into account. Moreover, time-dependent inelastic deformation rates can increase significantly with testing temperature T, especially at high homologous temperatures Th (= T/Tm, where T is the test temperature and Tm is the melting temperature, both in Kelvin) > 0.6, making the measurement of elastic stiffness on unloading more difficult.

2228

M. A. Monclús and J. M. Molina-Aldareguia

Figure 3b shows the load-displacement curve for indentations performed on Cu at 200
C, showing an increased creep deformation during the hold period, with respect to the room temperature result. If creep rates are not minimized during unloading, the contact area and elastic modulus determination can be seriously compromised; consequently, the loading histories need to be modified to include sufficiently longer hold times at peak load, to minimize the effect of creep during unloading [16]. Feng and Ngan [17] have suggested a correction term due to creep in the apparent contact compliance (=1/S) which has been successfully applied to correct high temperature properties data of Ni-based superalloys up to 400
C [18]. Depending on the application, some of the critical issues described will be more relevant than others. For instance, critical requirements for testing advanced alloys are oxidation minimization and time dependency, whereas for hard coatings, indenter durability and tip wear are more of a concern. In the case of ceramic coatings, creep is usually small, and using hold times between 10 and 30 s is sufficient to ensure testing accuracy at high temperatures; in addition, oxidation is not that critical, and typically tests are performed in a purged Ar atmosphere with a diamond tip or with a cBN tip. For other materials (e.g., bcc metals, solders, and polymers), considerable time dependency occurs at lower temperatures, and care in the experimental design should be taken to ensure accurate elastic modulus measurements. In order to validate data, it is recommended to calibrate the system using a test reference sample (e.g., fused silica, tungsten) at the testing temperature and when possible compare the data with measurements performed with other techniques. The temperature dependence of the elastic modulus of the indenter used should also be taken into account as its value is used to convert from indentation reduced modulus to Young’s modulus [14] (e.g., modulus of cBN decreases by  4% between RT and 950
C [7]). Micropillar compression involves pressing a flat punch indenter onto a micropillar with a well-defined geometry. Because of the uniaxial loading imposed on the micropillar, the stress and strain are easily computed from the load-displacement data after applying the necessary corrections for substrate deflection, pillar taper, and misalignment between the punch and top surface of the pillar [19]. Material strength parameters such as yield point and flow stress can thus be computed from the resulting stress-strain curves, providing more information on the constitutive behavior of the material. Even though the same problems as those described for the case of nanoindentation might arise regarding sample-tip interactions, the local stresses are greatly reduced between the indenter flat punch and the head of the pillar, which generally alleviates the phenomena. It is therefore possible to use diamond flat punches to compress ferrous metals and/or transition metals up to relatively high temperatures, especially in vacuum environments.

3.1

High Temperature Nanoindentation Studies

It is better to discuss separately the elevated temperature nanoindentation studies performed on metals and ceramics, not only due to the very different nature of these

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High Temperature Nanomechanical Testing

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materials but also due to the very different scopes of the studies that have been performed in each case.

3.1.1 Metals and Semimetals Several elevated temperature nanoindentation studies have focused on studying the effect of temperature on the hardness of metals. Due to the thermally activated nature of plastic deformation, temperature is expected to increase the dislocation nucleation rate (due to thermal fluctuations), the dislocation mobility, and the diffusion-mediated flow rate [20]. There are two main advantages associated with the use of nanoindentation to study thermally activated plastic flow. Firstly, only a small volume of material is needed for the test, so that either nanoindentation is the only available technique capable of measuring the mechanical properties of some materials (as it is the case for thin-films and some nanostructured metals), or it can be used in a high-throughput manner in a combinatorial material’s discovery approach. And secondly, but not less important, depending on the scale of the nanoindentation test with respect to the characteristic size of the microstructure, it can be used to decouple different deformation processes: from the onset of dislocation nucleation for very small indentations to collective dislocation dynamics when the indentation is large but smaller than the grain size, including the effect of grain boundaries when the indentation size encompasses several grains. Several nanoindentation works on the effect of temperature in each case can be found in literature. For instance, the effect of temperature (up to  0.2 Tm) on the very early stages of plasticity during low-load nanoindentation of fcc and bcc oxide-free metal surfaces has revealed interesting phenomena on the dynamics of dislocation nucleation. For example, indentations on pure (100) platinum (bcc) revealed that temperature promotes a monotonic decrease in the critical load for dislocation nucleation and that the serrated nature of plastic flow was more severe as temperature was raised from 20 to 200
C, indicating that these nucleation events were stress biased and thermally activated [21]. Nanoindentation of (001) tantalum (fcc) also exposed a change in the shape of the loading curves from one large pop-in marking the onset of plasticity at 25
C to multiple pop-ins at 200
C [22], due to the thermal activation of screw dislocations at temperatures above 0.15 Tm. Other nanoindentation studies have also shown that the variability of the critical load for dislocation nucleation is due to thermal fluctuations, while in the case of metallic glasses, for which the onset of plastic deformation occurs by the nucleation and propagation of a shear band, the variability in the critical load is dominated by structural fluctuations [23]. Higher up in the length scale where the indentation is governed by the collective behavior of dislocations, several studies have also focused on the temperaturedependent single crystal behavior, either by indenting single crystals or individual grains far from grain boundaries in polycrystalline metals. For instance, some high temperature nanoindentation studies have focused on studying the competition between plasticity and pressure-induced phase transformations in single crystal silicon. At room temperature, it is well known that upon loading Si suffers a phase transformation from the diamond structure to a metallic tetragonal Si phase, which back-transforms to amorphous Si upon unloading, giving rise to a well-known

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M. A. Monclús and J. M. Molina-Aldareguia

pop-out event [24]. However, several regimes have been observed at elevated temperature. While the room temperature behavior remained fairly constant up to 200
C, at higher temperatures the pop-out disappeared, and the elastic modulus decreased substantially, due to the formation of amorphous silicon during loading [25]. However, the hardness of Si dropped substantially for temperatures > 350
C, which was attributed to the transformation to another high-pressure metallic Si phase [26], while for even higher temperatures (between 500 and 650
C), thermally activated dislocation flow controlled the indentation process, without evidence of any pressure-induced phase transformations [10]. Another interesting application of nanoindentation at the single crystal level is to use the nanoindentation response of grains with varying crystallographic orientation, in combination with single crystal plasticity (SCP) simulations, to determine the critical resolved shear stress (CRSS) of the different slip and twinning systems in magnesium [27] . In another study, the temperature evolution of the CRSS of the different deformation modes was determined using a sphero-conical indenter in the temperature range of 25–300
C, demonstrating a transition in plasticity from basal slip and extension twinning-dominated flow at room temperature to basal and prismatic slip dominance at 300
C [28]. The case of single crystal indentation cannot be completely discussed without making reference to the indentation size effect (ISE). ISE manifests itself in many single crystals as an increase in hardness with decreasing depth of penetration and becomes important at depths less than approximately 1 μm, as a result of the larger contribution of strain gradients caused by the density of geometrically necessary dislocations [29]. Nanoindentation in Cu at 200
C has shown a significantly reduced ISE contribution, suggesting that thermal activation plays a major role in determining the length scale for plasticity [30]. Several elevated temperature studies can also be found in the regime where the nanoindentation imprint encompasses several grains, and therefore, plasticity is controlled by the polycrystalline behavior. Hot indentation hardness studies of polycrystalline aluminum (up to 100
C) [13], gold (up to 200
C) [25], and copper (up to 400
C) [31] have shown a decrease of hardness with temperature, consistent with their polycrystalline bulk behavior. The high temperature indentation of polycrystalline metal alloys (Ni-W [32]) has shown that the extent of the hardness drop depends on grain size, being less pronounced for finer grain sizes, although this observation cannot be extended to other material systems, where a higher hardness drop with temperature