Mechanics of Heterogeneous Materials 3031287436, 9783031287435

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Advanced Structured Materials

Holm Altenbach · Giovanni Bruno · Victor A. Eremeyev · Mikhail Yu. Gutkin · Wolfgang H. Müller   Editors

Mechanics of Heterogeneous Materials

Advanced Structured Materials Volume 195

Series Editors Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, Esslingen, Germany Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, Portugal Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

Common engineering materials are reaching their limits in many applications, and new developments are required to meet the increasing demands on engineering materials. The performance of materials can be improved by combining different materials to achieve better properties than with a single constituent, or by shaping the material or constituents into a specific structure. The interaction between material and structure can occur at different length scales, such as the micro, meso, or macro scale, and offers potential applications in very different fields. This book series addresses the fundamental relationships between materials and their structure on overall properties (e.g., mechanical, thermal, chemical, electrical, or magnetic properties, etc.). Experimental data and procedures are presented, as well as methods for modeling structures and materials using numerical and analytical approaches. In addition, the series shows how these materials engineering and design processes are implemented and how new technologies can be used to optimize materials and processes. Advanced Structured Materials is indexed in Google Scholar and Scopus.

Holm Altenbach · Giovanni Bruno · Victor A. Eremeyev · Mikhail Yu. Gutkin · Wolfgang H. Müller Editors

Mechanics of Heterogeneous Materials

Editors Holm Altenbach Fakultät für Maschinenbau Otto-von-Guericke-Universität Magdeburg, Sachsen-Anhalt, Germany Victor A. Eremeyev DICAAR University of Cagliari Cagliari, Italy Department of Civil and Environmental Engineering and Architecture University of Cagliari Cagliari, Italy Wolfgang H. Müller Institut für Mechanik TU Berlin Berlin, Germany

Giovanni Bruno Bundesanstalt fuer Materialforschung und -pruefung Berlin, Germany Mikhail Yu. Gutkin Institute for Problems in Mechanical Engineering Russian Academy of Sciences St. Petersburg, Russia ITMO University St. Petersburg, Russia Department of Mechanics and Control Processes Peter the Great St. Petersburg PolytechnicUniversity St. Petersburg, Russia

ISSN 1869-8433 ISSN 1869-8441 (electronic) Advanced Structured Materials ISBN 978-3-031-28743-5 ISBN 978-3-031-28744-2 (eBook) https://doi.org/10.1007/978-3-031-28744-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface I

Igor Sevostianov in his office at the New Mexico State University (Las Cruces, NM), 2020 (Photo from the archive of Elena Sevostianova)

Igor Borisovich Sevostianov was born on December 13, 1965, in Krasnoyarsk, Russia. He graduated from St. Petersburg State University (Russia) in 1988 and defended his Ph.D. in Solid Mechanics at St. Petersburg State University (Russia) in 1993. After that, he was appointed as a Visiting Scientist at Max-Planck Institute, Dresden, Germany (1993–1997); as Senior Research Associate at University of Natal, Durban, South Africa (1997–1998); as Senior Research Associate at Department of Mechanical Engineering, Tufts University, Medford, MA, USA (1998– 2001). Finally, he joined Department of Mechanical and Aerospace Engineering, New Mexico State University (NMSU), Las Cruces, USA, in 2001 as an Assistant Professor. Finally, he became Dwight L. and Aubrey Chapman Distinguished Professor of mechanical engineering at NMSU in 2014. Professor Sevostianov was appointed as a visiting professor at BAM, Berlin, Germany (2016–2019); Gdansk Polytechnic University, Poland (2018); Tomsk Polytechnic University, Russia (2015); University of Modena and Reggio Emilia, Italy

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(2014, 2015, 2018); TU Vienna, Austria, (2013); University of Lorraine, Nancy, France (2012). Professor Sevostianov was a member of the editorial boards of the following journals: International Journal of Engineering Science (Elsevier); Mathematical Methods in the Applied Sciences (Wiley); Studia Geotechnica et Mechanica (Sciendo) Reviews on Advanced Materials Science (RAS); International Journal of Theoretical and Applied Multiscale Mechanics (Inderscience); Nanomechanics Science and Technology (Begell); International Journal of Materials (NAUN); International Journal of Mechanics (NAUN); Journal of the Computational Engineering (Hindawi); World Journal of Methodology (Baishideng); Acta Mechanica et Automatica (Poland); Journal of Applied and Computational Mechanics (Iran); Vestnik of DSTU (Russia); Scientific Letters of Rzeszow University of Technology (Poland), PNRPU Mechanics Bulletin (Russia). Igor’s scientific interests are related to the following topics: • Micromechanics: quantitative characterization of microstructures, microstructure property relationships; cross-property connections; application to real-life problems. • Biomechanics: mechanical properties of hard tissue; viscoelastic properties of cells. • Advanced manufacturing: evaluation of effective properties and residual stresses in 3-D printed metals and ceramics. • Contact mechanics: contacting rough surfaces and granular materials. • Multi-physics phenomena in solids: mechanics of coupled fields, piezoelectric materials, and design of materials with isotropic zero thermal expansion. • Inverse problems: recovery of information on the microstructure of materials from the measurements of their elastic and electric properties. • Mechanics of viscoelastic solids: development of fraction-exponential operator technique. • Anisotropic solids: development of mathematical methods suitable to solve boundary value problems for anisotropic materials. Igor Sevostianov conducted research in micromechanics of materials, focusing on relations between material microstructure and the overall mechanical and physical properties. His most important scientific results include • Establishing the cross-property connection between different physical properties of heterogeneous materials. • Formulation and development of the Maxwell homogenization technique for elastic properties of materials. • Establishing replacement relations for elastic and conductive properties of materials with the same microstructure but different properties of inhomogeneities. • Formulation of the principles of quantitative characterization of microstructure.

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Igor Sevostianov in the New Mexico State University (Las Cruces, NM), 2021 (Photo from the archive of Elena Sevostianova)

• Development of thermodynamic methods for the characterization of microcracked materials. • Development of the methodology for calculation of the overall properties of heterogeneous materials with irregular microstructure. • Establishing a methodology for non-destructive evaluation of material performance from electric conductivity measurements. • Establishing a methodology for the evaluation of thermal properties of small particles. The applications of his results are in biomechanics (control of material properties of bone and dentin, design of materials for implants), geomechanics (evaluation of thermal and elastic properties of nonconsolidated rock), development of thermal barrier coatings (where the microstructure must be designed for the best combination of the conductive and elastic properties), and design of self-diagnosing composites (having polymer matrix filled with graphene particles). He had published more than 270 papers in peer-reviewed journals and a few monographs including Kachanov, M. and Sevostianov, I. (eds) Effective Properties of Heterogeneous Materials, Springer, 2013, and Kachanov, M. and Sevostianov, I. Micromechanics of Materials, with Applications, Springer, 2018. This volume of the Advanced Structured Materials Series is dedicated to the memory of our friend and colleague, Prof. Igor Sevostianov. It contains a selection of scientific papers prepared by his friends and colleagues from different countries.

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It is devoted to actual research in the mechanics of inhomogeneous materials and micromechanics. Magdeburg, Germany Berlin, Germany Cagliari, Italy Saint Petersburg, Russia Berlin, Germany December 2022

Holm Altenbach Giovanni Bruno Victor A. Eremeyev Mikhail Yu. Gutkin Wolfgang H. Müller

Preface II

I met Igor in Dresden in July 1995. I went there to work for a month in the Research Group on Mechanics of Heterogeneous Solids of the Max-Planck Society led by Prof. Wolfgang Pompe under models describing the diffusion of vacancies and the growth of pores during high-temperature oxidation of metals. My trip was organized by Prof. Alexey Romanov who was the supervisor of my M.Sc. and Ph.D. theses and had long-standing scientific ties with Prof. Pompe. It was he who introduced me to Igor on the very first day of my stay in Dresden. I remember very well how it was. I had just begun to look around my workplace when it was time for lunch, the door to the office opened, and Alexey appeared accompanied by a tall smiling guy with a beard, mustache, and curly hair. He was very open and cheerful, and his eyes beamed with joy. It seems to me that it was very typical for him to meet each new person with sincere friendliness, openness, and interest. We had lunch together, then met at work every day and spent a couple of weekends together. From Igor, I learned that he graduated from the Department of Elasticity Theory of the Faculty of Mathematics and Mechanics of the Saint Petersburg State University, and then postgraduate studies there, and that his supervisor in postgraduate studies was Prof. August Vakulenko, and the topic of his Ph.D. thesis was the Eshelby problem for a physically nonlinear ellipsoidal inclusion. Quite quickly after its defense in 1993, he came to Dresden to Prof. Pompe and was going to work there as long as there was such an opportunity. At that time, he worked under modeling the stress-strain state of a drying body and used the Marc finite element package, which was quite popular thereat. Although this topic was completely unfamiliar to me, we talked quite a lot and we quickly became friends. We were about the same age, Igor was 29, and I was 33, we had a largely similar education—Igor had a big bias in mathematics, and I was in physics, we both enjoyed doing science, and most importantly, we read in childhood the same books, watched the same movies, knew the same songs, so we perfectly understood each other. This understanding remained until our last meeting in St. Petersburg in June 2021, just over a month before his tragic death on August 3, 2021.

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Fig. 1 Igor Sevostianov in Costa Rica, 2021. All photos here are from the archive of Elena Sevostianova

The second time I came to Dresden was at the end of September of the same 1995, now for three months. At this time, I became even closer to Igor. His wife Lena came to Igor from St. Petersburg, and on weekends we spent a lot of time together. In my work at that time, it became necessary to use computer modeling in the Marc package, and Igor helped me a lot in building a calculation model. In general, Igor always surprised me with his openness and willingness to help. When I had problems with housing, he and Lena first helped me to find a room in a student hostel for a couple of weeks, and then offered to live with them in the part of the house that they rented from a German family. So the last month of 1995, before our return to Russia, we lived with Igor and Lena under the same roof. It was a great time, and I always remember it with pleasure.

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Fig. 2 Mark Kachanov, Kosta Markov, Igor Sevostianov, Viktor Berdichevsky, and Valeriy Levin (from left to right) at the Eighth International Symposium on Continuum Models and Discrete Systems, June 1995, Varna, Bulgaria

Fig. 3 Igor Sevostianov in his office at the New Mexico State University (Las Cruces, NM), 2018

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Fig. 4 Igor’s family—Igor, Lena, Vlad, and Ksenia (from left to right)—at Vlad’s Harvard graduation, 2019

After my departure from Dresden, I continued to correspond with Igor and occasionally met him during his brief periods in Russia. In 1998, he left for a year to South Africa, where he worked in Durban, at the School of Mechanical Engineering of the University of Natal, under the guidance of Prof. Victor Verijenko. There he continued his work on the Eshelby problem as applied to the process of manufacturing composite materials using the resin film infusion method. Judging by his letters of that time, he spent an unforgettable year in Durban and returned from there full of the most vivid impressions. Igor infected me with his interest in South Africa and recommended me to Prof. Verijenko to participate in a similar project. With his help, I made an application for a grant and received it, so that during the year, from April 2000 to March 2001, my wife Kristina and I worked at the same university with Prof. Verijenko. It was a very bright and eventful year. We have never regretted that with the help of Igor we visited Durban, and we will always be grateful to him for this. At that time, Igor, Lena, and their son Vlad were already living in the USA, in Boston, where Igor received a postdoc at Tufts University and worked under the guidance of Prof. Mark Kachanov. As far as I understand, the most fruitful period in Igor’s scientific career began there. The work done at that time on cross-property connections quickly brought him to the forefront of world-class mechanics. He continued these studies after receiving a permanent position at New Mexico State University in Las Cruces, where he and Lena had a daughter, Ksenia, and where Igor managed to

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create a group of students, graduate students, and postdocs who actively developed many promising areas of modern mechanics of porous and composite materials. Thus, one of his most distinguishing areas of research was an investigation of the effect of non-ellipsoidal inhomogeneities including irregular cracks and toroidal, polyhedral, superspherical, polyspherical, and helicoidal inclusions or voids on effective properties of materials. At the same time, Igor tried to find and investigate such problems that would be interesting and important not only for mechanics but also for materials scientists. For example, one of such problems was identifying crossproperty connection between the yield limit and electrical conductivity in zircaloy, subjected to neutron irradiation, with taking into account the dislocation density. In recent years, he was interested in the development of the theoretical description of viscoelastic properties of human tooth dentin, and we did a couple of joint works on this subject. In recent years, Igor’s scientific and organizational activities have become more and more intense. He spoke a lot at conferences, some of which he himself organized, lived and worked for months in Germany and Italy, and collaborated with many scientific groups around the world. He performed many works together with colleagues from Russia. We continued to meet at summer conferences in St. Petersburg, walked a lot around the city, and celebrated our meetings, and I saw that he remained as open to new ideas and new people as he was at the beginning of our acquaintance. It seemed that he was not at all tired of his oversaturated life, that he was full of energy and new plans and was ready to continue such a life for many, many more years. I am sure that it would have been so, and he would have managed to do a lot more, if not for his sudden and tragic death as a result of an accident. St. Petersburg, Russia

Mikhail Yu. Gutkin

In Memory of Igor Sevostianov

The first time I met Igor Sevostianov was during one of the school-conferences “Advances in Problems of Mechanics” (APM). At that time, these annual events were organized in Repino, a small municipal settlement of Saint Petersburg, Russia. Being particularly isolated from the center of the city, participants of the conference spend a lot of time together in quite an informal atmosphere. Being almost the same age and having almost the same interests in mechanics, as well as common friends, we spent some time together with Igor discussing interesting scientific and nonscientific topics. Moreover, Igor’s former supervisor Prof. August Vakulenko was a close friend of my former supervisor Prof. Leonid Zubov. So, we had a lot in common with Igor. After the first meeting in Repino, there were a lot of others, in Magdeburg, Bologna, Skolkovo, Tel Aviv, Rzeszów, Gda´nsk, and Rostov-on-Don. Igor was a very enthusiastic person who organized a lot of events in different places. He delivered his course on micromechanics at Gda´nsk University of Technology (GUT) supported through the GUT visiting professorship program. As a result, the author learned a lot about micromechanics, effective properties, and cross-property connections, from his course. Igor had also relatives and friends in Rostov-on-Don, my home Russian city, where he also delivered some lectures at Don State Technical University. Igor was a fantastic and influential scientist and person. He had a lot of plans for his next research, meetings, and common projects. He won again the next visiting professorship in GUT, and we planned to meet again in Italy, in Cagliari. Unfortunately, life is unpredictable; this sudden and tragic accident destroyed everything. We lost a brilliant colleague and friend.

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In Memory of Igor Sevostianov

Fig. 1 Igor Sevostianov in Gda´nsk, 2018. All photos here are by the author

In Memory of Igor Sevostianov

Fig. 2 Igor Sevostianov with Holm Altenbach, Repino, 2009

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In Memory of Igor Sevostianov

Fig. 3 With colleagues in Rzeszów, 2014. From the left are Yuri Petrov, Gennady Mishuris, Igor Sevostianov, Mark Kachanov, and the author

Fig. 4 Igor with Ph.D. students, Gda´nsk, 2018

In Memory of Igor Sevostianov

Fig. 5 In Tel Aviv, 2018

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Fig. 6 In SkolTech, Skolkovo, 2018

In Memory of Igor Sevostianov

In Memory of Igor Sevostianov

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Fig. 7 With colleagues in Rostov-on-Don, 2019. From the left are Sergei Aizikovich, Igor Sevostianov, the author, and Adair R. Aguiar

Cagliari, Italy Gda´nsk, Poland

Victor A. Eremeyev

Contents

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Micromechanical Modeling of Non-linear Stress–Strain Behavior of Polycrystalline Microcracked Ceramics . . . . . . . . . . . . . . Giovanni Bruno, Mark Kachanov, and Igor Sevostianov 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Materials and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 β-Eucryptite Subjected to Cyclic Tensile Loading of Increasing Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Aluminum Titanate Subjected to Compressive Interrupted Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Micromechanical Explanation and Modeling . . . . . . . . . . . . . . . . . 1.3.1 Uniaxial Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Uniaxial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shamrovskii’s Version of the Refined Dynamical Plate Theory . . . . . Igor V. Andrianov and Isaac Elishakoff 2.1 Introduction. Uflyand-Mindlin Theory: History of Question . . . . 2.2 Newton Polygon and Its Generalizations . . . . . . . . . . . . . . . . . . . . . 2.3 The Idea of Shamrovskii’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . 2.4 Shamrovskii’s Refined Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Brief Biography of A. D. Shamrovskii . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Resolved Multifractal Analysis of Electron Beam Induced Piezoelectric Polymer Fiber Dynamics: Towards Multiscale Thread-Based Microfluidics or Acoustofludics . . . . . . . . . E. L. Buryanskaya, O. V. Gradov, M. A. Gradova, V. V. Kochervinskii, and I. A. Maklakova

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Contents

3.1

From Charging of the SEM Samples to the Formation of SAW Potential Contrasts and Micromechanical Local Movements in Piezoelectric Fiber Samples, Induced by the Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Possibility of Designing Electron Beam-Controlled Acoustofluidics, Including Electron Beam-Controlled Thread-Based Micro-Acoustofluidics . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Methods for Studying the Dynamics of PVDF Fibers . . . . . . . . . . 3.4 Investigation of the Characteristic Dimensions of Piezoelectric Polymer Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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Simple Coarse-Grained Model of the Zebrafish Embryonic Aorta Suggesting the Mechanism Driving Shape Changes During Stem Cell Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dmitrii Chalin, Andrei Nikolaev, Evgeniy Sadyrin, Karima Kissa, and Ivan Golushko 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The HSC Production Process in Zebrafish . . . . . . . . . . . . . . . . . . . . 4.3 Model of the Dorsal Aorta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Averaging-Based Approach to Toughness Homogenisation for Radial Hydraulic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 G. Da Fies, M. Dutko, and D. Peck 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.1 The Radial Model with Inhomogeneous Toughness . . . . 72 5.2.2 Form of the Material Toughness . . . . . . . . . . . . . . . . . . . . 75 5.2.3 Parameterising the Fracture Regime . . . . . . . . . . . . . . . . . 77 5.2.4 Numerical Algorithm and Behaviour of the Key Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Comparison of Homogenisation Strategies . . . . . . . . . . . . . . . . . . . 84 5.3.1 The Maximum Toughness and Temporal-Averaging Approaches . . . . . . . . . . . . . . . . 87 5.3.2 Results for Balanced Toughness Distributions . . . . . . . . . 89 5.3.3 Results for Unbalanced Layering . . . . . . . . . . . . . . . . . . . . 95 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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Towards Multi-Angle Multi-Channel Optical Porometry and Scanning Electron Microscopic Porometry of LDPE Composites Including Geotechnical Biodegradable Ones . . . . . . . . . . Elena Grigorieva, Oleg Gradov, Margaret Gradova, and Irina Maklakova 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Materials and Sample Preparation . . . . . . . . . . . . . . . . . . . 6.2.2 Dynamical Multi-Channel Multi-Angle Optical Porometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Scanning Electron Microscopic Porometry . . . . . . . . . . . 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Dynamic Multi-Channel Multi-Angle Optical Porometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Scanning Electron Microscopic Porometry . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Deformations of Anisotropic Elastic Bodies with Distributed Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evgeniya V. Goloveshkina and Leonid M. Zubov 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Input Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Cylindrical Tube with Distributed Dislocations . . . . . . . . . . . . . . . 7.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Hydrostatic Compression . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Misfit Stress Relaxation at Boundaries of Finite-Length Tubular Inclusions Through the Generation of Prismatic Dislocation Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Yu. Gutkin, E. A. Mordasova, A. L. Kolesnikova, and A. E. Romanov 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A Numerical Determination of the Interactions Between Dislocations and Multiple Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . 159 Zhizhen Jiang, Kuanyu Liu, Kai Zhu, Pu Li, and Xiaoqing Jin 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

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9.3

Equivalent Inclusion Method for Inhomogeneity-Dislocation System . . . . . . . . . . . . . . . . . . . . . . 9.4 Interaction Energy and Force on Dislocation . . . . . . . . . . . . . . . . . 9.5 Numerical Implementation of the EIM . . . . . . . . . . . . . . . . . . . . . . 9.6 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 The Interaction of a Screw Dislocation with a SiC/Ti–6Al–4V Inhomogeneity . . . . . . . . . . . . . . . 9.6.2 Layered Inhomogeneities and Horizontally Distributed Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Layered Inhomogeneities and Vertically Distributed Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.4 Multiple Circular Inhomogeneities Around a Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.5 Multiple Elliptical Inhomogeneities Around a Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.6 Multiple Rectangular Inhomogeneities Around a Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: The Solution of the Screw and Edge Dislocations . . . . . . Appendix 2: The Elementary Solution for a Rectangular Inclusion . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Numerical Simulations of Interface Propagation in Elastic Solids with Stress Concentrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polina K. Kabanova, Aleksandr Morozov, Alexander B. Freidin, and Alexander Chudnovsky 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Basic Relations for a Two-Phase Elastic Solid . . . . . . . . . . . . . . . . 10.3 Numerical Procedure and Its Verification . . . . . . . . . . . . . . . . . . . . 10.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Evolution of the Interface Around a Circular Hole Under Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Phase Transformations Induced by a Stress Concentration at An Elliptical Hole . . . . . . . . . . . . . . . . . . 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Effect of Gravity on the Dispersion and Wave Localisation in Gyroscopic Elastic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Kandiah, I. S. Jones, N. V. Movchan, and A. B. Movchan 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Passive Gyroscopic System under Gravity . . . . . . . . . . . . . . . . . . . 11.2.1 Asymptotic Connection Between the Gyroscopic Pendulum and the Gyroscopic Flexural Beam . . . . . . . . . 11.2.2 Dimensionless System of Equations: General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164 166 168 171 171 174 178 178 184 187 187 190 191 197 201

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11.2.3 Trajectories of the Gyropendulum . . . . . . . . . . . . . . . . . . . 11.2.4 Gyropendulum Trajectories of Prescribed Shapes and Rotational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.5 Optimal Design of a Gyropendulum for Regular Polygonal Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Active Gyroscopic Systems Subjected to Gravity . . . . . . . . . . . . . 11.4 The Chain of Gyropendulums: Floquet-Bloch Waves . . . . . . . . . . 11.4.1 Floquet-Bloch Waves in a Chiral Chain Under Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Illustrative Examples of Floquet-Bloch Waves . . . . . . . . 11.5 The Effect of Gravity on the Dynamic Green’s Kernel . . . . . . . . . 11.5.1 Construction of the Dynamic Green’s Kernel . . . . . . . . . 11.5.2 The Dynamic Response for Special Regimes Such as Stop Bands and Pass Bands . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Illustration of the Effect of Gravity on the Elliptical Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Green’s Matrix and Localised Defect Modes for the Case of No Pre-tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Green’s Matrix for a Chiral Lattice with No Pre-tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Illustration of Exponentially Localised Defect Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Time-harmonic Trajectories and Their Perturbations Due to Gyricity . . . . . . . . . . . . . . . . . . . . . . . Appendix 2: Dispersion of Floquet-Bloch Waves and Two-Dimensional Motion of the Nodal Points . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Controlling the Structure and Properties of Metaland Polymer-Based Composites Fabricated by Combined 3D Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. G. Knyazeva, A. V. Panin, M. A. Anisimova, D. G. Buslovich, M. S. Kazachenok, and S. V. Panin 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Ti–6Al–4V-Based Metal-Matrix Composites Built by Wire-Feed Electron Beam Additive Manufacturing . . . . . . . . . 12.2.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Conclusions to Sect. 12.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Structure and Properties of 3D Printed PEEK-Based Composites for Antifriction Applications . . . . . . . . . . . . . . . . . . . . 12.3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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227 234 236 241 243 243 249 252 252 253 256 258 258 260 265 265 268 272

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277 279 279 280 282 287 288 288 289

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12.3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Comparison of the Characteristics of the “PEEK + 0.3%HA + 10%PTFE” Composites Fabricated by the HC and HDM Methods . . . . . . . . . . . . . . . . . . . . . . 12.3.5 Conclusions to Sect. 12.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Evolution of the Functional Properties of Composites During the Formation of Intermediate Phases Between the Matrix and Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Research on Fe-Based Composites with Titanium Carbide Inclusions upon Their Laser-Beam Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 The Behavior of the Ti–Al–C Composite Upon Its Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4 Mechanical Stresses Around the Diffusion Zone During the Synthesis of a Composite from Titanium and Carbon Powders . . . . . . . . . . . . . . . . . 12.4.5 Conclusions to Sect. 12.4 . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Analysis of the Periodicity Cell Problems for the Fiber-Reinforced Plate and Applications . . . . . . . . . . . . . . . . . . A. G. Kolpakov and S. I. Rakin 13.1 Homogenization Method as Applied to Fiber-Reinforced Plates. Linear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Elastic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Plates with Unidirectional Systems of Fibers . . . . . . . . . . 13.2.2 Plates with Unidirectional Systems of Channels . . . . . . . 13.2.3 Plates with the Cross Reinforcements . . . . . . . . . . . . . . . . 13.3 Thermoelastic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Plates with Unidirectional Systems of Fibers . . . . . . . . . . 13.3.2 Plates with the Cross Reinforcements . . . . . . . . . . . . . . . . 13.4 Non-Linear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Fiber-Reinforced Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Plates with Unidirectional Systems of Channels . . . . . . . 13.5 Applications of the Results of the Numerical Analysis . . . . . . . . . 13.5.1 The Boundary Layers on the Top/bottom Surfaces of Inhomogeneous Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Wrinkling the Top/bottom Surfaces of Inhomogeneous Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.3 The Representative Model of Multilayer Plate . . . . . . . . . 13.5.4 The Strength of the Composite Plate . . . . . . . . . . . . . . . . . 13.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14 Boundary Layers at the Interface of Layers of Unidirectional Fibers in Fibrous Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander G. Kolpakov, Igor V. Andrianov, and Sergey I. Rakin 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Contact and Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Calculation of Stress–strain State in a Fragment of a Composite Containing a Connection of Layers . . . . . . . . . . . . 14.5 Numerical Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Contact Problem for a Coating/Substrate Interface Crack Under Action of a Moving Punch. Statistical Model of Coating Delamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ilya I. Kudish, Avetik Sahakyan, Vahram Hakobyan, and Harutyun Amirjanyan 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Formulation of a Contact Problem for Double Coated Half-Plane with an Interface Crack . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Some Results for an Interface Crack . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Statistical Model of Coating Delamination . . . . . . . . . . . . . . . . . . . 15.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Prediction of Dissipation in Electronic Components by Computing Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yiming Liu, Bilen Emek Abali, and Victor Eremeyev 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Weak Form for Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Computational Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Integral Eshelby’s Formulas for Generalized Continuum and Couple-Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergey Lurie and Yury Solyaev 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Second Gradient Theory of Continuum Media with Additional Internal Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Clapeyron’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Eshelby’s Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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341 341 343 344 346 347 349 349

351

352 353 356 359 367 367 369 369 371 375 378 382 383 385 385 386 391 392 395 396

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18 Heterogeneous Contact Modelling and Analysis via Numerical Equivalent Inclusion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wanyou Yang, Kuanyu Liu, Zhizhen Jiang, Pu Li, Qinghua Zhou, and Xiaoqing Jin 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Contact Theory for Heterogeneous Material . . . . . . . . . . . . . . . . . . 18.2.1 Equivalent Inclusion Method in Contact Problems . . . . . 18.2.2 The Elementary Solution of the Elastic Field . . . . . . . . . . 18.2.3 Elementary Solution for Inclusion in Half-Space . . . . . . 18.3 Numerical Solutions for Contact of Heterogeneous Materials . . . 18.3.1 Numerical EIM for Multiple Inhomogeneities . . . . . . . . . 18.3.2 The Conjugate Gradient Method . . . . . . . . . . . . . . . . . . . . 18.3.3 Numerical Solutions for Distributed Inhomogeneities Under a Rough Surface . . . . . . . . . . . . . 18.4 Numerical Studies on Heterogeneous Contact . . . . . . . . . . . . . . . . 18.4.1 Smooth Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.2 Rough Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 Extended Application for Heterogeneous Contact . . . . . . . . . . . . . 18.5.1 Coated Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.2 Two Joined Quarter Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Response Primitive Functions of Half Space Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2: The Expression of the Primitive Functions in the Elementary Solution . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Effective Engineering Constants for Micropolar Composites with Imperfect Contact Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Rodríguez-Ramos, V. Yanes, Y. Espinosa-Almeyda, C. F. Sánchez-Valdés, J. A. Otero, F. Lebon, R. Rizzoni, M. Serpilli, S. Dumont, and F. J. Sabina 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Heterogeneous Problem Statement and Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Asymptotic Homogenization Method and Effective Engineering Moduli for Periodic Laminated Micropolar Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.1 Effective Engineering Moduli . . . . . . . . . . . . . . . . . . . . . . 19.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.1 Non-uniform Imperfect Interface . . . . . . . . . . . . . . . . . . . . 19.4.2 Uniform Imperfect Interface . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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399 402 403 405 408 409 409 410 414 416 416 420 426 426 429 430 432 434 446 449

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20 The Mixed Problems of Poroelasticity for Rectangular Domains . . . Natalya Vaysfeld and Zinaida Zhuravlova 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 The Statement of the Poroelasticity Problems for Domains of Rectangular Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Deriving the Exact Solution of the Stated Problem for the Poroelastic Finite Rectangular Domain . . . . . . . . . . . . . . . . 20.4 Deriving the Exact Solution of the Stated Problem for the Poroelastic Semi-infinite Rectangular Domain . . . . . . . . . . 20.5 The Behavior Features of Mechanical Characteristics and Pore Pressure in Rectangular Domains . . . . . . . . . . . . . . . . . . . 20.5.1 The Behavior Features of Mechanical Characteristics and Pore Pressure in Finite Rectangular Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.2 The Behavior Features of Mechanical Characteristics and Pore Pressure in Semi-infinite Rectangular Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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21 Pore-Fluid Filtration by Squeezing a Fluid-Saturated Poroelastic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir B. Zelentsov and Polina A. Lapina 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Statement of the Problem of Squeezing a Fluid-Saturated Poroelastic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Integral Equation of the Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Analytical Solution of the Two-Dimensional Integral Equation . . 21.5 Dynamics of Fluid Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2D Asymptotic Analysis of a Thin Elastic Beam with Density-Dependent Generalized Young’s Modulus . . . . . . . . . . . Barı¸s Erba¸s, Julius Kaplunov, and Kumbakonam R. Rajagopal 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Asymptotic Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Two-Term Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Refined 1D Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467 469 470 473 474

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479 479 480 483 484 484 487 489 495 497 498 501 501 504 506 507 509 511 511

Chapter 1

Micromechanical Modeling of Non-linear Stress–Strain Behavior of Polycrystalline Microcracked Ceramics Giovanni Bruno, Mark Kachanov, and Igor Sevostianov

Abstract We discuss the non-linear stress–strain behavior of microcracked polycrystalline ceramics under uniaxial tension and compression (displacement control). Micromechanics explanation and modeling of its basic features, such as non-linearity and hysteresis in stress–strain curves, are developed, with stable microcrack propagation and “roughness” of intergranular cracks playing critical roles in tension and crack sliding playing a critical role in compression. Experiments involving complex loading histories are explained, and the model is shown to reproduce the basic features of the observed stress–strain curves. Keywords Non-linearity · Stress–strain relations · Hysteresis · Tension · Ceramics · Rocks · Microcracking · Polycrystals

1.1 Introduction We explain the behavior of brittle polycrystalline materials (such as ceramics or rocks) possessing pre-existing intergranular microcracking, under uniaxial, displacement-controlled loading and unloading, both in tension and compression. Such existing microcracks appear during/after cooling from high temperatures, i.e. G. Bruno (B) BAM, Bundesanstalt für Materialforschung und -Prüfung, Unter den Eichen 87, 12205 Berlin, Germany e-mail: [email protected] University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany M. Kachanov Department of Mechanical Engineering, Tufts University, Medford, MA 02155, USA Nizhny Novgorod State University Named After Lobachevsky, Nizhny Novgorod 603155, Russian Federation I. Sevostianov Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_1

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after sintering in ceramics (Thomas and Stevens 1989), and are due to the thermal expansion mismatch, either between different phases in multiphase materials (such as aluminum titanate (Bruno et al. 2010a) or ceramic-ceramic composites (Kupsch et al. 2022; Laquai et al. 2019a)) or between different orientations of crystallites of the same phase (such as in aluminum titanate, Bruno et al. 2010b, cordierite, Bruno et al. 2010c). It is to be noted that single-phase ceramics or even porous ceramics hardly display such features (Bruno et al. 2011a, 2012), unless they undergo a phase transformation (Liens et al. 2020), which is not the subject of this chapter. Microcracks typically have complex shapes, and generally follow grain boundaries and other weak surfaces in the microstructure (depending on the crystallography of the grains or domains, Bruno et al. 2010a; Shyam et al. 2012a). Furthermore, the intergranular microcrack faces possess certain “roughness”, depending on the chemical and mechanical properties of the neighboring grains. Under tension, microcracks can grow stably, to most favorably oriented, larger microcracks (normal to the load direction) (Bruno et al. 2019). Under compression, the extra strain due to the microcrack presence is generated by crack sliding (Bruno et al. 2013; Bruno and Kachanov 2013, 2016). Both scenarios result in the non-linearity of the stress–strain curves. The non-linear stress–strain behavior in uniaxial compression tests on porous ceramics has been also studied in some detail by Pozdnyakova et al. (2009) and Bruno et al. (2010d). A similar behavior has been observed and analyzed by Ghassemi-Kakroudi et al. (2009) in bending tests on compact refractory castables, and by Babelot et al. (2011) on aluminum titanate. The non-linearity has been shown to increase with loading (see Bruno et al. 2012a, 2013). Note that, although microcrack propagation under displacement-controlled conditions is generally stable, yet another possible factor contributing to the stability is that, having to follow grain boundaries or other weak surfaces, microcracks may run into obstacles that hinder crack propagation (for example, intersections or pores). Remark. A similar behavior has been observed in rocks (Okubo and Fukui 1996; Heap and Faulkner 2008; Hawkes and Mellor 1973), concrete (Bocca et al. 1991), ferroelectric ceramics (Lynch et al. 1995), (glass- or ceramic-matrix) composites (Prewo 1986), and thermal barrier coatings (Liu et al. 2007). In both cases of tension and compression, the following features of the stress– strain curves should be highlighted: 1. In the first loading, the curve “softens” and becomes non-linear. Upon shifting to unloading, the slope of the curve increases noticeably (“stiffer” response). At full unloading, a certain residual strain remains. The first loading–unloading cycle exhibits substantial hysteresis; 2. In subsequent cycles, the loading–unloading curves become almost linear (little or no hysteresis) provided the peak load does not exceed the one in the first cycle; 3. If the peak load in the next cycle exceeds the one in the previous cycle, the loading curve becomes non-linear above this point (Heap et al. 2013). In the context of geological materials, some of these features have been observed starting from the 1970s. Hawkes and Mellor (1973) conducted tensile tests on Berea sandstone, Indiana limestone, and Barre granite and reported that at low loads

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Young’s modulus is similar in tension and compression, but decreases in tension at higher loads and increases in compression, up to the stage of incipient failure. Stimpson and Chen (1993) proposed a testing technique in which the moduli in both tension and compression can be measured on the same specimen and reported nonlinear behavior under the tensile loading of several rocks (halite, potash, granite, and limestone). Okubo and Fukui (1996) performed uniaxial tension tests on nine Japanese rocks and observed that the stress–strain curves in tension are non-linear; some of them are shown in Fig. 2a. Heap et al. (2013) observed the evolution of the elastic moduli of basalt with increasing microcrack density under displacementcontrolled loading. They also observed that, upon further cycling, microcracking only proceeds if the load exceeds the maximum load of all preceding cycles. Young et al. (2015) attributed the non-linearity in tension to the stable growth of microcracks, as also mentioned in Cooper et al. (2017). In the context of ceramics, Kroupa (1995) discussed similar non-linearities in thermally sprayed ceramic coatings under tension and attributed the decrease of Young’s modulus to the increase of microcrack density; he suggested a semi-empirical relation for the non-linearity. Sadowski and Samborski (2003) considered the nonlinear behavior of porous polycrystalline ceramics in tension and compression and related the intergranular character of crack propagation to smaller fracture surface energy of grain boundaries. Gao et al. (2013) discussed a possible micromechanism of non-linearity that involves frictional sliding on parts of zigzag cracks induced by tensile loads. Cooper et al. (2017) modeled the non-linearity of tensile stress– strain curves utilizing a modified differential scheme (Vavakin and Salganik 1975; McLaughlin 1977) and assuming microcrack extension as the main damage mechanism; the microcrack evolution parameters were chosen as a linear function of applied strain. The behavior under cyclic loading was not modeled. Moreover, it was assumed that the crack density remains unchanged upon unloading (linear unloading curve). This assumption actually implies that stiffnesses at both peak loading and the onset of unloading are the same, E = E peak (εmax ())—contrary even to their own experimental results. The present overview discusses micromechanics-based explanations and modeling of the non-linear and hysteresis features discussed above. In the case of compression, we explain the stress–strain curves by means of microcrack closure and frictional sliding, while in tension, we invoke the jamming of rough surfaces. The model also allows the estimation of microstructural quantities such as crack densities from the stress–strain curves, and the comparison to other possible ways of determining such microstructural features (e.g. scanning electron microscopy (SEM) images (Bruno and Kachanov 2013) or X-ray refraction radiography (Müller et al. 2018; Laquai et al. 2019b)). The quantitative model involves an empirical part that describes the increase of crack density on the applied strain, which cannot be derived analytically. The model developed hereafter is then applied to tension data on β-eucryptite and in compression to data on aluminum titanate.

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1.2 Materials and Experiments 1.2.1 β-Eucryptite Subjected to Cyclic Tensile Loading of Increasing Amplitude The reported experiments were done on β-eucryptite ceramics specimens. Their preparation and microstructure are reported in several works (Cooper et al. 2017; Müller et al. 2018; Bruno et al. 2012b; Pandey et al. 2014), and details are not reported here. In brief, we started with a glass precursor consisting of a non-stoichiometric mixture of Li2 O, SiO2 , and Al2 O3 yielding the chemical formula of the oxide as LiAlSiO4 (β-eucryptite). The glass was poured into large pads that were crystallized using titanium oxide (of less than 5% weight) as a nucleating agent. Two materials, with different grain sizes, LGS and MGS (large and medium grain sizes, with average grain sizes of 30 μm and 5 μm, respectively), were obtained by the following annealing treatments: 16 h at 1300 °C for LGS and 1 h at 1300 °C for MGS. Their typical microstructure is shown in Fig. 1.1. Prior to loading, the MGS material had a moderate level of microcracking, whereas the LGS material had a large density of microcracks. Uniaxial tension experiments were performed at ORNL (see Shyam et al. 2012b) on an in-house built micro tensile rig assembled on an optical bench and equipped with an optical microscope. The optical images were captured periodically (1 Hz acquisition frequency) and analyzed by standard digital image correlation (DIC) techniques, to calculate strain during loading. Rectangular specimens were machined and mounted to grips using a thermal forming adhesive. The uniaxial tests were performed at a constant cross-head displacement rate of 1 μm/s for both loading and unloading. Multiple loading and unloading cycles to ~25, 50, and 75% of failure

(a)

(b)

75 μm

75 μm

Fig. 1.1 Microstructure of the two β-eucryptite materials: MGS (a) and LGS (b), see also (Shyam et al. 2012b)

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Fig. 1.2 Stress–strain behavior of LGS and MGS β-eucryptite specimens subjected to cyclic loading. Solid and hollow symbols correspond to loading and unloading, respectively. Figure adapted from Bruno et al. (2019)

strength were performed. Figure 1.7 shows the stress–strain curves for the LGS (a) and MGS (b) specimens (Fig. 1.2). Note that Young’s modulus of the materials remains the same at the beginning of each cycle. Our hypothesis that the stiffness upon unloading matches the stiffness at the beginning of forward-loading is justified by the experimental data. Physically, it implies that propagating cracks get stuck upon unloading, but other existing cracks continue contributing to the overall compliance to the same amount as for the initially microcracked material.

1.2.2 Aluminum Titanate Subjected to Compressive Interrupted Loading Al2 TiO5 ceramics were synthesized by extrusion of a mixture consisting of Al2 O3 , TiO2 , SiO2 , SrCO3 , CaCO3 , pore formers, and binders (see Bruno et al. 2010e; Backhaus-Ricoult et al. 2010) followed by sintering for more than 6 h at a temperature in excess of 1450 °C. The raw materials were fed in the form of powders with nearly round shapes. The microstructure of two kinds of aluminum titanate (AT) for diesel particle filter (DPF) applications (with porosity of 38% and 50%, determined by an Autopore 9520 porosimeter, Micromeritics), as captured in a JEOL 3000 field emission gun scanning electron microscope, FEG-SEM, is shown in Fig. 1.3 (for details, see Bruno and Kachanov 2013; Chen et al. 2020). Such microstructures are similar to those observed earlier by Backhaus et al. (2010). The material contained three phases: AT (gray), alumina (black spots), and feldspar (white spots). Such phases were also identified by laboratory X-ray diffraction (on a PANalytical X’pert Pro System with Cu radiation). Microcracks appear at all scales, since AT grains are composed of many small crystallites, and should be regarded as “domains” (such as in the case of cordierite; see Bruno et al. 2010;

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(b)

Fig. 1.3 Microstructure of AT materials: a 38% porosity; b 50% porosity. For the description, see the text and refer to Bruno et al. (2019); Bruno and Kachanov (2013)

Shyam et al. 2012); they extend within and around grains, and their linear dimensions are typically in tens of microns (however up to over a hundred microns). Pores have generally larger dimensions than microcracks. The microstructure is approximately overall isotropic. Uniaxial compression tests (load-unload cycles) were conducted on specimens of non-commercial AT samples manufactured in the form of honeycombs at Corning Incorporated. Specimens were cut to the size 12 × 25 × 50 mm. An Instron test rig was used, equipped with a load cell of 50 kN with two tailored aluminum platens. An extensometer was used to track the sample macroscopic strain (along the axial, extrusion direction). The loading rates were 1000 N/min. It is to be noted that the experiment was done in situ during a neutron diffraction measurement campaign (see Bruno and Kachanov 2013), not reported here. Consequently, the force was held constant at selected loads for about 10 min., to acquire a statistically significant neutron diffraction signal. The stress–strain behavior under uniaxial compression is shown in Fig. 1.4 for both samples (38 and 50% porosity, AT 38 and AT50). Both load and unload branches are shown, indicated by arrows, and both strains and stresses are indicated as negative if compressive. The external stress was obtained from the applied load and the sample cross-sectional area upon correcting for the closed frontal area of the honeycomb structure (Bruno et al. 2010e). The peak stress was 38 MPa for AT38, and 19 MPa for AT50. The observations on the stress–strain curves can be summarized as follows: • At low stresses, the stress–strain curves are almost linear, i.e. the axial compression stiffness is almost constant; • At higher stresses, the curves become non-linear, indicating increasing stiffness under increasing compression; • There is a sharp increase in axial stiffness when switching from loading to unloading;

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Fig. 1.4 Macroscopic strain versus applied stress curves for aluminum titanate, for the 38% and 50% porosity specimens (AT38 and AT50). Loading and unloading curves are represented by arrows. The experimental error is smaller than 5% (relative)

• At the end of unloading, the axial stiffness approximately returns to its value at the beginning of forward-loading. Very importantly, the overall axial stiffness at all stages of deformation remains around 20 to 70 times smaller than Young’s modulus of the defect-free bulk material (see Bueno et al. 2008). This drastic decrease of the axial stiffness (tangent to the stress–strain curve) confirms the effect of the high density of microcracks (the effect of porosity on stiffness is much smaller). Interestingly enough, the lattice deformation is approximately linear elastic, within the data scatter, and displays no hysteresis. Moreover, it has been shown that non-microcracked but porous ceramics do behave linearly (see Bruno et al. 2011a, b). This confirms that the non-linearities are solely caused by microcracks. Based on these microstructural observations, we describe the non-linearities under compression as mainly caused by two micromechanisms: 1. Closure of microcracks, which explains stiffening under increasing load; 2. Frictional sliding on closed cracks, which explains hysteresis, as well as sharply higher stiffness at the onset of unloading (due to delayed “backsliding”).

1.3 Micromechanical Explanation and Modeling 1.3.1 Uniaxial Compression The differential scheme, which is commonly used for porous and microcracked materials, is applied. In the following, we will also account for porosity, since the

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investigated material (AT) was also porous (see below). The contribution of individual cracks (as well as pores) to the overall strains and compliances is proportional to their linear sizes cubed; therefore, only the largest cracks have to be accounted for. An (open) circular crack of radius a under uniaxial stress p in the direction normal to the crack, and a spherical pore of the same radius, under the same loading, produce the following extra strains, per reference volume (see, for example, Sadowski et al. 2003): crack = ε33

16(1 − v02 ) a 3 p 2π(1 − v0 )(9 + 5v0 ) a 3 p por e , ε33 = 3 V E0 7 − 5v0 V E0    

4.75at v0 =0.33

(1.1)

8.38at v0 =0.33

where E 0 and ν 0 are Young’s and Poisson’s ratios of the bulk defect-free material, respectively. We represent, as usual, the overall strains εij generated by applied stress σ ij and the overall compliances S ijkl as sums of the contributions of the defect-free bulk material (having compliances S 0 ijkl ), and those of pores and microcracks. We also distinguish between those cracks that remain open in the considered range of compressive loads, and closed cracks that experience frictional sliding: por es

εi j = εi0j + εi j

open cracks

+ εi j

por es

= (Si0jkl + Si jkl 

open por es

+ Si jkl 

sliding cracks

+ εi j

sliding cracks

+ Si jkl

) σkl 

ef f

Si jkl

= (Si0jkl + Si jkl )σkl

(1.2)

  For the isotropic bulk material Si0jkl = (1/4G 0 ) δik δ jl + δil δ jk − (ν0 /E 0 )δi j δkl . δij denotes Kronecker’s delta. In the differential scheme, pores and cracks are introduced in small increments, and the background material is homogenized after each step, i.e. replaced by a homogeneous material possessing the updated effective properties. The original version of the scheme was modified by MacLaughlin (1977) to account for the fact that pores introduced later produce a stronger effect than the ones introduced earlier since the volume of the solid material that remains available for adding porosity is reduced by the factor of 1–ψ (where ψ is the amount of porosity). Then, placing random porosity ψ into a porous material would have the same effect as placing porosity (1 − ψ)−1 dψ into the virgin material. This treatment yields the solution E e f f = E 0 (1 − ψ)C1

(1.3)

where C 1 is the so-called pore shape factor (Bruno et al. 2011a; Gibson and Ashby 1982). In the limiting case of cracks (ψ = 0), the modification does not apply (there is no “lost” volume) and one has to use the exponential solution E e f f = E 0 e−C1 ψ , with porosity changed to crack density ρ and the shape factor being the one for cracks.

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The final result is E e f f = E 0 (1 − ψ)C por e e−Ccrack ρ

(1.4)

It is assumed that cracks have circular (penny) shapes, of radii a(i) . Their concentration in the isotropic case of random orientations is characterized by the usual crack density parameter introduced by Bristow (1960): ρ=

1  (i)3 a i V

(1.5)

The crack density parameter was extended to elliptical cracks with identical eccentricity by Budiansky and O’Connell (Budiansky and O’Connell 1976) by replacing  a 3 → 2S 2 /π P where S and P are the ellipse’s area and perimeter. Remark. In the case of cracks of “irregular” shapes, the crack density parameter is not defined, but should be understood as the density of a certain fictitious, or equivalent, set of circular cracks that produces the same effect, with an inevitable shortcoming that it may not be explicitly linked to the actual geometrical features. The following considerations are relevant in this context: • Multiple flat (planar) cracks of irregular shapes can be replaced, with good accuracy, by certain equivalent sets of circular cracks provided the shape irregularities (deviations from circles) are random (Kachanov 1980); • Non-flatness of open, traction-free cracks can be ignored (in its effect on the overall compliance) provided it is small-to-moderate (Grechka and Kachanov 2006); • The effect of crack intersections is relatively small provided they are not so numerous as to render the material incoherent; • The 3-D crack geometry is often difficult to assess. This is particularly relevant for very thin cracks that may not be visible in SEM images. In addition, small partial contacts between crack faces may stiffen a crack substantially (Sevostianov and Kachanov 2002), and this factor is not easily identified in the images. In such cases, there is no choice but to treat the crack density parameter in the above-mentioned sense, i.e. as the density of a certain equivalent set of circular cracks. We have mentioned above that our model of the compression behavior of microcracked materials comprises microcrack sliding. For a closed frictionally sliding crack (with unit normal n), sliding is driven by the traction τn +μσn induced by the applied load on a plane of orientation n; τ n and σ n are the shear and normal tractions (σ n is negative if compressive) and μ is the friction coefficient. The situation is sketched in Fig. 1.5. In the considered axisymmetric case of uniaxial compression in the direction x 3 , it is convenient to specify the crack orientation by an angle φ between n and x 3 . In our case, crack growth is assumed to be stable or negligible at this stage. For multiple cracks, the range of crack orientations satisfying the condition τn + μσn > 0, i.e. where sliding takes place, has been identified under different

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Fig. 1.5 Sketch of the microcrack geometry n

x3 n

loading conditions by several authors; see, for Under uniaxial  example, (Walsh 1965).  compression stress p, we have τn +μσn = p sinϕcosϕ − μsin2 ϕ and the mentioned range is ϕ = (0, ϕ∗ ) wher e ϕ∗ ≡ arctanμ−1

  = 590 at μ = 0.6

(1.6)

Sliding starts in the entire range given by Eq. (1.6) from the beginning of loading. increases from The sliding displacement on a crack is proportional to τn + μσn —and 

zero at the endpoints of the range (9) to the maximal value of (1/2) 1 + μ2 − μ at the midpoint ϕ = (1/2)arctanμ−1 . The macroscopic strain due to sliding cracks is obtained by summation over all the cracks that slide, i.e. by integration over the range (1.6). To find the strain contributions of sliding cracks, we note that for  of area S, its strain  any flat crack contribution per reference volume V is εi j = bi n j + b j n i S/(2V ), where b is the average displacement discontinuity vector across the crack. For a circular crack, b is parallel to  the direction of traction τ + μσ with the proportionality coefficient  16a 1 − ν02 /[3(2 − ν0 )E 0 ] (see, for example, (Rice 1979)). Macroscopic strains due to sliding cracks are linear functions of applied stress p, since the range of sliding cracks is constant and τn + μσn on each crack is a linear function of p. Since n 3 = cosϕ, b3 = sinϕ and S = πa 2 , , summing contributions of multiple cracks of random orientations (and generally diverse sizes) one obtains, in the non-interaction approximation:

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sliding

ε33 − 

11

=

ϕ=ϕ∗

3 p 32(1 − v02 ) sin3 ϕ sin5 ϕ cos ϕ cos5 ϕ ρ sliding (1.7) − +μ − 3(2 − v0 ) 3 5 3 5 E0 ϕ=0   C3 =0.342 at μ=0.6,v0 =0.33

where the braces represent the result of integration over the range (1.6). The lateral strains are ε11 = ε22 = −ε33 /2 (no volume change due to sliding). Note that the density of sliding cracks is related to the overall density parameter ρ of thin cracks by the relation ρ sliding = (2ϕ∗ /π )ρ. In the present work, we assume μ = 0.6. Importantly, frictional sliding upon loading in uniaxial compression does not generate non-linearities in the predicted stress–strain curve; it only changes its slope. In fact, the orientation interval of sliding cracks given by Eq. (1.6) remains constant in forward-loading, and on each crack it is proportional to the applied stress. Therefore, the observed non-linearity under increasing uniaxial compression should be attributed to changes in the density of open and sliding microcracks under compression, i.e. by the reduction of ρ open and the increase of ρ sliding (caused by newly closed cracks). Conversely, we will see below that our model does predict the non-linearity of the stress–strain curve upon unloading, since in this stage the orientation interval of (back)sliding cracks does change as a function of the applied load. For open randomly oriented cracks under uniaxial loading, using results of Bristow (Bristow 1960) for non-interacting cracks, the braces have to be replaced by the term (10 − 3ν0 )/30 so that one obtains open cracks

ε33

=

16(1 − v02 )(10 − 3v0 ) open p ρ 45(2 − v0 ) E0   

(1.8)

C2 =1.71 at v0 =0.33

The open cracks contribute to the linear elastic reversible part of the stress–strain curves (similar to pores). Their strain contribution is substantially larger than that of sliding cracks—by a factor of five, at the same crack density. To obtain the effective axial modulus E eff , all the strain contributions due to the deformation of the matrix, to pores, and to sliding and open cracks have to be added up. Taking into account the interaction effect via the differential scheme as discussed above, one obtains   E0 = (1 − ψ)−C1 · exp C2 ρ open + C3 ρ sliding Ee f f

(1.9)

Upon unloading, cracks start sliding back only after some finite interval of load reduction. This is analogous to a frictionally sliding block that is pushed by a horizontal force and compresses a spring attached to a wall at the other end; when the force is reduced, the block starts sliding back only after a certain finite reduction

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of the force. In contrast with forward-loading, backsliding starts only on the most favorably oriented cracks and gradually expands to cover the entire range of cracks that have slid forward, as p is reduced. When the stress is reduced from its peak value p0 to a certain p, backsliding on a given crack starts when the elastic “restoring force” overcomes the (reduced) value of τ + μσ (in the spring-block analogy, this restoring force corresponds to the maximal compression of the spring at the peak stress):   0 τn + μσn0 − (τn + μσn ) = 0

(1.10)

This translates into the following range of orientations of backsliding cracks: ( p0 − p)sin2ϕ − μ( p0 + p)(1 − cos2ϕ) = 0

(1.11)

Backsliding starts as soon as the load is reduced on cracks that are close to vertical, i.e. to the endpoint φ ~ 0 of the range given by Eq. (1.6). As unloading proceeds, the range of backsliding cracks expands reaching the midpoint of the range given =   in Eq. (1.11) at p



2 2 2 2 p0 1 + μ − μ 1 + μ / 1 − μ + μ 1 + μ . At full unloading to p = 0, backsliding spreads to all the cracks that have slid forward; the specimen returns to its original state. As mentioned before, this expansion of the backsliding range leads to non-linearity of the stress–strain curve upon unloading—in contrast with the case of forward-loading. At the beginning of unloading, strains produced by backsliding are very small. This is due to two factors: (1) narrow interval of orientations for backsliding, and (2) small elastic restoring force. Thus, at the beginning of unloading, the axial stiffness is almost unaffected by closed cracks, and is given by   E0 = (1 − ψ)−C1 · exp C2 ρ open Ee f f

(1.12)

This explains the sharp change in the axial stiffness upon changing from loading to unloading. At different stages of the loading history, different micromechanisms of deformation are operative. We now relate the stress–strain curves to these micromechanisms. For the AT specimen with 38% porosity, the initial stage of loading displays an almost linear stress–strain curve. This indicates that neither crack closures nor frictional sliding occurs in significant amounts at this stage. As the applied stress increases, closure starts on crack orientations for which σn = p sin2 ϕ π e = 4(1−v satisfies the condition Pclosur 2 λ, where λ the initial aspect ratio λ = E0 0) (max initial opening)/(crack diameter ). The slope of the stress–strain curve— the axial stiffness E e f f /E 0 —is rather determined by the porosity ψ and by the density of all cracks that are open at this stage, ρ open . Hence, in applying Eq. (1.9) to this stage, we should set ρ sliding = 0 and interpret ρ open as ρ open, initial .

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At the endpoint of loading (peak load), E e f f /E 0 , as given by Eq. (1.15), is determined by the porosity, by the density of sliding cracks ρ sliding , and by the density of cracks that remain open at the peak load ρ open, peak (which is lower than ρ open, initial ); At the onset of unloading, the stress–strain curve is (nearly) linear, indicating that none of the mechanisms of inelasticity is operative to a significant extent. Indeed, (1) the strains produced by backsliding are negligible at this stage, and (2) the strains produced by the opening of the cracks that have been closed upon loading are insignificant as well, since their orientation range is quite narrow. The axial stiffness E e f f /E 0 is controlled by ψ and ρ open, peak ; we should again set ρ sliding = 0 in Eq. (1.9). The above considerations can be summarized as a set of three equations: ⎫  ⎧ ⎪ E 0 /E e f f star t C2 ρ open, initial ⎬ ⎨   −C open, peak 1 E 0 /E e f f peak load = (1 − ψ) · exp C2 ρ + C3 ρ sliding,   ⎪ ⎩ ⎭ open, peak C2 ρ E 0 /E e f f onset o f unload 

peak

(1.13) where the left-hand side represents the experimentally measured data and the righthand side the quantities to be calculated: crack densities at the beginning (open), at the top load (open and sliding), and at the beginning of unloading (open). We use the following values of constants: C 1 = 3.0 (the shape factor of pores corresponding to a mixture of interpenetrating solid spheres and interpenetrating round pores; see (Bruno et al. 2011a)), C 2 = 1.71 (open cracks; see above Eq. (11)), and C 3 = 0.35 (sliding cracks). The above equations yield the microcrack density parameters given in Table 1.1. An important observation is that the overall microcrack density at peak load ρ open, peak + ρ sliding, peak is higher than the density ρ open,initial of all the microcracks that behave as open ones in the initial loading interval. A possible explanation is that some of the cracks that slide at higher loads behave as closed at the initial loading stage. This is a confirmation that the applied loads (superimposed to the residual microstress in the material) induce additional damage (Bruno et al. 2010f) and in particular crack propagation (Müller et al. 2018), a mechanism that is not modeled here. As unloading proceeds, both backsliding and opening of the previously closed cracks start to produce a noticeable effect. Both factors contribute to a gradual decrease in axial stiffness. At the end of unloading, the axial stiffness is predicted to return to its value at the beginning of the forward-loading. Table 1.1 Estimated microcrack density parameters for the two samples, using Eq. (16)

Density parameter

Porosity (%) 38

50

ρ open, initial

1.51

1.15

ρ open, peak

0.86

0.79

ρ sliding, peak

1.59

1.50

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Note that in both samples, there is some residual strain at the end of the loading cycle, of about 1000 × 10–6 in AT 38% and 250 × 10–6 in AT 50%. This is most probably due to two factors: 1. Some microcrack propagation has taken place; 2. On some of the cracks, backsliding may have been only partial. Both phenomena are not modeled here. The theoretical framework developed here allows one to simulate the stress– strain curve provided the law of evolution of crack density is known. This has been demonstrated by David et al. (2012) and by Leplay et al. (2010). David et al. have used a highly simplified 2-D model involving cracks of identical size, while Leplay et al. have used some a priori assumed functional dependence of the axial modulus. However, the available microstructural information is insufficient for formulating such evolution laws, and therefore we limit our treatment to the determination of crack densities as key points of the stress–strain curve. Had we had the full microstructural information (for instance, stitching together SEM images over a significant fieldof-view (FoV)), we would be have been able to fully predict the stress–strain curve ideally without measuring it.

1.3.2 Uniaxial Tension The micromechanical explanation of the features described above in the tension case is based on two factors: (1) complex crack geometries that follow grain boundaries or weak surfaces, and (2) roughness of crack faces, with roughness profiles getting “mismatched” as cracks propagate, thus impeding the reversal of displacements of crack faces at unloading. The sketch in Fig. 1.6 illustrates the role of roughness. At the first loading cycle, the softening in forward-loading is due to microcracking; similarly, further softening upon cycling at a higher peak load is related to additional microcracking at loads above the previous peak. The behavior at unloading is related to the roughness of crack surfaces. Indeed, in forward-loading, relative displacements of crack faces comprise both the normal (opening) and the tangential (shear) components. If crack propagation occurs, then roughness profiles of crack faces get “mismatched” (the profiles shift with respect to one another). This prevents full reversal of the mentioned displacements upon unloading. This leads to a “stiffer” response at the beginning of unloading (as compared to the end of loading), to hysteresis, and to consequent residual strain. This phenomenon constitutes one of the basic features of the proposed micromechanical model. In subsequent cycles, the mismatched (due to crack growth) roughness profiles prevent movement of crack faces; the cracks are “stuck” in the positions reached at the (previous) peak load. This leads to almost linear stress–strain curves, as described in the introduction. Quantitative modeling of stress–strain curves requires a model for the effective elastic properties of a polycrystalline material (treated as homogenized isotropic material) that contains cracks. We again use the differential scheme that has been

1 Micromechanical Modeling of Non-linear Stress–Strain Behavior …

15

Fig. 1.6 Sketch illustrating the role of roughness of crack faces. In forward-loading, roughness profiles of crack faces get mismatched when non-linearity starts due to crack propagation (point 2); at unloading (point 4), the faces get “stuck” (their displacement at peak load C is locked)

shown to be relatively accurate for cracked solids (Saenger et al. 2006). This scheme, first formulated by Bruggeman (Bruggeman 1935, 1937) for the effective dielectric and elastic constants of a matrix with spherical inhomogeneities, was applied to the elastic properties of cracked solids in Vavakin and Salganik (1975); Hashin (1988); for the ellipsoidal inhomogeneities, the equations were given by McLaughlin (1977), who solved them explicitly for spherical inhomogeneities; this solution was further analyzed in Zimmerman (1985); Zimmerman (1991). In the isotropic case of randomly oriented inhomogeneities, we have two coupled differential equations for the effective bulk and shear moduli. If, however, we are interested in Young’s modulus only, then one can construct a simple approximate solution that has satisfactory accuracy if Poisson’s ratio of the material prior to loading is ν0 < 0.4 (Sevostianov et al. 2006); this is very often the case, especially in the case of porous ceramics (Bruno et al. 2011a). In the case of circular cracks (that are understood as fictitious cracks producing the same effect as the actual ones; see the Remark above), Young’s modulus has the form, see (Kachanov and Sevostianov 2018), E ≈ E 0 e−D0 ρ

(1.14)

   0) where D0 = − 16 1 − ν02 (10−3ν and subscript “0” refers to the material prior to 45 (2−ν0 ) loading; ρ denotes the increase of crack density under loading with respect to the initially microcracked state.

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In displacement-controlled loading, the crack density increases as ρ = ρ(ε). If this dependence is known, Eq. (1.14) gives Young’s modulus as a function of applied strain, E = E(ε), which is generally non-linear. Since crack density is understood as the density of an equivalent set of circular cracks, we will denote it in the following by R (rather than ρ for penny-shaped cracks) and call it a “generalized” crack density. We proceed as follows: • We observe that, for the dependence E = E(ε) to be constructed, the parameter R does not actually need to be geometrically defined; it is its dependence on applied strain that is needed; • We retain the structure of Eq. (1.14), with ρ → R:

E ≈ E 0 e−D0 R

(1.15)

The problem in the case of β-eucryptite reduces to formulating the dependence R = R(ε) where “” refers to the increment of microcrack density compared to its pre-existing level. We select this dependence to fit the experimentally observed stress–strain curves. This is achieved by taking 2

R(ε) = eaε − 1

(1.16)

where a is a fitting parameter that provides the best fit (we will see later that parameter a is generally grain-size-dependent). The values of E 0 and ν0 refer to the material prior to loading (they reflect the pre-existing level of microcracking); for β-eucryptite, they are E 0 = 24GPa and ν0 = 0.28 (see Cooper et al. 2017). The procedure of extracting R is illustrated in Fig. 1.7a, b. We combine Eqs. (1.15) and (1.16), varying the fitting parameter a to get the best agreement with the experimental data. Figure 1.7a shows the best fit of the experimental data. Figure 1.7b shows the extracted dependence R(ε).

(a)

(b)

Fig. 1.7 Procedure for the extraction of R from experimental data for β-eucryptite: best fit of the stress–strain curve (first cycle) (a); change in generalized crack density as a function of applied strain (b). (In the latter, possible subsequent cycles are indicated by the dashed lines.) Figures adapted from Bruno et al. (2019)

1 Micromechanical Modeling of Non-linear Stress–Strain Behavior …

17

It is to be noted that the exponential in Eq. (1.16) is a fitting function. The exponential character reflects the material behavior: first, at low applied load, cracks oriented normally to the load direction start to propagate slowly, and then crack growth becomes faster and involves a strongly non-linearly increasing number of crack orientations. Figure 1.8 shows the simulation of some of the stress–strain curves shown in Fig. 1.2. The forward-loading curve is simulated by formula σ = E(ε)(ε − εr es ) where E(ε) is given by Eq. (1.15) and εr es is taken as zero in the first cycle and from the data of Fig. 1.2 in the subsequent cycle. The unloading curve represents the linear elastic response corresponding to the locking of microcracks (due to roughness) at unloading so that at the beginning of unloading it holds σ = E 0 (ε − εr es ), i.e. this slope is the same as at the beginning of the forward-loading cycle. Unlike the case of compression, this hypothesis is assumed to hold throughout the whole unloading process; the hypothesis will be justified and discussed later. If we apply the developed model to the cyclic tensile loading of beta-eucryptite ceramics assuming increasing crack density in forward-loading and linear elastic unloading corresponding to “locked” microcracks, we obtain the R(ε) shown in Fig. 1.8. The values of E 0 were taken as 24 and 34 GPa (Cooper et al. 2017) for the LGS and MGS specimens, respectively—the difference being due to higher preexisting microcrack density in the LGS specimens. Poisson’s ratio was taken as ν0 = 0.28. 2 We used the same formula R(ε) = eaε − 1 as above, where the best fit for the LGS and MGS specimens is given by a = 85 and a = 35, respectively, indicating grain-size dependence of parameter a. Incidentally, the value of a is the same for LGS and the cordierite reported in Bruno et al. (2013). In fact, the grain sizes of that cordierite and β-eucryptite LGS are nearly the same, while that of MGS is smaller. The dependence of a on grain size is, however, a separate subject. Note that the vertical drop to zero for R at unloading reflects the locking of cracks resulting in their zero contribution to the overall compliance. Upon reloading, cracks start to contribute to the overall strain only when the previous peak is reached. Fig. 1.8 The first and the second loading–unloading cycles for MGS β-eucryptite, as predicted by the model Eq. (1.2) in comparison with the experimental data from Müller et al. (2018). Figures adapted from Bruno et al. (2019)

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(a)

G. Bruno et al.

(b)

Fig. 1.9 The microcrack density parameter in successive tensile loading–unloading cycles for βeucryptite, as predicted by the model Eq. (1.16), for a LGS; b MGS. Note the different deformation reached by the two materials. Figures adapted from Bruno et al. (2019)

Figure 1.9 shows the stress–strain behavior during each of the three-and-a-half cycles, for the LGS samples; it compares the simulated curves with the experimental data. Figure 1.10 contains similar information for the MGS specimens.

(a)

(b)

(c)

(d)

Fig. 1.10 Comparison of model predictions and experimental data for the third and fourth cycles for LGS (a and b) and MGS (c and d) specimens. Figures adapted from Bruno et al. (2019)

1 Micromechanical Modeling of Non-linear Stress–Strain Behavior …

19

1.4 Concluding Remarks We have proposed micromechanisms explaining the non-linear behavior of microcracked ceramics under compression and tension (the latter including cyclic loading) and modeled these mechanisms quantitatively. The main ideas are that (1) In compression, microcrack closure and sliding introduce extra (compressive) strain, so that a non-linear stress–strain curve appears. Upon unloading, sliding is activated in a delayed fashion, so that at the beginning of unloading the material behaves as extremely stiff, and a hysteresis appears. (2) In tension the non-linearity is related to intergranular (or along weak surfaces) crack propagation and the hysteresis is due to the roughness of crack faces that gets “mismatched” due to the propagation and thus impedes backward displacements of crack faces at unloading. The model is shown to be able to reproduce experimental loading and unloading stress–strain curves. It is also shown to reproduce the data on complex loading history. Note that the suggested micromechanisms may not be the only ones responsible for the observed stress–strain behavior. In particular, there is a possibility that under compression crack propagation occurs and under tensile loading, certain branches of zig-zag-shaped cracks experience local compressive conditions and may undergo frictional sliding. We also comment on the challenge of quantifying the crack density—that is encountered in many materials science applications. While we treat microcracks as penny-shaped, we emphasize that a “generalized crack density”—denoted by R— could be introduced by necessity to yield a certain measure of crack density, especially if the evolution of ρ with loading is of importance. We note, in conclusion, that the constructed model can be utilized to design a specific non-linear tensile behavior of brittle microcracked polycrystalline materials. Acknowledgements The authors acknowledge the fruitful discussions undertaken with the coauthors of their papers cited in this work, and in particular A. Shyam (ORNL, Oak Ridge, TN, USA) for the provision of the β-eucryptite experimental data.

References Babelot C, Guignard A, Huger M, Gault C, Chotard T, Ota T, Adachi N (2011) Preparation and thermomechanical characterization of aluminum titanate flexible ceramics. J Mater Sci 46:1211– 1219 Backhaus-Ricoult M, Glose C, Tepesch P, Wheaton B, Zimmermann J (2010) Aluminum titanate composites for diesel particulate filter applications. In: Narayan RCP (ed) Advances in bioceramics and porous ceramics III Ceramic engineering & science proceedings. American Ceramic Soc., pp 145–56 Bocca P, Carpinteri A, Valente S (1991) Mixed mode fracture of concrete. Int J Sol Struct 27:1139– 1153 Bristow JR (1960) Microcracks, and the static and dynamic ellastic constants of annealed and heady cold-worked metals. Br J Appl Phys 11:81–85

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Bruggeman DAG (1935) Berechnung verschiedener physikalisher Konstanten von heterogenen Substanzen. I. Dielectrizitätkonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann Physik Leipzig. 24:636–679 Bruggeman DAG (1937) Berechnung verschiedener physikalisher Konstanten von heterogenen Substanzen. III. Die elastische Konstanten der Quaiisotropen Mischkörper aus isotropen Substanzen. Ann Physik Leipzig 29:160–178 Bruno G, Kachanov M (2013) Porous microcracked ceramics under compression: micromechanical model of non-linear behavior. J Eur Ceram Soc 2013(33):2073–2085 Bruno G, Kachanov M (2016) Microstructure-property relations in porous microcracked ceramics: the possibilities offered by micromechanics. J Am Ceram Soc 99:3829–3852 Bruno G, Efremov AM, Wheaton BR, Webb JE (2010a) Microcrack orientation in porous aluminum titanate. Acta Mater 58:6649–6655 Bruno G, Efremov AM, Wheaton BR, Bobrikov I, Simkin VG, Misture S (2010b) Micro and macroscopic expansion of stabilized porous aluminium titanate. J Eur Ceram Soc 30:2555–2562 Bruno G, Efremov AM, Clausen B, Balagurov AM, Simkin VN, Wheaton BR et al (2010c) On the stress-free lattice expansion of porous cordierite. Acta Mater 58(6):1994–2003. https://doi.org/ 10.1016/j.actamat.2009.11.042 Bruno G, Efremov AM, Levandovskiy AN, Pozdnyakova I, Hughes DJ, Clausen B (2010d) Thermal and mechanical response of industrial porous ceramics. Mater Sci Forum 652:191–196 Bruno G, Efremov AM, Wheaton BR, Webb JE (2010e) Microcrack orientation in porous aluminum titanate. Acta Mater 2010(58):6649–6655 Bruno G, Efremov A, Wheaton B, Bobrikov I, Simkin VG, Misture S (2010f) Micro- and macroscopic thermal expansion of stabilized aluminum titanate. JEurCeramSoc. 30:2555–2562 Bruno G, Efremov AM, Levandovskiy AN, Clausen B (2011a) Connecting the macro and micro strain responses in technical porous ceramics: modeling and experimental validations. J Mater Sci 46:163–175 Bruno G, Efremov AM, An CP, Wheaton BR, Hughes DJ (2012a) Connecting the macro and micro strain responses in technical porous ceramics. Part II microcracking. J Mater Sci 47:3674–3689 Bruno G, Garlea OV, Muth J, Efremov AM, Watkins TR, Shyam A (2012b) Microstrain temperature evolution in β-eucryptite ceramics: measurement and model. Acta Mater 60:4982–4996 Bruno G, Kachanov M, Sevostianov I, Shyam A (2019) Micromechanical modeling of non-linear stress-strain behavior of polycrystalline microcracked materials under tension. Acta Mater 164:50–59 Bruno G, Efremov AM, Clausen B, Balagurov AM, Simkin VN, Wheaton BR, Webb JE, Brown DW (2010) On the stress-free lattice expansion of porous cordierite. Acta Materialia 58 1994–2003 Bruno G, Efremov AM, An C, Nickerson S (2011) Not all microcracks are born equal: thermal vs mechanical microcracking in porous ceramics. In: Widjaja S, Singh D (eds) Advances in bioceramics and porous ceramics IV—ceramic engineering & science proceedings (CESP). Amer Ceram Soc Bruno G, Efremov AM, An CP, Wheaton BR, Hughes DJ (2012) Connecting the macro and microstrain responses in technical porous ceramics. Part II: microcracking. J Mater Sci 47:3674–3689 Bruno G, Kilali Y, Efremov AM (2013) Impact of the non-linear character of the compressive stress–strain curves on thermal and mechanical properties of porous microcracked ceramics J Eur Ceram Soc 33:211–219 Budiansky B, O’connell RJ (1976) Elastic moduli of cracked solids. Int J Solids Struct 12:81–91 Bueno S, Hernández MG, Sánchez T, Anaya JJ, Baudín C (2008) Non-destructive characterisation of alumina/aluminium titanate composites using a micromechanical model and ultrasonic determinations Part I. Evaluation of the effective elastic constants of aluminium titanate. Ceram Int 34:181–188 Chen C, Müller BR, Prinz C, Stroh J, Feldmann I, Bruno G (2020) The correlation between porosity characteristics and the crystallographic texture in extruded stabilized aluminium titanate for diesel particulate filter applications. J Eur Ceram Soc 40:1592–1601

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Cooper RC, Bruno G, Wheeler MR, Pandey A, Watkins TR, Shyam A (2017) Effect of microcracking on the uniaxial tensile response of β-eucryptite ceramics: experiments and constitutive mode. Acta Mater 135:361–371 David EC, Brantut N, Schubnel A, Zimmerman RW (2012) Sliding crack model for nonlinearity and hysteresis in the uniaxial stress–strain curve of rock. Int J Rock Mech Mining Sci 52:9–17 Gao Z, Zimmerman JW, Kachanov M (2013) On microstructural mechanisms causing non-linear stress-strain behavior of porous ceramics under tension. Int J Fract 183:283–288 Gibson IJ, Ashby MF (1982) The mechanics of three-dimensional cellular materials. Proc Roy Soc Lond A382:43–59 Grechka V, Kachanov M (2006) Effective elasticity of fractured rocks: a snapshot of the work in progress. Geophysics 71(6):W45–W58 Hashin Z (1988) The differential scheme and its application to cracked materials. J Mech Phys Solids 36:719–734 Hawkes I, Mellor M (1973) Uniaxial testing in rock mechanics laboratories. Eng Geol 4:177–285 Heap MJ, Faulkner DR (2008) Quantifying the evolution of static elastic properties as crystalline rock approaches failure. Int J Rock Mech Mining Sci 45:564–573 Heap MJ, Vinciguerra S, Meredith PJ (2013) The evolution of elastic moduli with increasing crack damage during cyclic stressing of a basalt from Mt. Etna volcano. Tectonophysics 471:153–160 Kachanov M (1980) Continuum model of medium with cracks. J Eng Mech Div ASCE 106:1039– 1051 Kachanov M, Sevostianov I (2018) Micromechanics of materials, with applications. Springer Kakroudi MG, Huger M, Gault C, Chotard T (2009) Damage evaluation of two alumina refractory castables. J Eur Ceram Soc 29:2211–2218 Kroupa F (1995) Nonlinear behavior in compression and tension of thermally sprayed ceramic coatings. J Therm Spray Technol 16:84–95 Kupsch A, Laquai R, Muller BR, Paciornik S, Horvath J, Tushtev K et al (2022) Evolution of damage in all-oxide ceramic matrix composite after cyclic loading. Adv Eng Mater 24(6):2100763. https://doi.org/10.1002/adem.202100763 Laquai R, Gouraud F, Müller BR, Huger M, Chotard T, Antou G et al (2019a) Evolution of thermal microcracking in refractory ZrO2 -SiO2 after application of external loads at high temperatures. Materials 12(7):1017 Laquai R, Gouraud F, Müller BR, Huger M, Chotard T, Antou G et al (2019) Evolution of thermal microcracking in refractory ZrO2 -SiO2 after application of external loads at high temperatures. Mater 12:1017 Leplay P, Réthoré J, Meille S, Baietto MC (2010) Damage law identification of a quasi-brittle ceramic from a bending test using digital image correlation. J Eur Ceram Soc 30:2715–2725 Liens A, Reveron H, Douillard T, Blanchard N, Lughi V, Sergo V et al (2020) Phase transformation induces plasticity with negligible damage in ceria-stabilized zirconia-based ceramics. Acta Mater 183:261–273. https://doi.org/10.1016/j.actamat.2019.10.046 Liu Y, Nakamura T, Srinivasan V, Vaidya A, Gouldstone A, Sampath S (2007) Nonlinear elastic properties of plasma sprayed zirconia coatings and associated relationships to processing conditions. Acta Mater 55:4667–4678 Lynch CS, Hwang SC, McMeeking RM (1995) Micromechanical theory of the nonlinear behavior of ferroelectric ceramics. Proc SPIE 2427:300–312 McLaughlin R (1977) A study of the differential scheme for composite materials. Int J Eng Sci 15:237–244 Müller BR, Cooper RC, Lange A, Kupsch A, Wheeler M, Hentschel MP et al (2018) Stress-induced microcrack density evolution in β-eucryptite ceramics: experimental observations and possible route to strain hardening. Acta Mater 144:627–641 Okubo S, Fukui K (1996) Complete stress-strain curves for various rock types in uniaxial tension. Int J Rock Mech Min Sci & Geomech Abstr 33:549–556 Pandey A, Shyam A, Watkins TR, Lara-Curzio E, Stafford RJ, Hemker KJ (2014) The uniaxial tensile response of porous and microcracked ceramic materials. J Am Ceram Soc 97(3):899–906

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Pozdnyakova I, Bruno G, Efremov AM, Clausen B, Hughes D (2009) Stress-dependent elastic properties of porous cellular ceramics. Adv Eng Mater 11:1023–1029 Prewo KM (1986) Tension and flexural strength of silicon carbide fibre-reinforced glass ceramics. J Mater Sci 21:3590–3600 Rice JR (1979) Theory of precursory processes in the inception of earthquake rupture. Beitrage Geophysik Gerlands 88:91–121 Sadowski T, Samborski S (2003) On the different behaviour of porous ceramic poly crystalline materials under tension and compression stress state. In: Bathe KJ (ed) Computational fluid and solid mechanics. Elsevier Science, pp 615–618 Saenger EH, Krueger OS, Shapiro SA (2006) Effective elastic properties of fractured rocks: dynamic vs. static considerations. Int J Fract 139:569–576 Sevostianov I, Kováˇcik J, Simanˇcík F. (2006) Elastic and electric properties of closed-cell aluminum foams. Cross-property connection. Mater Sci Eng A 420:87–99 Sevostianov I, Kachanov M (2002) On the elastic compliances of irregularly shaped cracks. Int J Fract 114:245–257 Shyam A, Lara-Curzio E, Pandey A, Watkins TR, More KL (2012a) The thermal expansion, elastic and fracture properties of porous cordierite at elevated temperatures. J Am Ceram Soc 95:1682– 1691 Shyam A, Muth J, Lara-Curzio E (2012b) Elastic properties of β-eucryptite in the glassy and microcracked crystalline states. Acta Mater 60(16):5867–5876 Shyam A, Lara-Curzio E, Pandey A, Watkins TR, More KL (2012) The thermal expansion, elastic and fracture properties of porous cordierite at elevated temperatures. J Am Ceram Soc 1–10 Stimpson B, Chen R (1993) Measurement of rock elastic moduli in tension and in compression and its practical significance. Can Geotech J 30:338–347 Thomas HAJ, Stevens R (1989) Aluminium titanate—a literature review part 1: microcracking phenomena. Br Ceram Trans J 88:144–151 Vavakin AS, Salganik RL (1975) Effective characteristics of nonhomogeneous Media with isolated nonhomogeneities 58–66 Walsh JB (1965) The effect of cracks on the uniaxial elastic compression of rocks. J Geophys Res 70:399–411 Young G, Cai Z, Zhang X, Fu D (2015) An experimental investigation on the damage of granite under uniaxial tension by using a digital image correlation method. Opt Lasers Eng 73:46–52 Zimmerman RW (1985) The effect of microcracks on the elastic moduli of brittle materials. J Mater Sci Lett 4:1457–1460 Zimmerman RW (1991) Elastic moduli of solid containing spherical inclusions. Mech Mater 12:17– 24

Chapter 2

Shamrovskii’s Version of the Refined Dynamical Plate Theory Igor V. Andrianov

and Isaac Elishakoff

Abstract This study presents the refined theory of elastic plates developed by A. D. Shamrovskii. This theory has some important features that differ it from well-known Uflyand-Mindlin theory.

2.1 Introduction. Uflyand-Mindlin Theory: History of Question Timoshenko pointed out that when the beam oscillates, the contribution of the shear deformation exceeds, sometimes three times, that of the Bresse-Rayleigh inertia of rotation (Bresse 1859; Germain 1962; Grigolyuk and Selezov 1973; Lamb 1917; Lord 1889). The generalization of Timoshenko-Ehrenfest theory, which incorporates shear effect to plate deformation, has been developed in the 40’s, in static setting. Reissner (1944, 1945, 1975) presented a stress-based theory which accounts for shear effect in static setting (see detailed analysis of Reissner’s theory in Kromm (1953)). Reissner obtained the following PDE for the deflection of an isotropic plate: D2 W = p −

2h 2 (2 − υ) p, 5(1 − υ)

(2.1)

3

2h where D = 3(1−ν 2 ) ; 2h is the thickness of plate; W is the normal displacement of plate; E is Young’s modulus; ν is Poisson’s coefficient; p is the external load; 2 ∂2  = ∂∂X 2 + ∂Y 2 ; X, Y are the Cartesian coordinates.

I. V. Andrianov (B) Chair and Institute of General Mechanics, RWTH Aachen University, Eilfschornsteinstraße 18, 52062 Aachen, Germany e-mail: [email protected] I. Elishakoff Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_2

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Furthermore, Bolle (1947) also introduced a three-field kinematic theory (one deflection and two independent rotations), which is the natural generalization of the Bresse-Timoshenko beam model at the plate level. Related works on shear plate theories are the works of Hencky (1947) who developed the bending equations of a plate with shear effect, based on a three-field kinematic theory (one deflection and two independent rotations): D2 W = p −

2h 2 p. 3(1 − υ)

(2.2)

This theory can be treated as a generalization to the plate the Bresse theory based on assumption of shear correction coefficient being equal unity, k = 1. Uflyand (1948) derived the general static and dynamic plate equations with a three-field kinematics:   2 3D ∂ W 2ρh 3 ∂2W 2ρ 2 h 3 ∂ 4 W 2 1+ 3 + 2ρh + D W − 3 2h kG ∂t 2 ∂t 2 3kG ∂t 4 3 2 ρh ∂ p D p + = p− . (2.3) 2kGh 3kG ∂t 2 Here, t is the time; ρ is the material density; G is the shear modulus. Uflyand (Uflyand 1948) chose k = 2/3, as considered earlier by Timoshenko (Grigolyuk and Selezov 1973) for beam problems. Equation (2.3) derived by Uflyand has been also confirmed later by Mindlin (1951, 1960), through variational arguments. Mindlin also calibrated the shear correction factor from the exact antisymmetric vibration mode of thickness-shear vibration and found k=

π2 . 12

(2.4)

The calibration of the shear correction factor of the so-called Uflyand-Mindlin plate model from three-dimensional elasticity has been performed analytically by Hutchinson (2001), Wittrick (1987), and Stephen (1997) by comparing the natural frequencies of the three-dimensional solution with the one of the Uflyand-Mindlin plate model. Batista (2010) used some hypotheses for the construction of the same equations. They obtained the shear correction factor valid for plates: k=

5 . 6−ν

(2.5)

This shear correction factor may be also obtained from the paper by o Timoshenko (1922) (Grigolyuk and Selezov 1973) by considering the plane strain assumptions: 

5 ν 5(1 + ν) ⇒k= ν= and k = 1 − ν 6 + 5ν 6 − ν

(2.6)

2 Shamrovskii’s Version of the Refined Dynamical Plate Theory

25

Such a correspondence between the shear correction factor of the beam model and the one of the plate models was also observed by (Stephen 1997). (Hutchinson 2001) obtained this shear correction factor from the vibration analysis of a circular plate, whereas Wittrick (1987) considered the free vibration of a simply supported rectangular plate. Stephen (Stephen 1997) derived this value from plane strain exact Rayleigh-Lamb frequency equation for infinite plate. This shear correction factor was also numerically obtained by comparison of the three-dimensional elasticity solution with Uflyand-Mindlin plate model by (Srinivas et al. 1970) and (Dawe 1979) who obtained a shear correction factor k ≈ 0.88 for ν = 0.3 which is exactly the value given by the formulae (5). Goldenveizer et al. (1993) used an asymptotic expansion of the mixed displacement and stress field with respect to some small parameters linked to the relative depth of the plates (or shells) with shear effects. They also obtained the shear correction factor given by Eq. (2.5). More recently, Elishakoff et al. (2017) showed from an asymptotic expansion of the displacement field (solely) that the asymptotically based shear correction factor is the one given by Eq. (2.5). They also derived a truncated Uflyand-Mindlin plate model issued of the three-dimensional elasticity governing equations:   2 3D ∂ W 2ρh 3 ∂2W 1+ 3 + 2ρh = 0. D W − 3 2h kG ∂t 2 ∂t 2 2

(2.7)

It was shown that Eq. (2.7) is the plate analogy of the truncated Bresse-Timoshenko equation derived from asymptotic arguments; interested readers can consult with the book by (Vesnitskii 2001).

2.2 Newton Polygon and Its Generalizations Let’s briefly describe the technique used by Shamrovskii to build a refined plate theory. It can be interpreted as a generalized multidimensional computerized Newton polygon. Recall the Newton polygon algorithm for a one-dimensional case. Let F be an implicit function. F(ε, z) = 0, F(0, z 0 ) = 0 problem consisting in determining an explicit dependence z(ε) z = z 0 + z 1 εβ1 + z 2 εβ2 + . . . β1 < β2 < β3 < . . . Determination of exponents βi is a problem solved with the help of Newton polygon (the terms Newton diagram, Newton-Puiseux algorithm, etc. are also used) (Andrianov et al. 2014; Bruno 2000; Vainberg and Trenogin 1974). This method and

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its various generalizations is an important stage in the asymptotic analysis of ODEs and PDEs and the construction of various simplified theories (Andrianov et al. 2014; Shamrovskii 1979, 1997; Shamrovskii et al. 2004). Let us now show the generalization of Newton’s polygon to a two-dimensional case. As the toy example, consider the deformation of a membrane on a stiff WinklerFuss foundation, described by the equation:   ε wx x + w yy − w = 0, ε 0, 0 < r < l(t), ∂t r ∂r

(5.1)

where q(r, t) is the fluid flow rate. Integrating over the space–time domain yields the global fluid balance equation: 

L(t)

2π

 r (w(t, r ) − w(0, r )) dr =

0

t

q0 (τ ) dτ .

(5.2)

0

The (Newtonian) fluid flow is assumed to be laminar, and as such q is given by the Poiseuille equation: q=−

1 3∂p w , t > 0, 0 < r < l(t), M ∂r

(5.3)

where M = 12μ and μ is the fluid viscosity. In practical application, it is preferable not to use (5.3) as it degenerates at the crack tip (where w → 0 and dp/dr → −∞). Instead, we utilise the approach based on the speed equation, previously introduced and used in Linkov (2012), Wrobel and Mishuris (2015), Peck et al. (2018a, b). We begin by defining the fluid velocity: v=

1 ∂p q = − w2 , t > 0, 0 < r < l(t). w M ∂r

(5.4)

In the absence of fluid lag, and assuming the fluid leak-off is bounded, the fluid velocity matches that of the crack tip, yielding the speed equation:

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dL = v (t, L(t)) , t > 0. dt

(5.5)

Note that, while (5.4) still degenerates at the crack tip, we can utilise (5.5) to evaluate the fluid velocity at this point. As such, with proper application of the tip asymptotics, all issues related to the degeneration at the crack tip are eliminated. While the above equations did not require any modification to incorporate the variable fracture toughness, the remaining equations require some small modification. The crack extension is considered in terms of linear elastic fracture mechanics, taking the form of the Irwin criterion: K I (t) = K I c (L(t)) ,

(5.6)

where K I is the mode-I stress intensity factor, given by  K I (t) = 2

L(t) π

 0

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−

s2

s ds. L(t)

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It should be noted that criterion (5.6) means that the fracture does not stop expanding at any point, although its rate of growth can decrease close to zero. This can largely be assumed, as we do not decrease the pumping rate during the process. It may however have a small effect on the result during the first instance of the crack tip encountering a significantly tougher layer. Finally, the solid and fluid phases are related by the elasticity equation, which is taken in the form presented in Peck et al. (2018a):     r ∂p L 2 (t) − r 2 8L(t) L(t) s 4 , (t, s) ds +  K I C (L(t)) w(t, r ) = K , πE 0 L(t) L(t) ∂s E πL(t)  

 

w1 (t,r )

w2 (t,r )

(5.8) where ⎧      ⎨η E arcsin(η)| ξ 22 − E arcsin( η )| ξ 22 , η < ξ, η ξ η      K (η, ξ) = ⎩η E arcsin(η)| ξ 22 − E ξ 22 , η > ξ, η η

(5.9)

and E is the incomplete elliptic integral of the second kind. Note that in (5.8), the term w1 (t, r ) describes the effect of the (viscous) fluid pressure on the fracture walls, while w2 (t, r ) describes the impact of the material toughness. As such, the ratio of the size of these two terms can be used as a rough measure for whether the fracture is in the toughness-, transient- or viscosity-dominated regime at a specific point in time. This approach to parameterising the fracture regime will be used in the investigation to follow, similar to that utilised in Da Fies et al. (2022), with the details provided in Sect. 5.2.3.

5 Averaging-Based Approach to Toughness Homogenisation …

75

5.2.2 Form of the Material Toughness Equations (5.1)–(5.9) now provide a complete description of the relations between the various aspects of the radial hydraulic fracture in the case with (pre-defined) heterogeneous material toughness. The aim now is to investigate the effect of inhomogeneous toughness on the fracture behaviour, and the relative effectiveness of various proposed homogenisation strategies. For the sake of simplicity, the toughness is assumed to be axisymmetric in nature, such that the value of K I c (r ) is independent of the angle θ. The domain is assumed to consist of layered materials, such that the toughness is periodic in space, with period X . The distribution of the toughness over this period is primarily defined in terms of its maximum and minimum values 0 < K min < K max < ∞ where K min = min K I c (r ),

K max = max K I c (r ).

0≤r d L . ⎩ t (L) − t (L − d L) t (L−d L)

(5.18)

For completeness, we also include the progressive measure, denoted with subscript p, which is given by  K 2∗p (L)

=

1 t (L + d L)



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  K I2C L(ξ) dξ.

(5.19)

Examples of the homogenised values of K I c obtained by measures (5.16)–(5.19), alongside the maximum toughness, are provided in Fig. 5.13.

5.3.2 Results for Balanced Toughness Distributions With the model in place and the various measures established, we can now begin an investigation of the effectiveness of each homogenisation strategy for the fracture toughness. We start with the case of balanced toughness. Simulations were conducted for periodic toughness with each of the step-wise and sinusoidal distributions, for each of the regime pairings outlined in Table 5.3. These were compared with results

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Fig. 5.13 Results of the measures (5.16)–(5.19) for the periodic toughness distribution in Case 2 (toughness-transient regime) with balanced layering. Here we show a, b the initial behaviour; c, d the long-time behaviour. The toughness distributions are: a, c sinusoidal; b, d step-wise

obtained by simulating the same system, but utilising the toughness distribution obtained by each of the homogenisation strategies: the maximum toughness strategy and the temporal-averaging measures (5.16)–(5.19). The relative difference obtained for the key process parameters, the fracture (half-)length, l(t); the crack opening, w(t, 0) and the pressure at the mid-point, p(t, l(t)/2) are provided in Figs. 5.14, 5.15 and 5.16. The first case, when the fracture propagates in the toughness-toughness regime for almost the entire duration, is shown in Fig. 5.14. Here, the relative error of the maximum toughness strategy exceeds 10% for all three parameters. This improves over time, particularly for the step-wise distribution, but requires more than 102 seconds to decrease below 1% even for the fracture length, and does not achieve this until almost 104 seconds for the fluid pressure. The temporal-averaging measures fair slightly better in all cases, with the maximum relative error remaining just below 10% for the crack length over the entire duration, but exceeding it for the other parameters. However, it should be noted that the results for measures (5.16)–(5.19) are almost as accurate as a homogenisation strategy can be in this case, as the inherent oscillations of the parameters over time will always yield some difference (see Fig. 5.5–5.12), and the measures are already close to the ‘ideal’ average.

5 Averaging-Based Approach to Toughness Homogenisation …

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Fig. 5.14 Results for the toughness-toughness distribution with balanced layering (Case 1, see Table 5.3). Relative differences for the a, b fracture length L(t); c, d fracture opening at the injection point w(t, 0); e, f fluid pressure at the mid-point p(t, L(t)/2)

Meanwhile, the toughness-transient case provided in Fig. 5.15 sees a clear difference in relative error for the two forms of homogenisation strategy considered here. The maximum toughness strategy has a relative error close to or exceeding 10% for all the process parameters, while the temporal-averaging-based approaches remain below this threshold. Note that over long time (after 102 s), the different strategies all yield similar results to those for the toughness-toughness case, as the regime changes over time.

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Fig. 5.15 Results for the toughness-transient distribution with balanced layering (Case 2, see Table 5.3). Relative differences for the a, b fracture length L(t); c, d fracture opening at the injection point w(t, 0); e, f fluid pressure at the mid-point p(t, L(t)/2)

Finally, Fig. 5.16 provides the transient-viscosity case. In this instance, it is clear that all of the homogenisation strategies are reasonably effective, particularly during the early stages, with the error only increasing as the fracture regime changes. It is clear, however, that the temporal-averaging homogenisation strategy is visibly more effective, with the relative error never exceeding 1% for almost all process parameters over the entire duration (only figure (e), for the fluid pressure under sinusoidal toughness distribution, briefly exceeds it).

5 Averaging-Based Approach to Toughness Homogenisation …

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Fig. 5.16 Results for the transient-viscosity distribution with balanced layering (Case 3, see Table 5.3). Relative differences for the a, b fracture length L(t); c, d fracture opening at the injection point w(t, 0); e, f fluid pressure at the mid-point p(t, L(t)/2)

To summarise, the approaches based on temporal averaging remain consistently more effective than the maximum toughness strategy, although the difference is less than that seen for the KGD model. For the toughness-toughness case, and to some extent the toughness-transient case, the measures based on temporal-averaging approach the ideal limit of what can be achieved by any homogenisation strategy, with the error largely resulting from the inherent oscillation of the system parameters. For cracks with one region in the viscosity-dominated regime, the regime change over time for penny-shaped cracks does play a notable role, as can be seen clearly in

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Fig. 5.17 Relative difference in the fracture (half-)length L(t) for the toughness-transient distribution with unbalanced layering (Case 2, see Table 5.3). Relative differences for a, b h = 0.25 (see (5.11)); c, d h = 0.1, e, f h = 0.01

Fig. 5.16. As the regime changes to the transient-toughness case (see Table 5.2), the effectiveness of the maximum toughness strategy decreases significantly, although the relative error for the crack length remains below 4%. The temporal-averagingbased approach is notably less affected by this.

5 Averaging-Based Approach to Toughness Homogenisation …

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5.3.3 Results for Unbalanced Layering With the effectiveness of the various homogenisation strategies evaluated for the case where the differing material layers are evenly distributed, now we move onto the case where the maximum toughness layer represents a smaller portion of the total material. This degree to which the material layering in ‘unbalanced’ is dictated by the constant 0 < h < 1, defined in (5.11), and represents the extent to which the average toughness of the material (over space) is closer to the maximum toughness

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Fig. 5.19 Relative difference in the fluid pressure at the mid-point p(t, L(t)/2) for the toughnesstransient distribution with unbalanced layering (Case 2, see Table 5.3). Relative differences for a, b h = 0.25 (see (5.11)); c, d h = 0.1, e, f h = 0.01

(as h → 1) or the minimum toughness (as h → 0) of the heterogeneous material. We only consider the latter case, with h < 0.5. To obtain the fullest possible picture of the effect of the unbalanced layering on the effectiveness of the various homogenisation strategies, we split the investigation into two parts. First, we will analyse the effect of varying h when keeping the maximum and minimum toughness fixed (the toughness-transient case). Next, we will keep h fixed at an arbitrarily low value, h = 0.01 and examine the impact of the regime within which the crack is initially propagating on the homogenisation strategies (varying K max , K min ).

5 Averaging-Based Approach to Toughness Homogenisation …

5.3.3.1

97

Toughness-transient Layering with Unbalanced Toughness

We begin by examining the effect of varying degrees of unbalanced layering on the effectiveness of the different homogenisation strategies. The relative difference between the solution obtained for the periodic toughness distribution, taking h = 0.01, 0.1, 0.25, and those obtained using the various homogenisation strategies (maximum toughness, temporal averaging) is provided in Figs. 5.17, 5.18 and 5.19. From the figures, the effectiveness of the maximum toughness strategy does not appear to depend on the width of the layering, for either the sinusoidal or step-wise distributions, even in the case with a significant imbalance (h = 0.01). This is in line with the previous observations about the fracture behaviour for the unbalanced case, discussed in Sect. 5.2.4, and again we note that it may not hold when considering h > 0.5 (in fact, it is likely that the maximum toughness strategy would become almost optimal as h → 1). The same appears to hold true for the temporal-averaging-based approaches (5.16)–(5.19), with the unbalanced layering having only a negligible impact on the effectiveness of the homogenisation strategy in the toughness-transient case.

5.3.3.2

The Case with a Minimal Width Toughness Layer

Next, we examine the case where the average toughness of the heterogeneous material is almost identical to that of the minimum toughness, i.e. when the maximum toughness layer is so thin as to seem to be negligible. We do this by taking the fixed value of h = 0.01 (see (5.11)), and again investigating the relative difference between the process parameters for the periodic toughness simulation, and that estimated by the various homogenisation strategies. The resulting relative differences for the toughness-toughness, toughness-transient and transient-viscosity regimes are provided in Figs. 5.20, 5.21 and 5.22. It is clear from a comparison of Figs. 5.14 and 5.15 with Figs. 5.20 and 5.21 that the effect of unbalanced layering, even in the extreme example considered here, is negligible in both the toughness-toughness and (initially) toughness-transient regimes. This is in line with the results of the previous subsection, and the observations about the process behaviour given in Sect. 5.2.4. In the transient-viscosity case, however, the difference is far more significant, as can be seen when comparing Fig. 5.22 with Fig. 5.16. Alongside the expected ‘smoothing’ of the relative error (due to reduced oscillation of the system parameters), the error of the maximum toughness strategy does increase slightly, for instance, the relative error for the pressure exceeds 10% for the unbalanced layering, but not for the balanced layering (compare Fig. 5.16e,f with Fig. 5.22e,f). A similar impact can be seen for the temporal-averaging measures; however, the relative error for all parameters never exceeds 3% (where previously it was 2%). However, there is also a significant improvement in the effectiveness of measures (5.16)–(5.19) during the initial stages of the fracture, with the relative error for the aperture and pressure after the first period remaining below 0.1% for the first second,

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Fig. 5.20 Results for the toughness-toughness distribution with unbalanced layering (h = 0.01, Case 1, see Table 5.3). Relative differences for the a, b fracture length L(t); c, d fracture opening at the injection point w(t, 0); e, f fluid pressure at the mid-point p(t, L(t)/2)

where previously it was below 1%, while the error of the length only slightly exceeds 0.1% during this time. Combining the results of the previous two subsections, we can conclude that the temporal-averaging approach remains effective irrespective of the unbalanced layering. The maximum toughness strategy is typically only negligibly effected, except in the transient-viscosity-dominated regime where there is a slight decrease in effectiveness.

5 Averaging-Based Approach to Toughness Homogenisation …

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5.4 Conclusions A variety of homogenisation strategies for the material toughness have been considered, in the context of hydraulic fracture (HF) of rock with a periodic toughness distribution. The strategies considered took two different forms. The first was the maximum toughness strategy proposed in Dontsov and Suarez-Rivera (2021), while the second utilised the concept of temporal averaging previously proposed by the authors Da Fies et al. (2022), and provided here in (5.16)–(5.19). A wide range of peri-

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odic toughness distributions were considered, including differing fracture regimes, unbalanced material distributions and taking both step-wise and sinusoidal forms. The investigation was able to demonstrate the following: • The temporal-averaging approach to homogenising the fracture toughness was consistently more effective than the maximum toughness strategy for the radial model of hydraulic fracture (as has previously been demonstrated for plane strain HF (Da Fies et al. 2022)). The largest difference between the two was seen for

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a fracture starting in the transient-viscosity-dominated regime, while that for the toughness-toughness distribution was far smaller (see Figs. 5.14, 5.15 and 5.16). • The effectiveness of both measures improved with time (after the first 2–3 toughness periods) for the toughness-toughness and toughness-transient distributions, but not for the transient-viscosity distribution. Here, the transition from the viscosity-to-transient distribution caused an increase in the relative error of key process parameters for the maximum toughness strategy, while the temporal-averaging approach remained more consistent. • The results above also hold for the case with unbalanced layering, where the maximum toughness layer only makes up a small portion of the material. The likely physical explanation for this was outlined in Sect. 5.2.4. It was demonstrated that even in the case of an extreme imbalance (h = 0.01, see (5.11)), the only effect was a small increase in the relative error of the approximation, with the maximum toughness strategy more adversely affected than the temporal-averaging approach. It is, however, unlikely that these results would hold if the maximum toughness material instead made up the majority of the heterogeneous body (h > 0.5). • Interestingly, in the case with an extremely unbalanced layering of the rock strata, it was still the transition between viscosity- and toughness-dominated regimes which played the crucial role in determining the effectiveness of the homogenisation method. As such, the only significant effect of unbalanced layering on the homogenisation methods was through its influence on the fracture evolution (for cracks with one layer starting in the viscosity-dominated regime). It should be restated that the temporal-averaging homogenisation strategy utilised in this work is not dependent upon the periodic distributions examined here and should continue to be effective for disordered toughness distributions as well. One factor of the presented analysis that should be highlighted is that the principles behind the toughness homogenisation strategy outlined here are not particular to the hydraulic fracture process. This means that they should continue to provide an effective homogenisation strategy for any steady-state propagation process within a heterogeneous media. While these results indicate the effectiveness of the temporal-averaging homogenisation strategy for a system only experiencing toughness heterogeneity, there are a number of crucial effects which should still be considered for HF. Most notably, materials with toughness heterogeneity are also likely to experience heterogeneity of the elastic parameters (a significant challenge, see, e.g. Hossain et al. (2014)), the fluid leak-off and potential differences in the in situ stress (stress barrier, see, e.g. Dontsov (2022)). Additionally, the step-wise fracture advancement present even in homogeneous media (see, e.g. Cao et al. 2017) could have an even larger impact when coupled with the effect of toughness heterogeneity. The fact that the utilised measures (5.16)–(5.19) incorporate the instantaneous fracture velocity, v(L), is however promising in this regard. The dependence on the instantaneous velocity does however lead to the most significant open question regarding the proposed strategy. For this work, the results of existing numerical simulations were used to provide the instantaneous velocity used

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to compute the average toughness. In HF, however, such data is not typically available in real time. Developing an effective approach to computing (or approximating) measures (5.16)–(5.19) in the general case remains an open problem. Funding The authors have been funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement Effect Fact No 101008140 and the Welsh Government via Sˆer Cymru Future Generations Industrial Fellowship grant AU224. Acknowledgements MD acknowledges the Royal Academy of Engineering for the Industrial Fellowship. The authors would like to thank Prof. Gennady Mishuris for his continuous interest and helpful discussions during the course of this work.

References Aiuppa A, Bitetto M, Donne D, La Monica F, Tamburello G, Coppola D, Schiava M, Innocenti L, Lacanna G, Laiolo M, Massimetti F, Pistolesi M, Silengo M, Ripepe M (2021) Volcanic CO2 tracks the incubation period of basaltic paroxysms. Sci Adv 7(38):eabh0191 Bai M (2011) Improved understanding of fracturing tight-shale gas formations, volume all days of . In: SPE Oklahoma City oil and gas symposium/production and operations symposium. SPE140968-MS Brunet J, Pierrat B, Badel P (2021) Review of current advances in the mechanical description and quantification of aortic dissection mechanisms. IEEE Rev Biomed Eng 14:240–255 Caiulo A, Kachanov M (2010) On absence of quantitative correlations between strength and stiffness in microcracking materials. Int J Fract 164(4):155–158 Cao T, Milanese E, Remij E, Rizzato P, Remmers J, Simoni L, Huyghe J, Hussain F, Schrefler B (2017) Interaction between crack tip advancement and fluid flow in fracturing saturated porous media. Mech Res Commun 80:24–37 Charalambakis N (2010) Homogenization techniques and micromechanics. A survey and perspectives. Appl Mech Rev 63(3):030803 Criado F (2011) Aortic dissection: a 250-year perspective. Texas Heart Inst J 38(6):694–700 Da Fies G (2020) Effective time-space adaptive algorithm for hydraulic fracturing. Published PhD thesis, Aberystwth University Da Fies G, Dutko M, Mishuris G (2021) Remarks on dealing with toughness heterogeneity in modelling of hydraulic fracture. In: U.S. rock mechanics/geomechanics symposium. ARMA2021–2010 Da Fies G, Peck D, Dutko M, Mishuris G (2022) A temporal-averaging based approach to toughness homogenisation in heterogeneous material. Math Mech Solids (in press). https://doi.org/10.1177/ 10812865221117553 Dontsov E (2022) A continuous fracture front tracking algorithm with multi layer tip elements (multipel) for a plane strain hydraulic fracture. J Pet Sci Eng 217:110841 Dontsov E, Suarez-Rivera R (2021) Representation of high resolution rock properties on a coarser grid for hydraulic fracture modeling. J Pet Sci Eng 198:108144 Economides M, Nolte K (2000) Reservoir simulation. John Willey & Sons, Chichester Garagash D (2009) Scaling of physical processes in fluid-driven fracture: Perspective from the tip. In: Borodich F (ed) IUTAM symposium on scaling in solid mechanics. Springer, Dordrecht, Netherlands, pp 91–100 Garagash D, Detournay E (2000) The tip region of a fluid-driven fracture in an elastic medium. J Appl Mech 67:183–192

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Gültekin O, Hager S, Dal H, Holzapfel G (2019) Computational modeling of progressive damage and rupture in fibrous biological tissues: application to aortic dissection. Biomech Model Mechanobiol 18:1607–1628 Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(2):127–140 Hossain M, Hsueh C-J, Bourdin B, Bhattacharya K (2014) Effective toughness of heterogeneous media. J Mech Phys Solids 71:15–32 Kachanov M (1994) On the concept of damage in creep and in the brittle-elastic range. Int J Damage Mech 3(4):329–337 King G (2010) Thirty years of gas shale fracturing: what have we learned? volume all days of. In: SPE annual technical conference and exhibition. SPE-133456-MS Lecampion B, Zia H (2019) Slickwater hydraulic fracture propagation: near-tip and radial geometry solutions. J Fluid Mech 880 Linkov A (2012) On efficient simulation of hydraulic fracturing in terms of particle velocity. Int J Eng Sci 52:77–88 Nienaber C, Clough R, Sakalihasan N, Suzuki T, Gibbs R, Mussa F, Jenkins M, Thompson M, Evangelista A, Yeh J, Cheshire N, Rosendahl U, Pepper J (2016) Aortic dissection. Nat Rev Dis Primers 2:16053 Peck D, Da Fies G (2022a) Impact of the tangential traction for radial hydraulic fracture. Eur J Mech/A Solids 104896. https://doi.org/10.1016/j.euromechsol.2022.104896. Peck D, Da Fies G (2022b) Impact of the tangential traction for radial hydraulic fracture. ArXiV. arXiv:2210.00046 Peck D, Wrobel M, Perkowska M, Mishuris G (2018) Fluid velocity based simulation of hydraulic fracture: a penny shaped model-part I: the numerical algorithm. Meccanica 53:3615–3635 Peck D, Wrobel M, Perkowska M, Mishuris G (2018) Fluid velocity based simulation of hydraulic fracture: a penny shaped model-part II: new, accurate semi-analytical benchmarks for an impermeable solid. Meccanica 53:3637–3650 Peirce A, Detournay E (2008) An implicit level set method for modeling hydraulically driven fractures. Comput Methods Appl Math Eng 197:2858–2885 Peruzzo C, Cao D, Milanese E, Favia P, Pesavento F, Hussain F, Schrefler B (2019) Dynamics of fracturing saturated porous media and self-organization of rupture. Eur J Mech-A/Solids 74:471– 484 Peruzzo C, Simoni L, Schrefler B (2019) On stepwise advancement of fractures and pressure oscillations in saturated porous media. Eng Fract Mech 215:246–250 Savitski A, Detournay E (2002) Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions. Int J Solids Struct 39:6311–6337 Unwin H, Tuffen H, Phillips E, Wadsworth F, James M (2021) Pressure-driven opening and filling of a volcanic hydrofracture recorded by tuffisite at húsafell, iceland: a potential seismic source. Front Earth Sci 9 Wrobel M, Mishuris G (2015) Hydraulic fracture revisited: particle velocity based simulation. Int J Eng Sci 94:23–58 Wrobel M, Mishuris G, Piccolroaz A (2017) Energy release rate in hydraulic fracture: can we neglect an impact of the hydraulically induced shear stress? Int J Eng Sci 111(28–51):1

Chapter 6

Towards Multi-Angle Multi-Channel Optical Porometry and Scanning Electron Microscopic Porometry of LDPE Composites Including Geotechnical Biodegradable Ones Elena Grigorieva, Oleg Gradov , Margaret Gradova , and Irina Maklakova Abstract This report describes novel methods of multi-angle multi-channel optical porometry (both in lensless and non-lensless versions) and scanning electron microscopic porometry, indicating the feasibility of their technical and algorithmic integration within the framework of CLEM (Correlative Light and Electron Microscopy) by creating correlative light and electron porometry. The data on the reconstruction of the surface texture of a complex pore using an incoherent light source and the Sobel– Feldman operator is presented. Examples of optical density and luminance isolines obtained for spectrozonal multiplexing of porometric images using Orekhov’s technology are given. The image processing methods described in this report reconstruct and visualize the fillable volume of pores and the heterogeneity of their surface, which makes it possible to use the above methods as complementary data sources for volumetric porosimetry. Keywords LDPE · Porometry · SEM · CLEM · Multi-channel microscopy

6.1 Introduction Porometry and porosimetry of building materials are the most important areas of their qualimetry, which determine the micromechanical properties of the components of buildings and engineering structures (including masonry/brickwork Cultrone et al. 2007; Binda et al. 1997), their moisture resistance, resistance to freezing and heating, as well as to drying, and their air permeability/ventilation (Torres and Freitas 2007; Tian et al. 2014; Rawal 2014; Wardeh and Perrin 2006; Flores-Colen et al. 2016). E. Grigorieva · I. Maklakova IBCP RAS, Kosygina Str. 4, Moscow 119991, Russia O. Gradov (B) · M. Gradova · I. Maklakova FRC CP RAS, Kosygina Str. 4, Moscow 119991, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_6

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The above parameters determine the durability of buildings and the possibility of their restoration using new materials that differ from the standard ones in porosity (Ontiveros-Ortega et al. 2018). In addition, the porosity of the material plays a key role in the possibility of its atmospheric chemistry and hydrochemical/hydrothermal weathering (Delgado et al. 2015), leaching of the individual components of the material from the heterogeneous matrix (Makni et al. 2021), and its ability to work as a “chromatographic carrier” (such as zeolites) in ion exchange between the medium and the structural material (Nguyen et al. 2017). The structural advantages of “rigid materials” were the reason why the main group of construction (and potentially applicable in construction) materials studied using porometry was brick, stone, concrete (or reinforced composite) structures, as well as their precursors in the form of mineral aggregates (from clay to lime) or nodules (sand) (Gunnink 1991; Malab et al. 2017). At the same time, for polymer materials in natural conditions (soil, atmosphere, hydrosphere), such studies were performed much less frequently, usually limited to geopolymers (Nguyen et al. 2018) and geotextiles (Aydilek et al. 2005; Beena and Babu 2008; Sanyal 2017; Tang et al. 2020). However, in the first case, we are talking about a rigid geopolymer material (for example, silicates), and in the second, about biogenic fibers (for example, jute fibers). Therefore, it is not possible to talk about a trend in the field of soft matter physics in porometry of polymer construction materials (although, generally speaking, biogenic fibers studied by porometry in a number of papers should be referred to as soft matter Ashrafi et al. 2019; Qiu et al. 2019). At the same time, such polymers as LDPE (Low-Density Polyethylene) are very often used in the construction industry, both independently and as a component of composites (including those obtained from secondary raw materials/wastes) (Azeko et al. 2016; Tuna Kayili et al. 2020; Khan et al. 2021). Building blocks of various sizes are made of them or with their use, including bricks, facing panels, etc. (Khan et al. 2021; Gaggino 2012; Arulrajah et al. 2017; Zhang et al. 2018; Mohan et al. 2020; Dolores et al. 2020; Zulkernain et al. 2021). At the same time, since the 1990s, technologies are being developed using LDPE to improve the quality of not only asphalt (Kim et al. 1998, 1997; Murphy et al. 2001; Panda and Mazumdar 2002; Ho et al. 2006; Punith and Veeraragavan 2007, 2011; Al-Hadidy and Tan 2009; Othman 2010; Brovelli et al. 2014; Karmakar and Roy 2016; Formela et al. 2016; Addissie et al. 2018; Mazouz and Merbouh 2019; Hoque et al. 2019; Celauro et al. 2019; Ameri et al. 2021), but also road surfaces in general, which gives them flexibility and permeability (Al-Hadidy and Yi-qiu 2009; Kalantar et al. 2012; Jeong et al. 2011; Nadirov et al. 2020; Scholz and Grabowiecki 2007). However, it is obvious that permeability is achieved due to porosity, and not due to the properties of a solid monolithic material, since solid LDPE structures are used to create watertight geomembranes for desert regions (Park and Nibras 1993; Menaa et al. 2012). At best, such structures are designed in compliance with the gas permeability requirements (Stark and Choi 2005), at worst (for example, when using LDPE in greenhouses Emekli et al. 2016), conditions favor the accumulation of gas at the boundary between two media separated by the LDPE film, as in the “greenhouse effect”. Therefore, the following studies seem to be extremely necessary even for practical use:

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• in situ analysis of the processes of pore formation and collapse during LDPE heating and thermal cycling up to LDPE melting temperature • a similar analysis of biodegradable LDPE-containing composites (due to the fact that biodegradable compounds for construction purposes are popular now (Silva et al. 2014; García-González et al. 2020), although it is known that the damage to the buildings is induced by microorganisms, the growth of bacteria, algae, and fungi is a serious problem (Bech-Andersen 2006; Barberousse et al. 2007; Tanaca et al. 2011; Grigorieva et al. 2021); • analysis of the stability of the fibrous heat-insulating structures during melting (Manohar et al. 2000; Jerman et al. 2019; Barkhad et al. 2020) (for example, wood flour Väntsi and Kärki 2014; Ratanawilai and Taneerat 2018; Chanhoun et al. 2018; Hao et al. 2019; Ding et al. 2020) and the possibility of their application as visualizing agents of the particle flow, similar to those used earlier in PIV (Westerweel 1997; Grant 1997; Adrian 2005); • electron microscopic analysis of the size and shape of pores (since both SEM methods for analyzing biodegradable composites and polymer-containing structures (Ribas Silva 1995, 1997) and methodological principles of modifying doped coatings with an electron beam for LDPE-asphalt systems (Ahmedzade et al. 2014) have long been known), including those under different gas and vaporphase conditions in ESEM and ASEM modes (Gradov and Gradova 2016), as well as with the fast freezing for the analysis of the molecular structure of porous LDPE (Gradov and Gradova 2014); • sequential or parallel comparison of the results of optical and electron microscopy of LDPE-based structures, including 3D visualization with multi-angle and holographic registration (Gradov 2018a, b, 2019a, b; Gorchenev et al. 2019); • determination of the dynamics of porosity change by means of correlation spectral analysis with the determination of integral frequency characteristics and integral spatial characteristics (previously used by the authors for different systems, from mineral to biorelevant ones (Gradov et al. 2018a, b; Gradov et al. 2019). Dynamic porometry, during the polymer melting, will be considered in detail in our forthcoming paper. In this work, we just demonstrate the data/images of multi-angle porometry, in which the pores are static, and the registration time base corresponds to the change in the light source angle relative to the pores. In this case, the light source can be either incoherent or coherent, which determines data interpretation principles and data processing algorithms. The light receiving detector can be either lens or lensless, that is, either a microscope with an objective or a lensless one, especially a holographic lensless microscope. The latter is compatible with multi-angle porometry by definition.

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6.2 Materials and Methods 6.2.1 Materials and Sample Preparation Composite materials under investigation are based on the low-density polyethylene (LDPE) and wood flour or corn starch. To produce a porous structure of the composite material, a dispersed chemical porogen Hydrocerol BIF (endothermic foaming agent) is used. The gaseous products of hydrocerol thermal decomposition include CO2 and H2 O. The gas produced is distributed within the polymer melt in the form of pores formation, wall rupture, and merging of the cells. Compositions of polyethylene with hydrocerol were obtained by mechanical mixing in the melt using laboratory microrollers with heating. After mixing, the viscous mass was removed from the rollers and subjected to cooling at room temperature until solidification. The final samples in the form of plates were obtained by pressing a hardened crushed mass on a manual laboratory hydraulic press at a temperature of 120 °C.

6.2.2 Dynamical Multi-Channel Multi-Angle Optical Porometry The pore structure was studied using a setup including an inverted trinocular microscope BIOSTAR and UCMOS08000KPB digital USB camera based on 1/2.5 CMOS sensor with a resolution of 3264 × 2448 pixels equipped with Altami Studio software (v. 3.0). Registration was performed with an angular resolution (under illumination from different angles) and scanning the sample in depth (when moving the focus due to the vertical movement of the lens). Further digital processing of the video stream/image series consisted of the volume reconstruction from the serial visualizations, including decomposition by spectral or color channels. In particular, we used RGB binarization and posterization on 3–6 channels, taking into account the negative images. In this case, isoopaque and isophot maps were obtained for the 3D distribution of optical density. In particular, it was possible to cut off the background and analyze the morphology of a single micropore at different angles. Reconstruction of the surface texture of a complex pore when using a non-coherent light source was carried out using the gradient Sobel–Feldman operator. This allowed us to provide each projection angle or each depth of field/focusing value with an individual timecode, due to which, at the stage of data decoding, there was no need to use angular and linear displacement encoders.

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6.2.3 Scanning Electron Microscopic Porometry SEM porometry was carried out using a JEOL T-330A scanning electron microscope with the data collection using CANON 590IS digital camera synchronized with the microscope according to the scheme developed by Alexandrov (IBC RAS). For 3D porometry, it is recommended to use multi-angle and multi-axis rotation tables similar to those installed at TESLA BS-300 scanning electron microscopes and special cameras for reading analytical chips implemented on their basis (Gradov 2018a, b). For 2D porometry, one can use Altami Studio software (v. 3.0), which allows for obtaining extended morphometric data on the shape and size of the pores (Maklakova et al. 2021).

6.3 Results 6.3.1 Dynamic Multi-Channel Multi-Angle Optical Porometry Two examples of reconstruction and visualization of the pore shape of LDPE at different angles and focusing are shown in Figs. 6.1 and 6.2. Fragment of the multichannel (color-multiplexed) isoline reconstruction of a simpler pore in different modes, illustrating the principles of scanning multi-angle porometry, are shown in Figs. 6.3 and 6.4.

6.3.2 Scanning Electron Microscopic Porometry SEM-assisted porometry begins with the electron microscopy study of the porous material. Figure 6.5 shows SEM images of pores in foamed polyethylene containing 5 wt.% of hydrocerol (a, b), 5 wt.% of hydrocerol and 15 wt.% of the wood flour, fraction 0–80 µm (c, d), 5 wt.% of hydrocerol and 15 wt.% of the wood flour, fraction 0–200 µm (e, f). It can be seen that the pore size, shape, and surface texture depend on the nature and fraction of the additives in the polymer composite.

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Fig. 6.1 Reconstruction of the surface texture of a complex pore using an incoherent light source and the Sobel–Feldman operator. In the upper corner, the original images (screenshots) are given, and the timecode is overlaid on the image processing data (visualization option A)

6.4 Conclusions The variety of morphology and anisotropy of the pores in polymer composites makes the application of a multi-angle three-dimensional visualization approach necessary and uncontested. However, the existing methods of electron microscopy, except electron tomography, fail to provide such opportunities. Therefore, it is recommended to integrate multi-angle multi-channel optical porometry and scanning electron microscopic porometry into CLEM (correlative light and electron microscopy), resulting in the emergence of a new field of microscopic methods for morphological control of the polymer composites, including construction materials, based on SCLEP (spatially-correlated light and electron porometry).

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Fig. 6.2 Reconstruction of the surface texture of a complex pore using an incoherent radiation source and the Sobel–Feldman operator. In the upper corner, the original images (screenshots) are given with the timecode in the bottom (visualization option B)

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Fig. 6.3 Isoline mapping of the vertical scan (optical slicing) of the singular pore

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Fig. 6.4 Isoline mapping of the angular scan (optical slicing) of the singular pore

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Fig. 6.5 SEM images of pores in foamed polyethylene with 5 wt.% of hydrocerol (a, b), 5 wt.% of hydrocerol and 15 wt.% of the wood flour, fraction 0–80 µm (c, d), and 5 wt.% of hydrocerol and 15 wt.% of the wood flour, fraction 0–200 µm (e, f)

Acknowledgements The authors gratefully acknowledge the technical assistance of M.K. Filippov and P.L. Alexandrov in the restoration and automation of the SEM instrument.

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Chapter 7

Nonlinear Deformations of Anisotropic Elastic Bodies with Distributed Dislocations Evgeniya V. Goloveshkina and Leonid M. Zubov

Abstract We discuss the Lamé problem on large deformations of an elastic hollow circular cylinder made of an anisotropic material with continuously distributed dislocations. For the distribution of edge dislocations, an exact solution is found in an explicit analytical form. The interaction of dislocations with external and internal hydrostatic pressures is studied. It is shown that scalar densities of the dislocation density tensor can be arbitrary, including the Dirac delta function. This density is used to model dislocations concentrated on a cylindrical surface inside a cylinder. Keywords Nonlinear elasticity · Anisotropic medium · Circular cylinder · Dislocations · Exact solution

7.1 Introduction Consideration of a microstructure of a deformable body leads to various generalized models of a continuum. Among the generalized models, the theory of continuously distributed dislocations takes an important place; see, for example, Forest and Sedláˇcek (2003), Clayton et al. (2006), Forest (2008), Altenbach et al. (2011), Eremeyev et al. (2013), Altenbach and Eremeyev (2013), and Altenbach and Eremeyev (2014). The nonlinear continuum theory of dislocations is discussed in the works Kondo (1952), Bilby et al. (1955), Eshelby (1956), Kröner (1960), and Berdichevsky and Sedov (1967). Its further development can be traced in the papers Le and Stumpf (1996), Clayton (2011), Derezin and Zubov (2011), Teodosiu (2013), and Goloveshkina and Zubov (2019). Evgeniya V. Goloveshkina (B) Rostov State Transport University, Rostovskogo Strelkovogo Polka Narodnogo Opolcheniya Sq. 2, 344038 Rostov-on-Don, Russia e-mail: [email protected] Leonid M. Zubov Institute of Mathematics, Mechanics, and Computer Science of Southern Federal University, Milchakova Str. 8a, 344090 Rostov-on-Don, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_7

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The mechanical properties of a medium are largely determined by its symmetry. The symmetry of crystals and their physical anisotropy are associated with a structure of a crystal lattice. The presence of structural defects in the form of dislocations is a common phenomenon in a crystalline body. One of the mechanisms for the appearance of dislocations in crystals is their growth. Using dislocation distributions, it is possible to represent other defects in microstructured materials. Note that micromechanics and additive manufacturing, which are one of the applications of the dislocation theory, are an important part of the scientific activity of Prof. Igor Sevostianov (see, for example, Omari and Sevostianov 2016; Kachanov and Sevostianov 2018). It is advisable to use dislocation models to describe such phenomena as crystal growth, fatigue, fracture, plastic flow, and inelasticity. Crystals are characterized by significant anisotropy. Tungsten and aluminum are slightly anisotropic. Cubic and hexagonal crystals (for example, zinc, cadmium, and graphite) have more pronounced anisotropic properties. Parameters characterizing the elastic properties of various crystalline materials are presented in Huntington (1958), Theocaris and Philippidis (1992), Theocaris (1994), Hayes and Shuvalov (1998), and Cazzani and Rovati (2005). The elasticity of anisotropic materials is considered, for example, in the works Lekhnitskii (1963), Ostrosablin (1992), Kalinin and Bayuk (1994), Boulanger and Hayes (1995), Ting (2005), and Cazzani and Rovati (2003). Large deformations of some anisotropic bodies were investigated in Levin et al. (2018). In this work, we solve the Lamé problem for an elastic hollow circular cylinder made of an anisotropic material with dislocations. In what follows, we assume large deformations and consider continuously distributed dislocations. In fact, the number of dislocations in solids is rather large, so the latter assumption is reasonable. In other words, we discuss the problem under consideration within the framework of the continuum theory of continuously distributed dislocations. For the distribution of edge dislocations, an exact solution in an explicit analytical form is found. The interaction of dislocations with external and internal hydrostatic pressures is studied. It is shown that scalar densities of the dislocation density tensor, which determines the distribution of dislocations, can be arbitrary. For example, a scalar density of the form of the Dirac delta function corresponds to the distribution of dislocations over the surface inside an elastic body. We also investigate dislocations concentrated on a cylindrical surface inside the cylinder, as well as on the inner and outer surfaces. The obtained exact solution complements the few exact solutions given in Zelenina and Zubov (2009), Yavary and Goriely (2012), Zelenina and Zubov (2013), Zubov (2014), Zhbanova and Zubov (2016), Goloveshkina and Zubov (2019), and Goloveshkina and Zubov (2021) within the nonlinear continuum theory of dislocations, which make it possible to detect the quantitative and qualitative effects of deformation of bodies with dislocations. A number of exact solutions for an isotropic cylinder with dislocations were obtained in Zelenina and Zubov (2013) and Zubov (2020).

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7.2 Input Relations If dislocations with the tensor density α are distributed in the body, then the vector field R(r) does not exist, and the geometric relations for the deformation tensor C C = grad R(r)

(7.1)

should be replaced by the so-called incompatibility equation (Zubov 2004, 2011) curl C = α.

(7.2)

Here, α must satisfy the solenoidality condition div α = 0 .

(7.3)

In this case, tensor C is called the distortion tensor. If the tensor field of the dislocation density, α is assumed to be given, then the incompatibility equation (7.1), together with the equilibrium equation (Lurie 1990; Ogden 1997) divD = 0, (7.4) the constitutive equations of elastic material D(C) = dW (G)/dC,

G = C · CT ,

(7.5)

and the boundary conditions allows us to determine the tensor distortion field, and, consequently, the stress field in the body. In (7.4) and (7.5), D is the asymmetric Piola stress tensor, G is the metric tensor, and W is the specific strain energy. Mass forces are not taken into account hereinafter. Note that for α = 0, vector field R(r) does not exist. Operators of gradient, curl, and divergence in curvilinear coordinates q n = q n (x1 , x2 , x3 ), n = 1, 2, 3, of the reference configuration are expressed by the formulas (Lurie 1990; Eremeyev et al. 2018) ∂ ∂ , curl  = rn × n , n ∂q ∂q n ∂q ∂ , div  = rn · n , rn = ik ∂q ∂ xk grad  = rn ⊗

(7.6)

where  is an arbitrary differentiable tensor field of any order, ·, ⊗, and × are the dot, dyadic, and cross products, respectively.

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7.3 Cylindrical Tube with Distributed Dislocations Let us introduce in the reference configuration of a medium the cylindrical coordinates r , φ, and z, connected with the Cartesian coordinates by the relations x1 = r cos φ, x2 = r sin φ, x3 = z. As a vector basis, we will use the unit vectors er , eφ , and i3 , which are directed tangentially to the coordinate lines. Consider an elastic body in the form of a hollow circular cylinder with an outer radius r0 , an inner radius r1 , and an axis parallel to the vector i3 . The following formulas are valid er = i1 cos φ + i2 sin φ, eφ = −i1 sin φ + i2 cos φ . Let us assume that the dislocation density tensor is given in the following form: α = αr (r )er ⊗ er + α φ (r )eφ ⊗ eφ + α zφ (r )i3 ⊗ eφ + α z (r )i3 ⊗ i3 .

(7.7)

The functions αr (r ), α φ (r ), and α z (r ) are the scalar densities of screw dislocations in the radial, azimuthal, and axial directions, respectively, and the function α zφ (r ) is the scalar density of edge dislocations. Equation of solenoidality (7.3) applied to (7.7) leads to one equation d (r αr ) . (7.8) αφ = dr It follows from (7.8) that the solenoidality condition for the dislocation density tensor does not impose any restrictions on the functions α zφ (r ) and α z (r ), so they can be arbitrary. It makes sense to look for the distortion and stress tensor fields, according to (7.7), in the form D = Dr (r )er ⊗ er + Dφ (r )eφ ⊗ eφ + Dz (r )i3 ⊗ i3 + Dϕz (r )eϕ ⊗ i3 +Dzϕ (r )i3 ⊗ eϕ C = Cr (r )er ⊗ er + Cφ (r )eφ ⊗ eφ + C z (r )i3 ⊗ i3 + Cϕz (r )eϕ ⊗ i3

(7.9)

+C zϕ (r )i3 ⊗ eϕ .

(7.10)

Let us now consider the equilibrium of a cylindrical tube in the case where αr = α φ = α z = 0. The dislocation density tensor now has the form α = β(r )i3 ⊗ eφ and describes the distribution of edge dislocations whose axes are parallel to the cylinder axis i3 . The scalar dislocation density β(r ) is an arbitrary function. The distortion tensor becomes C = Cr (r )er ⊗ er + Cφ (r )eφ ⊗ eφ + λi3 ⊗ i3 .

(7.11)

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Here, it is assumed that C z = λ, where λ is a real value. The equilibrium equation (7.4) transforms into the equation Dr − Dφ dDr + = 0, dr r

(7.12)

and the incompatibility equation (7.2) takes the form d (rCφ ) − Cr = rβ(r ), β(r ) = i3 · α · eφ . dr

(7.13)

Consider as a nonlinearly elastic material an anisotropic material with the constitutive relations Levin et al. (2018) 1 ν21 ν31 Dr − Dφ − Dz , E1 E2 E3 ν12 1 ν32 Cφ − 1 = − Dr + Dφ − Dz , E1 E2 E3 ν13 ν23 1 Cz − 1 = − Dr − Dφ + Dz . E1 E2 E3 Cr − 1 =

(7.14) (7.15) (7.16)

The elastic constants E m and νsk are related to each other through the expressions E 1 ν21 = E 2 ν12 ,

E 2 ν32 = E 3 ν23 ,

E 3 ν13 = E 1 ν31 .

(7.17)

Taking into account that C z = λ = const, we express in (7.16) stress Dz in terms of Dr and Dφ as follows:   ν13 ν23 Dz = E 3 λ − 1 + Dr + Dφ . E1 E2

(7.18)

Substituting Dz from (7.18) into the relations (7.14) and (7.15), we get Cr = 1 + K Dr + L Dφ − ν31 (λ − 1), Cφ = 1 + L Dr + M Dφ − ν32 (λ − 1) .

(7.19) (7.20)

Here, we have introduced the notation 1 − ν32 ν23 1 − ν31 ν13 , M= , E1 E2 ν12 + ν32 ν13 ν21 + ν31 ν23 L=− =− . E1 E2

K =

(7.21) (7.22)

Both expressions of the right side of L are equal to each other according to the equalities (7.17).

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Then, according to (7.19) and (7.20), the incompatibility equation (7.13) can be transformed to the form  d   r 1 + L Dr + M Dφ − ν32 (λ − 1) − 1 − K Dr − L Dφ + ν31 (λ − 1) = rβ(r ) . dr (7.23) Given the connection (7.12) of the stress Dφ with the stress Dr , we obtain this equation in the form r2

  d2 d K (λ − 1)(ν31 − ν32 ) rβ(r ) D Dr = − + . (7.24) D + 3r + 1 − r r 2 dr dr M M M

Its solution is given by the expression  r  r r1λ r2λ ρ −λ1 β(ρ)dρ + ρ −λ2 β(ρ)dρ (λ1 − λ2 )M r1 (λ2 − λ1 )M r1 K K (λ − 1)(ν31 − ν32 ) , λ1 = −1 + , λ2 = −1 − . + K−M M M

Dr = C1 r λ1 + C2 r λ2 +

The integration constants C1 and C2 are determined from the boundary conditions that describe the action of the hydrostatic pressures p0 and p1 on the outer r = r0 and inner r = r1 surfaces of the cylinder, respectively Dr (r0 ) = − p0 λCφ (r0 ),

(7.25)

Dr (r1 ) = − p1 λCφ (r1 ) .

(7.26)

Parameter λ can be found from a given longitudinal force applied to the end face of the cylinder  Q = 2π

r0

Dz (r )r dr .

r1

7.4 Numerical Results Let us carry out a numerical analysis for the cylinder of dimensions r0 = 1, r1 = 0.9 with the scalar dislocation density α 32 = β0 r κ in the case η = 1, ν12 = 0.18, ν21 = 0.23, ν13 = 0.18, ν31 = 0.23, ν23 = 0.5, E 1 = 1, and λ = 0.9.

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7.4.1 Inflation Let us consider the inflation of the cylinder by the hydrostatic pressure p1 = 0.1. Numerical analysis is presented for the scalar dislocation density α 32 = β0 r κ with κ = 2. Stresses and distortions are plotted in Figs. 7.1, 7.2, 7.3 and Figs. 7.4, 7.5, respectively. Negative scalar dislocation densities decrease stresses in absolute value. The sign of the scalar density significantly affects the distortion values. With an increase in the parameter β0 , the distributions of the stress Dr and the distortion Cr over the cylinder thickness become more uniform, which is not the case for the stresses Dφ and Dz , as well as for the distortion Cφ . The distribution of dislocations over a cylindrical surface r = r∗ located inside the cylinder can be modeled with a scalar density β(r ) in the form β(r ) = β0 δ(r − r∗ ),

r1 ≤ r∗ ≤ r0 ,

(7.27)

where δ(r − r∗ ) is the Dirac delta function. The problem of dislocations concentrated on a spherical surface inside a nonlinear elastic ball is considered in the work Zhbanova and Zubov (2016).

Fig. 7.1 Stress Dr for inflation by pressure p1 = 0.1, κ = 2

Fig. 7.2 Stress Dφ for inflation by pressure p1 = 0.1, κ = 2

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Fig. 7.3 Stress Dz for inflation by pressure p1 = 0.1, κ = 2

Fig. 7.4 Distortion Cr for inflation by pressure p1 = 0.1, κ = 2

The case r∗ = 0.75 corresponds to the stresses and the distortions in Figs. 7.6, 7.7, 7.8 and Figs. 7.9, and 7.10, the distribution of dislocations over the inner surface r = r∗ = 0.5 of the cylinder corresponds to the stresses in Figs. 7.11, 7.12 and 7.13, and the distribution over the outer surface r = r∗ = 1 of the cylinder is in Figs. 7.14, 7.15, and 7.16. In the case of dislocations concentrated over the surface inside the cylinder, the conclusion that negative scalar dislocation densities decrease the stresses in absolute value is no longer valid. This is violated for the stresses Dφ and Dz . For the stress Dr and the distortion Cr , it can be seen that these quantities are independent of the dislocation density parameter β0 near the inner unloaded cylindrical surface. When dislocations are concentrated on the inner unloaded surface of the cylinder, negative scalar dislocation densities increase the stresses Dr and Dφ in absolute value and decrease Dz . With an increase in the parameter β0 , the stress distributions over the cylinder thickness become more uniform. If dislocations are concentrated on the outer loaded surface of the cylinder, a change in the parameter β0 , with the exception of a small neighborhood of the outer surface for Dφ and Dz , does not entail a change in the stresses in the cylinder. The dependence of the longitudinal force Q on the parameter λ for the cylinder r1 = 0.9 in the case κ = 2 is shown in Fig. 7.17.

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Fig. 7.5 Distortion Cφ for inflation by pressure p1 = 0.1, κ = 2

Fig. 7.6 Stress Dr for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 0.75

Fig. 7.7 Stress Dφ for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 0.75

The dependence of the applied pressure p1 (R1 ), where the designation R1 = r1 · Cφ (r1 ) is introduced, for the cylinder r1 = 0.9 at κ = 2 is plotted in Fig. 7.18, and that for the distribution of dislocations over the surface r∗ = 0.75 is shown in Fig. 7.19. In the absence of dislocations, the value R1 means the inner radius after deformation, and in the presence of dislocations, the dependence p1 (R1 ) can in some

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Fig. 7.8 Stress Dz for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 0.75

Fig. 7.9 Distortion Cr for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 0.75

Fig. 7.10 Distortion Cφ for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 0.75

way be considered as the effect of dislocations on the resistance of the body to the applied load p1 . Such a study for a nonlinear elastic ball with dislocations is presented in the work Goloveshkina and Zubov (2021). It can be seen from Fig. 7.18 that an increase in the dislocation parameter β0 decreases the pressure p1 .

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Fig. 7.11 Stress Dr for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 0.5

Fig. 7.12 Stress Dφ for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 0.5

Fig. 7.13 Stress Dz for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 0.5

7.4.2 Hydrostatic Compression Consider the compression of the cylinder by the hydrostatic pressure p0 = 0.1. For the scalar dislocation density α 32 = β0 r κ with κ = −1, the stresses and the distortions are shown in Figs. 7.20, 7.21, 7.22 and Figs. 7.23, and 7.24, respectively. For the scalar dislocation density α 32 = β0 r κ with κ = 2, the stresses and the distortions are shown in Figs. 7.25, 7.26, 7.27 and Figs. 7.28, and 7.29, respectively.

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Fig. 7.14 Stress Dr for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 1

Fig. 7.15 Stress Dφ for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 1

Fig. 7.16 Stress Dz for inflation by pressure p1 = 0.1, κ = 2, and distribution of dislocations over the surface r∗ = 1

Negative scalar dislocation densities decrease in absolute value the stress Dr , decrease on the inner surface, and increase on the outer surface the stresses Dφ and Dz , as well as the distortion Cr (the opposite is true for Cφ ). The case of dislocation distribution over the cylindrical surface r∗ = 0.75 inside the cylinder is shown in Figs. 7.30, 7.31 and 7.32 (the stresses) and Figs. 7.33, 7.34 (the distortions).

7 Nonlinear Deformations of Anisotropic Elastic Bodies with Distributed Dislocations Fig. 7.17 Dependence of longitudinal force Q on parameter λ for the cylinder r1 = 0.9, κ = 2

Fig. 7.18 Dependence p1 (R1 ) for cylinder r1 = 0.9, κ = 2

Fig. 7.19 Dependence p1 (R1 ) for cylinder r1 = 0.9 with dislocation distribution over surface r∗ = 0.75, κ=2

131

132 Fig. 7.20 Stress Dr for compression by pressure p0 = 0.1, κ = −1

Fig. 7.21 Stress Dφ for compression by pressure p0 = 0.1, κ = −1

Fig. 7.22 Stress Dz for compression by pressure p0 = 0.1, κ = −1

E. V. Goloveshkina and L. M. Zubov

7 Nonlinear Deformations of Anisotropic Elastic Bodies with Distributed Dislocations Fig. 7.23 Distortion Cr for compression by pressure p0 = 0.1, κ = −1

Fig. 7.24 Distortion Cφ for compression by pressure p0 = 0.1, κ = −1

Fig. 7.25 Stress Dr for compression by pressure p0 = 0.1, κ = 2

133

134 Fig. 7.26 Stress Dφ for compression by pressure p0 = 0.1, κ = 2

Fig. 7.27 Stress Dz for compression by pressure p0 = 0.1, κ = 2

Fig. 7.28 Distortion Cr for compression by pressure p0 = 0.1, κ = 2

E. V. Goloveshkina and L. M. Zubov

7 Nonlinear Deformations of Anisotropic Elastic Bodies with Distributed Dislocations Fig. 7.29 Distortion Cφ for compression by pressure p0 = 0.1, κ = 2

Fig. 7.30 Stress Dr for compression by pressure p0 = 0.1, κ = 2, and dislocation distribution over the surface r∗ = 0.75

Fig. 7.31 Stress Dφ for compression by pressure p0 = 0.1, κ = 2, and dislocation distribution over the surface r∗ = 0.75

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136 Fig. 7.32 Stress Dz for compression by pressure p0 = 0.1, κ = 2, and dislocation distribution over the surface r∗ = 0.75

Fig. 7.33 Distortion Cr for compression by pressure p0 = 0.1, κ = 2, and dislocation distribution over the surface r∗ = 0.75

Fig. 7.34 Distortion Cφ for compression by pressure p0 = 0.1, κ = 2, and dislocation distribution over the surface r∗ = 0.75

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7.5 Conclusion In this work, we have solved the problem of large deformations of an elastic hollow cylinder made of an anisotropic material, taking into account distributed dislocations. In the case of an axisymmetric distribution of edge dislocations, an exact solution is found. The stress state due to external and internal hydrostatic pressures is investigated, that is, the generalized Lamé problem is considered. The resistance of the cylinder to the applied load is studied. The plots for the longitudinal force are also presented. The dislocations in the cylinder are assumed to be continuously distributed. The dislocation density tensor described the distribution of screw and edge dislocations, for which it is shown that the scalar dislocation densities can be arbitrary, including the Dirac delta function. The discussed results show the influence of dislocation on deformations of solids.

References Altenbach H, Eremeyev VA (eds) (2013) Generalized continua: from the theory to engineering applications. CISM Courses and Lectures, Springer, Wien Altenbach H, Eremeyev V (2014) Strain rate tensors and constitutive equations of inelastic micropolar materials. Int J Plast 63:3–17 Altenbach H, Maugin GA, Erofeev V (eds) (2011) Mechanics of generalized continua, advanced structured materials, vol 7. Springer, Berlin Berdichevsky VL, Sedov LI (1967) Dynamic theory of continuously distributed dislocations. Its relation to plasticity theory. Prikl Mat Mekh 31(6):989–1006 Bilby BA, Bullough R, Smith E (1955) Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry. Proc R Soc Lond A Math Phys Eng Sci A231:263–273 Boulanger P, Hayes M (1995) On young’s modulus for anisotropic media. Trans ASME J Appl Mech 62(3):819–820 Cazzani A, Rovati M (2003) Extrema of young’s modulus for cubic and transversely isotropic solids. Intern J Solids Struct 40(7):1713–1744 Cazzani A, Rovati M (2005) Extrema of young’s modulus for elastic solids with tetragonal symmetry. Intern J Solids Struct 42(18/19):5057–5096 Clayton JD (2011) Nonlinear mechanics of crystals. Springer, Dordrecht Clayton JD, McDowell DL, Bammann DJ (2006) Modeling dislocations and disclinations with finite micropolar elastoplasticity. Int J Plast 22(2):210–256 Derezin SV, Zubov LM (2011) Disclinations in nonlinear elasticity. Ztsch Angew Math und Mech 91:433–442 Eremeyev VA, Lebedev LP, Altenbach H (2013) Foundations of micropolar mechanics. Springerbriefs in applied sciences and technologies, Springer, Heidelberg et al Eremeyev VA, Cloud MJ, Lebedev LP (2018) Applications of tensor analysis in continuum mechanics. World Scientific, New Jersey Eshelby JD (1956) The continuum theory of lattice defects. In: Seitz F, Turnbul D (eds) Solid State Phys, vol 3. Academic Press, New York, pp 79–144 Forest S (2008) Some links between Cosserat, strain gradient crystal plasticity and the statistical theory of dislocations. Phil Mag 88(30–32):3549–3563 Forest S, Sedláˇcek R (2003) Plastic slip distribution in two-phase laminate microstructures: dislocation-based versus generalized continuum approaches. Phil Mag 83(2):245–276

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Goloveshkina EV, Zubov LM (2019) Universal spherically symmetric solution of nonlinear dislocation theory for incompressible isotropic elastic medium. Arch Appl Mech 89(3):409–424 Goloveshkina EV, Zubov LM (2021) Spherically symmetric tensor fields and their application in nonlinear theory of dislocations. Symmetry 13(5):830 Hayes M, Shuvalov A (1998) On the extreme values of young’s modulus, the shear modulus, and poison’s ratio for cubic materials. Trans ASME J Appl Mech 65(3):786–787 Huntington HB (1958) The elastic constants of crystals. Solid State Phys 7:213–351 Kachanov M, Sevostianov I (2018) Micromechanics of materials, with applications. Springer, Cham Kalinin VA, Bayuk IO (1994) Thermodynamic limits on effective elastic modules of anisotropic rocks. Phys Solid Earth 30:10–17 Kondo K (1952) On the geometrical and physical foundations in the theory of yielding. In: Proceedings 2nd Japan National Congress of applied mechanics, Tokyo, pp 41–47 Kröner E (1960) Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch Ration Mech Anal 4:273–334 Le KC, Stumpf H (1996) A model of elastoplastic bodies with continuously distributed dislocations. Int J Plast 12(5):611–627 Lekhnitskii SG (1963) Theory of elasticity of an anisotropic elastic body (Holden-Day series in mathematical physics). Holden-Day, NY Levin VA, Zubov LM, Zingerman KM (2018) Multiple joined prestressed orthotropic layers under large strains. Int J Eng Sci 133:47–59 Lurie AI (1990) Nonlinear theory of elasticity. North-Holland, Amsterdam Ogden RW (1997) Non-linear elastic deformations. Dover, New York Omari M, Sevostianov I (2016) Evaluation of changes in microstructure and mechanical properties. Lambert Academic Publishing Ostrosablin NI (1992) The most restrictive bounds on change in the applied elastic constants for anisotropic materials. J Appl Mech Tech Phys 33(1):95–101 Teodosiu C (2013) Elastic models of crystal defects. Springer, Berlin Theocaris PS (1994) The limits of Poisson’s ratio in polycrystalline bodies. J Mater Sci 29(13):3527– 3534 Theocaris PS, Philippidis FP (1992) True bounds on Poisson’s ratio for transversely isotropic solids. J Strain Anal Eng Design 27(1):43–44 Ting TCT (2005) On anisotropic elastic materials for which young’s modulus e(n) is independent of n or the shear modulus g(n, m) is independent of n and m. J Elast 81(3):271–292 Yavary A, Goriely A (2012) Riemann-Cartan geometry of nonlinear dislocation mechanics. Arch Ration Mech Anal 205:59–118 Zelenina AA, Zubov LM (2009) Bending and twisting of nonlinear elastic bodies with continuously distributed dislocations. Vestn Yuzhn Nauchn Tsentr RAN 3(4):15–22 Zelenina AA, Zubov LM (2013) Nonlinear effects during the tension, bend, and torsion of elastic bodies with distributed dislocations. Doklady Phys 58(8):354–357 Zhbanova EV, Zubov LM (2016) The influence of distributed dislocations on large deformations of an elastic sphere. In: Naumenko K, Aßmus M (eds) Advanced methods of continuum mechanics for materials and structures, advanced structured materials, vol 60. Springer, Singapore, pp 61–76 Zubov LM (2004) Continuously distributed dislocations and disclinations in nonlinearly elastic micropolar media. Dokl Phys 49(5):308–310 Zubov LM (2011) The continuum theory of dislocations and disclinations in nonlinearly elastic micropolar media. Mech Solids 46(3):348–356 Zubov LM (2014) Spherically symmetric solutions in the nonlinear theory of dislocations. Doklady Phys 59(9):419–422 Zubov LM (2020) Universal solutions of nonlinear dislocation theory for elastic cylinder. Mech Solids 55(5):701–709

Chapter 8

Misfit Stress Relaxation at Boundaries of Finite-Length Tubular Inclusions Through the Generation of Prismatic Dislocation Loops M. Yu. Gutkin, E. A. Mordasova, A. L. Kolesnikova, and A. E. Romanov Abstract This work is devoted to the memory of our friend and colleague Igor Sevostianov, who did a great contribution to micromechanics of inhomogeneous solids. Elaboration on new models of elastic inclusions with non-ellipsoidal shapes was one of the main topics in his activity. Working under related problems in micromechanics of stress relaxation in various misfitting heterostructures, we suggest for this memorial book our recent work dealing with an elastic model of a finite-length tubular inclusion in an infinite matrix with the matrix inside its hole. We give and study an analytical solution for the stress fields of the inclusion and suggest a mechanism for the relaxation of these stresses through the formation of small rectangular prismatic dislocation loops in different places of the inclusion axial section. We determine and study the energy barriers and the critical conditions for the generation of the loops with special attention to the most preferred place of their nucleation within the inclusion wall and the most preferred shape of the loops. We show that (i) the most preferred place for loop generation is the region in the middle of the wall section at its inner boundary, (ii) the most preferable loops are elongated along this boundary, and (iii) the critical misfit value at which the generation of these loops becomes energetically favorable, decreases with the increase in the outer radius of the tubular inclusion and the ratio of its inner and outer radii and with a decrease in the inclusion height. Thus, the plane-ring-shaped inclusions of relatively larger radii and thinner walls are shown to be the least stable in case of this way of misfit stress relaxation.

M. Yu. Gutkin (B) · E. A. Mordasova · A. L. Kolesnikova Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, 61 Bolshoj Pr, V.O St. Petersburg 199178, Russia e-mail: [email protected] E. A. Mordasova e-mail: [email protected] A. L. Kolesnikova e-mail: [email protected] M. Yu. Gutkin · A. L. Kolesnikova · A. E. Romanov ITMO University, 49 Kronverksky Pr, St. Petersburg 197101, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_8

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Keywords Inclusions · Misfit stress · Stress relaxation · Dislocation loops · Analytical modeling

8.1 Introduction This paper is devoted to the memory of our friend and colleague Igor Sevostianov. Three of us (M.Yu.G., A.L.K., and A.E.R.) knew Igor very closely since 1995 when we spent some months in Dresden (Germany) working together as visiting researchers at the Research Group on Mechanics of Heterogeneous Solids of the Max-PlanckSociety led by Prof. W. Pompe. After that, we met Igor many times, spent together much time in various circumstances, and always knew him as a brilliant scientist with many new ideas and plans, a reliable and attentive friend, and a wonderful person in general. Starting from his post-graduate research (Sevostianov 1992), one of the main topics in Igor’s studies was the solution of inclusion and inhomogeneity problems in continuum mechanics with multiple applications to various porous and composite materials (Kachanov and Sevostianov 2018). In particular, he considered different non-ellipsoidal inhomogeneities such as irregular cracks (Mear et al. 2007; Kachanov and Sevostianov 2012) and toroidal (Radi and Sevostianov 2016; Krasnitskii et al. 2019), polyhedral (Trofimov et al. 2017), superspherical (Sevostianov et al. 2008; Chen et al. 2018), polyspherical (Lanzoni et al. 2020), and helicoidal (Trofimov and Sevostianov 2017) inclusions or voids, and studied their effect on the effective properties of materials. For many years, we have also dealt with the inclusion and inhomogeneity problems with special attention to mechanisms of misfit stress relaxation through the generation and development of various defect structures (see, for example, reviews (Gutkin et al. 2020a; Smirnov et al. 2020a) and some recent papers (Kolesnikova et al. 2018a, 2018b, 2021; Krauchanka et al. 2019; Mikaelyan et al. 2019; Smirnov et al. 2020b; Gutkin et al. 2020b; Chernakov et al. 2020; Romanov et al. 2020, 2021; Pomanov et al. 2022)). In particular, truncated spherical (Kolesnikova et al. 2018a, 2018b; Gutkin et al. 2020b), finite-length cylindrical (Kolesnikova et al. 2018b, 2021; Romanov et al. 2020, 2021; Pomanov et al. 2022) and infinite prismatic (Smirnov et al. 2020a, 2020b; Mikaelyan et al. 2019; Gutkin and I.A. Ovid’ko, A.G. Sheinerman, 2003) inclusions embedded in the infinite medium (Kolesnikova et al. 2018b), half-space (Gutkin 2003), planar nanolayers (Mikaelyan et al. 2019), spherical nanoparticles (Kolesnikova et al. 2018a, 2018b; Gutkin et al. 2020b), and cylindrical nanowires (Kolesnikova et al. 2018b, 2021; Smirnov et al. 2020b; Romanov et al. 2020, 2021; Romanov et al. 2022) have been in the focus of the research. In the present work, we extend our studies to the case of a finite-length tubular inclusion in an infinite matrix with the matrix inside its hole. To the best of our knowledge, nobody has dealt with it until now. However, tubular inclusions with different values of the aspect ratio, from very small as for ring quantum dots in semiconductor nanostructures (see Baghdasaryan et al. 2017 and references therein) to rather large as, for example, for

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tubular symplectic inclusions in olivine (Stevens et al. 2010), attach much attention in various fields of materials science. Encouraged by this interest, we give and study an analytical solution for the stress fields of the inclusion and suggest a mechanism for the relaxation of these stresses through the formation of small rectangular prismatic dislocation loops (PDLs) in different places of the inclusion axial section. The nucleation of such PDLs can be considered as the initial stage of stress relaxation in various nanoheterostructures (Gutkin and Smirnov 2014, 2015; Gutkin et al. 2015; Krasnitckii et al. 2018, 2019, 2020). We also determine the critical conditions for the onset of this relaxation process since such critical conditions are of great importance for many modern technologies, in particular, for semiconductor nanoheterostructures used in electronics, optoelectronics, and photonics (Gutkin et al. 2020a; Smirnov et al. 2020a). From a comparison of the critical conditions for different types of PDLs, we theoretically reveal the most preferred scenario of the misfit stress relaxation in the system.

8.2 Model Consider an elastically homogeneous model of a dilatational inclusion in the shape of a finite-length hollow tube in an infinite elastically isotropic matrix (Fig. 8.1). The model consists of two co-axial cylindrical domains D1 and D2 (D1 is the internal part of D2 , D1 ⊂ D2 ) of the same height h and with radii c1 and c2 . Let these domains be subjected to 3D homogeneous dilatational eigenstrains ε1∗ and ε2∗ , respectively. In the case of a tubular inclusion (D2 \D1 ) with the matrix inside the hole D1 , we put ε1∗ = − f and ε2∗ = f , where f is the eigenstrain caused by the mismatch in either the crystalline lattices or thermal extension coefficients of the inclusion and matrix materials. In the first case, for example, f = (ain − am )/am is the misfit strain for cubic crystalline lattices, ain and am are the lattice parameters of the tubular inclusion and matrix materials, respectively. Hereinafter, for the sake of definiteness, we consider the misfit strains and stresses by assuming that ain > am and hence f > 0. The elastic field of the tubular inclusion is a superposition of the fields of two cylindrical inclusions of finite length nested into each other (Kolesnikova et al. 2018b). Let us sequentially consider the regions of such a tubular inclusion and determine the stresses in them: (i) Within the D1 domain (and obviously at the same time within the D2 domain), when r < c1 , |z| ≤ h/2, the misfit stresses are given by the sum of the stress field inside finite-length cylindrical inclusions D1 and D2 in the matrix

1 ,in) 2 ,in) σi(in) = σi(D + σi(D , j j j

(8.1)

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z

Fig. 8.1 Model of a dilatational inclusion in the shape of a finite-length hollow tube in an infinite elastically isotropic matrix

c2

h/2

c1

0 D2

D1

y

̶ h/2

x 1 ,in) 2 ,in) where the stress fields σi(D and σi(D are determined by Kolesnikova et al. j j 2018b

k ,in) = σi(D j

G(1 + ν)εk∗ (Dk ,in) i j 2(1 − ν)

(8.2)

with k = 1, 2 (Dk ,in) = −J (1) (1, 0; 0) − J (2) (1, 0; 0) + J (1) (1, 2; 0) + J (2) (1, 2; 0) − 2, rr (8.3) c  k (1) k ,in) J (1, 1; −1) + J (2) (1, 1; −1)] + 1 , (D = −2 (8.4) ϕϕ r   k ,in) (D = 2 J (1) (1, 0; 0) + J (2) (1, 0; 0) − 2 , (8.5) zz

r(Dz k ,in) = −2 sgn(z)[J (1) (1, 1; 0) − J (2) (1, 1; 0)],

(8.6)

where G is the shear modulus, ν is the Poisson ratio  ∞ (G and ν are the same for the inclusion and matrix materials), J (l) (m, n; p) = 0 Jm (s)Jn (sr/ck ) exp(−sξl )s p ds is the Lipschitz-Hankel integral (Eason et al. 1955), Jm (t) and Jn (t) are the Bessel functions of the first kind; l = 1, 2, 3; ξ1 = (h/2 − |z|)/ck , ξ2 = (h/2 + |z|)/ck , and ξ3 = (|z| − h/2)/ck . (ii) Outside the D2 domain (and naturally outside the D1 domain), when |z| > h/2 or r > c2 , |z| ≤ h/2, the misfit stresses are given by the sum of stress fields outside finite-length cylindrical inclusions D1 and D2

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1 ,out) 2 ,out) σi(out) = σi(D + σi(D , j j j

(8.7)

1 ,out) 2 ,out) and σi(D are determined by Kolesnikova et al. where the stress fields σi(D j j 2018b

k ,out) = σi(D j

G(1 + ν)εk∗ (Dk ,out) i j 2(1 − ν)

(8.8)

with (D ,out) rr k =

⎧ ⎨

J (3) (1, 0; 0) − J (2) (1, 0; 0) + J (3) (1, 2; 0) + J (2) (1, 2; 0),

|z| > h/2

2 , ⎩ − 2ck − J (1) (1, 0; 0) − J (2) (1, 0; 0) + J (1) (1, 2; 0) + J (2) (1, 2; 0), |z| ≤ h/2, r > ck r2

(8.9) (Dk ,out) ϕϕ =

ck r

|z| > h/2 J (3) (1, 1; −1) − J (2) (1, 1; −1), , − J (1) (1, 1; −1) − J (2) (1, 1; −1), |z| ≤ h/2, r > ck (8.10)



|z| > h/2 −J (3) (1, 0; 0) + J (2) (1, 0; 0), , (1) (2) J (1, 0; 0) + J (1, 0; 0), |z| ≤ h/2, r > ck

(3) |z| > h/2 J (1, 1; 0) − J (2) (1, 1; 0), = −2 sgn(z) (1) . J (1, 1; 0) − J (2) (1, 1; 0), |z| ≤ h/2, r > ck

k ,out) (D zz

r(Dz k ,out)

2ck r



=2

(8.11)

(8.12)

(iii) In the wall of the tubular inclusion when c1 < r < c2 , |z| ≤ h/2, the stresses can be written as

1 ,out) 2 ,in) σi(wall) = σi(D + σi(D , j j j

(8.13)

where stress components 1 ,out) 2 ,in) are given by Eq. (8.8) with functions (8.9b)-(8.12b) at k = 1 and σi(D σi(D j j are stresses (8.2)–(8.6) at k = 2. and σi(out) in the axial Figure 8.2 shows the combined maps for stress fields σi(in) j j section of the tubular inclusion and around it for the case of c1 = 0.3c2 and h = 4c2 . It is seen that the stress components σrr and σr z are continuous across both the inner and outer side inclusion/matrix interfaces, while the stress components σzz and σr z are continuous across the top and bottom inclusion/matrix interfaces. All normal stresses are negative inside the inclusion material and reach rather high levels there; for example, ∼ 2G(1 + ν) f /(1 − ν) for the σϕϕ component (see Fig. 8.2b). At typical values of ν = 0.3 and f = 0.01, this gives ∼ 0.037G ≈ G/27 that is a

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very high stress magnitude. On the other hand, the shear stress σr z magnitude is not as high throughout the inclusion section, with the exception of the inclusion edges where it is singular. At a small distance from the edges, the shear stress magnitude reaches the value of ∼ 0.8G(1 + ν) f /(1 − ν) (see Fig. 8.2d) that roughly gives ∼ 0.015G ≈ G/68 for ν = 0.3 and f = 0.01. As it follows from the comparison of the normal and shear stress levels, one can expect for the first turn the relaxation mechanism that would diminish the normal stresses at the places of their concentration. To reveal these places in more detail, let us consider the normal stress plots shown in Fig. 8.3. They demonstrate the distribution of the stress components σϕϕ (Fig. 8.3a, b), σrr (Fig. 8.3c), and σzz (Fig. 8.3d) along the lines (x = 0, y, z = 0) and (x = 0, y, z = h/2 − 0) for h = 4c2 and different values of the ratio of the inner and outer radii of the inclusion: c1 /c2 = 0.1, 0.3, (b)

z/c2

z/c2

(a)

y /c2

z/c2

(d)

z/c2

(c)

y /c2

y /c2

y /c2

Fig. 8.2 Maps of stress components (a) σrr , (b) σϕϕ , (c) σzz , and (d) σr z in the axial section x = 0 of a tubular inclusion in an infinite matrix in the case of c1 = 0.3c2 and h = 4c2 . The stress values are given in units of G(1 + ν) f /(1 − ν). The thin dashed lines in (a) and (d) show the inclusion boundaries

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0.5, and 0.7. It is seen that the maximum compressive stresses σϕϕ and σzz act at the inner boundary of inclusion. With an increase in the radius c1 of the inclusion hole at a constant outer radius c2 of the inclusion, the σϕϕ and σzz magnitudes inside the inclusion noticeably increase, while in the matrix, both in the hole region and outside of the inclusion, they decrease. The maximum tensile stresses act in the matrix at the outer boundary of the inclusion. Naturally, as the distance from the inclusion increases, the magnitudes of all stress components drop. Inside the inclusion, the σϕϕ magnitude monotonously decreases in the radial direction from the inner boundary of the inclusion to the outer one for any value of the c1 /c2 ratio, and the σzz magnitude does the same at c1 /c2 ≥ 0.3, while it first decreases, reaches a minimum value, and then increases at c1 /c2 = 0.1. In contrast, the σrr magnitude shows just the opposite behavior: it monotonously increases in the radial direction inside the inclusion at c1 /c2 ≥ 0.3, while it first increases, reaches a maximum value, and then decreases at c1 /c2 = 0.1. Of greatest interest here is the obvious disproportion in the stress magnitudes inside the wall of the inclusion (at c1 < r < c2 ) and in the matrix inside the (a)

(b)

y /c2

y /c2

(d)

(c)

y /c2

y /c2

Fig. 8.3 Plots of the tubular inclusion stress components (a, b) σϕϕ , (c) σrr , and (d) σzz along the lines (a, c, d) (x = 0, y, z = 0) and (b) (x = 0, y, z = h/2 − 0) for h = 4c2 and different values of the ratio of the inner and outer radii of the inclusion: c1 /c2 = 0.1, 0.3, 0.5, and 0.7. The stress values are given in units of G(1 + ν) f /(1 − ν)

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z

Fig. 8.4 Small rectangular prismatic dislocation loops (PDLs-i, i = 1, 2, 3) of vacancy type in the regions of maximum compressive stresses within an axial section z = 0 of the inclusion wall

c2

h/2

c1 PDL-3

0

PDL-1

b D2

D1

PDL-2

z0 2s 2q

y0

y

̶ h/2

x inclusion hole (at r < c1 ). In fact, the first ones are much higher than the second ones. Based on the aforementioned observations, let us assume that the most probable mechanism of stress relaxation in the system is the nucleation of small prismatic dislocation loops (PDLs) of vacancy type in the regions of maximum compressive stresses (see Figs. 8.2 and 8.3). These regions are placed in the longitudinal sections of the inclusion, at the centers and corners of its inner and outer interfaces. Consider the conditions for the nucleation of small rectangular PDLs in three such characteristic regions in the inclusion section x = 0 as shown in Fig. 8.4. The corresponding PDLs are denoted as PDL-1, PDL-2, and PDL-3. The formation of such a PDLi (i = 1, 2, 3) is energetically favorable if the corresponding changes in the total energy Wi is negative. The energy change Wi can be represented by the sum Wi = Wst + Wc + Wint i ,

(8.14)

where Wst is the strain energy of a PDL, Wc is its core energy, and Winti is the energy of elastic interaction of the PDL-i with the stress inside the inclusion wall. The strain energy Wst is given by Gutkin et al. (2003)

 4q s K +s 4s q K +q − ln − 2(q + s − K ) , + s ln − ln Wst = 2Db q ln rc rc 2 K −q 2 K −s (8.15) 2

where D = G/[2π(1 − ν)], b = |b| is the Burgers vector magnitude of the PDLs, 2q × 2s are the PDL sizes, rc is the cut-off radius of the PDL stresses at the PDL lines, and K 2 = q 2 + s 2 .

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The core energy Wc is approximated by the standard way as (Gutkin and Smirnov, 2014) Wc = 2(q + s)Db2 .

(8.16)

The interaction energy Wint i is calculated as the work required for the generation (wall) given by Eq. (8.13) as follows: of a PDL in the inclusion stress field σϕϕ

c1 +2s

dz −q

Wint 2 = −b

c1



h 2

(wall) σϕϕ (x = 0, y, z)dy,

c2

dz

Wint 3 = −b



q

Wint 1 = −b

h 2 −2q

q



c2 −2s c2

dz −q

c2 −2s

(wall) σϕϕ (x = 0, y, z)dy,

(wall) σϕϕ (x = 0, y, z)dy.

(8.17)

(8.18) (8.19)

After the integration, the interaction energies can be represented by the following formulas:

  c1 1 ∞ 1+ν −2 4qs 1 + − J1 (κ)(P1 − P2 )κ dκ , Wint 1 = −bG f 1−ν c1 + 2s 2 0 (8.20)

  c2 1 ∞ 1+ν 4qs 1 + 2 1 − J1 (κ)(Q 1 − Q 2 )κ −2 dκ , Wint 2 = −bG f 1−ν 2 0 c2 − 2c2 s (8.21)

  ∞ c12 1+ν −2 4qs 1 + 2 − Wint 3 = −bG f J1 (κ)(R1 − R2 )κ dκ , 1−ν c2 − 2c2 s 0 (8.22) where  P1 = c1 α1

[π κ(c1 + s)H 0 (r1 κ) − 2c1 ]J1 (r1 κ) − κ(c1 + s)[π H 1 (r1 κ) − 2]J0 (r1 κ) 

− c1 [π κ H 0 (κ) − 2]J1 (κ) + c1 κ[π H 1 (κ) − 2]J0 (κ)

,

(8.23)

P2 = c2 α2 {[π κ(c1 + s)H 0 (r2 κ) − 2c2 ] J1 (r2 κ) − κ(c1 + s)[π H 1 (r2 κ) − 2]J0 (r2 κ) −[π c1 κ H 0 (r3 κ) − 2c2 ]J1 (r3 κ) + c1 κ[π H 1 (r3 κ) − 2]J0 (r3 κ)},

(8.24)

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Q1 R1 = = c1 β1 α1

 − [π κ(c2 − 2s)H 0 (d1 κ) − 2c1 ]J1 (d1 κ) + κ(c2 − 2s)[π H 1 (d1 κ) − 2]J0 (d1 κ) 

(8.25) + [π c2 κ H 0 (c2 κ/c1 ) − 2c1 ]J1 (c2 κ/c1 ) − c2 κ[π H 1 (c2 κ/c1 ) − 2]J0 (c2 κ/c1 ) ,  Q2 R2 = = c2 − [π κ(c2 − 2s)H 0 (d2 κ) − 2c2 ]J1 (d2 κ) + κ(c2 − 2s)[π H 1 (d2 κ) − 2]J0 (d2 κ) β2 α2  + c2 [π κ H 0 (κ) − 2]J1 (κ) − c2 κ[π H 1 (κ) − 2]J0 (κ)

.

(8.26)

Here r1 = 1+s/c1 , r2 = (c1 +s)/c2 , r3 = c1 /c2 , d1 = (c2 −2s)/c1 , d2 = 1−2s/c2 , α1 = [ex p(2κq/c1 ) − 1] ex p [−κ(h + 2q)/(2c1 )], α2 = [ex p(2κq/c2 ) − 1] ex p[−κ(h + 2q)/(2c2 )], β1 = [ex p(2κq/c1 ) − 1] ex p[−κ(h + 2q)/c1 ][ex p(κh/c1 ) + ex p(2κq/c1 )], β2 = [ex p(2κq/c2 ) − 1] ex p[−κ(h + 2q)/c2 ][ex p(κh/c2 ) + ex p(2κq/c2 )], H 0 (κ) and H 1 (κ) are the Struve functions. Thus, the energy terms figured in Eq. (8.14) are given by Eqs. (8.15), (8.16), and (8.20)-(8.26).

8.3 Results To compare the relative preferences in the generation of PDLs in different regions of the tubular inclusion (see Fig. 8.4) and to find the corresponding critical conditions in the system under study, let us consider the maps of the energy changes Wi in the space of the normalized sizes 2s/b and 2q/b of the PDLs for different values of the misfit parameter f at rc = b, ν = 0.3, and an exemplary set of typical inclusion sizes: c1 = 100b and c2 = h = 200b. Figure 8.5 shows four maps built numerically for PDL-1 with (a) f = 0.015, (b) 0.016, (c) 0.025, and (d) 0.039. It is seen from Fig. 8.5a-c that, at relatively low values of f (here at f ≤ 0.025), the nucleation of PDL-1 requires overcoming very high energy barriers whose height and extension decrease with an increase in f value. When f becomes high enough (here at f = 0.039), the energy barrier practically disappears (see Fig. 8.5d). The corresponding value of f can be treated as the critical misfit value f ci which is characteristic of the region of PDL generation at a given set of inclusion sizes. In the case under consideration (PDL-1, c1 = 100b and c2 = h = 200b), f c1 ≈ 0.039. For PDL-2 and PDL-3, the maps of the energy changes W2 and W3 are rather similar to that for PDL-1 (they are not shown here), and the critical misfit values are f c2 = 0.057 and f c3 = 0.062, respectively. Comparing these critical misfits for three types of the PDLs, one can see that, for the chosen set of inclusion sizes, PDL-1 needs the lowest value of f to be generated in a barrier-less manner, and therefore, it can be treated as the most preferable PDL in the system (Gutkin and Smirnov, 2014, 2015; Gutkin et al. 2015).

8 Misfit Stress Relaxation at Boundaries of Finite-Length Tubular …

2q/b

(b)

2q/b

(a)

149

2s/b (c)

2s/b

2q/b

2q/b

(d)

2s/b

2s/b

Fig. 8.5 Maps of the energy change W1 in the space of the normalized sizes 2s/b and 2q/b of the PDL-1 at c1 = 100b, c2 = h = 200b, rc = b, and ν = 0.3 for different values of the misfit parameter f : (a) 0.015, (b) 0.016, (c) 0.025, and (d) 0.039. The energy change is given in units of Db3

On the other hand, the relative preference in the generation of the three kinds of PDLs can also be extracted from the comparison of the energy gains |Wi ( f )| caused by the PDL generation when the PDLs reach some small conditional sizes at f > f ci . For example, for the PDL sizes 2q = 2s = 4b and the misfit value f = 0.07, these energy gains are |W1 | ≈ 37.1Db3 , |W2 | ≈ 24.4Db3 , and |W3 | ≈ 19.8Db3 . Thus, from the comparison of the critical misfit values f ci and the energy gains |Wi | for the PDLs under discussion, one can conclude that PDL-1 is more preferable to PDL-2 and PDL-3 because it can nucleate at the lowest misfit, and its generation results in the highest energy gain. In its turn, PDL-3 is characterized by the highest critical misfit value and the lowest energy gain, and therefore, it is the least preferable among the PDLs under study.

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It is of interest to consider how the above conclusions are sensitive to the shapes of PDLs and their inclusion. To this end, let us use the following formula: Wi = 0,

(8.27)

to calculate the critical misfit value f ci , f ci =

Wst + Wc , −Wint i / f

(8.28)

in dependence on the PDL aspect ratio, λ = q/s, and the inclusion sizes c1 , c2 , and h. Following the approach in Gutkin and Smirnov (2014, 2015); Gutkin et al. (2015), we take three characteristic shapes of the PDLs: (i) elongated normally to the cylindrical interface (N-PDLs) with λ < 1, (ii) square (S-PDLs) with λ = 1, and (iii) elongated along the cylindrical interface (A-PDLs) with λ > 1. Thus, one can analyze 9 cases with PDLs of three different shapes nucleated in three different regions: N-PDLs-i, S-PDLs-i, and A-PDLs-i, with i = 1, 2, and 3. Since the numerator in Eq. (8.28) does not depend on the region of PDL nucleation, while the denominator does depend on it, it is reasonable to trace how the interaction energy magnitude |Wint i | changes with the ratios λ and t = c1 /c2 for different values of c2 . Figure 8.6 shows the dependence of |Wint i | on t for different values of c2 , s, and λ at h = c2 , ν = 0.3, and i = 1, 2, and 3. The general result is that A-PDLs interact with the inclusion stress noticeably stronger than N- and S-PDLs, independent of the region of their nucleation. Among A-PDLs-i, the A-PDLs-1 demonstrates the highest values of the interaction energy magnitude in the entire range of t. Therefore, the generation of A-PDLs-1 can be considered the most preferable mechanism of stress relaxation in the inclusion of the 9 variants under study. It is also seen from Fig. 8.6 that all PDLs are characterized by an increase in |Wint i | with an increase in the ratio t for a given value of c2 . For PDLs-2 and -3, this increase accelerates with increasing t over its entire range (see Fig. 8.6b, c). For PDLs-1 (see Fig. 8.6a), such an increase slows down with increasing t. At the same time, in some cases, the curves increase over the entire range of t values with a tendency to reach saturation (the curves for N- and S-PDLs-1 at c2 /b = 100 and 150), while in the rest, they reach a maximum and then start to decrease. This behavior of the |Wint i | curves can naturally be explained by the fact that with increasing t, the σϕϕ stress level increases in most of the rectangular section of the inclusion wall, however, its peak value near the inner surface of the inclusion decreases slightly (see Fig. 8.3a). From the point of view of practical applications, of great interest is the dependence of the interaction energy magnitude |Wint i | on the size of the inclusion while maintaining its proportions. In the context of the discussion of Fig. 8.6, it means the position of the curves for |Wint i | in dependence on c2 for different values of t. In the case of PDLs-1, |Wint 1 | increases with increasing c2 (Fig. 8.6a). This can be explained as follows. In these calculations, it was assumed that c2 = h, and therefore, the change in the stress state in the inclusion section of interest can be obtained by simply scaling the map for σϕϕ in Fig. 8.2b. In this case, the size of a PDL-1 remains

8 Misfit Stress Relaxation at Boundaries of Finite-Length Tubular …

(a)

c2/b = 200 150

λ=2

|Wint 1 |

100 200 150 100

|Wint 2 |

200 150 100

c2/b = 200 200 150

λ = 0.5

λ=2 λ=1

150 100 100

100

150 200

λ = 0.5

t

c2/b = 100

|Wint 3 |

λ=1

t

(b)

(c)

151

100 100

λ=2

150

200 150 150

200

200

λ = 0.5 λ=1

t Fig. 8.6 Dependence of the interaction energy magnitude |Wint i | on the ratio t for different values of c2 , s, and λ at h = c2 , ν = 0.3, and i equal to (a) 1, (b) 2, and (c) 3. The loop sizes are s = 5b and q = 10b for A-PDLs, s = q = 6b for S-PDLs, and s = 10b and q = 5b for N-PDLs. The insets at curves illustrate the shapes and positions of PDLs in the longitudinal section of the inclusion wall in accordance with the i and λ values. The interaction energy magnitude is given in units of G f (1 + ν)/(1 − ν)

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constant, which means that, with an increase in the sectional area of the inclusion wall, the fixed area occupied by the PDL-1 will occur under ever higher levels of σϕϕ (Fig. 8.2b). Accordingly, the value of |Wint 1 | will also grow. In the case of PDLs-3, the situation is reversed—the value of |Wint 3 | decreases with increasing c2 (Fig. 8.6c). The explanation of this effect is completely similar to the previous one, with the only difference that, here, as the inclusion wall section increases, the fixed area occupied by a PDL-3 will occur under lower and lower levels of σϕϕ (Fig. 8.2b). Hence, the value of |Wint 3 | will also drop with the growth of c2 . The case of PDLs-2 (Fig. 8.6b) is the most difficult to interpret, since here both an increase and a decrease in |Wint 2 | with increasing c2 are observed. Growth is typical for A- and S-PDLs-2 at lower values of t, a decrease is characteristic of them at higher values of t and for N-PDLs-2 in the entire range of t. To explain these effects, it is necessary to conduct a detailed study of the σϕϕ stress distribution near the corner point of the inclusion wall section and the change in this distribution with a change in t, which is beyond the scope of this work. Let us now turn to the consideration of the dependence of the critical misfit f c of the formation of PDLs of different types and shapes on the ratio t (Fig. 8.7). As is seen from Fig. 8.7, monotonically decreasing dependences f c (t) are typical for all PDLs. In the case of PDLs-1, the rate of fall decreases with increasing t, while in the cases of PDLs-2 and -3, it increases. For PDLs-1, the critical misfit f c decreases with an increase in the outer radius of the inclusion c2 , which is in good agreement with the increase in the absolute value of the interaction energy |Wint 1 | (Fig. 8.6a) and is explained by an increase in the stress σϕϕ level in the area where the PDLs-1 are formed. At a given misfit value f , PDLs-1 can be generated when the inequality f > f c (t) is satisfied. Then the following three options are possible: The horizontal straight line corresponding to f passes below the curve f c (t); in this case, the nucleation of PDLs-1 is impossible. (ii) Such a straight line passes above the curve f c (t); in this case, the nucleation of PDLs-1 is possible at any value of t. (iii) Such a straight line intersects the curve f c (t) at some point t = tc ; then the nucleation of PDLs-1 is possible if the ratio t exceeds this critical value: t > tc . From the plots in Fig. 8.7, it can be seen that, with an increase in f and c2 , the critical value tc decreases. Thus, with an increase in f and c2 , the range of values of t (in other words, the range of inclusion wall thicknesses) increases, in which the formation of PDLs-1 can occur. (i)

For PDLs-3, the critical misfit f c increases with an increase in the outer radius of the inclusion c2 , which is in good agreement with the drop in |Wint 3 | (Fig. 8.6c) and is explained by a decrease in the stress σϕϕ level in the area where the PDLs-3 are formed. All three options discussed above for PDLs-1 are possible in this case as well. The main difference here is that, when implementing the third option, an increase in c2 will lead not to a decrease, but to an increase in the critical value tc . In this case, the range of t, in which the formation of PDLs-3 can occur, will increase with increasing f and decrease with increasing c2 .

8 Misfit Stress Relaxation at Boundaries of Finite-Length Tubular …

(a)

c2/b = 200 150 100

200

153

150

fc

100 150 200

100

λ = 0.5 t

(b)

c2/b = 200

100 150 200

150

100

fc

100 150 200

λ=1 t

(c)

c2/b = 200

100 150

150

100

fc

200 100 150 200

λ=2 t Fig. 8.7 Dependence of the critical misfit f c on the ratio t for different values of c2 , s, and i at h = c2 , ν = 0.3, and λ equal to (a) 0.5, (b) 1, and (c) 2. The loop sizes are s = 5b and q = 10b for A-PDLs, s = q = 6b for S-PDLs, and s = 10b and q = 5b for N-PDLs. The insets at curves illustrate the shapes and positions of PDLs in the longitudinal section of the inclusion wall in accordance with the i and λ values

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For PDLs-2, all the same scenarios are possible, the third of which leads to the appearance of a critical value of tc . However, with regard to the dependence of tc on c2 , the situation becomes somewhat more complicated and is largely determined by the type of PDLs. In the case of N-PDLs-2 (Fig. 8.7a), the conclusions drawn for PDLs-3 are valid. In the cases of S-PDLs-2 (Fig. 8.7b) and A-PDLs-2 (Fig. 8.7c), there are two intervals on the scale of t, t < t∗ and t > t∗, where t∗ is the point of intersection of the curves f c (t) plotted for two different values of c2 . In the interval t < t∗, the conclusions made for the PDLs-1 are valid, in the interval t > t∗, the conclusions made for the PDLs-3 are valid. As is seen from Figs. 8.7b, c, in the cases of S-PDLs-2 and A-PDLs-2, we have t∗ ≈ 0.4 and 0.6, respectively. Note that the conclusions made above are reliably established for those values of the model parameters for which the curves f c (t) were plotted and studied. This is especially true for the conclusions and estimates made for those PDLs-2, for which there is a reversal of the dependence of the critical misfit f c on c2 at the point t = t∗; in particular, this concerns the numerical estimates of the value of t∗. From the plots shown in Fig. 8.7, it can be seen that the smallest value of the critical misfit f c is typical for the A-PDL-1 which is formed in the middle of the inner boundary of the inclusion (at z 0 = 0) and is elongated along this boundary (see Fig. 8.7c). Let us consider how the dependence f c (t) obtained for this type of PDLs changes with a variation in the height h of the inclusion. Figure 8.8 shows that the critical misfit f c increases with increasing inclusion height that is explained by some decrease in the σϕϕ stress level in the inclusion wall near the internal boundary of the inclusion and some increase in the gradient of σϕϕ in the wall along the inclusion radius (these results are not shown here). At h = 0.3c2 , the value of f c is the smallest of the three variants (h/c2 = 0.3, 1.5, 3.0) under consideration. Thus, a decrease in h leads to a decrease in the stability of the inclusion against the generation of the A-PDLs-1.

h/c2 = 3.0

fc

Fig. 8.8 Dependence of the critical misfit f c on the ratio t for the formation of an A-PDL-1 with sizes s = 5b and q = 10b at c2 = 200b, ν = 0.3, and different values of the aspect ratio h/c2

1.5 0.3

t

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155

8.4 Summary and Conclusions An elastic model of a finite-length tubular inclusion in an infinite matrix with the matrix inside its hole is suggested. The stress fields of the inclusion caused by a threedimensional uniform dilatational eigenstrain (misfit strain) are found and analyzed in detail. It is shown that inside the inclusion, in the hole filled with the matrix, the level of normal radial and hope stresses is much (by an order of magnitude) lower than in the wall of the inclusion. With an increase in the radius of the inner hole, the hope stress increases in the wall and decreases outside of it. A mechanism for the relaxation of these stresses due to the formation of small rectangular prismatic dislocation loops (PDLs) in different places of the inclusion axial section is proposed. The critical conditions for the generation of the PDLs are studied with special attention to the determination of the most preferred place of the inclusion wall section for the PDL nucleation and the most preferred shape of the PDLs. It is shown that (i) the most preferred place for PDL generation is the region in the middle of the wall section at its inner boundary, (ii) the most preferable PDLs are those elongated along this boundary, and (iii) the critical misfit at which the generation of these PDLs becomes energetically favorable, decreases with the increases in the outer radius of the tubular inclusion and the ratio of its inner and outer radii and with a decrease in the inclusion height. Thus, the plane-ring-shaped inclusions of relatively larger radii and thinner walls are the least stable with respect to this way of misfit stress relaxation.

References Baghdasaryan DA, Hayrapetyan DB, Sarkisyan HA, Kazaryan EM, Pokutnyi SI (2017) Exciton states and direct interband light absorption in the ensemble of toroidal quantum dots. J Nanophoton 11(4):046004 Chen F, Sevostianov I, Giraud A et al (2018) Combined effect of pores concavity and aspect ratio on the elastic properties of a porous material. Intern J Sol Struct 134:161–172 Chernakov AP, Kolesnikova AL, Gutkin MYu, Romanov AE (2020) Periodic array of misfit dislocation loops and stress relaxation in core-shell nanowires. Int J Eng Sci 156:103367 Eason G, Noble B, Sneddon IN (1955) On certain integrals of Lipschitz-Hankel type involving products of Bessel functions. Philos Trans R Soc London–Ser A 247:529–551 Gutkin MYu, Kolesnikova AL, Romanov AE (2020a) Nanomechanics of stress relaxation in composite low-dimensional structures. In: Altenbach H, Öchsner A (eds), Encycl Contin Mech, Springer, Berlin, Heidelberg, p 1778–1799 Gutkin MYu, Ovid’ko IA, Sheinerman AG (2003) Misfit dislocations in composites with nanowires. J Phys: Condens Matter 15:3539–3554 Gutkin MYu, Kolesnikova AL, Mikheev DS, Romanov AE (2020b) Misfit stresses and their relaxation by misfit dislocation loops in core-shell nanoparticles with truncated spherical cores. Europ J Mech A Solids 81:103967 Gutkin MYu, Smirnov AM (2014) Generation of rectangular prismatic dislocation loops in shells and cores of composite nanoparticles. Phys Solid State 56(4):731–738 Gutkin MYu, Smirnov AM (2015) Initial stages of misfit stress relaxation in composite nanostructures through generation of rectangular prismatic dislocation loops. Acta Mater 88:91–101

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Gutkin MYu, Krasnitckii SA, Smirnov AM, Kolesnikova AL, Romanov AE (2015) Dislocation loops in solid and hollow semiconductor and metal nanoheterostructures. Phys Solid State 57(6):1177–1182 Kachanov M, Sevostianov I (2012) Rice’s internal variables formalism and its implications for the elastic and conductive properties of cracked materials, and for the attempts to relate strength to stiffness. J Appl Mech 79:031002 Kachanov M, Sevostianov I (2018) Micromechanics of Materials, with Applications. Springer Kolesnikova AL, Gutkin MYu, Krasnitckii SA, Smirnov AM, Dorogov MV, Serebryakova VS, Romanov AE, Aifantis EC (2018a) On the elastic description of a spherical Janus particle. Rev Adv Mater Sci 57(1/2):246–256 Kolesnikova AL, Gutkin MYu, Romanov AE (2018b) Analytical elastic models of finite cylindrical and truncated spherical inclusions. Int J Solids Struct 143:69–72 Krauchanka MYu, Krasnitckii SA, Gutkin MYu, Kolesnikova AL, Romanov AE (2019) Circular loops of misfit dislocations in decahedral core-shell nanoparticles. Scr Mater 167:81–85 Krasnitskii S, Trofimov A, Radi E et al (2019) Effect of a rigid toroidal inhomogeneity on the elastic properties of a composite. Math Mech Solids 24:1129–1146 Kolesnikova AL, Romanov AE, Gutkin MYu, Bougrov VE (2021) Multi-step dilatational inclusion in an elastically isotropic cylinder. Mater Phys Mech 47(5):697–705 Krasnitckii SA, Kolomoetc DR, Smirnov AM, Gutkin MYu (2018) Misfit stress relaxation in composite core-shell nanowires with parallelepipedal cores by rectangular prismatic dislocation loops. J Phys: Conf Ser 993:012021 Krasnitckii SA, Smirnov AM, Mynbaev KD, Zhigilei LV, Gutkin MYu (2019) Axial misfit stress relaxation in core-shell nanowires with hexagonal core via nucleation of rectangular prismatic dislocation loops. Mater Phys Mech 42(6):776–783 Krasnitckii SA, Smirnov AM, Gutkin MYu (2020) Axial misfit stress relaxation in core-shell nanowires with polyhedral cores through the nucleation of misfit prismatic dislocation loops. J Mater Sci 55:9198–9210 Lanzoni L, Radi E, Sevostianov I (2020) Effect of spherical pores coalescence on the overall conductivity of a material. Mech Mater 148:103463 Mear ME, Sevostianov I, Kachanov M (2007) Elastic compliances of non-flat cracks. Intern J Sol Struct 44:6412–6427 Mikaelyan KN, Gutkin MYu, Borodin EN, Romanov AE (2019) Dislocation emission from the edge of a misfitting nanowire embedded in a free-standing nanolayer. Int J Solids & Structures 161:127–135 Romanov AE, Kolesnikova AL, Gutkin MYu, Bougrov VE (2022) Elastic interaction of quantum disks in hybrid QD/NW structures. Techn Phys Lett 48(1):34–36 Radi E, Sevostianov I (2016) Toroidal insulating inhomogeneity in an infinite space and related problems. Proc Roy Soc A: Math Phys Eng Sci 472:20150781 Romanov AE, Kolesnikova AL, Gutkin MYu, Dubrovskii VG (2020) Elasticity of axial nanowire heterostructures with sharp and diffuse interfaces. Scripta Mater 176(2):42–46 Romanov AE, Kolesnikova AL, Gutkin MYu (2021) Elasticity of a cylinder with axially varying dilatational eigenstrain. Int J Solids Struct 213:121–134 Sevostianov IB (1992) The stressed-strained state of inhomogeneous media with physically nonlinear discrete phase. PhD Thesis, St. Petersburg State University, St. Petersburg, in Russian Sevostianov I, Kachanov M, Zohdi T (2008) On computation of the compliance and stiffness contribution tensors of non ellipsoidal inhomogeneities. Intern J Sol Struct 45:4375–4383 Smirnov AM, Krasnitckii SA, Rochas SS, Gutkin MYu (2020a) Critical conditions of dislocation generation in core-shell nanowires: a review. Rev Adv Mater Technol 2(3):19–43 Smirnov AM, Krasnitckii SA, Gutkin MYu (2020b) Generation of misfit dislocations in a core-shell nanowire near the edge of prismatic core. Acta Mater 186:494–510 Stevens MR, Bell DR, Buseck PR (2010) Tubular symplectic inclusions in olivine from the Fukang pallasite. Meteorit Planet Sci 45(5):899–910

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Chapter 9

A Numerical Determination of the Interactions Between Dislocations and Multiple Inhomogeneities Zhizhen Jiang, Kuanyu Liu, Kai Zhu, Pu Li, and Xiaoqing Jin

Abstract The interactions between inhomogeneities and dislocations may affect the mechanical properties of heterogenous materials remarkably. However, the previous analytical solutions accounting for the interactions are mainly limited to inhomogeneities with idealized geometries. In this chapter, an efficient iterative computational scheme is presented for evaluating the elastic fields and consequently the interaction energy due to a screw/edge dislocation interacting with an arbitrarily shaped inhomogeneous inclusion, by employing the numerical equivalent inclusion method (NEIM) in conjunction with the two-dimensional fast Fourier transform (2D-FFT) technique. The effectiveness of the proposed method is illustrated by several examples, including an inhomogeneity with complex boundary, layered inhomogeneities of various shapes, and multiple inhomogeneities. Keywords Inhomogeneity · Inhomogeneous inclusion · Screw dislocation · Edge dislocation · Numerical equivalent inclusion method (NEIM) · Fast Fourier transform (FFT)

9.1 Introduction Heterogeneous materials (Sevostianov and Kachanov 2002; Sevostianov et al. 2009; Cortes et al. 2010) have broad applications in various engineering fields, including aerospace construction (Soutis et al. 2015; Katnam et al. 2013; Nicolas et al. 2016), nuclear (Li et al. 2013), new energy technologies (Amanieu et al. 2014; Mu et al. 2022), to name a few. In addition, common materials generally have heterogeneous properties at the micro-scale, due to the existence of inhomogeneities, cavities, or second phases. Thus, as evidenced in Sevostianov’s profound works (Sevostianov and Z. Jiang · K. Liu · K. Zhu · X. Jin College of Aerospace Engineering, Chongqing University, Chongqing, China P. Li (B) School of Science, Harbin Institute of Technology, Shenzhen, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_9

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Kachanov 2000, 2007; Kachanov and Sevostianov 2005, 2018; Giraud and Sevostianov 2013), investigating heterogeneous materials from the micromechanical point of view could be meaningful and indispensable. Dislocation is one of the main defects in crystalline materials. The dislocation arrangement and mobility at the micro-scale may remarkably affect the mechanical properties of materials (Indenbom and Lothe 2012; Hull and Bacon 2011), e.g., ductility, toughness, and yield strength, and could play a significant role in plastic deformation and strain hardening (Yuan et al. 2019; Ters and Shehadeh 2020; Schoeck et al. 1956). However, the motion of a dislocation can be enhanced or hindered by the micro-defects or inhomogeneities resulting from materials and manufacturing processes, particularly when the inhomogeneities are located in the vicinity of the dislocation (Zhong et al. 2018; Zheng et al. 2020; Svoboda et al. 2019; Li et al. 2019). The investigation of the interactions between dislocations and inhomogeneities is a classical topic in solid mechanics and could contribute to a better understanding of the strengthening and hardening mechanisms. In particular, the interaction energy is considered to be of fundamental importance in the study of the trapping mechanism and the motion of dislocations (Indenbom and Lothe 2012; Dundurs 1969). There are two basic types of dislocations, the edge dislocation and the screw dislocation. The first analytical treatment of the force on a screw dislocation close to a grain boundary or a surface film was provided by Head (1953). Later, Dundurs (1969, 1967) proposed an analytical solution to the problem of a screw dislocation interacting with a circular inhomogeneity. Meguid and his associates extended Dundurs’ work (1967) to an elliptical inhomogeneity (Gong and Meguid 1994) and investigated the interaction energy in piezoelectric materials (Deng and Meguid 1999). A closedform solution for the stress field arising from a screw dislocation interacting with coated circular fibers was derived by Xiao and his colleagues (Xiao and Chen 2000, 2002; Xiao et al. 2004). Xiao et al.’s three-phase cylindrical model (Xiao and Chen 2000) was also adopted to study the effect of the image force on screw dislocation (Wang and Zhou 2012; Liu et al. 2004; Feng et al. 2011; Fang et al. 2008). As for the cases of edge dislocation, Dundurs and Mura (1964) first presented a closed-form solution to the elastic field of an edge dislocation located outside a circular inhomogeneity, and some related studies on circular inhomogeneity were subsequently performed (Kachanov and Sevostianov 2018). Furthermore, Stagni and Lizzio (1981) considered the effect of the traction-free surface and analytically solved the interaction between an edge dislocation and an elliptical inhomogeneity (Stagni and Lizzio 1983), but their solutions in the Laurent series tend to be complicated to apply. Thus, in order to circumvent the inconvenience of dealing with the infinite series (Stagni and Lizzio 1981, 1983), Santare and Keer (1986) presented an analytical study of the interaction between an edge dislocation and a rigid elliptical inhomogeneity using Muskhelishvili’s complex variable techniques (Muskhelishvili 1953). With this method, Chen et al. (2011) investigated the elastic field of a coated elliptical inhomogeneity that interacted with an edge dislocation. When the cavities are considered as a type of special inhomogeneity, Dai (2018) presented solutions for an edge dislocation interacting with a circular hole inside a semi-infinite plane,

9 A Numerical Determination of the Interactions Between Dislocations …

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which is an advantageous supplement to their counterpart of the full-plane problem (Dundurs and Mura 1964). A list of classical solutions for the interactions between inhomogeneities and dislocations are summarized in Table 9.1, where the complex variable technique developed by Muskhelishvili (Muskhelishvili 1953) was utilized in most of the studies. These works on dislocation-inhomogeneity interactions are shown to be advantageous for the treatment of a few specific geometries (Dundurs 1969; Li and Shi 2002), such as a circle or ellipse. However, it is essential to note that both the shapes and distributions of the inhomogeneities could be arbitrary in reality; while many of the inhomogeneities, e.g., precipitates martensite (Mura 1982), may contain initial eigenstrains. The quantitative determination of the interaction energy in problems involving arbitrarily shaped and distributed inhomogeneities, as well as inhomogeneous inclusions, has received less attention in the literature. In addition to Muskhelishcili’s complex potential method, Eshelby’s equivalent inclusion method (EIM) (Eshelby and Peierls 1957) is another powerful tool for solving the dislocation-inhomogeneity interaction problems. The EIM was originally proposed by Eshelby for solving three-dimensional (3D) ellipsoidal inhomogeneities and can also be effective for handling two-dimensional (2D) plane inhomogeneity problems (Jin et al. 2014; Li et al. 2021a). As noted by Eshelby and Peierls (1957), the EIM could circumvent mathematical complexities, but analytical endeavors are still challenging except for idealized geometries; as a result, research efforts have spread to numerical studies. The numerical implementation of Eshelby’s EIM theory (Eshelby and Peierls 1957) is termed as the numerical equivalent inclusion method (NEIM) and has been demonstrated to be stable and robust for any material combinations in plane inhomogeneity problems (Zhou et al. 2015). In the implementation of the NEIM, the inhomogeneity is discretized into a number of elementary subdomains with proper equivalent eigenstrains to be determined via an iterative scheme. In 1987, Hutchinson (Hutchinson 1987) proposed an approximate method to investigate the crack tip shielding caused by profuse micro-cracking in brittle solids. Hutchinson’s method is regarded as the 0th-order approximation (Li et al. 2019, 2021a; Zhou et al. 2015), since only the effect of far field loading is taken into consideration while the effect of the local disturbance due to the equivalent eigenstrains is neglected. By employing Hutchinson’s approximate method, Li and his associates (Li et al. 2011; Shi and Li 2006; Yang and Li 2003) studied the interactions between the crack/dislocation and inhomogeneities. However, this approximation method may lead to inevitable and appreciable errors (Dundurs 1967; Dundurs and Mura 1964), especially for the field points located in the vicinity of the dislocation (Li et al. 2019, 2021a; Zhou et al. 2015). In recent years, Jin and his co-workers (2019, 2021a, b) further developed the NEIM and provided an efficient, flexible, and accurate alternative approach for dealing with dislocationinhomogeneity interactions in either plane or anti-plane elasticity. It is seen that the NEIM exhibits an improvement over Hutchison’s approximate method, in that the latter ignores a coupling term and hence lacks adequate iterations in determining the unknown eigenstrains.

1964

1965

1967

1983

1983

1986

1987

1991

1991

1994

Durdurs and Mura (1964)

Dundurs and Sendeckyj (1965)

Dundurs (1967)

Stagni and Lizzio (1983)

Warren (1983)

Fukuzaki and Shioya (1986)

Fukuzaki and Shioya (1987)

Luo and Chen (1991)

Worden and Keer (1991)

Gong and Meguid (1994)

Elliptical

Annular

Circular

Circular

Circular

Elliptical

Elliptical

Circular

Circular

Circular

Shape of inhomogeneities

1

1

1

2

2

1

1

1

1

1

Number of inhomogeneities O

Infinite plane; edge dislocation

Infinite plane; screw dislocation

✓

✓

Infinite plane; edge dislocation

✓

Arranged in the order of inhomogeneity-edge dislocation-inhomogeneity

✓

Three-phase composite; edge dislocation

Arranged in the order of inhomogeneity-inhomogeneity-edge dislocation

✓

Infinite plane; edge dislocation

Infinite plane; screw dislocation

✓

Infinite plane; edge dislocation

Infinite plane; edge dislocation

✓

✓

Notes

✓

✓

✓

I

Relative position between inhomogeneities and dislocations

Where the symbol “I” stands for the dislocation located inside the inhomogeneity, while “O” represents the outside

Year

Author

Table 9.1 Some classical analytical solutions for the interaction between inhomogeneities and dislocations

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In the present chapter, a numerical method is presented to study the interaction energy of dislocations with arbitrarily shaped inhomogeneities. A brief discussion of the EIM and NEIM for solving inhomogeneous inclusion problems is presented first. From an energy point of view, the fundamental equations of the interaction energy between a dislocation and an inhomogeneous inclusion are then established. With the assistance of the NEIM and the fast Fourier transform (FFT) algorithms, an effective iterative computational scheme is proposed for problems involving inhomogeneity and inhomogeneous inclusions. Subsequently, parametric studies are conducted for various material combinations to demonstrate the effectiveness of the proposed scheme. In addition, the stress field due to a dislocation interacting with layered inhomogeneities as well as multiple inhomogeneities of diverse shapes are investigated, while the interaction energy and force on dislocation are discussed in some cases. Note that the solutions to a screw or edge dislocation in a homogeneous medium are provided in Appendix 1, and the elementary solutions of a rectangular inclusion in an infinite plane or anti-plane are given in Appendix 2.

9.2 Problem Formulation In two-dimensional elasticity, consider an edge or screw dislocation located at a position (ξ, η) near an arbitrarily shaped inhomogeneity , which is assumed to be perfectly bonded to the surrounding matrix and subjected to initial eigenstrains εiPj . As shown in Fig. 9.1, the infinite matrix with elastic moduli Ci jkl is denoted as region “1”, and the subdomain, , with elastic moduli Ci∗jkl is designated as region “2”. The corresponding subscripts “1” and “2” on field quantities are henceforth adopted in reference to the matrix and inhomogeneity, respectively. Fig. 9.1 Schematic of a dislocation interacting with an arbitrarily shaped inhomogeneous inclusion, , subjected to initial eigenstrains εiPj

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9.3 Equivalent Inclusion Method for Inhomogeneity-Dislocation System The equivalent inclusion method (EIM) is proposed innovatively by Eshelby 1959 and Eshelby and Peierls (1957), and its applications for solving an inhomogeneity problem in three-dimensional or plane elasticity have been extensively discussed in literature, cf. Jin et al. (2014), Zhou et al. (2015), Zhang et al. (2018) among others. In the theory of the EIM (Eshelby and Peierls 1957; Jin et al. 2014), the disturbance resulting from the existence of inhomogeneity can be regarded as eigenstresses caused by the corresponding homogeneous inclusion with properly selected eigenstrains εi∗j . Therefore, the inhomogeneous inclusion problem can be solved by superposing a homogeneous material solution and an equivalent inclusion solution (Li 2021a, c; Jin et al. 2014), as shown in Fig. 9.2. The consistency condition may be established inside the subdomain  as  0   0  P P ∗∗ + Sklmn εmn − ε P + εkl = Ci jkl εkl + Sklmn εmn + εkl − εkl Ci∗jkl εkl

(9.1)

0 where εkl denotes the strain solution due to dislocation in a homogenous material (cf. Fig. 9.2a); Sklmn is the so-called Eshelby tensor, which depends on the inclusion shape; εkl is the strain disturbance and is determined from the equivalent eigenstrains, ∗ i.e., εi j = Si jkl εkl ; εklP is the initial eigenstrains due to such inelastic deformations ∗∗ satisfy the as thermal expansion or lattice mismatch, while the total eigenstrains εkl ∗∗ P ∗ ∗ relation εkl = εkl + εkl . In Eq. (9.1), the eigenstrains εkl are the only unknown to be determined, theoretically through solving the complex tensorial equation. Alternatively, in light of the compliance moduli for the matrix Mi jkl and the inhomogeneity Mi∗jkl , the governing equation of EIM, i.e., Eq. (9.1), can be recast into the following form according to the consistency condition of the strains (Li 2021a, c; Jin et al. 2014):

(a)

(b)

Fig. 9.2 Schematic of the equivalent inclusion method: a An homogeneous material field only due to a dislocation; b A disturbance field caused by the corresponding equivalent inclusion. Note that εi∗∗j = εiPj + εi∗j

9 A Numerical Determination of the Interactions Between Dislocations …

    P P Mi∗jkl σkl0 + Tklmn εmn + σkl = Mi jkl σkl0 + Tklmn εmn + σkl + εi∗j

165

(9.2)

where σkl0 is the dislocation solution for stress; Tklmn is the stress Eshelby tensor; and ∗ . Note that the solutions σkl is the eigenstress caused by the unknown eigenstrain εkl of the screw and edge dislocations are detailed in Appendix 1. In Eq. (9.2), the compliance moduli for an isotropic material with Young’s modulus E and Poisson’s ratio ν can be expressed as Mi jkl

  δik δ jl + δil δ jk (1 + ν)/2 − νδi j δkl = E

(9.3)

Equation (9.2) can be further simplified as   P εi∗j = Mi jkl σkl0 + Tklmn εmn + σkl

(9.4)

where Mi jkl is the compliance difference between the inhomogeneity and the matrix, i.e., Mi jkl = Mi∗jkl − Mi jkl . Note that Eq. (9.4) can be applied to the case of a general 3D inhomogeneous inclusion as well. For an edge dislocation interacting with an inhomogeneous inclusion, only the in-plane components in the governing equations of EIM are retained, and the corresponding compliance moduli difference Mi jkl can be presented in matrix form: ⎡ ⎤ m 1111 m 1122 0   κ1 + 1 ⎣ (9.5) Mi jkl = 0 ⎦ m 2211 m 2222 4μ1 (1 + α) 0 0 m 1212 where the components of m i jkl are expressed as m 1111 = m 2222 = α m 1122 = m 2211 = (2β − α) m 1212 = 4(α − β)

(9.6)

with Dundurs’ parameters α and β being α=

μ2 (κ1 − 1) − μ1 (κ2 − 1) μ2 (κ1 + 1) − μ1 (κ2 + 1) ,β = μ2 (κ1 + 1) + μ1 (κ2 + 1) μ2 (κ1 + 1) + μ1 (κ2 + 1)

(9.7)

Note that in Eq. (9.7), μ is the shear modulus with μ = E/2(1 + ν), and Kolosov’s constant, κ is

κ = 3 − 4ν, in plane strain , in plane stress κ = 3−ν 1+ν

(9.8)

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For a screw dislocation interacting with an inhomogeneous inclusion, the governing equation of EIM in Eq. (9.4) becomes 

∗ ε31 ∗ ε32



μ1 − μ2 = 2μ1 μ2



  p  0 σ31 T3131 T3132 2ε31 σ31 + + p 0 σ32 T3231 T3232 2ε32 σ32

(9.9)

where σkl0 is the initial homogeneous stress for screw dislocation. When the initial eigenstrains vanish in Eq. (9.1), the governing equation can be applied to an inhomogeneity problem,  0   0  ∗ ∗ ∗ + Sklmn εmn + Sklmn εmn − εkl = Ci jkl εkl Ci∗jkl εkl

(9.10)

For a dislocation pile-up interacting with multiple inhomogeneous inclusions, the corresponding stress consistency conditions may be established inside each inhomogeneity: ⎡ )⎣ Ci∗(I jkl

M 

0(J ) εkl

+

= Ci jkl ⎣

p(I ) Sklmn εmn −

p(I ) εkl



⎤ + εkl ⎦

i=1,i= I

j=1

⎡

N  

M  j=1

0(J ) εkl

+

N  

p(I ) Sklmn εmn

−

p(I ) εkl



⎤ + εkl −

∗(I ) ⎦ εkl ,

I = 1, 2, ..I, .., N

i=1,i= I

(9.11) where the numbers of inhomogeneous inclusions and dislocations are defined as N and M, respectively.

9.4 Interaction Energy and Force on Dislocation The interaction energy from the case depicted in Fig. 9.1 can be obtained by considering two sub-problems (Li 2021a): (1) an inhomogeneity interacting with a dislocation, and (2) the effect of the initial eigenstrains, εiPj , on the dislocation. For the first sub-problem, the strain energy, W , per unit length in the x3 -direction can be taken as the work required to inject the dislocation (Li 2021a, c; Dundurs 1967): 1 W = 2



 0  σi j + σi j u i,0 j d D

(9.12)

D

where D indicates the material system and u i,0 j is the displacement gradient due to the dislocation. When the material is homogeneous, the energy, W0 , caused by the dislocation in the absence of inhomogeneity is

9 A Numerical Determination of the Interactions Between Dislocations …

W0 =

1 2

167

 σi0j u i,0 j d D

(9.13)

D

Therefore, the interaction energy, W1 , between the dislocation and the inhomogeneity could be written as W1 = W − W0 =

1 2

 σi j u i,0 j d D

(9.14)

D

To simplify the interaction energy in Eq. (9.14), it is essential to note that   0   ∗ u i, j = σi0j u i, j − εi∗j σi j u i,0 j = Ci jkl u k·l − εkl

(9.15)

where u i, j is the displacement gradient owing to the inhomogeneity. Also, in view of the divergence theorem as well as the traction-free condition at the remote boundary and equilibrium condition, one can obtain 

 σi0j u i, j d D = D

 σi0j u i n j d S −

S

σi0j, j u i d D = 0

(9.16)

D

With the aid of Eqs. (9.15) and (9.16), Eq. (9.14) can be simplified as W1 = −

1 2



σi0j εi∗j d

(9.17)



For the second sub-problem, when the initial eigenstrains, εiPj , are superimposed on the strain field described by the inhomogeneity and the dislocation, the interaction energy, W2 , can be obtained through the work done on the dislocation (Dundurs 1967):     ∼ 0 (9.18) W2 = σ i j u i, j d D = − σi0j εiPj + εi∗P j d 

D ∼

∼

where for the homogeneous inclusion and σ i j =  σ i j ∗is the  stress ∗P ; εi∗P − εkl Ci jkl u k,l − εkl j denotes the equivalent eigenstrains due to the initial eigenstrains εiPj . According to Eqs. (9.17) and (9.18), the resulting interaction energy for the dislocation interacting with the inhomogeneous inclusion may be obtained by the superposition of W1 and W2 : (Li 2021a) W = −

1 2

 

σi0j εi∗j d −

 

  σi0j εiPj + εi∗P j d

(9.19)

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Note that when the initial eigenstrains are not considered, i.e., εiPj = 0, Eq. (9.19) can be applied to an inhomogeneity problem: W = −

1 2



σi0j εi∗j d

(9.20)



On the other hand, taking into account the discontinuity  of displacement    across the dislocation and using the relation, σi j u i,0 j = σi j u i0 , j − σi j, j u i0 = σi j u i0 , j , Eq. (9.20) could be recast as W =

1 2

 σi j bi n j d

(9.21)

where bi is termed as the Burgers vector and represents the displacement jump across the slit; denotes dislocation line starting from the core to infinity, and hence the domain of integration need to be truncated when Eq. (9.21) is implemented for numerical calculations. The general formula of interaction energy in Eq. (9.19) is established from an energy point of view. Note that unlike the infinite domain of integration in Eq. (9.12), the computational domain in Eq. (9.19) is finite and only confined to the subdomain . Therefore, the computational time and the memory space could be remarkably reduced when one employs the present numerical method illustrated in Sect. 9.5. Accordingly, the force exerted on the dislocation is defined as the negative gradient of the interaction energy with respect to the dislocation position (ξ, η): F(ξ, η) = −

∂ W − → ∂ W − → i − j ∂ξ ∂η

(9.22)

− → − → where i and j are unit vectors in the Cartesian coordinates. Note that a positive force implies the inhomogeneity repels the dislocation; otherwise, the dislocation is attracted by the inhomogeneity.

9.5 Numerical Implementation of the EIM The inhomogeneities with an idealized shape, such as ellipse and circle, have been comprehensively investigated (Jin et al. 2009, 2011, 2014, 2017; Zhou et al. 2015). In order to deal with an arbitrarily shaped inhomogeneity, numerical implementation of the EIM (NEIM) is presented next. The computational domain, encompassing all the inhomogeneities under consideration, is numerically discretized into N x × N y rectangular elements of the same size (Fig. 9.3). With the superposition principle, the resultant elastic stresses caused by the inhomogeneities can be obtained by summing all the contributions from each

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Fig. 9.3 Schematic of the numerical EIM for inhomogeneities of any arbitrary shapes and a typical rectangular   element centered  at x10 , x20 with its sides of length 1 × 2 parallel to the coordinate axes

rectangular equivalent inclusion element, which should be sufficiently small so that the eigenstrains inside are assumed to be uniform. Based on the elementary solutions supplied in Appendix 2, the stresses at field point (I, J ) are obtained as σi j (I, J ) =

Ny Nx  

  ∗    I ,J Ti jkl I − I  , J − J  εkl

(9.23)

I  =1 J  =1

where 1 ≤ I ≤ N x , 1 ≤ J ≤ N y . Since the above equation has a 2D discrete convolution structure, the fast Fourier transform (FFT) algorithm (Liu et al. 2012; Sun et al. 2020) can be adopted to reduce the tremendous computational burden. In light of the FFT, Eq. (9.23) can be rewritten as   ∼∗   σi j (I, J ) = F F T −1 T˜i jkl I − I  , J − J  ε kl I  , J 

(9.24)

where 1 ≤ I, I  ≤ 2N x , 1 ≤ J, J  ≤ N y . The hat “~” denotes the Fourier transform and F F T −1 represents the inverse Fourier transform. The elastic energy for inhomogeneities of arbitrary shapes may be determined as (cf. Eq. 9.17) 1 W = − 2

 

x  1 =− σi j (I, J )εi∗j (I, J ) 2 I =1 J =1

N

σi0j εi∗j d

Ny

(9.25)

In view of the superposition principle, the resultant energy may be obtained by summing the contributions from each element. In order to obtain the interaction energy between the inhomogeneities and the ∗ in dislocations, the key step is to determine the suitable equivalent eigenstrains εkl each rectangular element. In this section, a numerical scheme of the NEIM for solving the interaction energy and force on dislocation due to the dislocations interacting with inhomogeneities is proposed (Fig. 9.4).

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Fig. 9.4 Flowchart of the present computational scheme for solving the interaction energy and force on dislocation due to a dislocation interacting with inhomogeneous inclusions

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As demonstrated in the flowchart (Fig. 9.4), the present scheme starts from a pre-processing step of initializing the material parameters Mi jkl , Mi∗jkl , and input the initial eigenstrains εiPj and homogeneous material solution σi0j . The next step is to set the initial iterative parameters, where the disturbance terms σi(1) j and the 0th equivalent ∗(0) eigenstrain εi j are assumed to be zero. Then the iteration loop is performed to update the equivalent eigenstrains, until the given accuracy constraints are achieved or the maximum allowed number of iterations has been attained. Note that the superscript K denotes the K -th iteration, and L is used to record the position of dislocation. If the convergence criteria are not satisfied, the disturbance solutions will be re-evaluated according to the refreshed equivalent eigenstrains. The conclusion of the iteration is invoked to obtain the interaction energy and the force on dislocation through Eqs. (9.25) and (9.22). Theoretically, a fine mesh, more iterations, or both, can lead to satisfactory accuracy (Li et al. 2019, 2021a, b).

9.6 Results and Discussions In this section, comprehensive numerical studies are conducted to examine the effectiveness of the computational scheme proposed in Sect. 9.5. The following cases are investigated numerically: (1) SiC/Ti–6Al–4V inhomogeneity with complex boundary; (2) Layered inhomogeneities of diverse shapes; (3) Multiple inhomogeneities in different shapes. Note that due to space limitations, only the screw dislocations and inhomogeneities are discussed here. Nevertheless, based on the corresponding solutions, which have been well documented in Appendices A and B, the interactions between edge dislocations and inhomogeneities can also be solved in a similar manner.

9.6.1 The Interaction of a Screw Dislocation with a SiC/Ti–6Al–4V Inhomogeneity SiC or Ti–6Al–4V reinforcing particles are widely used in aircraft structure materials for enhancing the mechanical properties, as well as extending the service life of aerospace systems (Li et al. 2016; Yan et al. 2008). It has been reported that crack initiation is generally observed in the vicinity of inhomogeneities (Heinz and Eifler 2016; Murakami 2002). In the case studies, Ti–6Al–4V and SiC (see Table 9.2 for material parameters) are chosen as inhomogeneity materials, while the interaction energy and the force on dislocation will be studied next. Consider a screw dislocation located at (ξ, 0) having Burgers vector b3 = 1, while an elliptical inhomogeneity with semi-axes a1 and a2 is centered at the origin (Fig. 9.5). Note that the mesh is set to be 512 × 512 and the allowed maximum

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Table 9.2 Computational parameters of an inhomogeneity and the corresponding matrix

Shear modulus, μ (GPa) Poisson’s ratio, ν

Matrix

SiC

Ti–6Al–4V

80.8

176.7

41.0

0.30

0.16

0.34

iteration number is chosen as 5, which tends to be sufficient as demonstrated in Li et al. (2019, 2021a, b). Figure 9.6 demonstrates the variation of the normalized interaction energy with dislocation position ξ/a1 varying along the x1 -axis ranging from 1.1 to 1.5, and the ratio of semi-axes a2 /a1 is selected as 0.25, 0.5, 1.0, 2.0, 4.0. It could be found that as the screw dislocation moves towards the inhomogeneity, the interaction energy and the force on dislocation both rise with gradually increasing slope. In addition, when the position of dislocation is fixed, the interaction energy also increases for the larger ratio of semi-axes. The existing analytical solutions are mainly restricted to the inhomogeneity with a specific shape, e.g., ellipse and circle (cf. Table 9.1); but in engineering practice, the heterogenous materials commonly contain distributed inhomogeneities

Fig. 9.5 Schematic of an elliptical inhomogeneity interacting with a screw dislocation

(a)

(b)

Fig. 9.6 a The normalized interaction energy between an elliptical inhomogeneity and a screw dislocation; b The normalized force due to an elliptical inhomogeneity on a screw dislocation

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173

of arbitrary shape. Thus, an inhomogeneity, whose shape function is r (m) (ψ) = a0 + 2an cos(mψ), is considered, as shown in Fig. 9.7. In the present numerical, a0 is set as 1, the shape coefficient, an , is chosen as 0.05, and rugosity of the inhomogeneity boundary, m, is selected as 4 or 10. As shown in Fig. 9.8, SiC inhomogeneity, which is stiffer than the matrix, repels the screw dislocation, while Ti–6Al–4V inhomogeneity is softer and attracts the dislocation. It could be expected that when the inhomogeneity becomes a pore-type defect, the dislocation annihilation will arise, since the nearby dislocations move into the cavity through the attractive force, as observed by Alquier et al. (2005). Besides, the interaction energy and the force on dislocation are quite different for the inhomogeneity having various shapes, especially when the dislocation approaches the inhomogeneity.

Fig. 9.7 Schematic of an inhomogeneity with complex boundary interacting with a screw dislocation

(a)

(b)

Fig. 9.8 a The normalized interaction energy between an inhomogeneity with complex boundary and a screw dislocation; b The normalized force due to an inhomogeneity with complex boundary on a screw dislocation

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9.6.2 Layered Inhomogeneities and Horizontally Distributed Dislocations The interaction between dislocations and heterogeneous microstructures (e.g., voids, second phases, twins, and cracks) plays an important role in the context of material design and strength evaluation. Particularly in the vicinity of dislocation sources, the presence of an inhomogeneity may lead to the piling up of dislocations. The stress concentration near such a dislocation pile-up may further triggers a number of undesirable phenomena such as phase transformation and initiation of Frank-Read sources (Voskoboinikov et al. 2007). In the following case studies, screw dislocations near a layered circular inhomogeneity are uniformly distributed along the x1 -axis (Fig. 9.9). The first dislocation to the right of the inhomogeneity is located at (1.5a, 0). The interval between two adjacent dislocations is defined as = 0.2a. The component of Burgers vector in each dislocation is set as b3 = 1. Poisson’s ratios for both the inhomogeneity and the surrounding matrix are selected as 0.3. In order to implement the proposed algorithm, a computational domain, −2r ≤ x1 , x2 ≤ 2r , containing the layered inhomogeneity is meshed by 512 × 512 rectangular elements, and the allowed maximum iterative number is set as 5. Note that r = a in all these cases, unless otherwise indicated. As shown in Fig. 9.9, the inhomogeneity is centered at the origin of the Cartesian coordinates. The shear modulus of matrix is μ1 , while the shear moduli from the outer layer to the inner core are denoted as μc1 , μc2 and μc3 . The corresponding radius is defined as Rc1 = a, Rc2 = 0.8a, Rc3 = 0.6a, respectively. The dimensionless disturbance stresses for different combinations of shear moduli are reported in Fig. 9.10. From the results, it can be found that the maximum stress exhibits on the inside of each layer, which means the stress-induced phase transformation is more likely to be produced in these locations. Compared with the softer outer layer in Fig. 9.10c–d, the harder outer layer (cf. Fig. 9.10a–b) may increase the disturbance stresses. Figure 9.11 demonstrates a three-layer elliptical inhomogeneity interacting with multiple screw dislocations. The shear moduli of the layers from the outer to the inner are represented by μe1 , μe2 , and μe3 , respectively. The corresponding major and minor semi-axis of the layers are defined as (a, 0.5a), (0.8a, 0.4a), Fig. 9.9 Schematic of the interaction of a three-phase circular inhomogeneity with multiple screw dislocations, which are distributed uniformly along the x1 -axis

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(a)

(b)

(c)

(d)

175

Fig. 9.10 Contour plots of the normalized disturbance stresses of a layered circular inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the x1 -axis. For the case that μc1 = 1.5μ1 , μc2 = μ1 , μc3 = 0.75μ1 : stress components a σ31 and b σ32 . For the case that μc1 = 0.75μ1 , μc2 = μ1 , μc3 = 1.5μ1 : stress components c σ31 and d σ32

(0.6a, 0.3a), respectively. Compared with the results of the layered circular inhomogeneity (Fig. 9.10), an obvious stress concentration can be found at the tip of elliptic heterogeneous structure, as shown in Fig. 9.12. For both harder and softer outermost layer, the maximum absolute values of the stresses occur at the interface between the outermost and the middle layers. Fig. 9.11 Schematic of the interaction of a three-phase elliptical inhomogeneity with multiple screw dislocations, which are distributed uniformly along the x1 -axis

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(a)

(b)

(c)

(d)

Fig. 9.12 Contour plots of the normalized disturbance stresses of a layered elliptical inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the x1 -axis. For the case that μe1 = 1.5μ1 , μe2 = μ1 , μe3 = 0.75μ1 : stress components a σ31 and b σ32 . For the case that μe1 = 0.75μ1 , μe2 = μ1 , μe3 = 1.5μ1 : stress components c σ31 and d σ32

As shown in Fig. 9.13, a two-phase rectangular inhomogeneity interacting with multiple screw dislocations is also studied. The rectangular inhomogeneity is centered at the origin of the Cartesian coordinates. The outer and inner rectangles have dimensions 2a × 2a and 0.8a × 0.8a, respectively. The corresponding shear moduli are denoted as μr 1 , and μr 2 , respectively. It can be found from Fig. 9.14 that the disturbance stress σ32 exhibits a jump across the interfaces between the outer layer and the surrounding matrix/inner core, when the field point varies along the x1 -axis.

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Fig. 9.13 Schematic of a two-phase rectangular inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the x1 -axis

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Fig. 9.14 Contour plots of the normalized disturbance stresses of a layered rectangular inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the x1 -axis. For the case that μr 1 = 1.5μ1 , μr 2 = 0.75μ1 : stress components a σ31 and b σ32 . For the case that μr 1 = 0.75μ1 , μr 2 = 1.5μ1 : stress components c σ31 and d σ32

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Fig. 9.15 Schematic of a layered circular inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the line x1 = 1.5a

9.6.3 Layered Inhomogeneities and Vertically Distributed Dislocations Consider a three-phase circular inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the line x1 = 1.5a with the interval between two adjacent dislocations being = 0.2a, as shown in Fig. 9.15. The other parameters not listed are identical to those in Sect. 6.1. Compared with the results in Fig. 9.10, the maximum disturbance stress component in the present case (Fig. 9.16) is approximately 30% larger. In summary, the dislocation-inhomogeneity interaction effects may have a significant impact on the motions of the dislocations, and the relevant studies may shed light on a better understanding of certain strengthening and hardening mechanisms of heterogeneous engineering materials. As for a two-phase rectangular inhomogeneity (Fig. 9.17), a larger disturbance stress may also be found, as evidenced in Fig. 9.18.

9.6.4 Multiple Circular Inhomogeneities Around a Dislocation In this section, the interaction of a screw dislocation with multiple circular inhomogeneities is studied. As shown in Fig. 9.19, a screw dislocation is placed at the origin of the Cartesian coordinates. Each inhomogeneity is labeled by a number, e.g. 1 through 4. The shear moduli of four inhomogeneities corresponding to the label number are denoted as μc1 , μc2 , μc3 , and μc4 , respectively. All the inhomogeneities have the same radius, r = a, and are uniformly distributed inside a computational domain, −2a ≤ x1 , x2  2a. The dimensionless stresses of four circular inhomogeneities interacting with the screw dislocation are studied in Fig. 9.20. To illustrate the effect of the homogeneous stresses on the heterogeneous structures, this work also reported the resultant solutions, cf. Fig. 9.20a, c. Due to the singularities arisen from the dislocation, the resultant stresses inside a region of radius 0.25a centered at the origin are removed. It

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Fig. 9.16 Contour plots of the normalized disturbance stresses of a layered circular inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the line x1 = 1.5a. For the case that μc1 = 1.5μ1 , μc2 = μ1 , μc3 = 0.75μ1 : stress components a σ31 and b σ32 . For the case that μc1 = 0.75μ1 , μc2 = μ1 , μc3 = 1.5μ1 : stress components c σ31 and d σ32 Fig. 9.17 Schematic of a two-phase rectangular inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the line x1 = 1.5a

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Fig. 9.18 Contour plots of the normalized disturbance stresses of a layered rectangular inhomogeneity interacting with multiple screw dislocations, which are distributed uniformly along the line x1 = 1.5a. For the case that μr 1 = 1.5μ1 , μr 2 = 0.75μ1 : stress components a σ31 and b σ32 . For the case that μr 1 = 0.75μ1 , μr 2 = 1.5μ1 : stress components c σ31 and d σ32 Fig. 9.19 Schematic of four circular inhomogeneities interacting with a screw dislocation, which is located at the origin of the Cartesian coordinates

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is interesting to show that the magnitude of disturbance stress caused by stiff impurities and dislocation stress have the same sign (Fig. 9.20a, b), while an opposite sign is found for the case of soft inhomogeneity (Fig. 9.20c, d). The results demonstrate that the presence of a stiff or soft inhomogeneity may increase or decrease the resultant stresses, respectively. The stress-induced phase transformations tend to be occurred around the stiff inhomogeneities. The present work also studies the combinations of different elastic constants. The absolute value of the maximum disturbance stress is found near the stiff inhomogeneity. Compared with the results in Fig. 9.21a, the combination of the shear moduli in Fig. 9.21c can effectively reduce the stresses caused merely by the inhomogeneities. Figure 9.22 demonstrates the variation of the normalized interaction energy and the force versus dislocation position ξ/a varying along x1 -axis. The radius of circular inhomogeneities is 0.9a here. The shear moduli of four inhomogeneities are identical

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Fig. 9.20 Contour plots of the normalized stress of four circular inhomogeneities interacting with a screw dislocation, which is located at the origin of the Cartesian coordinates. For the case that μc1 = μc2 = μc3 = μc4 = 2μ1 : a resultant stress σ31 ; b disturbance stress σ31 . For the case that μc1 = μc2 = μc3 = μc4 = 0.5μ1 : c resultant stress σ31 ; d disturbance stress σ31

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Fig. 9.21 Contour plots of the normalized stress of four circular inhomogeneities interacting with a screw dislocation, which is located at the origin of the Cartesian coordinates. For the case that μc1 = μc2 = 1.5μ1 , μc3 = μc4 = 0.75μ1 : a disturbance stress σ31 ; b disturbance stress σ32 . For the case that μc1 = μc3 = 0.75μ1 , μc2 = μc4 = 1.5μ1 : c disturbance stress σ31 ; d disturbance stress σ32

and the ratio of shear moduli μc /μ1 is set to be 1.5, 2, 3, 4, 6, and 8, respectively. It can be readily observed that as the dislocation moves along x1 -axis, the normalized interaction energy firstly increased then decreased; accordingly, the dislocation is attracted but then repelled by inhomogeneities. Besides, Fig. 9.22 reveals that the interaction effects become stronger with increasing μc /μ1 . Furthermore, multiple circular layered inhomogeneities interacting with a screw dislocation are studied, as shown in Fig. 9.23. The radius of the outer layer is set as a; the major and minor semi-axis of the inner core are defined as r1 and r2 , respectively. The shear moduli of each layered inhomogeneity are assumed as μi j , where i represents inhomogeneity number, while j = 1 and j = 2 denote the outer layer and inner core, respectively. The results of the interaction of layered inhomogeneities with a screw dislocation are reported in Fig. 9.24. From the disturbance solutions in Fig. 9.24a–c, the stresses are minimal for the combination of soft outer layer and stiff inner core in Fig. 9.24b.

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Fig. 9.22 a The normalized interaction energy for a screw dislocation interacting with four circular inhomogeneities and b the corresponding normalized force on the screw dislocation. The ratio of shear moduli μc /μ1 is set to be 1.5, 2, 3, 4, 6, and 8

Fig. 9.23 Schematic of four layered inhomogeneities interacting with a screw dislocation, which is located at the origin of the Cartesian coordinates

When the inner core varies from a circle (Fig. 9.24b) to an ellipse (Fig. 9.24d), stress concentration can be found at the tip of each elliptical inhomogeneity, and the maximum resultant stresses increase slightly. The normalized interaction energy and the force on dislocation are investigated as well (Fig. 9.25). For the combination of stiff outer layer and soft inner core, the overall trends are similar to that in Fig. 9.22; while the inhomogeneities with soft outer shell and stiff inner core have an opposite effect on the dislocation.

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Fig. 9.24 Contour plots of the normalized stress of four layered inhomogeneities interacting with a screw dislocation, which is located at the origin of the Cartesian coordinates. For cases a μ11 = μ21 = μ31 = μ41 = 1.5μ1 , μ12 = μ22 = μ32 = μ42 = 0.75μ1 ; or b μ11 = μ21 = μ31 = μ41 = 0.75μ1 , μ12 = μ22 = μ32 = μ42 = 1.5μ1 , the disturbance stress σ31 when r1 = r2 = 0.8a. For the cases that μ11 = μ31 = μ22 = μ42 = 1.5μ1 , μ12 = μ32 = μ21 = μ41 = 0.75μ1 , the disturbance stress σ31 , when c r1 = r2 = 0.8a or d r1 = 0.4a, r2 = 0.8a

9.6.5 Multiple Elliptical Inhomogeneities Around a Dislocation In this section, case studies are performed for four elliptical inhomogeneities interacting with a screw dislocation, which is placed at the origin of the Cartesian coordinates, as shown in Fig. 9.26. Each inhomogeneity is labeled by a number, and the major and minor semi-axes of all the inhomogeneities are a and a/2, respectively. The shear moduli of the four inhomogeneities are denoted sequentially as μe1 , μe2 , μe3 , and μe4 . The distance between the inhomogeneities 1 and 2 (or 3 and 4) is defined as . Compared with Fig. 9.27a, b, the disturbance stresses in Fig. 9.27c, d decrease gradually with the increase of the distance , which indicates the interaction

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Fig. 9.25 a The normalized interaction energy for a screw dislocation interacting with four layered inhomogeneities and b the corresponding normalized force on the screw dislocation. The radius of the outer layer is set as 0.9a and r1 = r2 = 0.7a. The shear moduli of four layered inhomogeneities are set to be μ11 = μ21 = μ31 = μ41 = μc1 and μ12 = μ22 = μ32 = μ42 = μc2 . The ratio of shear moduli μc1 /μ1 and μc2 /μ1 is set to be 1.5 and 0.75, respectively (blue line); while in another case, the ratio of shear moduli μc1 /μ1 and μc2 /μ1 is set to be 0.75 and 1.5, respectively (red line)

Fig. 9.26 Schematic of four elliptical inhomogeneities interacting with a screw dislocation, which is located at the origin of the Cartesian coordinates

between the inhomogeneities is weakened. For both stiff and soft inhomogeneities in Fig. 9.27c–f, the maximum and minimum disturbance stresses are located at either the boundary or the tip of the ellipses.

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Fig. 9.27 Contour plots of the normalized stress of four elliptical inhomogeneities interacting with a screw dislocation, which is located at the origin of the Cartesian coordinates. For the case that μe1 = μe2 = μe3 = μe4 = 2μ1 , = a: a disturbance stress σ31 ; b disturbance stress σ32 . For the case that μe1 = μe2 = μe3 = μe4 = 2μ1 , = 2a: a disturbance stress σ31 ; b disturbance stress σ32 . For the case that μe1 = μe2 = μe3 = μe4 = 0.75μ1 , = 2a: e disturbance stress σ31 ; f disturbance stress σ32

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9.6.6 Multiple Rectangular Inhomogeneities Around a Dislocation The interaction of four square inhomogeneities with a screw dislocation is studied in this section. As shown in Fig. 9.28, each rectangular inhomogeneity has the same side length 1.4a and is labeled by a number. The shear moduli of four inhomogeneities are denoted as μr 1 , μr 2 , μr 3 , and μr 4 , respectively. The corresponding results are discussed in Fig. 9.28. Compared with the case of circular or elliptical inhomogeneities, the stress concentration at the sharp corner can be easily found. For different combinations of shear moduli, the component σ31 of the disturbance stresses distributed inside the inhomogeneities is shown to be negative. When the shear modulus of the inhomogeneities decreases gradually from 1 to 4, the maximum and minimum disturbance stresses can be found at one corner of inhomogeneity “4” (Fig. 9.29e, f). Additionally, the variation of the normalized interaction energy and force are depicted in Fig. 9.30.

9.7 Conclusions The interaction between dislocations and inhomogeneities is of importance for crack growth and strain hardening in heterogeneous materials. The related topic could play a significant role in material design and strength evaluation. However, the existing solutions are mainly applicable to limited geometries and the problem may become analytically intractable for complex configurations. On the other hand, due to the inherent singularities involved in the dislocation solution, computational obstacles would be encountered for commercial finite element software. By employing the numerical equivalent inclusion method (NEIM) in conjunction with the fast Fourier transform (FFT) based algorithms, this work presents an effective iterative numerical scheme for evaluating the interaction energy and the Fig. 9.28 Schematic of four rectangular inhomogeneities interacting with a screw dislocation, which is placed at the origin of the Cartesian coordinates

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Fig. 9.29 Contour plots of the normalized disturbance stress of four rectangular inhomogeneities interacting with a screw dislocation, which is located at the origin of the Cartesian coordinates. The stress component σ31 for the case that a μr 1 = μr 2 = μr 3 = μr 4 = 2μ1 ; and b μr 1 = μr 2 = μr 3 = μr 4 = 0.75μ1 . For the case that μr 1 = μr 2 = 1.5μ1 , μr 1 = μr 2 = 0.75μ1 : c stress component σ31 ; d stress component σ32 . For the case that μr 1 = 1.5μ1 , μr 2 = 1.25μ1 , μr 3 = 0.75μ1 , μr 4 = 0.5μ1 : c stress component σ31 ; d stress component σ32

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Fig. 9.30 a The normalized interaction energy for a screw dislocation interacting with four rectangular inhomogeneities and b the corresponding normalized force on the screw dislocation. The side length of each rectangular inhomogeneity increases to 1.8a. The shear moduli of inhomogeneities are set to be μr 1 = μr 2 = μr 3 = μr 4 = μr and the ratio of shear moduli μr /μ1 is set to be 1.5, 2, 3, 4, 6, and 8

force on dislocation. The computational domain encompassing multiple arbitrarily shaped inhomogeneities as well as inhomogeneous inclusions could be discretized into a number of rectangular patches. With the superposition principle, the resultant elastic energy and interaction energy may be conveniently obtained by summing the contributions from each element. The proposed method is shown to be an efficient and flexible alternative for handling dislocation-inhomogeneity interactions. According to the complex cases (i.e., layered inhomogeneities of diverse shapes, multiple inhomogeneities in different shapes) investigated thoroughly in this work, it is shown that the interaction energy, the force on dislocation, and the resultant stress field are closely related to the material parameters, spatial distributions, and shapes of the inhomogeneities. In summary, the dislocation-inhomogeneity interaction effects may have a significant impact on the motions of the dislocations, and the relevant studies may shed light on a better understanding of certain strengthening and hardening mechanisms of heterogeneous engineering materials. Acknowledgements The work is supported by National Science Foundation of China (Grant Nos. 52205192, 11932004 and 51875059), and Graduate Research and Innovation Foundation of Chongqing, China (Grant No. CYS21011). X.J would like to acknowledge the support from Chongqing City Science and Technology Program (Grant No. cstc2020jcyj-msxmX0850).

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Appendix 1: The Solution of the Screw and Edge Dislocations Edge Dislocation Solution in an Infinite Homogeneous Plane In the Cartesian coordinate system, an edge dislocation in an infinite homogeneous   medium is located at x1 , x2 and has the Burgers vector components b1 ,b2 . The induced stresses σi0j at the field point (x1 , x2 ) are given by Hills et al. (1998) and Li (2021c) ⎡ ⎤ ⎤ 0 G 111 G 211  σ11 2μ 1 ⎣ G 122 G 222 ⎦ b1 ⎣σ0 ⎦ = 22 b2 π (κ1 + 1) 0 σ12 G 112 G 212 ⎡

(9.26)

where κ is Kolosov’s constant and κ = (3 − ν)/(1 + ν) for plane stress, with Poisson’s ratio ν; μ is the shear modulus; and the influence coefficients G i jk are     ξ2 3ξ12 + ξ22 ξ1 ξ12 − ξ22 , G 211 = G 111 = − 4 4  2 r 2  2 r 2 ξ2 ξ1 − ξ2 ξ2 ξ1 + 3ξ2 G 122 = , G 222 = 4 4 r  2   2r 2 2 ξ1 ξ1 − ξ2 ξ2 ξ1 − ξ2 G 112 = , G 212 = (9.27) r4 r4  in which ξ1 = x1 − x1 , ξ2 = x2 − x2 , and r = ξ12 + ξ22 . Note that the first subscript of the above coefficients corresponds to the Burgers vector, and the remaining two indices are associated with the stress components.

Screw Dislocation Solution in an Anti-plane Shear Problem For an anti-plane shear problem, the in-plane displacements, u 1 , u 2 are both zero while the out-of-plane displacement, u 3 , is the only component of displacements and independent of x3 , i.e., u 1 = u 2 = 0, u 3 = f (x1 , x2 )

(9.28)

Thus, the corresponding non-vanishing stresses are σ31 = μ1

∂u 3 ∂u 3 , σ32 = μ1 ∂ x1 ∂ x2

(9.29)

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It is well-known that a representative example of the anti-plane shear problem is a screw dislocation, which is created by making a slit from the core of the dislocation to infinity. According to Dundurs’ convention (Dundurs 1969, 1967), the positive side is determined as the right side of the cut from the core to the infinitely remote point, and the displacement jump across the slit is represented by the Burgers vector (0, 0, b3 ).   For a straight screw dislocation located at x1 , x2 , the elastic distortion at the field point (x1 , x2 ) can be written as (Li 2021c) u 01

=

u 02

=

0, u 03

  ξ2 b3 arctan = 2π ξ1

(9.30)

in which ξ1 = x1 − x1 and ξ2 = x2 − x2 . Therefore, the strains can be determined (Li 2021c): 0 ε31 =

−b3 ξ2 b3 ξ1 0  2  2  , ε32  = 2 4π ξ1 + ξ2 4π ξ1 + ξ22

(9.31)

By applying Hooke’s law or substituting Eq. (9.30) into Eq. (9.29), the corresponding stress components are obtained as follows (Li 2021c): 0 = σ31

−μ1 b3 ξ2 μ1 b3 ξ1 0  2   , σ32  = 2 2π ξ 1 + ξ2 2π ξ 21 + ξ22

(9.32)

Note that the  strains and stresses at the field point are inverse to the relative distance, i.e., ξ12 + ξ22 , and therefore tend to infinity as the dislocation core is approached. In this particular case x  = x2 = 0, the above solution can be reduced and has been documented in Li and Wang (2008).

Appendix 2: The Elementary Solution for a Rectangular Inclusion A Notation for the Elementary Solution     with its sides of length 1 Consider a rectangular inclusion centered at x10 , x20 and 2 parallel to the coordinate axes. Accordingly, the inclusion domain  can be determined as (Li 2021c)

  1  1 2  2   , x10 + , x2 ∈ x20 , x20 +  : x1 ∈ x10 − − 2 2 2 2

(9.33)

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Based on the method of Green’s function, the potentials associated with the inclusion problems can be defined as follows:   x2 −x 20 + 2 /2 x 1 −x 10 + 1 /2

f =

G(ξ1 , ξ2 , ξ3 )dξ1 dξ2 dξ3  x2 −x20 − 2 /2

(9.34)

 x1 −x10 − 1 /2

where G(ξ1 , ξ2 ) represents Green’s function, which only depends  relative   on the , x20 ; ξ1 and ξ2 locations of the field point (x10 , x20 ) and excitation source point x10 are ξ1 = x1 − x1 , ξ2 = x2 − x2

(9.35)

Once F(ξ1 , ξ2 ) is an antiderivative of G(ξ1 , ξ2 ) and satisfies ∂ 2 F(ξ1 , ξ2 ) = G(ξ1 , ξ2 ) ∂ξ1 ∂ξ2

(9.36)

Then Eq. (9.34) can be evaluated as f = F(x11 , x21 ) − F(x11 , x22 ) + F(x12 , x22 ) − F(x12 , x21 )

(9.37)

where

  − 1 /2, x21 = x2 − x20 − 2 /2 x11 = x1 − x10   x12 = x1 − x10 + 1 /2, x22 = x2 − x20 + 2 /2

(9.38)

For simplicity of presentation, a notation for the elementary solution is introduced:     , x20 ; 1 , 2 f = F(x1 , x2 ) x10

(9.39)

which have been discussed in detail in Jin et al. (2016). Note that the above notation illustrates the convolution structure between the center of the rectangle and the field point, and this feature is essential to the implementation of the FFT-related numerical computations.

The Elastic Field of a Rectangular Inclusion in an Infinite Plane For a rectangular inclusion in plane elasticity, the displacement components can be presented in matrix form as follows (Li 2021c):

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u1 u2



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W111 W122 W133 W211 W222 W233

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⎡ ∗ ⎤ ε11 ∗ ⎥ W112 ⎢ ε ⎢ 22 ⎥ ∗ ⎦ W212 ⎣ ε33 ∗ 2ε12

(9.40)

where the coefficient Wi jk is determined as Wi jk =

   −1  , x20 ; 1 , 2 wi jk (x1 , x2 ) x10 8π(1 − ν)

(9.41)

in which w111 = −(3 − 2ν)ξ2 ln(ξ12 + ξ22 ) − 4(1 − ν)ξ1 arctan w122 = (1 − 2ν)ξ2 ln(ξ12 + ξ22 ) − 4νξ1 arctan w133 = −2νξ2 ln(ξ12 + ξ22 ) − 4νξ1 arctan

ξ2 ξ1

w211 = (1 − 2ν)ξ1 ln(ξ12 + ξ22 ) − 4νξ2 arctan

(9.42) (9.43)

ξ2 ξ1

(9.44)

w112 = −(1 − 2ν)ξ1 ln(ξ12 + ξ22 ) − 4(1 − ν)ξ2 arctan

ξ1 ξ2

ξ1 ξ2

w222 = −(3 − 2ν)ξ1 ln(ξ12 + ξ22 ) − 4(1 − ν)ξ2 arctan w233 = −2νξ1 ln(ξ12 + ξ22 ) − 4νξ2 arctan

ξ2 ξ1

(9.46) ξ1 ξ2

ξ1 ξ2

w212 = −(1 − 2ν)ξ2 ln(ξ12 + ξ22 ) − 4(1 − ν)ξ1 arctan

(9.45)

(9.47) (9.48)

ξ2 ξ1

(9.49)

On account of the compatibility equations, strain components can be derived as (Li 2021c) ⎡

⎤ ⎡ ε11 S1111 ⎢ ε22 ⎥ ⎢ S2211 ⎢ ⎥ ⎢ ⎣ ε33 ⎦ = ⎣ S3311 2ε12 2S1211

S1122 S2222 S3322 2S1222

where the coefficient Si jkl is determined as

S1133 S2233 S3333 2S1233

⎤⎡ ∗ ⎤ S1112 ε11 ⎢ ε∗ ⎥ S2212 ⎥ ⎥⎢ 22 ⎥ ∗ ⎦ S3312 ⎦⎣ ε33 ∗ 2S1212 2ε12

(9.50)

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Si jkl =

   −1  si jkl (x1 , x2 ) x10 , x20 ; 1 , 2 2π (κ + 1)

(9.51)

in which κ denotes Kolosov’s constant, and s1111 = −(1 + κ) arctan

ξ2 2ξ1 ξ2 − 2 ξ1 ξ1 + ξ22

(9.52)

s1122 = −(3 − κ) arctan

ξ2 2ξ1 ξ2 + 2 ξ1 ξ1 + ξ22

(9.53)

s1112 =

1 2ξ 2 (1 − κ) ln(ξ12 + ξ22 ) + 2 1 2 2 ξ1 + ξ2

s2211 = −(3 − κ) arctan

ξ1 2ξ1 ξ2 + 2 ξ2 ξ1 + ξ22

(9.55)

s2222 = −(1 + κ) arctan

ξ1 2ξ1 ξ2 − 2 ξ2 ξ1 + ξ22

(9.56)

s2212 =

1 2ξ 2 (1 − κ) ln(ξ12 + ξ22 ) + 2 2 2 2 ξ1 + ξ2

s1211 = − ln(ξ12 + ξ22 ) + s1222 = − ln(ξ12 + ξ22 ) + s1212 =

(9.54)

(9.57)

2ξ12 + ξ22

(9.58)

2ξ22 ξ12 + ξ22

(9.59)

ξ12

  ξ2 2ξ1 ξ2 1 ξ1 arctan − + arctan + κ) (1 2 ξ1 ξ2 ξ12 + ξ22

(9.60)

For the plane strain cases, one has s1133 = −4v arctan

ξ2 ξ1

(9.61)

s2233 = −4v arctan

ξ1 ξ2

(9.62)

s1233 = −2v ln(ξ12 + ξ22 )

(9.63)

s3311 = s3322 = s3333 = s3312 = 0

(9.64)

When the plane stress problems are considered, one has

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s3311 = −

ξ1 4v arctan 1+ν ξ2

(9.65)

s3322 = −

ξ2 4v arctan 1+ν ξ1

(9.66)

2v ln(ξ12 + ξ22 ) 1+v   ξ1 4 ξ2 arctan + arctan =− 1+ν ξ2 ξ1 s3312 =

s3333

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s1133 = s2233 = s1233 = 0

(9.67) (9.68) (9.69)

Subsequently, the stress components can be derived by Hooke’s law as (Li 2021c) ⎤ ⎡ T1111 σ11 ⎢ σ22 ⎥ ⎢ T2211 ⎥ ⎢ ⎢ ⎣ σ33 ⎦ = ⎣ T3311 σ12 T1211 ⎡

T1122 T2222 T3322 T1222

T1133 T2233 T3333 T1233

⎤⎡ ∗ ⎤ ε11 T1112 ∗ ⎥ ⎢ ⎥ T2212 ⎥⎢ ε22 ⎥ ∗ ⎦ ⎣ ⎦ T3312 ε33 ∗ T1212 2ε12

(9.70)

where the coefficient Ti jkl is determined as Ti jkl =

   −μ  ti jkl (x1 , x2 ) x10 , x20 ; 1 , 2 π (κ + 1)

(9.71)

in which t1111 = 4 arctan

ξ1 2ξ1 ξ2 − 2 ξ2 ξ1 + ξ22

(9.72)

2ξ1 ξ2 ξ12 + ξ22

(9.73)

t1122 = t2211 = t1212 =

t1112 = t1211 = − ln(ξ12 + ξ22 ) + t2222 = 4 arctan

2ξ12 + ξ22

ξ12

ξ2 2ξ1 ξ2 − 2 ξ1 ξ1 + ξ22

t2212 = t1222 = − ln(ξ12 + ξ22 ) +

2ξ22 ξ12 + ξ22

(9.74) (9.75)

(9.76)

When the plane strain conditions are assumed, one has t1133 = t3311 = 4ν arctan

ξ1 ξ2

(9.77)

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t2233 = t3322 = 4ν arctan t3333

ξ2 ξ1

(9.78)

  ξ1 ξ2 = 4 arctan + arctan ξ2 ξ1

t1233 = t3312 = −2ν ln(ξ12 + ξ22 )

(9.79) (9.80)

On the other hand, when the stress plane case is considered, one has t1133 = t3311 = t2233 = t3322 = t3333 = t1233 = t3312 = 0

(9.81)

The Elastic Field of a Rectangular Inclusion in an Anti-plane Shear Problem According to the discussion of the anti-plane shear problem in Appendix “Screw Dislocation Solution in an Anti-plane Shear Problem”, in like manner, it is possible to derive an explicit solution for the elastic field produced by a rectangular inclusion in anti-plane elasticity. Hence, the corresponding displacement component can be obtained as (Li 2021c)  ∗   ∗ T u 3 = W331 W332 2ε31 2ε32

(9.82)

where the coefficient Wi jk is determined as Wi jk =

   1  wi jk (x1 , x2 ) x10 , x20 ; 1 , 2 4π

(9.83)

in which w331 = 2ξ1 arctan

ξ2 + ξ2 ln(ξ12 + ξ22 ) ξ1

(9.84)

w332 = 2ξ2 arctan

ξ1 + ξ1 ln(ξ12 + ξ22 ) ξ2

(9.85)

Afterward, the strain components can be derived as (Li 2021c) 

ε31 ε32



 =

S3131 S3132 S3231 S3232

where the coefficient Si jkl is determined as



∗ 2ε31 ∗ 2ε32

(9.86)

9 A Numerical Determination of the Interactions Between Dislocations …

Si jkl =

   1  si jkl (x1 , x2 ) x10 , x20 ; 1 , 2 8π

197

(9.87)

in which s3131 = 2 arctan

ξ2 ξ1

(9.88)

  s3132 = s3231 = ln ξ12 + ξ22 s3232 = 2 arctan

(9.89)

ξ1 ξ2

(9.90)

Consequently, the stress components are presented as follows (Li 2021c): 

σ31 σ32





T3131 T3132 = T3231 T3232



∗ 2ε31 ∗ 2ε32

(9.91)

where the coefficient Ti jkl is determined as Ti jkl =

   −μ  , x20 ; 1 , 2 ti jkl (x1 , x2 ) x10 4π

(9.92)

in which  ξ1 t3131 = 2 arctan ξ2   2 t3132 = t3231 = − ln ξ1 + ξ22 



t3232

ξ2 = 2 arctan ξ1

(9.93) (9.94)

 (9.95)

References Alquier D, Bongiorno C, Roccaforte F, Raineri V (2005) Interaction between dislocations and He-implantation-induced voids in GaN epitaxial layers. Appl Phys Lett 86(21):211911 Amanieu H-Y, Rosato D, Sebastiani M, Massimi F, Lupascu DC (2014) Mechanical property measurements of heterogeneous materials by selective nanoindentation: application to LiMn2 O4 cathode. Mater Sci Eng A 593:92–102 Chen FM, Chao CK, Chen CK (2011) Interaction of an edge dislocation with a coated elliptic inclusion. Int J Solids Struct 48(10):1451–1465 Cortes P, Sevostianov I, Valles-Rosales DJ (2010) Mechanical properties of carbon nanotubes reinforced composites: experiment and analytical modeling. Int J Fract 161(2):213–220

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Chapter 10

Numerical Simulations of Interface Propagation in Elastic Solids with Stress Concentrators Polina K. Kabanova, Aleksandr Morozov, Alexander B. Freidin, and Alexander Chudnovsky Abstract Stress-induced phase transformations in elastic solids with circular or elliptical holes as stress concentrators are considered. The evolution of the interface is described by a kinetic equation that relates the velocity of the interface with a configurational force equal to the jump of the normal component of the Eshelby stress tensor. Kinetics of propagation of a planar interface is studied analytically and numerically to verify the developed numerical procedure. Then the interface propagation in the vicinity of elliptical holes with various ratios of the semi-axes is analyzed basing on numerical modeling. It is studied how the shape of the hole and the thickness of the new phase layer affect the distribution of the configurational force along the interface. It is demonstrated how the stress concentration generated by the hole may induce a phase transformation even at external stress, at which no phase transformations occur in the absence of a stress concentrator. Keywords Stress-induced phase transformations · Interface propagation · Configurational force · Eshelby stress tensor · Numerical simulations · Stress concentrators

P. K. Kabanova · A. B. Freidin (B) Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia e-mail: [email protected] P. K. Kabanova e-mail: [email protected] A. Morozov Faculty of Mechanical Engineering, Institute of Mechanics, Berlin Institute of Technology, Berlin, Germany e-mail: [email protected] A. Chudnovsky Fracture Mechanics and Materials Durability Laboratory, Civil & Materials Engineering Department, The University of Illinois at Chicago, Chicago, IL 60607-7023, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_10

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10.1 Introduction Crack growth may be accompanied by the appearance of a zone of material with changed properties in the vicinity of a crack tip—a “process zone” (PZ). The crack propagation, in this case, is affected by the interaction between the crack and the process zone. A process zone may have various physical nature. In shape memory alloys, it is a domain of a new phase—the phase transformation from austenite to martensite is driven by stresses at the tip of the crack (see, e.g., McKelvey and Ritchie 1999; Robertson and Ritchie 2007; Wilkes and Liaw 2000). Using the method of in situ optical digital image correlation technique, it is possible to monitor the strain field and the configuration of the interface at the tip of a propagating crack (Daly et al. 2007). The combination of an in situ scanning electron microscopy (SEM) with electron backscatter diffraction (Roth et al. 2009) made it possible to study the evolution of a new phase in the vicinity of the tip during fatigue experiments. The phase transformation zones around the crack were also observed in ceria-zirconium alloy (Hannnink and Swain 1989) and antiferroelectric ceramics (Tan et al. 2014). The process zone may be also produced by the multiple appearances of new phase domains as it is in the case of transformation toughening in ceramics (McMeeking and Evans 1982). Inhomogeneous nucleation of new phase near stress concentrators and crack tip turns the problem in the direction of micromechanics of heterogeneous materials (see, e.g., Kachanov and Sevostianov 2005, 2018). Another example of a changed zone near a stress concentrator is a crazing zone. It was observed, for example, in the vicinity of a stationary crack in polystyrene under tension (Bevis and Hull 1970) and in the vicinity of the notch under cyclic loading (Botsis et al. 1987). When crazing density near the notch reached a certain level, crack initiation occurred. Then during fatigue crack propagation, a crazing zone surrounding the crack grew together with a crack. A crazing zone was also observed near a circular hole due to the stress concentration generated at the hole by the approaching crack (see also Chudnovsky 2014). Note that crazing, as well as the martensite phase transformation, is accompanied by transformation strain which is produced by orientational rearrangements of polymer molecules, and the crazing zone is bounded by the surface of strain discontinuity similar to the interfaces in the case of phase transformations. This motivates a formal consideration of the crazing zone as a domain of a new phase (Freidin 1989). Various approaches to PZ modeling as a new phase domain were offered. A limiting phase transformation surface in stress space can be defined similar to yield surfaces in plasticity with the use of various criteria and correction factors (see, e.g., Baxevanis et al. 2012; Birman 1998; Freed and Banks-Sills 2007; Laydi and Lexcellent 2015; Yi and Gao 2000; Yi et al. 2001). In particular, in Maletta and Furgiuele (2010) Irwin’s plastic zone model and bilinear stress-strain relation are used to evaluate the size of the transformation region. In Hazar et al. (2016), a comparison of shapes and sizes of the transformation zones obtained with different criteria is presented.

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Other models are based on considerations of stress-induced phase transformations within the framework of the phase-field approach, e.g., Chen (2002), Levitas et al. (2010), Mamivand et al. (2013), and Rezaee-Hajidehi and Stupkiewicz (2020) (see also review Levitas 2021). The phase-field approach implies considering the interface as a transition layer of finite thickness where material properties vary smoothly. The changes in properties are specified by introducing order parameters, the evolution of which is described by additional Ginzburg-Landau-type equations. In Bulbich (1992), Boulbitch and Toledano (1998), Boulbitch and Korzhenevskii (2020), Boulbitch and Korzhenevskii (2020), and Boulbitch and Korzhenevskii (2021), the phase-field approach was developed for studying the geometry of the phase transformation zone at the crack tip and the so-called wake zone located behind the crack tip. In the present paper, we implement a sharp-interface approach. The thickness of the interface is neglected. The constitutive equations are written in each phase separately and an additional kinetic or equilibrium condition is formulated for the interface. The kinetic equation is formulated basing on the dissipation inequality in the spirit of the mechanics of configurational forces (Gurtin 2000; Kienzler and Herrmann 2000; Maugin 2010). It was shown that the configurational force driving the interface is equal to the jump of the normal component of the Eshelby stress tensor (see, e.g., Wilmanski 1998; Silhavy 1997). In the case of the equilibrium interface in an elastic solid, by the Gibbs variational principle, this jump is zero (see Grinfeld 1991 and reference therein). Then a kinetic equation can be formulated in the form of the dependence of the normal component of the interface velocity on the configurational force (see, e.g., Abeyaratne and Knowles 2006 and reference therein). This results in a coupled problem for which stresses and strains depend on the interface position, and the interface position and velocity are affected by stresses and strains as they are presented in the configurational force. Such an approach was used for analytical and numerical modeling of the interface propagation, see, e.g., Mueller and Gross (1998), Mueller et al. (2006), Gross et al. (2002, 2003), Abeyaratne and Knowles (2006), Le (2007), Freidin (2019), and stressaffected chemical reaction front propagation with a configurational force defined by the combinations of the Eshelby stress tensors of the reaction constituents (Freidin et al. 2014, 2016, 2022; Morozov et al. 2019; Morozov 2021). The present paper is focused on numerical modeling of the interface propagation in the vicinity of nonsingular stress concentrators and can be considered as one of the first steps for further modeling of a crack and interface joint propagation. In Sect. 10.2, we give a brief review of basic relations describing the stress-strain state of a two-phase solid with a propagating interface. In Sect. 10.3, the description of a numerical procedure is given and the solution of the problem of the propagation of a planar interface is considered analytically and numerically to verify the numerical procedure. Section 10.4 contains the results of numerical simulations of the interface propagation in the vicinity of circular and elliptical holes. Special attention is paid to the case of external loading at which the new phase domain does not appear without stress concentrations and to the distribution of the configurational force along the interface. Conclusions are formulated in Sect. 10.5.

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10.2 Basic Relations for a Two-Phase Elastic Solid A problem of a two-phase elastic solid with a propagating interface is reduced to finding displacement u(x) and describing the kinetics of the interface , i.e., finding its position in dependence on time. Further, super- or subscripts “−” and “+” refer the values to a material being in the “−” (initial) and “+” (current) phase states, respectively, and  is the interface separating the phases (Fig. 10.1). The displacement is smooth and in a quasi-static case has to satisfy the equilibrium equation x∈ /:

∇ ·σ =0

(10.1)

with the boundary conditions, displacement, and traction continuity conditions across the unknown propagating interface : x∈:

[[u]] = 0, [[σ ]] · n = 0,

(10.2)

where σ is the stress tensor, n is the unit normal to the interface directed outward to the phase “+”, and [[...]] denotes the jump of a value from phase “−” to phase “+”: [[...]] = (...)+ − (...)− . An additional constitutive relation—a kinetic equation—is to be used to describe the kinetics of the interface . We take a kinetic equation in the form of the dependence of the normal component v of the interface velocity on the configurational force χ , which is equal to the jump of the normal component of the Eshelby stress tensor, see, e.g., Wilmanski (1998), Abeyaratne and Knowles (2006), Gurtin (2000): χ = n · [[b]] · n,

(10.3)

where in the case of small strains b = f E − ∇u · σ , and f is the volume density of the Helmholtz free energy, E is the second-rank unit tensor. The configurational force may be also represented in the form χ = [[ f ]] − σ : [[ε]].

Fig. 10.1 A schematic representation of a two-phase body

(10.4)

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Phase transformations in elastic solids are considered further. Constitutive relations for the phases have the form: σ ± = C± : (ε − ε ∗± ),

(10.5)

where ε is the strain tensor, C± are the stiffness tensors, and ε∗± are the strains in stress-free phase states. Then the volume density of the Helmholtz free energy of a body undergoing phase transformations is f (ε) = min{ f − (ε), f + (ε)},

(10.6)

1 f ± (ε) = f 0± + (ε − ε ∗± ) : C : (ε − ε ∗± ), 2

(10.7)

where f 0± are the free energy densities in stress-free states (chemical energies). In the case of phase equilibrium of elastic phases (Grinfeld 1982, 1991) χ = 0.

(10.8)

Then a kinetic equation can be taken in the form v = −κχ ,

(10.9)

where κ > 0 is the kinetic coefficient. Relationships (10.1)–(10.9) define the coupled problem of a two-phase body with a propagating interface, when the stress-strain state depends on the position of the interface, and the interface velocity depends on the stress-strain state. It is assumed that ε ∗− = 0, then ε ∗+ = ε tr is the transformation strain (eigenstrain). By (10.4), (10.5), and (10.7), the expression for configurational force can be written as 1 1 (10.10) χ = γ − σ − : ε − + σ + : (ε + − ε tr ) − σ ± : (ε + − ε − ), 2 2 where γ = f 0+ − f 0− is referred to further as the energy parameter. In the case of isotropic phases C± = λ± E ⊗ E + 2μ± I, where I is the fourth-rank unit tensor, λ± , μ± are the Lamé parameters.

(10.11)

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10.3 Numerical Procedure and Its Verification In a finite element modeling, it is assumed that the interface passes along the edges of the elements, and the remeshing of the geometry takes place on each time step. The developed numerical procedure does not describe the nucleation of the new phase, therefore the existence of some initial position of the interface is assumed. The finite element simulation for one time iteration contains the following steps: 1. 2. 3. 4.

Set the geometry of the body and the current position of the interface. Find stresses and strains at the interface. Calculate configurational force at the interface by (10.10). Find the position of the interface at the next time step, according to kinetic equation (10.9).

To verify the numerical procedure, we consider a plane strain problem of a twophase elastic plate with a cross-section x ∈ [−L , L], y ∈ [0, H ] under uniaxial tension in y-direction (Fig. 10.2). ϑ tr We assume that εtr = (E − ez ⊗ ez ), where ez is the out-of-plane x y unit 2 vector. The boundary conditions are u y | y=0 = 0, u y | y=H = u 0 , u x |x=±L = 0,

(10.12)

where u x and u y are the components of the displacement vector. In the numerical simulation, a half of the plate with x ∈ [0, L] is considered with the symmetry condition u x |x=0 = 0. The interface  is the plane y = h. Then the displacement in the layers occupied by phases “+” and “−” takes the form (10.13) u = u y± e y , where e y is the unit vector of the axis y,

Fig. 10.2 A schematic representation of a two-phase body with a planar interface

10 Numerical Simulations of Interface Propagation in Elastic Solids …

u y± = A± y + B± ,

207

(10.14)

and the coefficients A± and B± are found from the boundary conditions (10.12) at y = 0 and y = H and the continuity of the displacement u y and traction σ y across the interface y = h. It follows that (λ− + 2μ− )ε0 + (1 − ζ )(λ+ + μ+ )ϑ tr , (1 − ζ )(λ+ + 2μ+ ) + ζ (λ− + 2μ− ) (λ+ + 2μ+ )ε0 − ζ (λ+ + μ+ )ϑ tr , A− = (1 − ζ )(λ+ + 2μ+ ) + ζ (λ− + 2μ− ) B+ = 0, B− = (ε0 − A− )H, A+ =

(10.15) (10.16) (10.17)

where ζ = h/H and ε0 = u 0 /H . Then stresses and strains can be derived as the functions of ζ . Further, we consider the case for which [[λ]] = 0 and [[μ]] = 0. Then the expression of the configurational force takes the form F2 (10.18) χ = F1 + , ζ˜ = ζ − g, ζ˜ 2 where g=

q+ , [[q]]

F1 = γ +

p+ ϑ tr 2

2

  p+ 1− , [[q]]

F2 =

2 q− q+  [[q]]ε0 − p+ ϑ tr , 3 2[[q]] (10.19)

p+ =λ+ + μ+ , q± = λ± + 2μ± .

(10.20)

Kinetic equation (10.9) takes the form κ d ζ˜ = − χ (ζ˜ ). dt H

(10.21)

The equilibrium position ζeq = h eq /H of the interface in dependence on external strain ε0 can be found from the equilibrium condition (10.8) that, by (10.18), is reduced to the form ζ˜ 2 = f 2 , (10.22) where f 2 = −F2 /F1 . If the solution of Eq. (10.22) exists, then f 2 > 0. Then integrating (10.21) gives the dependence ζ˜ = ζ˜ (t) in the form  √ √ f 2  f 2 + ζ˜  κ F1 t =ζ˜ − ln  √ −  + f3 ,  f 2 − ζ˜  H 2

(10.23)

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where f 3 is defined by the initial condition ζ˜ (0) = ζ (0) − g. The Lamé parameters which correspond to Young’s moduli and Poisson’s ratios of the phases given in Table 10.1 are λ+ = 10 GPa, λ− = 40 GPa, μ+ = 26 GPa, and μ+ = 34 GPa. Then [[q]] < 0 and, by (10.19), F1 > 0 (since γ > 0) and F2 < 0. Thus, indeed, f 2 > 0. In connection with the choice of parameters, note that the equality of the configurational force to zero is only a necessary condition for energy minimization. The equilibrium interface perpendicular to the tension direction may be stable or unstable depending on material parameters. Stability analysis of the interface is out of the scope of the present paper, however, we refer to the linear stability analysis developed in Eremeev et al. (2003), Fu and Freidin (2004), and Yeremeyev et al. (2007), and recently studied numerically in Morozov et al. (2019) and Morozov (2021) for similar problems. Restrictions on the choice of the elastic moduli in the stability context also come from the construction of the so-called phase transition zones (PTZ) formed in a strain space by all strains which can exist at the equilibrium interfaces in a given material (Morozov and Freidin 1998; Freidin 2007). Various parts of the PTZ boundary correspond to various orientations of the interface with respect to eigenvectors of the strain tensor. The instability of two-phase deformations was not found if strains at the interfaces corresponded to the external boundaries of the PTZ (see, e.g., Freidin et al. 2006; Yeremeyev et al. 2007; Vilchevskaya et al. 2013). Then it was proved that belonging of the strains at the interface to the external PTZ boundaries is a necessary stability condition (Grabovsky and Truskinovsky 2011, 2013). From this point of view, if the considered interface is stable, the strains at the interface belong to that part of the PTZ boundary which corresponds to the interfaces perpendicular to the maximal tension. Material parameters given in Table 10.1 correspond to the interface stability from both aforementioned points of view. The dependence of the dimensionless equilibrium position of the planar interface ζeq = h eq /H on external strain is shown in Fig. 10.3. With the material parameters taken, the dependence is almost linear. The dependence allows us to find the range of deformations at which equilibrium interfaces can exist in the plate at the specified parameters. For modeling the kinetics of the planar interface propagation, the external tension is chosen to be ε0 = 0.02. The initial position of the interface is h 0 = 0.1H . Then, the equilibrium position of the interface is ζeq = 0.517. Numerical simulation of the interface evolution together with analytically obtained time-dependence of the interface position and the equilib-

Table 10.1 Material parameters

Phase “+” Phase “−”

GJ m3

E, GPa

ν

ϑ tr

γ,

59.2 86.4

0.139 0.27

0.01 0

0.013

κ,

m4 J·s

1 · 10−11

10 Numerical Simulations of Interface Propagation in Elastic Solids … Fig. 10.3 Dependence of the dimensionless equilibrium position of the planar interface on external strain

209

1.0

0.8

eq

0.6

0.4

0.2

0.0 0.010

0.015

0.020

0.025

0.030

0

Fig. 10.4 Dimensionless position of the interface from numerical simulation and analytical solution (blue solid line) and equilibrium position of the interface from analytical solution (green dashed line)

0.6

eq 0.5 0.4 0.3 0.2 0.1

Equilibrium position (analytical) Numerical and analytical solutions

0.0

0

1

2

3

4 ×104

time, s

rium position of the interface are shown in Fig. 10.4. The coincidence of the curves obtained numerically and analytically and the smooth convergence of the interface to the equilibrium position indicate the correctness of the numerical implementation.

10.4 Numerical Results and Discussion 10.4.1 Evolution of the Interface Around a Circular Hole Under Tension The interface evolution in the body with a circular hole is considered for an elastic body subjected to external tensile strain ε0 = 0.02 (Fig. 10.5). As mentioned above, this strain admits the existence of equilibrium planar interface at ζeq = 0.517 in the

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Fig. 10.5 A schematic representation of a two-phase body with a circular hole

Fig. 10.6 Distribution of the configurational force along the circular interface around the circular hole for various thicknesses of the new phase layer

0

0.5

×107 No hole r = 0.3rΓ r = 0.4rΓ r = 0.5rΓ r = 0.6rΓ

0.0 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0

0

π 12

π 6

π 4

π 3

5π 12

π 2

body without a hole (see Fig. 10.3). The influence of the circular hole on the interface propagation is explored by varying the radius r of the hole at a given initial interface radius r = 0.01L, i.e., by varying the thickness of the new phase layer around the hole. The distributions of the configurational force along the interface, in dependence of the polar angle ϕ, for various distances of the interface from the hole are shown in Fig. 10.6. Recall that due to the kinetic equation (10.9), the radius of the interface increases at negative values of the configurational force, and decreases otherwise. If r = 0 (no hole), then χ < 0 at all points of the interface. This reflects the tendency of the new phase to grow and evolve to the equilibrium planar position. Similar negative values of the configurational force are observed along the interface around relatively small holes with r = 0.3 r , r = 0.4 r . However, the configurational force can change its sign on the part of the interface if the hole radius increases, as it is observed at r = 0.5 r and r = 0.6 r . The interface evolution around the circular hole for various initial thicknesses of the new phase layer is presented in Fig. 10.7. Qualitatively, the interface evolution is

10 Numerical Simulations of Interface Propagation in Elastic Solids …

No hole

r = 0.3rΓ

r = 0.4rΓ

211

r = 0.5rΓ

Fig. 10.7 The evolution of interface for different cases of thickness of new phase layer: no hole; r = 0.3 r ; r = 0.4 r ; r = 0.5 r (red line—initial interface position)

similar to the case of the absence of a hole if the holes are relatively small. The new phase domain grows in all ϕ-directions at early iterations if r = 0.3 r and r = 0.4 r . If the thickness of the initial new phase layer decreases further, then some part of the interface moves toward the hole and the reverse phase transformation occurs, as it is at r = 0.5 r . To quantify the evolution of the shape of the new phase domain, we show the time dependencies of dimensionless variables ρx = X/L (Fig. 10.8) and ρ y = Y/H (Fig. 10.9) where X and Y are the coordinates of the intersection of the interface with axes x and y, respectively. The increase of the hole radius (decreasing the thickness of the initial new phase circular layer) accelerates the interface propagation in xdirection, but retards the propagation in y-direction. Moreover, if r = 0.5 r then ρ y and, thus, the thickness of the new phase domain in y-direction decreases during considered time interval. Fig. 10.8 Evolution of the shape of the new phase domain around the circular hole. The thicknesses of the domain along x-axis for various initial thicknesses

0.030 0.028 0.026 0.024

No hole r = 0.3rΓ r = 0.4rΓ r = 0.5rΓ

0.022 0.020 0.018 0.016 0.014 0.012 0.010 0.0

0.2

0.4

0.6

time, s

0.8

1.0

×102

212 Fig. 10.9 Evolution of the shape of the new phase domain around the circular hole. The thicknesses of the domain along y-axis for various initial thicknesses

P. K. Kabanova et al.

0.018 0.016

No hole r = 0.3rΓ r = 0.4rΓ r = 0.5rΓ

0.014 0.012 0.010 0.008 0.006 0.0

0.2

0.4

0.6

0.8

time, s

1.0

×102

10.4.2 Phase Transformations Induced by a Stress Concentration at An Elliptical Hole The effect of the hole shape on the distribution of the configurational force along the interface is studied by the example of an interface around a stress concentrator in the form of an elliptical hole under uniaxial tensile strain (Fig. 10.10). The external strain is ε0 = 0.005. According to Fig. 10.3, only the initial phase is possible at such external loading and there is no two-phase equilibrium, and therefore, no planar interface in the body. The horizontal semi-axis of the ellipse is a = 0.009L. The reference thickness of the new phase layer around the hole is d0 = 0.001L = 1/9 a. The distributions of the configurational force along the interface for various ratios of semi-axes a and b of the ellipse are shown in Fig. 10.11. The interface points are parametrized by angle ϕ. It is found that for ratios a/b = 1 and a/b = 2, the configurational force is positive at the entire interface. This leads to the interface

Fig. 10.10 A schematic representation of a two-phase body with an elliptical hole

0

10 Numerical Simulations of Interface Propagation in Elastic Solids … Fig. 10.11 Distribution of the configurational force along the interface for different ratios of the semi-axes of an elliptical hole

213

×107 1.4 1.2 1.0 0.8 0.6 a/b = 1

0.4

a/b = 2

0.2

a/b = 3

0.0

a/b = 6 a/b = 9

−0.2

0

π 12

π 6

π 4

π 3

5π 12

π 2

propagation backward to the hole and disappearance of the new phase domain over time. An increase in the ratio of semi-axes of the ellipse may lead to the change of the sign of the configurational force at the part of interface, as it is shown for a/b = 3, a/b = 6, and a/b = 9. An increase in the stress concentration at the tip of the elongated elliptical hole contributes to the interface propagation in a way that the new phase domain grows forward, even in case of insufficient external strains for the phase transition in a body without a hole. As in the case of the circular hole, the distribution of the configurational force along the interface also depends on the thickness d of the new phase layer, i.e., on how far the interface is from the stress concentrator. The distribution of the configurational force along the interface for different thicknesses of the new phase layer is shown in Fig. 10.12 for the elliptical hole with a/b = 3 under tensile strain ε0 = 0.005 at which the phase transformation can take place only in the presence of a stress concentrator. If the thickness is small, then the configurational force is negative at a small part of the interface in front of the ellipse tip and positive along other parts, as it is for the thicknesses d = 0.25d0 and d = 0.5d0 . At thicknesses greater than d0 , the configurational force is positive and the new phase boundary propagates toward the hole. Note that, due to the dissipation inequality and kinetic equation (10.9), the interface can move “forward” only if the configurational force is negative, but if the configurational force is positive, then the following cases are allowed: the interface moves “backward” or does not move. The latter is the so-called false equilibrium (see, e.g., Prigogine and Defay 1954). In this case, it can be expected that the metastable region of the new phase may remain behind the tip of the growing stress concentrator.

214 Fig. 10.12 Distribution of the configurational force along the interface for different thicknesses d of the new phase layer for the elliptical hole with a/b = 3, and d0 is the reference thickness

P. K. Kabanova et al.

1.5

×107

1.0 0.5 0.0 0.25d0 0.5d0 d0 2d0

−0.5 −1.0

0

π 12

π 6

π 4

π 3

5π 12

π 2

10.5 Conclusion Within the frames of mechanics of configurational forces, a numerical approach to modeling the interface propagation was developed and verified. Then the evolution of the interface near stress concentrators in the form of circular and elliptical holes was investigated as preliminary steps for further modeling of the joint propagation of the crack and the interface. The variety of the interface behaviors in dependence on the external strain, the distance between the interface and the concentrator, and the form of the concentrator were demonstrated based on examinations of the configurational force distribution along the interface. In the case of a circular hole, it was studied how the ratio of the radii of the hole and the new phase domain, i.e., the thickness of the new phase layer, affects the initial kinetics of the interface propagation. In the case of the phase transformation in the vicinity of an elliptical hole, the effects of the ratio of the semi-axes of the ellipse and the thickness of the new phase layer were examined. It was studied how the elliptical hole can induce interface propagation even at external strain, which is insufficient for the existence of a new phase in the absence of the stress concentrator. It has been shown that the construction of the distribution of the configurational force along the interface can be an effective tool for predicting interface evolution. Acknowledgements P. K. Kabanova and A. B. Freidin acknowledge the support of the Russian Science Foundation (Grant No. 19-19-00552- ).

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Grinfeld MA (1991) Thermodynamic methods in the theory of heterogeneous systems. Longman, New York Gross D, Kolling S, Mueller R, Schmidt I (2003) Configurational forces and their application in solid mechanics. Eur J Mech-A/Solids 22(5):669–692 Gross D, Mueller R, Kolling S (2002) Configurational forces—morphology evolution and finite elements. Mech Res Commun 29(6):529–536 Gurtin ME (2000) Configurational forces as basic concepts of continuum physics. Springer, New York, Berlin, Heidelberg Hannnink RHJ, Swain MV (1989) Metastability of the martensitic transformation in a 12 mol% Ceria-Zirconia alloy: I, deformation and fracture observations. J Am Ceram Soc 72(1):90–98 Hazar S, Anlas G, Moumni Z (2016) Evaluation of transformation region around crack tip in shape memory alloys. Int J Fract 197(1):99–110 Kachanov M, Sevostianov I (2005) On quantitative characterization of microstructures and effective properties. Int J Solids Struct 42(2):309–336 Kachanov M, Sevostianov I (2018) Micromechanics of materials, with applications, vol 249. Springer Kienzler R, Herrmann G (2000) Mechanics in material space with application to defect and fracture mechanics. Springer, New York, Berlin, Heidelberg Laydi MR, Lexcellent C (2015) Determination of phase transformation surfaces around crack tip in shape memory alloys. In: MATEC web of conferences, vol 33. EDP Sciences, p 02010 Le KC (2007) On kinetics of hysteresis. Continuum Mech Thermodyn 18:335–342 Levitas VI (2021) Phase transformations, fracture, and other structural changes in inelastic materials. Int J Plast 140:102914 Levitas VI, Lee D-W, Preston DL (2010) Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations. Int J Plast 26(3):395–422 Maletta C, Furgiuele F (2010) Analytical modeling of stress-induced martensitic transformation in the crack tip region of nickel-titanium alloys. Acta Mater 58(1):92–101 Mamivand M, Zaeem MA, El Kadiri H (2013) A review on phase field modeling of martensitic phase transformation. Comput Mater Sci 77:304–311 Maugin GA (2010) Configurational forces: thermomechanics, physics, mathematics, and numerics, 1st edn. Chapman and Hall/CRC, London, New York McKelvey AL, Ritchie RO (1999) Fatigue-crack propagation in Nitinol, a shape-memory and superelastic endovascular stent material. J Biomed Mater Res Off J Soc Biomater Jpn Soc Biomater Aust Soc Biomater Korean Soc Biomater 47(3):301–308 McMeeking RM, Evans AG (1982) Mechanics of transformation-toughening in brittle materials. J Am Ceram Soc 65(5):242–246 Morozov A (2021) Numerical and analytical studies of kinetics, equilibrium, and stability of the chemical reaction fronts in deformable solids. PhD thesis, Technische Universitét Berlin Morozov AV, Freidin AB, Müller WH (2019) Stability of chemical reaction fronts in the vicinity of a blocking state. PNRPU Mech Bull 3:58–64 Morozov NF, Freidin AB (1998) Phase transition zones and phase transformations of elastic solids under different stress states. Proc Steklov Mathe Inst 223:220–232 Mueller R, Gross D (1998) 3D simulation of equilibrium morphologies of precipitates. Comput Mater Sci 11(1):35–44 Mueller R, Gross D, Lupascu DC (2006) Driving forces on domain walls in ferroelectric materials and interaction with defects. Comput Mater Sci 35(1):42–52 Prigogine I, Defay R (1954) Chemical thermodynamics. Longmans, Green, London Rezaee-Hajidehi M, Stupkiewicz S (2020) Phase-field modeling of multivariant martensitic microstructures and size effects in nano-indentation. Mech Mater 141:103267 Robertson SW, Ritchie RO (2007) In vitro fatigue-crack growth and fracture toughness behavior of thin-walled superelastic Nitinol tube for endovascular stents: a basis for defining the effect of crack-like defects. Biomaterials 28(4):700–709

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Chapter 11

Effect of Gravity on the Dispersion and Wave Localisation in Gyroscopic Elastic Systems A. Kandiah, I. S. Jones, N. V. Movchan, and A. B. Movchan

Dedicated to the memory of Professor Igor Sevostianov

Abstract This paper presents new work on gravity-induced wave motion of gyroscopic systems composed of gyropendulums connected by elastic springs. Classification of trajectories of a single gyropendulum is given, followed by the Floquet-Bloch analysis of the dispersion and localisation for waves within a periodic gyroscopic chain. We construct Green’s matrices to identify regimes of propagating modes and wave localisation, which correspond to elliptical motions of the nodal elements. The waveforms subjected to gravity for localised defect modes are discussed in addition to the effect of no pre-tension along the chiral chain. The analytical results are accompanied by illustrative examples.

Supplementary Material Electronic supplementary material, which includes illustrative videos for the figures 11.16, 11.17, 11.18, 11.20 and 11.23, is available online, at URL: https://datacat. liverpool.ac.uk/id/eprint/2213 and DOI: https://doi.org/10.17638/datacat.liverpool.ac.uk/2213. Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-28744-2_11. A. Kandiah (B) · I. S. Jones · N. V. Movchan · A. B. Movchan University of Liverpool, Department of Mathematical Sciences, Liverpool L69 7ZL, UK e-mail: [email protected] I. S. Jones e-mail: [email protected] N. V. Movchan e-mail: [email protected] A. B. Movchan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_11

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11.1 Introduction The notion of chirality in physics and mechanics has been well established in the classical literature: we refer to the book by Lord Kelvin (1894), as well as classical papers Jaggard et al. (1979); Moore (2010); Pendry et al. (2004); Rechtsman et al. (2013); Hibbins et al. (2005). In particular, in mechanics, chirality may be introduced by gyroscopic spinners connected to a ‘master structure’, which also includes elastic constituents. An important practical work was conducted by Foucault (1851) who studied the mechanical spinner in the context of the demonstration of the rotation of the Earth with the pendulum experiment. On the practical side, the spinner has played a vital role in dozens of technologies over many decades, and in particular in the design of inertial navigation systems Roitenberg (1975); Andreev (1966); Ishlinskii (1976, 1963); Scarborough (1958); Wrigley et al. (1969); Pitman and Trimmer (1962). The concept of chiral flexural elements was introduced in D’Eleuterio and Hughes (1983, 1987), where the notion of beams with additional stored angular momentum, known as gyrobeams, was considered. In this paper, we analyse a system consisting of a pendulum connected to a gyroscopic spinner, which we call a gyropendulum. The book Webster (1904) includes illustrations based on the tracings of the gyropendulum trajectory. A class of boundary conditions, which are derived in Carta et al. (2018), for a gyro-hinge connecting a gyroscopic spinner to a rod, are used in this paper. We consider the linearised problem for the case when the angle of nutation and its derivatives are small, and we show that an appropriate choice of the initial conditions may lead to periodic motion in the system. The motivation for the development of chiral systems was due to some challenging tasks in the aviation and maritime industries (see Roitenberg (1975); Andreev (1966); Ishlinskii (1976, 1963); Scarborough (1958); Wrigley et al. (1969); Pitman and Trimmer (1962)). The book by Gray (1918), and the historical account of Lord Kelvin’s work, described by Whittaker (see Whittaker (1910)), provide theoretical discussions and ideas on physical phenomena involving the rotational motion of complex systems. Several illustrations of gyroscopic effects and stability have been presented in D’Eleuterio and Hughes (1983, 1987); Yamanaka et al. (1996), and mathematical models of vibrations of chiral elastic multi-structures were studied in Nieves et al. (2018). Discrete models considered in Carta et al. (2017); Brun et al. (2012); Carta et al. (2017); Brun et al. (2014); Nieves et al. (2017, 2016) analyse vibrations and localised waveforms in such multi-structures. The models can then be used to negate the unwanted vibrations generated in the structure. The motion of a single gyroscopic spinner, studied in the literature (see Goldstein et al. (1951)), is expressed as a system of non-linear differential equations. When the spinner is connected to a solid, such as a rod or elastic beam, the gyroscopic spinner creates a response which incorporates precession and nutation. The gyroscopic action may also combine with the elastic vibration of the structure (see, for example, Carta and Nieves (2021); Carta et al. (2019)). The precession equations of motion of the gyropendulum in the kinematic interpretation, as well as the Lyapunov stability of the

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equations, were discussed in Chelnokov (1980). Geometrically chiral elastic systems have been analysed in Tallarico et al. (2017); Bahaloo and Li (2019); Chaplain et al. (2019); an important class of gyroscopic chiral elastic systems, supporting vortex waves, was studied in Carta et al. (2019); Brun et al. (2012); Garau et al. (2018). The wave coupling in the continuum case for vibrations of a chiral elastic system comprising gyroscopic spinners attached to an infinite elastic rod was considered in Jones et al. (2022), where the connection with the Klein-Gordon equations in the case of chiral degeneracy was also highlighted. The model described here considers the effect of chiral coupling for a waveguide system subjected to gravity. We analyse the effect of gravity on the wave propagation of an infinite periodic system of equally spaced pendulums connected by springs, with a gyroscopic spinner attached at the tip of each rod (see Jones et al. (2020)). Here, the gyroscopic chiral elements bring the coupling between the transverse and longitudinal displacements. We give a systematic analysis of waves in such chiral elastic chains under gravity. Due to the presence of pass bands, partial pass bands and stop bands, the dispersion of Floquet-Bloch waves in a chiral chain is of interest. We show that the physical system can support Floquet-Bloch waves in addition to localised waveforms, depending on the choice of frequency regimes. The structure of the paper is as follows. The passive gyroscopic system in the context of gyroscopic boundary conditions and its motion under gravity are discussed in Sect. 11.2. Section 11.3 deals with active gyroscopic systems for the case of timeharmonic motion in the presence of gravity, where the gyricity of the system is proportional to the radian frequency of vibrations. In Sect. 11.4, we study clusters of gyropendulums and analyse their dynamic response as well as Floquet-Bloch waves. Green’s matrices for total pass bands, partial pass bands and stop bands are constructed analytically in Sect. 11.5 for a chiral elastic chain under gravity. Section 11.6 presents the case of a chiral chain with no pre-tension and the discussion of exponentially localised defect modes, followed by the concluding remarks.

11.2 Passive Gyroscopic System under Gravity In this section, we address the linearised equations of motion of the gyroscopic pendulum. The gyroscopic system under gravity has no external energy flux, and will be referred to as the passive gyroscopic system. For such a system, the energy is conserved. The connection between the gyroscopic boundary conditions studied in Nieves et al. (2018) and the motion of the gyropendulum is discussed here. We introduce the classification of quasi-periodic trajectories traversed by the gyropendulum and show an example of optimal design in the case of periodic motion, where the motion of the gyropendulum approximates a regular n-sided polygon. Consider a rigid massless rod of length L , positioned along the positive z-axis such that 0 ≤ z ≤ L. At z = 0, the rod is suspended from a pivot so that it can swing freely under gravity, and at z = L the rod is connected to a thin uniform disc of mass

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(a)

(b)

Fig. 11.1 (a) A gyropendulum consisting of a gyroscopic spinner connected at the tip of a rod. The rod is hinged at its base which is located at the origin O of the fixed coordinate system O x yz. (b) The gyroscopic spinner in the local coordinate system O  x  y  z  , which moves with the spinner as it nutates through an angle θ, precesses through an angle φ and spins through an angle ψ. The axes of the rod and the spinner are assumed to be aligned at any instant of time

m and radius R, which acts as a gyroscopic spinner (see Fig. 11.1a). In this case, the spinner is axisymmetric, with the centre of mass placed at the end of the rod. For 0 ≤ z ≤ L, the transverse displacements u(z, t) and v(z, t) are linear functions of z u(z, t) = zU (t), v(z, t) = zV (t), (11.1) where U (t) and V (t) are time-dependent non-dimensional coefficients associated with the transverse displacement components. The angles θ , φ and ψ, shown in Fig. 11.1, are the angles of nutation, precession and spin, respectively. In the linearised model, it is assumed that the nutation angle and its derivatives are small   j  d θ (t)    (11.2)  dt j   1, 0 ≤ j ≤ 2. Here, similar to Carta et al. (2018); Nieves et al. (2018), we neglect the second-order ˙ which is referred to as terms with respect to θ and define the quantity Y = φ˙ + ψ, gyricity. The positive direction of spin is chosen as in Fig. 11.1, i.e. counterclockwise relative to the z  −axis, where (x  , y  , z  ) are the local coordinates (see Fig. 11.1). According to Gray (1918), the differential equations for the transverse displacement components at z = L have the form         I1 mgL U (t) U¨ (t) U˙ (t) 0 + YR ˙ + = , V (t) 0 V¨ (t) V (t) I0 I0

(11.3)

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where the 90◦ rotation matrix, which couples U (t) and V (t), is given by 

 0 1 R= . −1 0

(11.4)

In the system of equations (11.3), g is the acceleration due to gravity, and I0 is the transverse moment of inertia, relative to the pivot point at z = 0, of the rigid rod combined with the spinner with respect to the x  −axis, which is assumed to be the same as for the y  −axis; the quantity I1 is the moment of inertia of the spinner relative to its z  −axis. The solutions of the system (11.3) are oscillatory with the radian frequencies which satisfy the following equation: I02 ω4 − (2Lgm I0 + I12 Y2 )ω2 + m 2 L 2 g 2 = 0.

(11.5)

In this approximation, we assume that the length of the arm of the pendulum is much larger compared to the radius of the spinner so that   R2  m L 2. I0 = m L 2 1 + 4L 2

(11.6)

In particular, when the spinner is represented by a uniform thin disc of radius R, we have 1 (11.7) I1 = m R 2 , 2 and the system (11.3) becomes         YR 2 g U (t) U¨ (t) U˙ (t) 0 + R + = . 0 V¨ (t) V˙ (t) 2 L2 L V (t)

(11.8)

Equations (11.3) are analogous to those which describe the vibrations of an electron in a magnetic field as pointed out in Kelvin and Tait (1867); Gray (1918), in the context of the gyrostatic analogue of the Lorentz force (see Lorentz (1906)). However, we will not pursue this comparison further here. The positive direction of spin is counterclockwise as we look from the positive direction of the z  −axis. We note that the positive direction of spin used in Gray (1918) is opposite to the one used here (hence, the replacement of Y by −Y). In the model of the elastic flexural gyroscopic multi-structure of Nieves et al. (2018), the moment of inertia I0 is computed relative to the pivot point at z = L, whereas, in the model of the gyropendulum, I0 is evaluated relative to the pivot point at z = 0 (see Sect. 11.2.1).

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11.2.1 Asymptotic Connection Between the Gyroscopic Pendulum and the Gyroscopic Flexural Beam In the earlier paper Nieves et al. (2018), the gyroscopic boundary conditions were derived and analysed for an elastic flexural beam connected to a gyroscopic spinner. As Young’s modulus of the elastic material of the beam tends to infinity, the beam becomes a rigid rod. Here, we explain this transition in the context of the gyroscopic and inertial characteristics of the mechanical system. Firstly, as the beam is assumed to be massless, the governing equations for the transverse flexural displacements become u I V (z, t) = 0, v I V (z, t) = 0.

(11.9)

The chiral boundary conditions representing the balance of moments at the end of the beam (z = L) have been derived and analysed in Nieves et al. (2018). These boundary conditions couple the displacements u and v and have the form − I0∗ u¨  (L , t) − I1 Yv˙  (L , t) − mgu(L , t) = E J u  (L , t),

(11.10)

− I0∗ v¨  (L , t) + I1 Yu˙  (L , t) − mgv(L , t) = E J v  (L , t).

(11.11)

Here, m, E and J denote the mass of the spinner, Young’s modulus and second moment of the cross-section of the beam, respectively. The quantity I0∗ is the moment of inertia about the principal transverse axes relative to the local coordinate system at the end of the beam, with the origin at O  . The boundary conditions for u and v (and their derivatives) characterising the balance of forces at z = L are (see Nieves et al. (2018)) m u(L ¨ , t) − E J u  (L , t) = 0,

(11.12)

m v(L ¨ , t) − E J v  (L , t) = 0.

(11.13)

At z = 0, the hinged end boundary conditions of the beam are given by u(0, t) = v(0, t) = 0,

(11.14)

u  (0, t) = v  (0, t) = 0.

(11.15)

The solutions to system (11.9) are cubic functions of z. Using the boundary conditions (11.14) and (11.15) at the hinged end, we write the solutions of (11.9) in the form u(z, t) = U1 (t)z 3 + U2 (t)z, v(z, t) = V1 (t)z 3 + V2 (t)z.

(11.16)

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We note that the coefficients U1 (t), U2 (t), V1 (t) and V2 (t) are time-dependent. Substituting (11.16) into the chiral boundary conditions (11.10)–(11.11) and the boundary conditions representing the balance of forces (11.12)–(11.13), we obtain −I0∗ (3L 2 U¨ 1 (t) + U¨ 2 (t)) − I1 Y(3L 2 V˙1 (t) + V˙2 (t)) −mgL(L 2 U1 (t) + U2 (t)) = 6E J LU1 (t), −I0∗ (3L 2 V¨1 (t) + V¨2 (t)) + I1 Y(3L 2 U˙ 1 (t) + U˙ 2 (t)) −mgL(L 2 V1 (t) + V2 (t)) = 6E J L V1 (t),

(11.17)

m L(L 2 U¨ 1 (t) + U¨ 2 (t)) − 6E J U1 (t) = 0, m L(L 2 V¨1 (t) + V¨2 (t)) − 6E J V1 (t) = 0. The general solution to system (11.17) refers to the roots of the equation  6m J L 3   L 6 m2  ∗ 2 8 − (I0∗ + m L 2 )I0∗ ω6 + (I0∗ mgL + I12 Y2 )ω4 (I0 ) ω − I12 Y2 ω6 + 2 E E

  + 9J 2 (I0∗ + m L 2 )2 ω4 − (2Lgm(I0∗ + m L 2 ) + I12 Y2 )ω2 + m 2 L 2 g 2 = 0. (11.18) In the case of a gyropendulum, we consider the solution for the gyroscopic flexural beam in the limit as E → ∞. In this case, four roots of equation (11.18) tend to infinity, while the remaining four roots tend to the solution of the limit equation (I0∗ + m L 2 )2 ω4 − (2Lgm(I0∗ + m L 2 ) + I12 Y2 )ω2 + m 2 L 2 g 2 = 0,

(11.19)

where I0∗ in the above equation (11.19) and I0 in (11.3) are related by the ‘Parallel Axis Theorem’ (see Goldstein et al. (1951); Abdulghany (2017)) for moments of inertia I0 = I0∗ + m L 2 . Thus, equation (11.19) is equivalent to (11.5) for the case of the gyropendulum. We note that in the limit as E → ∞, the functions U1 and V1 in (11.16) vanish, and this makes (11.16) to be equivalent to (11.1) in the case of the gyropendulum. This explains the asymptotic link between the flexural gyroscopic beam and the gyropendulum in the limit when the stiffness of the beam tends to infinity.

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11.2.2 Dimensionless System of Equations: General Solution In this section, we discuss the general motion of the gyropendulum for a variety of different initial conditions at the end of the rod (z = L). Introducing the dimensionless variables t˜ = Yt, Γ1 = I1 /I0 and Γ2 = mgL/I0 Y2 in the system (11.3), we obtain           d 2 U˜ 0 1 d U˜ U˜ 0 + Γ + Γ = , (11.20) 1 2 2 ˜ ˜ −1 0 d t˜ V 0 V˜ d t˜ V where U˜ (t˜) = U (t˜/Y) and V˜ (t˜) = V (t˜/Y). Assuming solutions of the form U˜ = A exp(i ω˜ t˜), V˜ = B exp(i ω˜ t˜), where ω˜ = ω/Y, and substituting into the system (11.20), we obtain      2 ˜ 1 A −ω˜ + Γ2 i ωΓ 0 = . −i ωΓ ˜ 1 −ω˜ 2 + Γ2 B 0

(11.21)

To find non-trivial solutions of the system (11.21), we require the following solvability condition: (11.22) ω˜ 4 − (Γ12 + 2Γ2 )ω˜ 2 + Γ22 = 0. The above equation has four roots which are given by     1 1 Γ1 − Γ12 + 4Γ2 , ω˜ 1 = − Γ1 + Γ12 + 4Γ2 , ω˜ 2 = 2 2     1 1 2 ω˜ 3 = − Γ1 + Γ1 + 4Γ2 , ω˜ 4 = Γ1 + Γ12 + 4Γ2 . 2 2

(11.23)

Taking into account that Γ1 > 0, Γ2 > 0, we note that ω˜ 1 and ω˜ 2 are negative while ω˜ 3 and ω˜ 4 are positive. By choosing the positive eigenfrequencies ω˜ 3 and ω˜ 4 , we write the corresponding eigenvectors u3 = (1, i)T and u4 = (1, −i)T , respectively. The general solution of (11.20) can be expressed as U = c1 u4 ei ω˜ 4 t˜ + c2 u3 ei ω˜ 3 t˜ + c3 u4 e−i ω˜ 4 t˜ + c4 u3 e−i ω˜ 3 t˜,

(11.24)

where (U˜ , V˜ )T = Re(U) and the bar denotes the complex conjugate. The four coefficients c j , j = 1, 2, 3, 4, can be obtained from the four initial conditions U˜ (0) = U˜ 0 , V˜ (0) = V˜0 ,

d U˜ (0) = U˙˜ 0 , d t˜

d V˜ (0) = V˙˜0 , d t˜

(11.25)

where U˜ 0 , V˜0 , U˙˜ 0 and V˙˜0 are given values of the normalised initial displacements and initial velocities. Then, solving the system of equations (11.20) with the initial conditions (11.25), the displacement components of the gyropendulum have the form

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      U˜ cos(ω˜ 4 t˜) − sin(ω˜ 4 t˜) ˜ ˜ + A2 = A1 sin(ω˜ 4 t˜) cos(ω˜ 4 t˜) V˜ + A˜ 3 where



   cos(ω˜ 3 t˜) sin(ω˜ 3 t˜) ˜ + A4 , − sin(ω˜ 3 t˜) cos(ω˜ 3 t˜)

V˙˜0 + ω˜ 3 U˜ 0 −U˙˜ 0 + ω˜ 3 V˜0 , A˜ 2 = , A˜ 1 = ω˜ 3 + ω˜ 4 ω˜ 3 + ω˜ 4 −V˙˜0 + ω˜ 4 U˜ 0 U˙˜ 0 + ω˜ 4 V˜0 A˜ 3 = , A˜ 4 = . ω˜ 3 + ω˜ 4 ω˜ 3 + ω˜ 4

(11.26)

(11.27)

To obtain time-harmonic solutions with the radian frequencies ω˜ 3 or ω˜ 4 defined in (11.23), the initial conditions should satisfy the relations U˙˜ 0 = (−1) j−1 ω˜ j V˜0 , V˙˜0 = (−1) j ω˜ j U˜ 0 ,

(11.28)

for j = 3 or j = 4. In this case, the gyropendulum follows a circular trajectory. For the case of non-zero gyricity (Y = 0), when the constraints (11.28)  are satisfied, the radius of the circular motion of the gyropendulum tip, at z = L, is U˜ 02 + V˜02 .

11.2.3 Trajectories of the Gyropendulum In this section, we classify the typical trajectories of the gyropendulum, which depend on the parameters Γ1 and Γ2 and the initial conditions at z = L (see Sect. 11.2.2). We show that the gyropendulum motion can be used to approximate a regular polygon, where the initial conditions can be chosen such that the gyropendulum trajectory encloses the exact area of the polygon. Appendix 1 includes illustrations of periodic trajectories and their perturbations due to gyricity. In particular, it includes an example linked to simple pendulum trajectories. Interesting descriptions of trajectories of Foucault’s pendulum, based on a class of hypocycloids, were provided in Bromwich (1914). In the text below, we describe all trajectories of the linearised gyropendulum, in full generality, periodic and non-periodic, smooth and non-smooth, with the formal classification provided.

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11.2.3.1

Example 1: Non-zero Initial Displacements and Zero Initial Velocities

˜ t˜) = (U˜ (t˜), V˜ (t˜))T . For the case of non-zero initial disFor convenience, we let Q( placements and zero initial velocities, the displacement components of the gyropendulum satisfy the following identity: U˜ 2 + V˜ 2 =

2 ˜ |Q(0)| (ω˜ 2 + ω˜ 42 + 2ω˜ 3 ω˜ 4 cos((ω˜ 3 + ω˜ 4 )t˜)), (ω˜ 3 + ω˜ 4 )2 3

(11.29)



Γ12 + 4Γ2 . For ω˜ 3 = ω˜ 4 , the maximum distance of the gyropen ˜ dulum trajectory from the origin will not exceed |Q(0)| = U˜ 02 + V˜02 . When either the moment of inertia in the principal vertical direction is zero (I1 = 0) or the gyricity is absent (Y = 0), the two frequencies are equal, ω˜ 3 = ω˜ 4 . Typical trajectories of the gyropendulum for non-zero initial displacements and zero initial velocities are demonstrated in Fig. 11.2. We observe that the trajectories do not pass through the origin since ω˜ 3 = ω˜ 4 , and as Γ2 increases the trajectories approach, but do not cross, the origin. where ω˜ 3 + ω˜ 4 =

11.2.3.2

Example 2: Zero Initial Displacements and Non-zero Initial Velocities

In the case of zero initial displacements and non-zero initial velocities, the trajectories at the tip of the rod, z = L , satisfy the relation ˙˜ 2 2|Q(0)| (1 − cos((ω˜ 3 + ω˜ 4 )t˜)). U˜ 2 + V˜ 2 = (ω˜ 3 + ω˜ 4 )2

(11.30)

It follows that the motion of the gyropendulum will pass through the origin at times t˜ = 2π k/(ω˜ 3 + ω˜ 4 ), for k ∈ N. In Fig. 11.3, we present typical trajectories for the case of zero initial displacements and non-zero initial velocities. We note that the gyropendulum trajectory passes through the origin with time period 2π/(ω˜ 3 + ω˜ 4 ), and increasing Γ2 results in narrower loops.

11.2.3.3

Classification of Trajectories

The trajectories of the gyropendulum may have different shapes, including loops, self-intersecting loops and cusps, and they are dependent on the initial conditions for the given values of Γ1 and Γ2 . Here, we give the full classification of different types of trajectories subject to the choice of initial conditions.

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Fig. 11.2 Gyropendulum trajectories with initial conditions U˜ 0 = 0.1, V˜0 = 0.1, U˙˜ 0 = 0 and V˙˜0 = 0

The examples discussed in Sects. 11.2.3.1 and 11.2.3.2 have already provided illustrations of trajectories containing cusps and loops passing through the origin. We note that these trajectories can be achieved with initial conditions which do not coincide with those of Sects. 11.2.3.1 and 11.2.3.2. There are cases of trajectories, which do not include the origin, as well as trajectories represented by self-intersecting loops that we would like to discuss here. Let us introduce the scalar quantity H (t˜) to characterise the orientation of the motion of the gyropendulum: ˙˜ t˜), ˜ t˜) × Q( H (t˜)k = (U˜ (t˜)V˙˜ (t˜) − V˜ (t˜)U˙˜ (t˜))k = Q(

(11.31)

where k is the basis vector along the z-axis. The vector H (t˜)k denotes the vector product of the normalised displacement and the velocity at the end of the rod at time t˜. We say that at time t˜ the gyropendulum has a positive H -orientation if H (t˜) > 0, and that the gyropendulum has a negative H -orientation if H (t˜) < 0. We say that

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Fig. 11.3 Gyropendulum trajectories with initial conditions U˜ 0 = 0, V˜0 = 0, U˙˜ 0 = 0.0001 and V˙˜0 = 0.0001. The values of ω˜ 3 and ω˜ 4 correspond to the radian frequencies given in Fig. 11.2 (in parts (a), (b), (c), (d), (e) and (f), respectively)

the motion is of positive H -orientation if H (t˜) > 0 for all t˜, except for a countable set of values of t˜ = t˜j , where H (t˜j ) = 0. Similarly, we say that the motion is of negative H -orientation if H (t˜) < 0 for all t˜, except for a countable set of values of t˜ = t˜j , where H (t˜j ) = 0. As shown in Fig. 11.4, for the case of self-intersecting loops, the motion of the gyropendulum changes from the positive H -orientation to the negative H orientation and vice versa. On the other hand, the two examples, presented in Fig. 11.5, show the motion of the constant orientation: the case of cusps, shown in Fig. 11.5a, 11.5b, corresponds to the negative H -orientation, whereas the case of non-self-intersecting loops in Fig. 11.5c, 11.5d corresponds to the positive H orientation.  We note that H (t˜) is a smooth periodic function, with the period 2π/ Γ12 + 4Γ2 , given by

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(b)

Fig. 11.4 (a) Gyropendulum trajectory with parameter values Γ1 = 0.0016 and Γ2 = 0.0001 (with ω˜ 3 = 0.0092 and ω˜ 4 = 0.0108) and initial conditions U˜ 0 = 0.1, V˜0 = 0.1, U˙˜ 0 = 0, V˙˜0 = 0.0001. (b) The corresponding periodic smooth function H (t˜). We note that the parameters Γ1 and Γ2 and the initial conditions satisfy the relation described in Class 1 (see (11.33))

     ˙˜ 2 2 ˜ Γ1 |Q(0)| 1 − cos t˜ Γ12 + 4Γ2 − Γ2 |Q(0)|    +H (0) 4Γ2 + Γ12 cos t˜ Γ12 + 4Γ2      ˙˜ T sin t˜ Γ 2 + 4Γ , ˜ · Q(0) +Γ1 Γ12 + 4Γ2 Q(0) 2 1

H (t˜) =

1 Γ12 +4Γ2

and the critical points t˜ = t˜k , where H  (t˜k ) = 0, are 

t˜k = (Γ12 + 4Γ2 )−1/2 − arctan



 ˙˜ T ) Γ 2 + 4Γ ˜ (Q(0) · Q(0) 2 1



˙˜ 2 − Γ |Q(0)| 2 − Γ H (0) ˜ |Q(0)| 2 1

 + kπ ,

for k ∈ N with t˜k ≥ 0. The product of the values of the function H (t˜) evaluated at two consecutive critical points determines the type of trajectory of the gyropendulum. It has the form    ˙˜ 2 2 ˜ Γ1 |Q(0)| − 2H (0) Γ1 |Q(0)| + 2Γ2 H (0) H (t˜k )H (t˜k+1 ) = − . (11.32) Γ12 + 4Γ2 The classification of the trajectories is linked to the initial value H (0) and to the interval ˙˜ 2 2 ˜ /2Γ2 , Γ1 |Q(0)| /2). Υ = (−Γ1 |Q(0)| Class 1 incorporates the cases when H (0) ∈ Υ , class 2 corresponds to H (0) ∈ ∂Υ and class 3 includes all configurations where H (0) ∈ / Υ . The corresponding description of the trajectories is as follows: Class 1 : self-intersecting loops of variable H -orientation occur when

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˙˜ 2 2 ˜ − Γ1 |Q(0)| /2Γ2 < H (0) < Γ1 |Q(0)| /2.

(11.33)

The motion of the gyropendulum corresponds to the alternation between positive H -orientation and negative H -orientation and vice versa. A representative trajectory with self-intersecting loops and the corresponding graph of H (t˜) are shown in Fig. 11.4. Class 2 : (a) cusps appear when ˙˜ 2 H (0) = −Γ1 |Q(0)| /2Γ2 .

(11.34)

The trajectories of the gyropendulum, which contains cusps, do not traverse the origin, as shown in Sect. 11.2.3.1. In this case, the motion of the gyropendulum generally follows either a positive H -orientation or a negative H -orientation, where the orientation does not change throughout the trajectory. A typical trajectory involving cusps and the corresponding graph of H (t˜) are shown in Fig. 11.5a, b, respectively. In this particular case, the motion of the gyropendulum is of negative H -orientation. (b) loops passing through the origin satisfy the condition 2 ˜ H (0) = Γ1 |Q(0)| /2.

(11.35)

The general motion of the gyropendulum is either of positive H -orientation or of negative H -orientation, with no cusps present. In our particular case, we have H (t˜) ≥ 0 as shown in Fig. 11.5d, so that the gyropendulum has a positive H -orientation. Examples of trajectories are shown in Fig. 11.3 and Fig. 11.5c. Class 3 : Trajectories in this class do not pass through the origin, and are additionally characterised by the vector product of the normalised velocity and acceleration components at the end of the rod at time t˜, that is, ˙˜ t˜) × Q( ¨˜ t˜) = (U˜˙ (t˜)V¨˜ (t˜) − V˜˙ (t˜)U¨˜ (t˜))k = Y (t˜)k. Q(

(11.36)

(a) smooth curves, representing the motion of negative H -orientation, with H (t˜) < 0, occur when ˙˜ 2 H (0) < −Γ1 |Q(0)| /2Γ2 .

(11.37)

In this case, the periodic function Y (t˜), defined in (11.36), changes sign. The trajectory of the gyropendulum does not go through the origin and there are no cusps. Furthermore, the displacement and velocity components do not vanish, and a typical trajectory is shown in Fig. 11.6a, where the corresponding function H (t˜) is negative as shown in Fig. 11.6b. The parametric

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(a)

(b)

(c)

(d)

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Fig. 11.5 (a) Cusp trajectory of the gyropendulum and (b) the function H (t˜) with parameter values Γ1 = 0.0016 and Γ2 = 0.0001 (with ω˜ 3 = 0.0092 and ω˜ 4 = 0.0108) and initial conditions U˜ 0 = −0.0416, V˜0 = 0, U˙˜ 0 = 0.001 and V˙˜0 = 0.005. (c) The gyropendulum trajectory consisting of loops passing through the origin and (d) the function H (t˜) with parameter values Γ1 = 0.0016 and Γ2 = 0.0001 (with ω˜ 3 = 0.0092 and ω˜ 4 = 0.0108) and initial conditions U˜ 0 = 0.01, V˜0 = 0.08, U˙˜ 0 = 0 and V˙˜0 = 0.00052. The conditions for such trajectories are discussed in Class 2 above (see (11.34) and (11.35))

plot in Fig. 11.6c for the normalised velocity field (U˙˜ (t˜), V˙˜ (t˜))T shows selfintersecting loops , which corresponds to the change of sign of Y (t˜). (b) smooth curves, representing the motion of positive H -orientation, with H (t˜) > 0, occur when 2 ˜ /2. H (0) > Γ1 |Q(0)|

(11.38)

A representative trajectory of the gyropendulum for this case is shown in Fig. 11.6d; the corresponding graph of the positive function H (t˜) is shown in Fig. 11.6e. In Fig. 11.6f, we show the parametric plot for the normalised velocity field (U˙˜ (t˜), V˙˜ (t˜))T . In this case, the function Y (t˜) maintains the same sign, and in both Fig. 11.6a and 11.6d, the H -orientation of the motion remains the same, at all times. However, the parametric curves describing the velocities (Fig. 11.6c and 11.6f) differ significantly, according to the behaviour of the function Y (t˜) above.

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 11.6 (a) Smooth trajectory of the gyropendulum and (b) the function H (t˜) with parameter values Γ1 = 0.0016 and Γ2 = 0.0001 (with ω˜ 3 = 0.0092 and ω˜ 4 = 0.0108) and initial conditions U˜ 0 = −0.0135, V˜0 = 0, U˙˜ 0 = 0.0005 and V˙˜0 = 0.0005; (c) the corresponding parametric plot of the normalised velocity field with self-intersecting loops. (d) The gyropendulum smooth trajectories and (e) the function H (t˜) with parameter values Γ1 = 0.0016 and Γ2 = 0.0001 (with ω˜ 3 = 0.0092 and ω˜ 4 = 0.0108) and initial conditions U˜ 0 = 0.06, V˜0 = 0.05, U˙˜ 0 = 0 and V˙˜0 = 0.0005. (f) The corresponding parametric plot of the normalised velocity field with smooth loops of constant H orientation. The trajectories above are obtained when the conditions (11.37) or (11.38) in Class 3 are satisfied

11.2.4 Gyropendulum Trajectories of Prescribed Shapes and Rotational Symmetry Can a gyropendulum be designed so that it traces a prescribed periodic trajectory possessing a degree of rotational symmetry? To answer this question, the parameters Γ1 and Γ2 need to be determined together with knowledge of how to set the gyropendulum in motion, i.e. the initial conditions. For polygonal shapes, this may be done to a very good degree of approximation and two examples will be given here. The solution in (11.26) may be regarded as a Fourier series with two frequency components and without the frequency-independent term. This may be fitted to the given shape.

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(a)

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(b)

Fig. 11.7 The approximations to the prescribed: (a) square orbit and (b) triangular orbit. The calculated parameters to generate these orbits are (a) Γ1 = 2, Γ2 = 3, U0 = 0.10189, V0 = 0, U˙0 = 0, V˙0 = −0.1528 and (b) Γ1 = 1, Γ2 = 2, U0 = −0.04936, V0 = −0.02850, U˙0 = −0.01140, V˙0 = 0.01974

The frequency ω˜ 3 in (11.26) is prescribed as any positive integer and ω˜ 4 is chosen as an integer multiple of ω˜ 3 . The Fourier coefficients may be determined and, through (11.26), the required initial conditions may be found. The parameters Γ1 and Γ2 are determined from the chosen frequencies through (11.23). For convenience, the solution (11.26) may be written in the complex form z(t˜) = c1 e−i ω˜ 3 t˜ + c2 ei(n−1)ω˜ 3 t˜,

(11.39)

where z(t˜) = U (t˜) + i V (t˜) and ci , i = 1, 2, are complex constants, and n > 1 is a given positive integer. Two examples are shown in Fig. 11.7. In Fig. 11.7a, a prescribed square (n = 4) may be seen together with its approximating trajectory. In view of the rotational symmetry of the square, the choice ω˜ 4 = (n − 1)ω˜ 3 with n − 1 = 3 is made. Given that ω˜ 4 may be any integer, it is chosen here as ω˜ 4 = 3 in order to result in positive values of the physical parameters Γ1 and Γ2 . The Fourier coefficients in this case are √ √ 8 2 8 2 c1 = 2 , c2 = − 2 . π 9π

(11.40)

The consequent values of the parameters Γ1 and Γ2 together with the initial conditions are shown in the figure caption. There is very good agreement between the two shapes considering a two-component truncated Fourier series only is available from the analytical solution. In Fig. 11.7b, a prescribed triangle (n = 3) is shown together with its approximating trajectory. In view of the rotational symmetry of the triangle, ω˜ 4 = (n − 1)ω˜ 3 with n − 1 = 2 and the choice ω˜ 4 = 2 is made. The Fourier coefficients in this case are

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c1 = −

9 √ 9 √ ( 3 + i), c2 = − ( 3 + i). 2 4π 16π 2

(11.41)

The consequent values of the parameters Γ1 and Γ2 together with the initial conditions are shown in the figure caption. Note that the origin of the coordinates is taken at the centroid of the given triangle since no frequency-independent term is available in the Fourier series. Again very good agreement between the two shapes is seen. It may be noted that the area enclosed by a trajectory is given by S=



  1  2π , ˜ ˜ H ( t )d t   2 0

where H (t˜) is defined in (11.31). For the examples in Fig. 11.7, the relative differences between the area of the prescribed shape and the area enclosed by the gyropendulum orbit are 0.6% for the square and 1.0% for the triangle. On the basis of this measure, there is very good agreement between the prescribed shape and its associated orbit for both the square and the triangle. This ability to design a gyropendulum and prescribe initial conditions to enable it to move in a good approximating trajectory to a prescribed shape with rotational symmetry may have useful practical application. It is noted that this method is applicable to any shape which has rotational symmetry; in particular, the segments do not have to be linear.

11.2.5 Optimal Design of a Gyropendulum for Regular Polygonal Approximations In this section, we use a method, different from Sect. 11.2.4, to look at periodic trajectories of the gyropendulum in the case when they are close to regular polygons. We note that in the complex plane, the Christoffel-Schwarz conformal map of the unit disc into the exterior of a regular n-sided polygon has the form (see, for example, Savin (1968))   1 + Wn (ζ ) , (11.42) z=C ζ where Wn (ζ ) = + +

2 n−2 ζ n−1 + 2 ζ 2n−1 n(n − 1) n (2n − 1)

(n − 2)(2n − 2) 3n−1 (n − 2)(2n − 2)(3n − 2) 4n−1 ζ ζ + 2n 3 (3n − 1) 12n 4 (4n − 1)

(n − 2)(2n − 2)(3n − 2)(4n − 2) 5n−1 ζ + . . . , n = 3, 4, 5, . . . , 60n 5 (5n − 1)

(11.43)

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and C is a positive real constant. We will consider the truncated series which consists of only the first two terms in (11.42). In particular, when ζ = ei t˜ is a point on the unit circle, we have  z = C e−i t˜ +

 2 ei(n−1)t˜ , n = 3, 4, 5, . . . . n(n − 1)

(11.44)

Hence, we observe that with the choice of ω˜ 3 = 1 and ω˜ 4 = n − 1, and with the appropriately chosen initial conditions we may be able to obtain a trajectory approximating a regular n−sided polygon. We note that ω˜ 3 and ω˜ 4 determine the dimensionless positive parameters Γ1 and Γ2 (see (11.23)). Setting z = U˜ (t˜) + i V˜ (t˜), it follows that the corresponding initial conditions, which satisfy (11.26) and (11.44), are given by     2 + n(n − 1) 2 ˙ ˙ ˜ ˜ ˜ ˜ , V0 = 0, U0 = 0, V0 = C −1 . U0 = C n(n − 1) n

(11.45)

In the above equations, C defines the size of the polygonal approximation. Here, we choose C so that the circumscribed circle relative to the approximate polygon, defined by formula (11.44), has the given radius R. We note that R = U˜ 0 in (11.45), and hence we have Rn(n − 1) R(n − 1)(2 − n) , U˜ 0 = R, V˜0 = 0, U˙˜ 0 = 0, V˙˜0 = . 2 + n(n − 1) 2 + n(n − 1) (11.46) In the illustrations, presented here, we show periodic trajectories of the gyropendulum that approximate regular polygons with rounded corners. These are similar to Sect. 11.2.4, but we focus specifically on regular polygonal shapes. We prescribe the radius R of the circumscribed circle to be 0.1 in the computation presented in Fig. 11.8, where four examples are shown. The initial conditions, corresponding to trajectories in Fig. 11.8, satisfy the relations (11.45) and (11.46) for the appropriate value of n. The functions U˜ (t˜) and V˜ (t˜) for each part of Fig. 11.8 are given by C=

    1 1 (a) U˜ (t˜) = C cos(t˜) + cos(2t˜) , V˜ (t˜) = C − sin(t˜) + sin(2t˜) , 3 3     1 1 (b) U˜ (t˜) = C cos(t˜) + cos(3t˜) , V˜ (t˜) = C − sin(t˜) + sin(3t˜) , 6 6     1 1 (c) U˜ (t˜) = C cos(t˜) + cos(4t˜) , V˜ (t˜) = C − sin(t˜) + sin(4t˜) , 10 10 and     1 1 (d) U˜ (t˜) = C cos(t˜) + cos(5t˜) , V˜ (t˜) = C − sin(t˜) + sin(5t˜) . 15 15

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(a)

(b)

(c)

(d)

Fig. 11.8 The polygonal approximations of the (a) triangular orbit, (b) square orbit, (c) pentagonal orbit and (d) hexagonal orbit. The calculated parameters to generate these orbits are (a) n = 3, C = 0.075, Γ1 = 1, Γ2 = 2, U˜ 0 = 0.1, V˜0 = 0, U˙˜ 0 = 0, V˙˜0 = −0.025, (b) n = 4, C = 0.0857, Γ1 = 2, Γ2 = 3, U˜ 0 = 0.1, V˜0 = 0, U˙˜ 0 = 0, V˙˜0 = −0.0429, (c) n = 5, C = 0.0909, Γ1 = 3, Γ2 = 4, U˜ 0 = 0.1, V˜0 = 0, U˙˜ 0 = 0, V˙˜0 = −0.0545 and (d) n = 6, C = 0.0938, Γ1 = 4, Γ2 = 5, U˜ 0 = 0.1, V˜0 = 0, U˙˜ 0 = 0, V˙˜0 = −0.0625. The circumscribed circle of radius 0.1 is shown by dashed lines

11.2.5.1

Additional Use of the Function H ( t˜) in the Evaluation of the Area

In Sect. 11.2.3.3, we introduced the function H (t˜) (see (11.31)) that was used for the classification of the trajectories traversed by the gyropendulum. Here, we show that this function can additionally be used to compare the exact regular polygons with the approximate polygonal shapes associated with the motion of the gyropendulum. Namely, the area of the approximate polygon associated with the trajectory of the gyropendulum is given by the formula

11 Effect of Gravity on the Dispersion and Wave Localisation …

 

 

 



 ˙˜ 1 2 − Γ |Q(0)| 2 Γ |Q(0)| S = 21  02π H (t˜)d t˜ = 2 ˜ 2(Γ12 +4Γ2 )  1

239

   Γ12 sin(2π Γ12 +4Γ2 )  +H (0) 8π Γ2 + Γ12 +4Γ2



˙˜ T ˜ · Q(0) +Γ1 Q(0)





  sin(2π Γ12 +4Γ2 )  2π − Γ12 +4Γ2

  2 1 − cos(2π Γ1 + 4Γ2 ) .

(11.47)

Taking into account that for the approximation of a regular n-sided polygon, we have Γ1 = n − 2, Γ2 = n − 1, and hence Γ12 + 4Γ2 = n 2 , we deduce S=



π  n 2 (n

    ˙˜ 2 2 ˜ − 2) |Q(0)| + 4(n − 1)H (0). − (n − 1)|Q(0)|

We use the area S as the functional, which we would like to evaluate when comparing the exact and approximate polygonal shapes. In this case, the relative error is defined by    S − Sexact   , (11.48)  S  exact where Sexact is the area of the exact polygon. Given the radius R1 of the circumscribed circle for the exact polygon, we have Sexact

  2π n R12 sin . = 2 n

(11.49)

Here, assuming that the constant C from (11.42) is given, the radius R1 is defined by   R1 = C 1 + Wn (1) ,

(11.50)

where the truncated approximation (11.43) of Wn (ζ ) is used. In particular, for n = 4, the exact square and the corresponding approximation of the square are plotted in Fig. 11.9, along with the circumscribed circles for each respective trajectory. This example presents the trajectory of the gyropendulum which is close to a square shape with rounded corners. The exact square is plotted using the formula (11.42) when ζ = ei t˜, and the trajectory of the approximate square is plotted using (11.44). The radius R1 of the circumscribed circle for the exact square is given by (11.50), with C defined in (11.45)–(11.46), and the radius R of the circumscribed

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Fig. 11.9 The exact square plot (dashed line) and its approximation (solid line) with the circumscribed circles for each case (dotted lines). The inner circle has radius R = 0.1, and the outer circle has radius R1 = 0.1026. The calculated parameters to generate the exact square are n = 4 and C = 0.0857; and for the approximate square we have n = 4, C = 0.0857, Γ1 = 2, Γ2 = 3, U˜ 0 = 0.1, V˜0 = 0, U˙˜ 0 = 0, V˙˜0 = −0.0429

circle for the approximate square is defined in (11.46). It is noted that R1 > R. In the case, shown in Fig. 11.9, the relative error (11.48) of this approximation does not exceed 0.5%.

11.2.5.2

Optimisation Relative to the Area Traversed by the Gyropendulum

We note that the area S of the approximate polygon (see (11.47)) can also be written in the equivalent form, which uses the mapping (11.44), as   2  2 1 . S = π |C|2 1 − n n−1

(11.51)

It is useful to observe that for the case of the polygon defined by the conformal map (11.42), the difference between S and Sexact is always positive:  S − Sexact = π |C|2

(n − 2)2 (n − 2)2 (n − 1)2 (n − 2)2 (n − 1)2 (3n − 2)2 + + 4 6 n (2n − 1) n (3n − 1) 36n 8 (4n − 1)

(n − 2)2 (n − 1)2 (3n − 2)2 (2n − 1)2 + ... + 225n 10 (5n − 1)

 > 0, n = 3, 4, 5, . . .

(11.52)

which implies that the approximation (11.51) provides an upper bound for the area of the regular polygon, which is also confirmed by the numerical illustration shown in Fig. 11.9. If the mapping (11.44) is replaced by

11 Effect of Gravity on the Dispersion and Wave Localisation …

 μn i(n−1)t˜ , n = 3, 4, 5 . . . , z = C e−i t˜ + e n−1

241

(11.53)

then the quantity μn > 2/n can be chosen to design the gyropendulum so that it traverses the approximate polygonal trajectory, which bounds the area S = Sexact . Namely,  (n − 2)2 (n − 2)2 (n − 1)2 4 + 4 + μn = (n − 1) 2 n (n − 1) n (2n − 1) n 6 (3n − 1) +

(n − 2)2 (n − 1)2 (3n − 2)2 36n 8 (4n − 1)

(n − 2)2 (n − 1)2 (3n − 2)2 (2n − 1)2 + ... + 225n 10 (5n − 1)

 1/2 , n = 3, 4, 5, . . . .

(11.54)

With the correction coefficient μn in place, the initial conditions (11.46) should be replaced by R(n − 1) R(n − 1)(μn − 1) , U˜ 0 = R, V˜0 = 0, U˙˜ 0 = 0, V˙˜0 = . n − 1 + μn n − 1 + μn (11.55) In this case, the area S bounded by the approximate polygonal trajectory of the gyropendulum is such that S = Sexact for any n−sided regular polygon. In Fig. 11.10, we show that the ratio of the correction coefficient μn and the original coefficient 2/n in (11.44) remains greater than 1 for all n ≥ 3 and has a finite limit which is also greater than 1: C=

nμn → A > 1 as n → ∞; A ≈ 1.11. 2

(11.56)

11.3 Active Gyroscopic Systems Subjected to Gravity Sometimes, it may be beneficial to have a controlled gyropendulum, whose gyricity is adjusted to the vibration of some ambient system. Compared to the normalised system, discussed in Sect. 11.2.2, we do not introduce a time-scaling here. The gyricity parameter Y, characterising the gyroscopic spinner, is now linked to the radian frequency of vibrations of the entire system (see, for example, Brun et al. (2012); Carta et al. (2014)). In this case, in equations (11.3), we assume that the complex displacement vector has the representation

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Fig. 11.10 The plot of the ratio of the correction coefficient μn to 2/n (solid line). The dashed line shows the asymptotic limit of the ratio, which is greater than 1

    I1 U (t) −iωt U Y = αω, =e with V (t) V I0

(11.57)

where α is a gyricity control parameter. We introduce the notation G=

g mgL = , I0 L

(11.58)

where we use I0 = m L 2 (see (11.6)). Substituting equation (11.57) into the system of equations (11.3), we then obtain  2     −ω + G −iαω2 U 0 = . (11.59) iαω2 −ω2 + G V 0 To obtain non-trivial solution of equation (11.59), we require that (1 − α 2 )ω4 − 2Gω2 + G 2 = 0.

(11.60)

The two positive eigenfrequencies are given by  ω1 =

G , ω2 = 1+α



G , where 0 ≤ |α| < 1. 1−α

(11.61)

It is noted that the above two frequency values are related to the two dispersion branches for a chain of gyropendulums which are analysed in Sect. 11.4. In particular, the values (11.61) also appear as the radian frequencies of standing waves (see (11.70) in Sect. 11.4.1). The corresponding eigenvectors are given by (U (1) , V (1) )T = (i, 1)T , (U (2) , V (2) )T = (−i, 1)T .

(11.62)

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The displacements, corresponding to the eigenvectors (11.62) and eigenfrequencies (11.61), represent circular trajectories with the parametric form  Re (ie−iω1 t , e−iω1 t )T = (sin(ω1 t), cos(ω1 t))T , in the clockwise direction, and  Re (−ie−iω2 t , e−iω2 t )T = (− sin(ω2 t), cos(ω2 t))T , in the counterclockwise direction.

11.4 The Chain of Gyropendulums: Floquet-Bloch Waves The results in Sect. 11.2 represent the transient motion of the linearised gyroscopic system for a single gyropendulum. We would like to follow the idea of the earlier work Jones et al. (2020) and study the interaction between gyroscopic elements within a periodic infinite system. The new feature in our current model, compared to Jones et al. (2020), is the presence of the gravity term in the governing equations. We will show that this additional term leads to a significant change in the overall dynamic response of the system. Formally, we apply the Fourier transform in time (with the radian frequency being denoted by ω), and assume that the chiral system is active so that, in a similar way to that in Jones et al. (2020), the gyroscopic coupling is represented by a term proportional to ω2 . We consider an infinite periodic chain of gyropendulums positioned at x = n, n ∈ Z, and connected by massless springs of elastic stiffness c1 > 0 (see Fig. 11.11). The notation U(n) = (U (n) (t), V (n) (t))T is used for the vector of the amplitude at the nodal point x = n. The additional forcing is provided due to gravity and the interaction with the nearest neighbours within the chain. Similar to Jones et al. (2020), we also allow for pre-tension in the springs, which is characterised by an effective transverse stiffness c2 ≥ 0.

11.4.1 Floquet-Bloch Waves in a Chiral Chain Under Gravity Here, we consider an active system, representing a periodic chain of gyroscopic pendulums with the chirality parameter α and amplitude U(n) of the time-harmonic motion with radian frequency ω. Active gyroscopic systems were considered in the earlier papers Jones et al. (2020); Brun et al. (2012), but without the presence of gravity. Although the formulation for the Floquet-Bloch waves may appear to be similar to the one of Jones et al. (2020), we show that the contribution from the gravity term

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Fig. 11.11 Chain of gyropendulums connected by elastic springs

brings a significant change in the dispersion properties, as well as the description of the standing waves. The displacement amplitude vectors of the inertial nodal points of the gyro-elastic chain satisfy the following vector equation: − mω2 U(n) = C(U(n−1) + U(n+1) − 2U(n) ) + iαmω2 RU(n) − mGU(n) , (11.63) where G is the normalised gravity parameter (see (11.58)), the rotation matrix R is given by (11.4) and the stiffness matrix   c1 0 . C= 0 c2 We assume that the spatial period between neighbouring nodal points is a, and impose the Floquet-Bloch condition (see, for example, Kittel (2004)): U(n+1) = eika U(n) , where k is the wave number. The system (11.63) is reduced to the form   (mG − mω2 )I − 2(cos(ka) − 1)C − iαω2 R U(n) = 0.

(11.64)

Introduce the dimensionless variables:  m 1 (n) ˜ c2 mG (n) ˜ U = U , k = ka, ω˜ = ω , c˜ = , G˜ = , a c1 c1 c1

(11.65)

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where the quantities with the symbol ‘∼’ are dimensionless. Equation (11.64) becomes (dropping the ‘∼’ for convenience) ω2 M (α)U(n) + C (k, c, G)U(n) = 0,

(11.66)



 1 iα M (α) = , −iα 1

where



and C (k, c, G) = −

 2(1 − cos k) + G 0 . 0 2c(1 − cos k) + G

We note that the matrix C is diagonal, and depends on the gravity parameter G. The parameter α characterises the effect of the gyroscopic action produced by the spinners, and the non-negative parameter c measures the degree of pre-tension in the spring. To obtain non-trivial solutions of (11.66), it is required that   det ω2 M (α) + C (k, c, G) = 0, which is equivalent to (1 − α 2 )ω4 − 2((c + 1)(1 − cos k) + G)ω2 + G 2 + 2(1 − cos k)(c + 1)G + 4c(1 − cos k)2 = 0.

(11.67)

This gives two dispersion branches defined by  1/2 , ω± = Q ± (α, c, k, G)

(11.68)

where

Q ± (α, c, k, G) =

  1 G + (1 − cos k)(c + 1) ± (G + (1 − cos k)(c + 1))2 (1 − α 2 ) 1/2  . −(1 − α 2 )(G 2 + 2G(1 − cos k)(c + 1) + 4c(1 − cos k)2 )

For G = 0, expression (11.68) is equivalent to the dispersion relations studied in Jones et al. (2020). However, for G > 0 there are significant changes in the wave dispersion. The most notable one is the presence of the band gap within a finite interval adjacent to ω = 0. This did not occur in the earlier work Jones et al. (2020). In addition, interior band gaps and partial pass bands may occur, as discussed in detail below.

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Illustrations of the Frequency Dependence on Chirality, Pre-tension and Gravity Parameters

To show the change in the dynamic response of the chiral system, with the varying values of the gravity and chirality parameters, we present the results of the test computations in Figs. 11.12 and 11.13. The radian frequency ω, as a function of k and G, is illustrated in Fig. 11.12 for c = 0 and c = 0.6. In Fig. 11.12, we observe that the radian frequency increases with increasing values of the gravity parameter G. For the case of no pre-tension (c = 0), the presence of gravity introduces an additional mode with non-zero frequency as observed in Fig. 11.12a. If c = 0 and G = 0, there is a single mode with non-zero frequency, independent of the chirality α, which agrees with the results in Jones et al. (2020). The variation of ω as functions of k and α for c = 0.5 and two values of G are shown in Fig. 11.13. For a chain of pendulums subjected to gravity, with pre-tension but without spinners (α = 0), the two dispersion branches are given by   (11.69) ω+ = G + 2(1 − cos k), ω− = G + 2c(1 − cos k).

(a)

(b)

Fig. 11.12 The graphs of ω± as functions of k and G for α = 0.4; (a) c = 0 and (b) c = 0.6

(a)

(b)

Fig. 11.13 The graphs of ω± as functions of k and α for c = 0.5; (a) G = 1 and (b) G = 10

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We note that when G > 0, there is a finite width band gap adjacent to ω = 0 (see Fig. 11.13). When α = 0 and k = 0, it follows that ω+ = ω− for the pre-tension parameter range 0 ≤ c < 1. Introducing the non-zero chirality parameter α, we observe that ω+ > ω− for any k provided that G > 0 and 0 ≤ c < 1; in this case, there is an additional band gap between the two dispersion curves.

11.4.1.2

Frequencies of Standing Waves

Figure 11.13 shows the surfaces representing the frequencies ω = ω± (α, c, k, G) as functions of the wave number k and the chirality α. When k = 0 (the gyropendulums move synchronously with no force exerted along the connecting links), the frequencies in equation (11.68) have the form     G G   , ω+  . (11.70) = = ω−  k=0 k=0 1+α 1−α When α = 0, the frequencies in (11.70) are identical and depend only on the gravity parameter G. We also note that equations (11.70) agree with (11.61), obtained for a single active gyropendulum. Similarly, when k = π (or k = −π ), the eigenfrequencies are given by   ω± 

 k=π

=

G + 2(c + 1) ±



(G + 4)(G + 4c)α 2 + 4(1 − c)2 1 − α2

 21 .

(11.71)

The values (11.70) and (11.71) give the frequencies of the standing waves within the gyro-elastic chain, and also represent the boundaries of important regions on dispersion diagrams discussed in the text below.

11.4.1.3

Partial Pass Bands, Total Pass Bands and Stop Bands

We classify the dispersion regimes with the use of the two branches defined in (11.68) for all 0 < α < 1 and 0 < c < 1. In particular, we consider the difference ω+ |k=0 − ω− |k=π (see (11.70), (11.71) and Fig. 11.14). We note that ω+ |k=0 is the minimum value associated with the upper dispersion curve, and ω− |k=π is the maximum value associated with the lower dispersion curve. The term ‘total pass band’ corresponds to a radian frequency ω where both types of Floquet-Bloch waves corresponding to the branches (11.68) occur, i.e. ω+ |k=0 ≤ ω ≤ ω− |k=π . We use the term ‘partial pass band’ for frequencies ω where only one type of Floquet-Bloch waves corresponding to the two branches (11.68) occurs. For frequencies, where all waveforms are evanescent, the term ‘stop band’ is used.

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Fig. 11.14 Dispersion diagrams for (a) γ = 0, (b) γ = 0.5, (c) γ = 1 and (d) γ = 1.5. In the calculations, the following parameters are chosen: c = 0.6, α = 0.6 and G = 0, 0.5, 1 and 1.5 (in parts (a), (b), (c) and (d), respectively)

The representative dispersion diagrams are shown in Fig. 11.14, where the dispersion regions are observed for c = 0.6 and α = 0.6. We identify the regimes by the difference ω+ |k=0 − ω− |k=π and the values of the non-negative parameter γ = Gα(c + 1)/(4c(1 − α)). These are as follows: • 0 ≤ ω+ |k=0 ≤ ω− |k=π is equivalent to 0 ≤ γ ≤ 1 (see Fig. 11.14a–c); when γ = 0, we observe one stop band, one pass band and one ‘partial pass band’, when 0 < γ < 1, we have two stop bands, one pass band and two ‘partial pass bands’, and for γ = 1 the dispersion diagram is shown in Fig. 11.14c. • ω+ |k=0 > ω− |k=π > 0 is equivalent to γ > 1; in this case, there are three stop bands and two ‘partial pass bands’ (see Fig. 11.14c). Here, the variation of γ occurs due to the gravity parameter G for fixed values of the pre-tension and chirality parameters. Introducing a non-zero gravity parameter yields a zero frequency band gap adjacent to ω = 0. For increasing values of G, such that γ > 1, we observe an additional intermediate stop band and no total pass bands as shown in Fig. 11.14d. The dynamics of the parameter γ in the classification of the dispersion regions is discussed in Appendix 2.

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11.4.1.4

249

Nodal Trajectories for Floquet-Bloch Waves

For any fixed value of the Floquet parameter k, the nodal trajectories producing Floquet-Bloch waves are all of the same elliptical shape. Namely, the displacement vector can be written in the form   (n) ikna −iωt , (11.72) = Re U e e u(n) ± ± where the eigenvectors of the amplitude are given by (see (11.64)–(11.66)) U(n) ±

 =

T iα Q ± (α, c, k, G) ,1 . G − Q ± (α, c, k, G) + 2(1 − cos k)

(11.73)

We note that Q ± (α, c, k, G) are both real since 0 < α < 1, 0 < c < 1 and G ≥ 0. Thus, the first component of the eigenvector (11.73) is purely imaginary and hence the resulting elliptical trajectories have axes aligned parallel and perpendicular to the chain. The trajectories of the nodal points satisfy the equation 

Q ± (α, c, k, G) − G − 2(1 − cos k) (n) u1 α Q ± (α, c, k, G)

2

2

+ u (n) = 1. 2

(11.74)

The major axes of the two elliptical modes align either parallel or perpendicular to the chain depending on the coefficients in (11.74).

11.4.2 Illustrative Examples of Floquet-Bloch Waves Both the common shape of each of the nodal trajectories and the phase shift between neighbouring nodes will determine the nature of the Floquet-Bloch wave in any situation. Here, we assume that the spacing between neighbouring nodal points is a = 1, and we fix the values of the pre-tension c = 0.2 and chirality α = 0.2. Three examples are presented: • Fig. 11.16 shows standing waves corresponding to k = 0, where the nodal points follow circular trajectories, and the chain itself moves as a rigid solid (the distance between neighbouring nodal points is always unity). • Fig. 11.17 represents standing waves corresponding to k = π , where the nodal points follow elliptical trajectories, with the eccentricity depending on the orientation of the motion (either clockwise or counterclockwise) and gravity; the phase shift is also observed when comparing the motion of neighbouring nodal points. • Fig. 11.18 corresponds to the case of propagating waves, of positive group velocity, when k = π/2; the eccentricity of the elliptical trajectories changes for different orientations of the motion and different values of the gravity parameter G.

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Fig. 11.15 The dispersion curves ω = ω± for the parameter values c = 0.2, α = 0.2 and G = 10

For the choice of parameters c = 0.2, α = 0.2 and G = 10, the dispersion diagram in Fig. 11.15 shows two curves ω = ω± (α, c, k, G) defined by (11.68). The points of interest are labelled as A = (0, 3.536), B = (0, 2.887), C = ( π2 , 3.760), D = ( π2 , 3.032), E = (π, 3.997) and F = (π, 3.140). In this example, the spinners rotate in the counterclockwise direction, and hence the frequency values ω+ correspond to the counterclockwise motion of the gyropendulums, and the values ω− correspond to the clockwise motion of the gyropendulums. The change in the shape of elliptical trajectories, depending on c, α, G and the choice of the branch ω = ω± , is discussed in Appendix 2. In particular, it is shown that an increase in G, while other parameters remain unchanged, leads to a decrease in the eccentricity of the elliptical trajectories. In these illustrations of motion of chains of gyropendulums, we emphasise two main points: • when k = 0, two standing waves occur due to the presence of gravity (when G = 0 there are no standing waves within the chain for k = 0); • in the presence of gravity, the orientation of the major axes and eccentricity of the elliptical trajectories of the nodal points change for different values of the wave number k (in the absence of gravity, the orientation and eccentricity of the elliptical trajectories are independent of k). When k = 0, the nodal trajectories are circular and the motion of the trajectories of the infinite chiral chain is shown in Fig. 11.16. In this particular case, the motions corresponding to ω+ and ω− (see (11.70)) have the same trajectories, but opposite orientations.

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Fig. 11.16 The circular trajectories of an infinite chiral chain for the wave number k = 0 corresponding to standing waves; the ‘+’ mode is associated with counterclockwise motion of the nodal points, and the ‘−’ mode is associated with the clockwise motion of the nodes

(a)

(b)

Fig. 11.17 The elliptical trajectories for an infinite chiral chain for the case k = π (corresponding to standing waves); (a) the major axes of ellipses for the ‘+’ mode are aligned along the chain and (b) the major axes of ellipses for the ‘−’ mode are perpendicular to the chain. The arrows show the direction of motion of the nodal points around the ellipses (counterclockwise motion for the ‘+’ mode and clockwise motion for the ‘−’ mode)

(a)

(b)

Fig. 11.18 The mode shapes for an infinite chiral chain for k = π/2; (a) the ‘+’ mode and (b) the ‘−’ mode. The arrows show the direction of motion of the nodal points around the ellipses (counterclockwise motion for the ‘+’ mode and clockwise motion for the ‘−’ mode)

For the case when k = ±π , the common nodal trajectories are elliptical, and the group velocity is equal to zero (see Fig. 11.15). For such a standing wave, we also observe the phase shift between the motions of neighbouring nodal points, as illustrated in Fig. 11.17. For the case of k = π/2, the nodal trajectories are elliptical and neighbouring nodes are phase shifted. The group velocities corresponding to the points C and D in Fig. 11.15 are positive. The elliptical motion of nodal points along the chain is shown in Fig. 11.18.

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11.5 The Effect of Gravity on the Dynamic Green’s Kernel Green’s kernels for a gyro-elastic chain, without gravity, have been analysed in Jones et al. (2020). It is emphasised that this analysis is essential for understanding the waves generated by external forces, as well as the dynamic response of the locally perturbed gyro-elastic chain. Here, we consider new features brought into the model through gravity terms. In particular, the presence of gravity is linked to special regimes associated with localised waveforms which are discussed in this section. Illustrative examples of trajectories of the nodal points in an infinite chiral chain are given in the text below.

11.5.1 Construction of the Dynamic Green’s Kernel Consider a chiral chain of gyropendulums, as discussed in Sect. 11.4, but with a timeharmonic force ( f 1 , f 2 )T e−iωt acting at the central node (n = 0). Using the same dimensionless variables, the governing equations are given by   (G − Ω)U1(n) = U1(n+1) + U1(n−1) − 2U1(n) + iαΩU2(n) + f˜1 δn0 ,   (G − Ω)U2(n) = c U2(n+1) + U2(n−1) − 2U2(n) − iαΩU1(n) + f˜2 δn0 ,

(11.75)

where Ω = ω2 and f˜i = f i /ac1 . Taking the discrete Fourier Transform of equations (11.75) with respect to the Fourier variable k, we obtain the solution  (n) Gi j f˜j , i = 1, 2, (11.76) Ui(n) = j=1,2

where Green’s matrix is given by Gi(n) j =

1 2π



π

−π

Gˆi j eikn dk,

(11.77)

with

Gˆi j =

  1 G − Ω + 2c(1 − cos k) iαΩ , −iαΩ G − Ω + 2(1 − cos k) σ (α, Ω, c, k, G) (11.78)

where   σ (α, Ω, c, k, G) = det ΩM (α) + C (k, c, G) .

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Green’s matrix is then written as G (n) = F(n, Ω, α, c, G) [(G − Ω)I + iαΩR)] − F(n, Ω, α, c, G)diag{c, 1}, (11.79) where I is the 2 × 2 identity matrix, R is given in (11.4) and

π eikn 1 dk, (11.80) 2π −π σ (α, ω, c, k, G)

1 π (cos k − 1)eikn dk F(n, Ω, α, c, G) = (11.81) π −π σ (α, ω, c, k, G) = F(n + 1, Ω, α, c, G) − 2F(n, Ω, α, c, G) + F(n − 1, Ω, α, c, G).

F(n, Ω, α, c, G) =

It is important to note that, compared to Jones et al. (2020), the function F(n, Ω, α, c, G) is now dependent on gravity, which brings significant changes in the dynamic response of the elastic gyroscopic system, as discussed below.

11.5.2 The Dynamic Response for Special Regimes Such as Stop Bands and Pass Bands Although we are not considering Floquet-Bloch waves in this section, and our prime objective is the analysis of the dynamic response of the elastic system due to the action of a point force applied at the origin, there is an apparent connection between the choice of the frequency of vibrations and the analytical structure of the gravitydependent dynamic Green’s matrix. Green’s matrix will take different analytical forms depending on whether the forcing frequency is in the pass band, partial pass band or stop band. The function F(n, Ω, α, c, G) may be written as F(n, Ω, α, c, G) = =

1 4π c

0



1 4π c(η+ − η− )

π 0

π

cos(kn) dk (cos k − η+ )(cos k − η− )

cos(kn)dk − cos k − η+

0

π

(11.82)

 cos(kn)dk , cos k − η−

where (c + 1)(Ω − G) ∓ η± = 1 −



Ω 2 (4α 2 c + c2 − 2c + 1) + G 2 (c − 1)2 − 2GΩ(c − 1)2 . 4c

(11.83)

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We introduce b± = η1± , and study the behaviour of |b± (Ω, α, c, G)|. By examining the sign of |b± | − 1, the value of the integral (11.82), and hence the function F(n, Ω, α, c, G), is determined. The typical behaviour of |b± | as a function of Ω is shown in Fig. 11.19 for α = 0.4 and c = 0.6, where the effect of the gravity parameter G on the dispersion regions is also demonstrated. For G < 2.25, there is one pass band present while for G > 2.25, there are no total pass bands as discussed in Sect. 11.4.1.3. For the sake of convenience, we use the terms ‘Stop band’, ‘Partial pass band’ and ‘Total pass band’ when choosing the frequency of vibrations, according to the diagrams in Fig. 11.19.

11.5.2.1

Exponentially Localised Vibrations

This case corresponds to the stop band regime when |b± | < 1. The forcing frequency is in the stop band, and hence the components of Green’s matrix are exponentially localised. The function F(n, Ω, α, c, G) in the stop band may be written as ⎡

− 1 ⎣ b F(n, Ω, α, c, G) = + − 4c(η − η ) 1 − (b− )2

−

11.5.2.2

b

+

1 − (b+ )2





1−



1 − (b− )2 b−

|n|

|n| ⎤  + 2 1 − 1 − (b ) ⎦(11.84) . b+

Combination of Exponentially Localised and Propagating Waveforms

Here, we consider the regime of the partial pass band when |b+ | > 1 and |b− | < 1, or |b+ | < 1 and |b− | > 1. The additional partial pass band region is a special feature that is introduced by the gravity parameter that was not present in Jones et al. (2020). We first consider the case when |b+ | > 1 and |b− | < 1. To evaluate the ‘+’ integral, we consider a suitable contour integral (see Jones et al. (2020)). Taking into account the radiation condition at infinity, the function F(n, Ω, α, c, G) in the partial pass band is given by ⎡ ⎤  |n|  − i|n|θ + − )2 1 − b 1 − (b 1 ie ⎣ ⎦, F(n, Ω, α, c, G) = − 4c(η+ − η− ) b− sin(θ + ) 1 − (b− )2

(11.85) where

cos θ ± = η± .

(11.86)

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Fig. 11.19 Left column: the functions |b± (Ω)| for α = 0.4, c = 0.6 and a range of values of G. The singularities of b± (Ω) correspond to the values of Ω where η± vanish. Right column: the corresponding dispersion diagrams for the chiral chain with the same parameter values

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The first term in (11.85) represents the evanescent solution corresponding to the eigenfrequency ω− and the second term represents the propagating solution associated with the eigenfrequency ω+ . Similarly, for |b+ | < 1 and |b− | > 1, we can represent the function F(n, Ω, α, c, G) in the partial pass band as ⎡  |n| ⎤  i|n|θ − + + )2 1 − ie 1 − (b 1 b ⎦, ⎣ F(n, Ω, α, c, G) = − 4c(η+ − η− ) sin(θ − ) b+ 1 − (b+ )2

(11.87) where the first term in (11.87) represents the propagating solution corresponding to the eigenfrequency ω− and the second term represents the evanescent solution corresponding to the eigenfrequency ω+ .

11.5.2.3

Outgoing Propagating Waves

In the regime of the total pass band, the point force applied at the origin generates two types of outgoing propagating waves. In this region |b± | > 1, and both ‘±’ integrals lead to propagating solutions. The function F(n, Ω, α, c, G) is given in the total pass band by 1 F(n, Ω, α, c, G) = + 4c(η − η− )



 − + iei|n|θ iei|n|θ − . sin(θ − ) sin(θ + )

(11.88)

Both terms in (11.88) represent propagating solutions that obey the radiation condition at infinity. Numerical illustrations, presented in the next section, can be used for the physical interpretation of the analytical representation of the displacement generated by a time-harmonic point force applied at the origin. The presence of gravity is essential for this framework, as the regime of the low-frequency stop bands did not exist in the system without gravity. It is also essential to note that an additional low-frequency partial pass band occurs when gravity is introduced in the analysis.

11.5.3 Illustration of the Effect of Gravity on the Elliptical Trajectories In Fig. 11.20, we give examples of chiral waveforms produced by a point source positioned at the central nodal point (n = 0), for different values of the excitation frequency ω and different values of the gravity parameter G. We also refer to Fig. 11.19 which shows the structure of the stop bands, total pass bands and partial pass bands for Floquet-Bloch waves in the periodic chiral chain subjected to gravity. The force

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Elliptical trajectories for G = 1

257

Elliptical trajectories for G = 3

Fig. 11.20 Elliptical trajectories for a chiral chain with c = 0.6, α = 0.4 and a range of values of the forcing frequencies associated with the dispersion regions for G = 1 and G = 3

amplitude is chosen to be ( f˜1 , f˜2 )T = (0.1, 0.6)T , and the chirality and pre-tension parameters are defined by α = 0.4 and c = 0.6, respectively. In Fig. 11.20a, b, we show two exponentially localised waveforms corresponding to G = 1 and G = 3 for frequencies in the first stop band region shown in Fig. 11.19 (adjacent to ω = 0). Here, we note that the widths of the first stop bands increase with the increase of the gravity parameter G.

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In Fig. 11.20c, d, the frequencies of the applied force are chosen to generate a propagating wave. In both cases, G = 1 and G = 3, the trajectories of the nodal points are elliptical, but compared to the Floquet-Bloch waves (see Sect. 11.4.1), the orientation of the major axes changes along the chain. In both illustrative examples shown in Fig. 11.20c,d, the frequency values correspond to a partial pass band, and according to Sect. 11.5.2.2, the change of the orientation of the major axes of the ellipses is clearly visible in the vicinity of the central node, and it converges to the constant major axis orientation away from the point of application of the external force. In Fig. 11.20e, f, we show the qualitative difference between the case of G = 1 and G = 3. Namely, when G = 1 and ω = 1.5, the forced vibrations correspond to the regime of the total pass band (as in Fig. 11.19) and hence generate the propagating waveforms, whereas when G = 3 there are no total pass bands at all for any frequency. On the contrary, the case of G = 1 does not possess any stop band regimes within the range from 0.845 to 2.680, and no exponentially localised forms are envisaged within this regime. As soon as the gravity input has changed, i.e. G = 3, the forced vibration at ω = 2.1 shows the evanescent waveform that is consistent with the results in Sect. 11.5.2.1. Figure 11.20g, h corresponds to vibrations at higher frequencies associated with the second partial pass band (see Fig. 11.19). In this sense, there is a similarity with Fig. 11.20c, d, but the orientations of the major axes of the ellipses and their sizes are different. Figure 11.20i, j shows the exponentially localised waveforms in the higher frequency stop bands, for G = 1 and G = 3, and they are consistent with the results in Sect. 11.5.2.2.

11.6 Green’s Matrix and Localised Defect Modes for the Case of No Pre-tension In this section, the special case of no transverse pre-tension (c = 0) in the infinite chiral chain is discussed. The corresponding Green’s matrix and exponentially localised defect modes in the stop band regimes are analysed.

11.6.1 Green’s Matrix for a Chiral Lattice with No Pre-tension In the absence of pre-tension, two regimes are observed: partial pass bands and stop bands. Introducing a non-zero gravity parameter G, two modes with non-zero frequencies are obtained. For simplicity, we refer to partial pass bands as pass bands and introduce

11 Effect of Gravity on the Dispersion and Wave Localisation … ±

Ω =

G+2±

259



α 2 (G 2 + 4G) + 4 . 1 − α2

(11.89)

Given that Ω = ω2 , pass bands are observed for frequencies in the range G/(1 + α) < Ω < Ω − and G/ (1 − α) < Ω < Ω + . The stop bands, where no propagating waves exist, correspond to the frequencies within the intervals 0 < Ω < G/(1 + α), Ω − < Ω < G/(1 − α) and Ω > Ω + . We introduce the function

π eikn 1 dk, (11.90) I (n, Ω, α, G) = − 4π(G − Ω) −π cos k − N where N =1+

(1 − α 2 )Ω 2 − 2ΩG + G 2 , 2(G − Ω)

(11.91)

for G = Ω. Then, Green’s matrix is given by ⎛

(G − Ω)I (n, Ω, α, G)

⎜ G (n) = ⎜ ⎝ −iαΩ I (n, Ω, α, G)

iαΩ I (n, Ω, α, G)

⎞

⎟ ⎟. ⎠ (G − Ω + 2)I (n, Ω, α, G) −I (n + 1, Ω, α, G) − I (n − 1, Ω, α, G) (11.92) We present the expressions for the integral (11.90) in the three stop band cases, which occur when |N | > 1, as follows: 1. The first stop band occurs in the frequency range 0 < Ω < G/(1 + α) with ⎡

|n| ⎤  2−1 N N − 1 ⎢ ⎥  I (n, Ω, α, G) = ⎣ ⎦ , for 1 < N < 1 + G/2. 2(G − Ω) N 2−1

(11.93) 2. The second stop band occurs when • Ω − < Ω < G with ⎡ |n| ⎤ √ 2−1 N + N 1 ⎢ ⎥ I (n, Ω, α, G) = − √ ⎣ ⎦ , for N < −1, 2(G − Ω) N 2−1 (11.94) • or G < Ω < G/(1 − α) with

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Fig. 11.21 The plot of the function N for α = 0.7 and G=1

⎡ |n| ⎤ √ 2 N − N −1 1 ⎢ ⎥ I (n, Ω, α, G) = √ ⎣ ⎦ , for N > 1. 2(G − Ω) N 2−1 (11.95) +

3. The third stop band occurs when Ω > Ω with ⎡ |n| ⎤ √ 2−1 N N + 1 ⎥ ⎢ I (n, Ω, α, G) = − √ ⎦ , for N < −1. ⎣ 2(G − Ω) N 2−1 (11.96) For the case when G = Ω in the second stop band region, we have % I (n, Ω, α, G) =

0, for n = 0, − α21Ω 2 , for n = 0.

(11.97)

The above functions I (n, Ω, α, G), in the stop bands, are exponentially localised as functions of n. In Fig. 11.21, we present a typical plot of the function N (see (11.91)). The boundaries of the stop band regions, which occur when |N | > 1, can be identified with the use of the shaded regions. We note the presence of the point of singularity at Ω = G (dashed vertical line). The singular point is located in the second stop band region, which corresponds to the region where Ω − < Ω < G/(1 − α).

11.6.2 Illustration of Exponentially Localised Defect Modes For the sake of illustration, we introduce a defect by assuming zero gyricity in the central node (n = 0) and allowing for the change of mass at the same node, i.e. while

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all gyropendulums for n = 0 have the same mass m and the same chirality α, the central node (n = 0) has the mass m + M > 0 and the zero gyricity A + α = 0. In this manner, M and A = −α are perturbation parameters, and the central node can be viewed as a non-gyroscopic pendulum.

11.6.2.1

The Governing Equations for the Defect Mode

The governing equations for the waveform, representing the defect mode, are (G − ω2 )(m + Mδn0 )U1(n) = c1 (U1(n+1) + U1(n−1) − 2U1(n) ) + iα(1 − δn0 )ω2 U2(n) , (G − ω2 )(m + Mδn0 )U2(n) = c2 (U2(n+1) + U2(n−1) − 2U2(n) ) − iα(1 − δn0 )ω2 U1(n) . (11.98) We introduce the dimensionless variables defined in (11.65) together with M M˜ = > −1, m

(11.99)

where the quantity with the symbol ‘∼’ is dimensionless. Here, we consider the case of zero pre-tension (c = 0); due to the presence of gravity, this framework is different from the analysis of Jones et al. (2020). Equations (11.98) then become (dropping the ‘∼’ for convenience) (G − Ω)U1(n) = U1(n+1) + U1(n−1) − 2U1(n) + iαΩU2(n) +(M(Ω − G)U1(n) − iαΩU2(n) )δn0 , (G −

Ω)U2(n)

=

−iαΩU1(n)

+ (M(Ω −

G)U2(n)

+

(11.100)

iαΩU1(n) )δn0 ,

where Ω = ω2 . The apparent total force, associated with the perturbation of mass and gyricity and the change in the inertia, acting on the central nodal point is ˜f = (M(Ω − G)I − iαΩR)U(0) .

(11.101)

Using (11.76), we deduce ' & I − G (0) (M(Ω − G)I − iαΩR) U(0) = 0.

(11.102)

The solvability condition has the form (G − Ω)2 det(G (0) )M 2 + (G − Ω)Tr(G (0) )M (0) ) + 1 = 0, − α 2 det(G (0) )Ω 2 + 2αΩIm(G12

(11.103)

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where G (n) is given in (11.92). Equation (11.103) is a quadratic in M provided Ω = G, with roots given by ±

M =−

Tr(G (0) ) ±



(0) 2 (0) (0) 2 4(αΩdet(G (0) ) − Im(G12 )) + (G11 − G22 )

2(G − Ω)det(G (0) )

. (11.104)

When Ω = G, equation (11.103) degenerates to (0) ) + 1 = 0. − α 2 det(G (0) )Ω 2 + 2αΩIm(G12

(11.105)

We note that as follows from (11.92) and (11.97) (the latter corresponds to the case where Ω = G), the equation (11.105) is independent of the mass perturbation M.

11.6.2.2

Possible Frequency Regimes for Defect Modes

In Fig. 11.22a, we present the dispersion curves for the case of no pre-tension, where three stop band regions are observed. The radian frequency regions of the stop bands introduced by the gravity parameter satisfy 1. 0 < Ω < G/(1 + α) in the first stop band. 2. Ω − < Ω < G/(1 − α) in the second stop band. 3. Ω > Ω + in the third stop band, where Ω ± are defined in (11.89). In this case, assuming Ω = G, Green’s matrix G (0) for the central node depends on the chirality, gravity and frequency parameters (see (11.92)). Let R = (G − (1 + α)Ω)(G + (α − 1)Ω)((1 − α 2 )Ω 2 − (2G + 4)Ω + G 2 + 4G), (11.106) then using (11.91) and (11.92), equation (11.103) may be written as √ (11.107) (−sign(G − Ω) R + α 2 Ω 2 + (Ω − G)2 M)(M + 1) = 0, in the stop band region where N < −1, and √ (sign(G − Ω) R + α 2 Ω 2 + (Ω − G)2 M)(M + 1) = 0,

(11.108)

in the stop band region where N > 1. We note that M + 1 > 0, and that R vanishes when Ω = (ω± )2 , where ω± are defined by (11.70) and (11.71). When Ω < G, the expressions (11.107) and (11.108) lead to −

√ R + α 2 Ω 2 + (Ω − G)2 M = 0, for N < −1,

(11.109)

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(a)

263

(b)

Fig. 11.22 (a) The dispersion diagram for the case of no pre-tension (c = 0). (b) The plot of the translational mass perturbation of the central node with the associated frequencies in the third stop band region where N < −1. Both figures are plotted for α = 0.7 and G = 1

and

√

R + α 2 Ω 2 + (Ω − G)2 M = 0, for N > 1.

(11.110)

In the first stop band, which corresponds to the frequency range 0 < Ω < G/(1 + α) with 1 < N < 1 + G/2, there are no solutions for M in the interval where M > −1. We consider exponentially localised modes corresponding to frequencies in the second and third stop bands. The solution in the second stop band occurs when Ω = G, and the eigenmode is independent of M. In the third stop band, we give an illustrative example in the text below, where M is determined within the interval −1 < M < 0.

11.6.2.3

Evanescent Waveforms

The three identified stop bands are shown in Fig. 11.22, and the corresponding results for these stop bands are as follows: • As verified by the analysis of (11.103), there are no real roots M > −1 for the first stop band, where 0 < Ω < G/(1 + α). • In the second stop band Ω − < Ω < G/(1 − α), and we identify the special evanescent waveform at the frequency such that Ω = ω2 = G. We also note that this mode exists for any perturbation of mass M > −1. In other words, the waveform represents the transverse motion of a simple pendulum. In the absence of the pre-tension, the displacements of all other nodes are zero, i.e. U(n) = 0 for n = 0. The corresponding waveform is shown in Fig. 11.23a. • In the third stop band Ω > Ω + , and we identify the evanescent waveforms where M and Ω are related by

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Fig. 11.23 The trajectories of an infinite chiral chain for c = 0, α = 0.7 and G = 1. (a) Ω = 1, (b) Ω = 11.56 and M = −0.706

 M=

 2 I (1, Ω, α, G) −1 − 1, I (0, Ω, α, G) G−Ω

(11.111)

where I (n, Ω, α, G) is defined by (11.96). Here, we note that M > −1, and the graph of M versus Ω is shown in Fig. 11.22b. The corresponding evanescent waveform is shown in Fig. 11.23b, and the decay of this evanescent mode is characterised by the ratio      I (n + 1, Ω, α, G)    = N + N 2 − 1 < 1 for N < −1.   I (n, Ω, α, G) 

(11.112)

We note that the decay rate, given by (11.112), is independent of n. In Fig. 11.23, the trajectories of seven nodes are shown, (n = −3, −2, −1, 0, 1, 2, 3). In part (a) of this figure, we display the localised defect mode where only the central nodal point is moving transversely, while all other nodal points are at rest; the corresponding radian frequency of vibrations for this defect mode is such that ω2 = Ω = G = 1, and it is located in the second stop band region of Fig. 11.22a. The amplitude eigenvector of the central node is given by U(0) = (0, 0.5)T , which is found from (11.102). In Fig. 11.23b, we show another exponentially localised defect mode where the mass perturbation is M = −0.706 and the corresponding radian frequency is such that ω2 = Ω = 11.56, and it is located in the third stop band (see Fig. 11.22a). Here, the relation between the defect frequency in the third stop band and the mass perturbation of the central non-gyroscopic pendulum is defined by (11.111), and the amplitude eigenvector is given by U(0) = (0.5, 0)T . The motion of the central node is parallel to the chain as shown in Fig. 11.23b, and the major axes of the elliptical trajectories of the neighbouring nodes are aligned with the chain.

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11.7 Concluding Remarks In this paper, the dynamics of passive and active chiral systems have been studied and the analytical descriptions of the gyroscopic pendulum and a chain of gyropendulums have been presented. In the introductory part of the paper, we have given a full classification of trajectories for passive gyroscopic systems subjected to gravity. This part also includes special illustrative examples of an optimal design, where periodic trajectories of the gyropendulum are approximated by regular polygons. In the active gyroscopic systems, subjected to gravity, where the motion is assumed to be time-harmonic, the gyricity is linked to the radian frequency, and the trajectories associated with the motion of the spinner are of an elliptical shape. For the infinite periodic waveguide, consisting of active gyropendulums, the description of chiral elastic waves, which includes dynamic Green’s matrices and dispersion properties of the Floquet-Bloch waves, has been presented. The discussion also includes analysis of the forced problem and exponentially localised waveforms. Compared to the comprehensive analysis of the chiral periodic systems without gravity, presented in Jones et al. (2020), we have identified new dispersion regimes and the new structure of Green’s kernels. We emphasise that in the presence of gravity, the wave dispersion and wave localisation change compared to elastic systems without gravity. The effect of gravity becomes especially important in the case of an elastic chain, connecting gyroscopic spinners, with no longitudinal pre-tension: the illustrative examples shown here explain the nature of the exponentially localised defect modes, which differ from those in the absence of gravity. The generalisations of the new concepts discussed in this paper are applicable to a wide variety of physical problems involving chiral lattice systems. Acknowledgements A. K. gratefully acknowledges the financial support of the EPSRC through the Maths DTP grant EP/V52007X/1, project reference 2599756. I. S. J. is grateful to the Department of Mathematical Sciences, University of Liverpool and the Liverpool Research Centre for Mathematics and Modelling for the provision of the research infrastructure and computational facilities.

Appendix 1: Time-harmonic Trajectories and Their Perturbations Due to Gyricity In the case of a standard pendulum, with no gyricity (Y = 0 rad/s), we always observe a time-harmonic trajectory of an elliptical shape (which may also degenerate into a straight line segment or a circular shape); in this non-chiral case, the displacement components are given by   (   (   ( mgL I0 mgL U (t) U0 U˙ 0 = cos + , t t sin V V (t) V˙0 I0 mgL I0 0

(11.113)

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where (U0 , V0 )T and (U˙ 0 , V˙0 )T are the initial displacements and the initial velocities, respectively (see Sect. 11.2 for the definitions of the parameters). A question can be raised: is it possible to set a time-harmonic motion of a gyropendulum? The answer is affirmative, and it is linked to additional constraints on the initial displacements and initial velocities. When the gyricity is present, the representation (11.113) has to be replaced by       cos(ω4 t) − sin(ω4 t) U (t) + A2 = A1 sin(ω4 t) cos(ω4 t) V (t)  + A3

   cos(ω3 t) sin(ω3 t) + A4 , − sin(ω3 t) cos(ω3 t)

(11.114)

where V˙0 + ω3 U0 −U˙ 0 + ω3 V0 , A2 = , ω3 + ω4 ω3 + ω4 −V˙0 + ω4 U0 U˙ 0 + ω4 V0 A3 = , A4 = , ω3 + ω4 ω3 + ω4 A1 =

(11.115)

and ω3 = −

I1 Y −



I12 Y2 + 4I0 mgL 2I0

, ω4 =

I1 Y +



I12 Y2 + 4I0 mgL 2I0

. (11.116)

We note that the gyricity introduces two eigenfrequency branches, such that ( ω3
0 provided that 0 < α < 1 and 0 < c < 1. In particular, the sign of the function D(c, α, γ ) is determined by the sign of ω+ |k=0 − ω− |k=π . We also mention that for all admissible 0 < α < 1 and 0 < c < 1, D(c, α, γ ) is a monotonically increasing function of γ , and we have +

• D(c, α, γ ) < 0 if and only if 0 ≤ γ < 1. • D(c, α, γ ) = 0 if and only if γ = 1. • D(c, α, γ ) > 0 if and only if γ > 1. We note that when G = 0, while 0 < α < 1 and 0 < c < 1, we have γ = 0, ω+ |k=0 = 0 and ω− |k=π > 0.

Trajectories Associated with the Two-Dimensional Motion of the Nodal Points As discussed in Sect. 11.4.2, the trajectories of the nodal points, associated with the Floquet-Bloch waves in the periodic chain of gyropendulums, have elliptical shapes. In this appendix, we give illustrations of these trajectories for different values of pre-tension parameter c, chirality parameter α, gravity parameter G and the wave number k.

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Fig. 11.26 The circular trajectories of the nodal points for the case k = 0. The values of the other parameters are c = 0.2, α = 0.8 with G = 1 and G = 10. The counterclockwise trajectories are associated with the ‘+’ mode, and the clockwise trajectories are associated with the ‘−’ mode. Both trajectories are circular

Standing Waves First, we describe standing waves obtained for k = 0. The nodal points follow circular trajectories, as shown in Fig. 11.26 (also see Sect. 11.4.2) for the parameter values of c = 0.2, α = 0.8 and two values of the gravity parameter: G = 1 and G = 10. The spinner rotation is assumed to be counterclockwise. The values ω+ and ω− correspond to frequencies of motions, where nodal points traverse the circle in the counterclockwise and clockwise directions, respectively; we also note that ω+ > ω− (see (11.70)). As shown in Fig. 11.26, when G increases, both ω+ and ω− increase, as well as ω+ − ω− . When k = π , the trajectories of all the nodal points are elliptical, the group velocity is zero and the phase shift occurs between the motions of neighbouring nodal points (also see Sect. 11.4.2). Figure 11.27 shows the representative shapes of the trajectories for different values of the pre-tension, chirality and gravity parameters. This illustration also shows the competition between the gravity G on the one hand and the pre-tension and chirality on the other. This is reflected in the orientation of the major axes of the ellipses and in the eccentricity of the elliptical trajectories. In particular, when c, α and G all increase, the eccentricity is reduced.

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Fig. 11.27 The mode shapes for the nodes of the chiral chain for a selection of values of c, α and G with k = π . The counterclockwise trajectories are associated with the ‘+’ mode, and the clockwise trajectories are associated with the ‘−’ mode

Propagating Waves In the above examples of Sect. 11.7, we show the standing waves where the trajectories of the nodal points within the chain are circular or elliptical. Here, we give an illustration for the case of k = π/2, where the group velocity of the Floquet-Bloch waves is positive. Hence, this case corresponds to a propagating wave. The formula (11.74) shows that the trajectories of all nodal points remain elliptical, but their shapes are now dependent on the wave number k (this is not the case when G = 0). Several representative shapes, together with the corresponding frequencies ω± , are shown in Fig. 11.28. Similar to Sect. 11.7, we note that increasing simultaneously c, α or G decreases the eccentricity of the elliptical trajectory of the nodes. It is also observed that both ω+ and ω− increase when G increases, while other parameters are fixed. The comparison of Figs. 11.27 and 11.28 also shows that the eccentricities of the ellipses are different for standing and propagating waves (i.e. they are dependent on the wave number k).

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Fig. 11.28 The mode shapes for the nodes of the chiral chain for a selection of values of c, α and G with k = π2 . The counterclockwise trajectories are associated with the ‘+’ mode, and the clockwise trajectories are associated with the ‘−’ mode

References Abdulghany AR (2017) Generalization of parallel axis theorem for rotational inertia. Am J Phys 85:791–795 Andreev VD (1966) Teoriya Inertsial’noy Navigatsii (Theory of inertial navigation). Nauka, Moscow Bahaloo H, Li Y (2019) Micropolar modeling of auxetic chiral lattices with tunable internal rotation. J Appl Mech 86(4):041002. https://doi.org/10.1115/1.4042428 Bromwich T (1914) On the theory of Foucault’s pendulum, and of the gyrostatic pendulum. Proc London Math Soc 2:222–235 Brun M, Giaccu GF, Movchan AB, Movchan NV (2012) Asymptotics of eigenfrequencies in the dynamic response of elongated multi-structures. Proc R Soc Lond A 468:378–394 Brun M, Jones IS, Movchan AB (2012) Vortex-type elastic structured media and dynamic shielding. Proc R Soc Lond A 468:3027–3046 Brun M, Giaccu GF, Movchan AB, Slepyan LI (2014) Transition wave in the collapse of the San Saba Bridge. Front Mater 1, Article 12. https://doi.org/10.3389/fmats.2014.00012 Carta G, Nieves MJ (2021) Analytical treatment of the transient motion of inertial beams attached to coupling inertial resonators. J Eng Math 127:1–25 Carta G, Brun M, Movchan AB, Movchan NV, Jones IS (2014) Dispersion properties of vortex-type monatomic lattices. Int J Solids Struct 51:2213–2225 Carta G, Jones IS, Movchan NV, Movchan AB, Nieves M (2017b) Gyro-elastic beams for the vibration reduction of long flexural systems. Proc R Soc Lond A 473:20170136

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Carta G, Nieves MJ, Jones IS, Movchan NV, Movchan AB (2018) Elastic chiral waveguides with gyro-hinges. Quart J Mech Appl Math 71:157–185 Carta G, Nieves MJ, Jones IS, Movchan NV, Movchan AB (2019) Flexural vibration systems with gyroscopic spinners. Philos Trans R Soc A 377:20190154 Carta G, Jones IS, Movchan NV, Movchan AB (2019) Wave polarization and dynamic degeneracy in a chiral elastic lattice. Proc R Soc Lond A 475:20190313 Carta G, Giaccu GF, Brun M (2017a) A phononic band gap model for long bridges. The ‘Brabau’ bridge case. Eng Struct 140:66–76 Chaplain GJ, Makwana MP, Craster RV (2019) Rayleigh-Bloch, topological edge and interface waves for structured elastic plates. Wave Motion 86:162–174 Chelnokov IN (1980) On the theory of the gyropendulum. J Appl Math Mech 44:703–708 D’Eleuterio GMT, Hughes PC (1983) Dynamics of gyro-elastic continua. In: 24th structures, structural dynamics and materials conference. 02 May 1983–04 May 1983, Lake Tahoe, NV, USA https://doi.org/10.2514/6.1983-826 D’Eleuterio GMT, Hughes PC (1987) Dynamics of gyroelastic spacecraft. J Guid Control Dyn 10:401–405 Foucault JBL (1851) Démonstration physique du mouvement de rotation de la terre au moyen du pendule. Comptes Rendus des Séances de L’Académie des Sciences 32:135–138 Garau M, Carta G, Nieves MJ, Jones IS, Movchan NV, Movchan AB (2018) Interfacial waveforms in chiral lattices with gyroscopic spinners. Proc R Soc Lond A 474:20180132 Goldstein H, Safko J, Poole C (1951) Classical mechanics. Addison-Wesley, San Francisco, Boston, New York Gray A (1918) A treatise on gyrostatics and rotational motion: theory and applications. Macmillan and Co. Ltd, London Hibbins AP, Evans BR, Sambles JR (2005) Experimental verification of designer surface plasmons. Science 308:670–672 Ishlinskii AY (1963) Mechanics of gyroscopic systems. Izdatel’stvo Akademii Nauk SSSR, Moscow Ishlinskii AY (1976) Orientation, gyroscopes, and inertial navigation. Nauka, Moscow Jaggard DL, Mickelson AR, Papas CH (1979) On electromagnetic waves in chiral media. Appl Phys 18:211–216 Jones IS, Movchan NV, Movchan AB (2020) Two-dimensional waves in a chiral elastic chain: dynamic green’s matrices and localised defect modes. Quart J Mech Appl Math 73:305–328 Jones IS, Movchan NV, Movchan AB (2022) Chiral waves in structured elastic systems: dynamics of a meta-waveguide. Quart J Mech Appl Math 75:63–89 Kelvin L (1894) The Molecular Tactics of a Crystal. Clarendon Press, Oxford Kelvin L, Tait PG (1967) Treatise on natural philosophy. Clarendon Press, Oxford Kittel C (2004) Introduction to solid state physics, 8th edn. Wiley, New York Lorentz HA (1906) Versuch einer Theorie der Electrischen und Optischen Erscheinungen in Bewegten Körpern. B. G. Teubner, Leipzig Moore JE (2010) The birth of topological insulators. Nature 464:194–198 Nieves MJ, Mishuris GS, Slepyan LI (2016) Analysis of dynamic damage propagation in discrete beam structures. Int J Solids Struct 97:699–713 Nieves MJ, Mishuris GS, Slepyan LI (2017) Transient wave in a transformable periodic flexural structure. Int J Solids Struct 112:185–208 Nieves MJ, Carta G, Jones IS, Movchan AB, Movchan NV (2018) Vibrations and elastic waves in chiral multi-structures. J Mech Phys Solids 121:387–408 Pendry JB, Martin-Moreno L, Garcia-Vidal FJ (2004) Mimicking surface plasmons with structured surfaces. Science 305:847–848 Pitman GR, Trimmer JD (1962) Inertial guidance. Am J Phys 30:937–937 Rechtsman MC, Zeuner JM, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A (2013) Photonic Floquet topological insulators. Nature 496:196–200 Roitenberg IN (1975) Gyroscopes. Nauka, Moscow

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Chapter 12

Controlling the Structure and Properties of Metal- and Polymer-Based Composites Fabricated by Combined 3D Methods A. G. Knyazeva, A. V. Panin, M. A. Anisimova, D. G. Buslovich, M. S. Kazachenok, and S. V. Panin Abstract The chapter includes three parts. The first one presents the research results on the microstructure and properties of a metal-matrix composite fabricated by the fusion of the Ti–6Al–4V wire and the TiC powder using electron beam additive manufacturing. The TiCx /Ti–6Al–4V composites were characterized by the uniform distribution of TiC globular eutectic particles in the titanium matrix. The segregation of the TiC eutectic phase particles along the boundaries of primary β grains caused reducing their dimensions with rising the TiC volume fraction. In the TiC8% /Ti– 6Al–4V and TiC20% /Ti–6Al–4V samples, primary β-phase grain sizes ranged from 30 to 100 μm. Inside them, martensitic α-phase plates were observed in addition to the TiC eutectic phase particles, which distribution density rose as the TiC volume fraction increased. The main phases in the TiCx /Ti–6Al–4V composites were the α-Ti, β-Ti and TiC ones. The β-Ti volume fraction varied within 3–5% regardless of the TiC contents. According to the X-ray diffraction analysis data, rising the TiC volume fraction was accompanied by increasing the lattice parameters of the α-Ti solid solution due to the presence of carbon atoms. At the TiC contents less than 10%, the levels of residual compressive stresses varied in the range from 0.7 to 0.9 GPa, weakly depending on its volume fraction. In the TiC20% /Ti–6Al–4V sample, the valued of residual compressive stresses was 1.5 GPa. Enhancing the TiC volume fraction in the TiCx/Ti–6Al–4V composites caused rising the microhardness values of both matrix and eutectic particles. Their maximum levels values (6400 and 8400 MPa, respectively) were found in the TiC20% /Ti–6Al–4V sample. The TiCx /Ti– 6Al–4V composites were also characterized by the greater tensile strength values but lower ductility compared to those of the Ti–6Al–4V alloy sample, fabricated by the same EBAM method. The TiC5% /Ti–6Al–4V sample possessed the maximum ultimate tensile strength of 1040 MPa. At the TiC volume fractions of 8% and more, the TiCx /Ti–6Al–4V composites experienced almost no plastic strains and brittle fracture occurred when applied stresses exceeded their yield points. The second part reports patterns of the structure formation and their effect on the tribological properties of A. G. Knyazeva (B) · A. V. Panin · M. A. Anisimova · D. G. Buslovich · M. S. Kazachenok · S. V. Panin Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_12

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3D-printed composites based on polyetheretherketone (PEEK) filled with nanoparticles of hydroxyapatite (HA) and polytetrafluoroethylene (PTFE). Compared with neat PEEK, its simultaneous loading with HA and PTFE deteriorated the composite structure to some extent. However, the wear rate level was greatly reduced and microabrasive damages to both steel and ceramic counterparts were eliminated by facilitating the transfer film formation. In addition to the self-lubricating effect of the formed composite structure, another (probable) reason for such a protection of the steel counterpart was the shielding effect of a transfer film from the standpoint of suppressing tribological oxidative processes during its interaction with PEEK. The slight lowering of the physical and mechanical properties of the composite fabricated by 3D printing, compared with hot-compressed one, was associated with the specifics of the additive manufacturing process. In this case, the interlayer adhesion had been reduced and the complete internal space filling had not been provided during the layer-by-layer formation of the composite macrostructure due to the decrease in the melt flow rate after loading PEEK with HA nanoparticles. Finally, the influence of forming intermediate phases between a matrix and inclusions on the evolution of the functional properties of composites is shown. The methods of micromechanics and the reactive diffusion theory have been applied for assessing changes in the functional properties of both Fe- and Ti–Al–C composites during their synthesis.

Dedicated to Professor Igor Sevostianov The reported investigations were possible due to the collaboration of the authors in the project preparation for the competition of the Ministry of Science and Higher Education of the Russian Federation (P 220 No. 2020-220-08-6706 “Control of the structure and residual stresses of composites for high-temperature applications obtained by combined 3D technologies”, in which Professor Igor Sevostianov acted as a leading scientist. The project received support from the Russian Government and it was assumed that Professor Sevostianov would head a laboratory, created for carrying out world-class investigations on the mentioned topic. Moreover, his fruitful scientific collaboration with Dr. Anna Knyazeva, which had begun much earlier, enabled to identify priority research areas, discuss the obtained results, and make plans for the future. The vast experience of Professor Sevostianov in the field of micromechanics of materials, including those fabricated by additive manufacturing, testified that functional properties of composites for high-temperature applications are determined by typical features of their structure and residual stresses (Azarmi and Sevostianov 2019; Sevostianov and Kachanov 2015; Sevostianov et al. 2018; Eshkabilov et al. 2021; Mishurova et al. 2020). Within the project framework, it was supposed to study both metal- and polymer-based ones. Unfortunately, the death of Professor Sevostianov made it impossible to implement these ideas. Nevertheless, the authors tried to present some patterns of the structure formation of such composites and its effect on their properties. In particular, the mutual and often competing influence of both technological and structural factors was considered. Also, a theoretical

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approach was developed for analyzing related processes occurring upon additive manufacturing procedures.

12.1 Introduction Composites possess unique properties that are not typical of conventional materials. One kind of the composite includes a metal matrix (Ti, Al, Fe, Ni, Cu, Mg, Ni, Co and their alloys) reinforced with high-strength/dispersed refractory fibers/particles (SiC, Al2 O3 , TiC, TiB2 , B4 C, etc.), not soluble in the base material. Some other types, so called non-metallic ones, are based on polymer, carbon or ceramic matrices loaded with reinforcing fillers such as glass, carbon, boron, organic fibers, metal wires and dispersed particles of different nature and ratios (Ashby 2011). Functional properties of the composites depend on their composition, structure, interaction between matrices and fillers at the interfaces, as well as many other factors. Advanced additive manufacturing procedures involve loading fabricated composite products with reinforcing particles in order to change their structure and, respectively, improve mechanical characteristics. Predicting these parameters is not an easy task. Typically, such properties are analyzed by performing mechanical tests, while both integral characteristics of a composite as a whole and data on individual phases are necessary for computer simulation of the material behavior. In general cases, the results of theoretical investigations devoted to the study of the functional properties of the composites can be divided into three groups (Dvorak 2012; Yin and Zhao 2018; Markov and Preziosi 2001; Kachanov and Sevostianov 2018), based on the following assumptions: • about effective inclusions (T. Mori, K. Tanaka, etc.); • about an effective matrix (K. Wakashima, M. Otsuka, S. Umekawa; C. Budiansky, etc.); • about three phases (A. Riccardi, F. Montheillet). These approaches are very approximate, because real properties of interfacial layers are not constant. They can both vary within ranges typical for the formed phases and differ from them greatly. The most significant achievements in this field have been reported by some current research teams led by Marcus Lasar (Germany, Technological University, Darmstadt), Gérard A. Maugin (France, Pierre and Marie Curie University, Paris), Elias C. Aifantis (USA, Michigan Technological University, Michigan), and X.-L. Gao (USA, University of Texas, Dallas). These scientists specialize in non-classical models of continuous media, non-local and gradient theories of elasticity and plasticity, and simulation of media with defect fields (dislocations, disclinations). Reviews of some major results can be found, for example, in Srinivasa and Reddy (2017), Shaat et al. (2020), Fafalis et al. (2012), Aifantis (2021). Another challenge inherent in additive manufacturing procedures is reliable predictions of forming structures and residual stresses in the face of a limited amount of data on both process and post-treatment patterns, or even in their absence. Many

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authors pay a lot of attention to this issue, using commercial software packages and models of different complexity levels for computer simulation (Rong et al. 2018; Bartlett and Li 2019; Vora et al. 2015; Mishurova et al. 2017; Chen et al. 2022; Tamanna et al. 2022; Zou et al. 2022, etc.). However, no calculated and verified results have been reported yet that have considered all scales of occurring physical phenomena, as well as variations in their characteristic durations. Both physical phenomena, which determine the structure formation and mechanisms responsible for residual stresses, are different for metal- and polymer-based composites. Some ways of solving the mentioned issues were suggested by Professor Sevostianov. One of the proposed ideas is to design and fabricate multilevel composites by additive manufacturing methods. Despite the widespread use of the term “multilevel” and a fairly large number of publications in this area, not so many real results have been obtained. The production of the multilevel composites means purposeful alignment of their structure at several scale levels under non-stationary rapidly-changing conditions of additive manufacturing. In such ultra-fast processes, controlling of the composite properties is impossible without post-treatment, the joint implementation of which with additive manufacturing procedures implies the term “combined 3D methods”. Currently, it is precisely such combinations that are applied, beginning from trivial (but effective) annealing, or the same treatment under isostatic pressing conditions, and ending with blade processing aimed at reducing roughness, or even at giving the final (finishing) shapes and sizes of such products. This chapter includes three parts, partially filling some gaps in knowledge about the identified scientific areas: 1. The research results are reported on the microstructure and properties of a metalmatrix composite fabricated by fusion of the Ti–6Al–4V wire and the TiC powder using electron beam additive manufacturing. To date, a lot of papers devoted to the use of these materials have been published. In addition, 3D printers designed for printing with the Ti–6Al–4V powder are commercially produced. Despite this, the issues of the formation of such hierarchically organized composites remain open, and even more so, any industrial procedures of their fabrication have not been developed and implemented yet. 2. Patterns of the structure formation and their effect on the tribological properties of 3D-printed composites based on polyetheretherketone (PEEK) filled with nanoparticles of hydroxyapatite (HA) and polytetrafluoroethylene (PTFE). For antifriction applications, thermoplastic parts and 3D printed products need to be reinforced/loaded with hardening and solid lubricating fillers. This significantly modifies the molten filament laying process and introduces features into the subsequent solidification one, as well as transforms the formation of interfacial boundaries and the macrostructure homogeneity in the composite as a whole. 3. The influence of forming intermediate phases between a matrix and inclusions on the evolution of the functional properties of composites. This issue is relevant, since almost all published papers on the results of such investigations are based on data from structural studies of already fabricated samples. The relationship

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between both physical and chemical processes that determine the structure of the composites at different scale levels and their functional properties remains unclear.

12.2 The Ti–6Al–4V-Based Metal-Matrix Composites Built by Wire-Feed Electron Beam Additive Manufacturing 12.2.1 Problem Definition In the last few decades, metal matrix composites based on the Ti–6Al–4V alloy reinforced with high-strength ceramic particles (such as TiC, TiB and SiC) have attracted more and more attention due to the wide prospects of their use in the aerospace industry (Zhu et al. 1999). The combination of the viscous titanium matrix with hard and rigid ceramic particles provides improved both mechanical and tribological properties of such composites, which are retained at high temperatures. In this case, titanium carbide is one of the most preferred reinforcing materials due to its great stability and compatibility with the matrix, close thermal expansion coefficients, as well as high both hardness and Young’s modulus levels (Kim et al. 2002). Conventional methods for producing metal matrix composites are hot compression and sintering of powder mixtures. Currently, there is also a huge interest in their fabrication by additive manufacturing methods that enable to build complex-shaped volumetric products according to a three-dimensional computer model by layer-bylayer material deposition (Wang et al. 2008). The most common one for obtaining the TiC/Ti–6Al–4V composites is the laser melting deposition (LMD) method (Pouzet et al. 2016). Its disadvantage is great residual stresses due to high cooling rates after powder melting (Attar et al. 2017). More promising procedures for such purposes are based on electron beam melting (EBM) of powder mixtures (Katz-Demyanetz et al. 2019). Unlike the LMD method, the electron beam melting process takes place in a vacuum. This makes it possible to limit thermal stresses in parts due to elevated processing temperatures, as well as to form more suitable microstructures of titanium alloys and their composites, which are characterized by high chemical activity at elevated temperatures. To date, the effect of the additive manufacturing parameters on patterns of the formation of the microstructure and phase composition of the TiC/Ti–6Al–4V composites has been studied in detail (Wei et al. 2011; Liu and Shin 2017; Ochonogor et al. 2012). In turn, they also depend on hardness (Mahamood et al. 2013), strength (Wang et al. 2018; Li et al. 2017) and tribological properties (Candel et al. 2010). It has been found that there is an optimal content of TiC particles that provides the maximum increase in the strength characteristics of the composites. Thus, the ultimate tensile strength of the TiC(5 vol%)/Ti6Al4V composite reaches 1250 MPa, but its elongation significantly decreases down to 1.5% in this case. Further increase in the TiC volume fraction gradually reduces these parameters, because unmelted TiC

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particles and their large dendritic phases contribute to the formation of pores and cracks. Wire-feed electron beam additive manufacturing (EBAM) procedures are of particular interest, allowing to fuse Ti–6Al–4V wires with TiC powders (Wang et al. 2007a, b, c; Panin et al. 2021). The key EBAM advantage is almost 100% efficiency in the use of consumables, as well as extremely high layer-by-layer deposition rates (Ding et al. 2015). In turn, the main drawback of the EBAM samples built from the Ti–6Al–4V alloy is their low strength because of large sizes of primary β-phase grains. However, they are successfully refined by loading them with high-strength ceramic particles. At the same time, mechanisms of the formation of the microstructure and phase composition of such TiC/Ti–6Al–4V composites still remain unclear. Also, the optimal process parameters and the TiC powder volume fraction have not been determined yet, which would make it possible to obtain well-bonded composites with improved strength characteristics. The purpose of this part of the chapter is to study the effect of the volume fraction of TiC particles on the microstructure and mechanical properties of the TiC/Ti–6Al–4V composites fabricated by electron beam fusion of the Ti–6Al–4V wire and the TiC powder.

12.2.2 Experimental Samples (25 mm long, 25 mm wide and 70 mm high) of the TiCx /Ti–6Al–4V composites were built using a “6E400” modified welding machine (NPK TETA LLC, Tomsk, Russia) equipped with a plasma-cathode electron-beam gun. The Ti–6Al–4V alloy wire with a diameter of 1.6 mm and the TiC powder (commercially available) with an average particle size of 1 μm (Fig. 12.1a) were used as feedstock. For enhancing the flowability of the fine TiC powder and its more even distribution in a molten pool during the EBAM process, the Ti–6Al–4V powder with an average particle diameter within 100 μm (Fig. 12.1b) had been additionally mixed with TiC particles using an “AGO-2” planetary ball mill before the composite fabrication. The obtained TiCx /Ti–6Al–4V flake-shaped powder mixture is shown in Fig. 12.1c, d. The EBAM fabrication of the TiCx /Ti–6Al–4V composites were performed at an accelerating voltage of 30 kV. The distance between the electron-beam gun and the Ti–6Al–4V substrates (base plates with dimensions of 150 × 150 × 10 mm) was 630 mm. The electron-beam current was gradually decreased from 24 down to 17 mA during the EBAM process. The Ti–6Al–4V wire was fed from the front side at an angle of 35° to the substrate surface at a feed speed of 2 m/min. The substrate was moved at a speed of 5 mm/s. Distances between beads of the same layer were ~3 mm, while their heights were about 3.2 mm. After completing the EBAM process, the samples were cut out from the substrates using an electric discharge machine. In the composites, the TiC volume fractions varied from 1 up to 20% by weight, which was reflected in the x subscript designation of the TiCx /Ti–6Al–4V composites. For metallographic studies, specimens were prepared by ion milling of rectangular plates using an “Ion Slicer EM-09100IS” setup (JEOL, Japan). Surfaces of the ground

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Fig. 12.1 SEM-micrographs of the TiC (a) and Ti–6Al–4V (b) powders, as well as the TiCx /Ti– 6Al–4V flake-shaped powder mixture (c, d)

and polished samples were preliminary etched with Kroll’s reagent (2% HF, 2% HNO3 , and 96% H2 O). The composite microstructures were investigated using a “JEM 2100” transmission electron microscope (JEOL, Japan) equipped with an “Inca ACT-X” energy-dispersive X-ray spectroscopy (EDX) detector (Oxford Instruments, USA). Micrographs were captured with both secondary and backscattered electrons for a clear demonstration of the lamellar α-phase morphology of the (α + β) dualphase alloy. X-ray diffraction analysis of the samples was performed using a “Shimadzu XRD6000” diffractometer (Shimadzu, Japan) with Bragg–Brentano focusing in the (θ –2θ ) scanning mode (θ was the angle between the beam and the reflecting atomic plane). X-ray diffraction patterns were registered using CuKα radiation at a wavelength of 1.540598 Å and a scan speed of 1.2°/min. Residual stresses were assessed using the sin2 ψ technique (Noyan and Cohen 1987). The Vickers hardness tests were carried out at the lateral sample surfaces at a load of 50 g and an exposure duration of 10 s. Uniaxial quasi-static tension tests were conducted using an “INSTRON 5582” electromechanical testing machine (Instron Gmbh, Germany) at room temperature and a loading rate of 0.3 mm/min. Before the mechanical tests, dumb-bell specimens with a gage section of 3 × 1 × 20 mm in size had been cut out by the electric discharge machine.

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12.2.3 Results and Discussion The microstructure and phase composition. The microstructure of the TiC1% /Ti– 6Al–4V sample consisted of equiaxed primary β-phase grains with an average size of 500 μm (Fig. 12.2a). Inside such grains, martensite α -phase plates were observed, grouped into packets with a transverse size of 20–30 μm (Fig. 12.2b). The transverse size of individual plates did not exceed 1 μm. Obviously, the presence of the carbide phase had not affected the melt pool cooling rate in the EBAM process and, respectively, had not determined the dimensions of such plates. However, globular particles of the TiC eutectic phase with an average size of about 3 μm were isolated both along the boundaries and inside the primary β-phase grains during the melt pool solidification due to the low solubility of carbon in titanium (in the temperature range of 1233–1877 K, the limiting solubility had varied from 1.16 to 0.55%). The formation of the TiC eutectic phase indicated the complete dissolution of TiC particles in the Ti–6Al–4V matrix. An increase in the TiC volume fraction had not only suppressed growing columnar primary β-phase grains, but also caused lowering of their dimensions in the TiCx /Ti– 6Al–4V composites. It was concluded from the analysis of Fig. 12.3 that sizes of the primary β-phase grains in the TiC5% /Ti–6Al–4V sample varied from 200 up to 350 μm. Obviously, TiC particles had prevented growing the primary β-phase grains upon the melt pool solidification. Figure 12.3b clearly reflected chains of globular particles of the TiC eutectic phase, segregated both along the boundaries and inside the primary β-phase grains. Nevertheless, both sizes and volume fractions of the TiC eutectic phase particles increased significantly. In the TiC8% /Ti–6Al–4V sample, the microstructure had been formed with smaller equiaxed primary β-phase grains, sizes of which varied from 30 up to 100 μm (Fig. 12.4a). Such a refinement was also associated with the presence of the TiC eutectic phase particles, precipitated along the boundaries of the primary β-phase grains (Fig. 12.4b).

Fig. 12.2 SEM micrographs of the microstructure of the TiC1% /Ti–6Al–4V sample

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Fig. 12.3 SEM micrographs of the microstructure of the TiC5% /Ti–6Al–4V sample

Fig. 12.4 SEM micrographs of the microstructure of the TiC8wt% /Ti–6Al–4V sample

Finally, the TiC volume fraction increase up to 20% did not result in a further decrease in the average size of primary β-phase grains in the TiC20% /Ti–6Al–4V sample (Fig. 12.5a). In addition to the TiC eutectic phase particles, both TiC dendrites and agglomerates of primary TiC particles up to 30 μm in their size were found inside such grains and along their boundaries (Fig. 12.5b). Figure 12.6 shows the results of EDS analysis at the TiC20% /Ti–6Al–4V sample surface. According to these data, the carbon content in the TiC eutectic phase particles was 5%, which corresponded to the Ti–C state diagram. The results of X-ray diffraction analysis showed that the TiCx /Ti–6Al–4V composites consisted of the α-Ti, β-Ti and TiC phases (Fig. 12.7). The β-Ti volume fraction varied within 3–5%, regardless of the TiC phase contents. It should be noted that their similarly low levels were characteristic of the Ti–6Al–4V alloy samples fabricated by additive manufacturing procedures due to a low vanadium content in the β phase as a β-stabilizing element (Liu and Shin 2019). Another reason for the low β-Ti volume fraction was the presence of carbon. It was an α-phase stabilizer that increased the α-Ti crystal lattice stability but compensated for the role of vanadium as a β-stabilizer in the β-Ti lattice. The presence of carbon atoms that occupied octahedral pores in the α-Ti solid solution was confirmed by the higher c/a

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Fig. 12.5 SEM micrographs of the microstructure of the TiC20% /Ti–6Al–4V sample

Fig. 12.6 A SEM micrograph and EDX maps of titanium, vanadium, aluminium and carbon in the TiC20% /Ti–6Al–4V sample. The EDX maps correspond to the SEM micrograph region marked by the red border rectangle

axial ratio values compared to the published data for the hcp titanium lattice (c/a = 0.15870 nm). As followed from the data summarized in Table 12.1, the c/a values continuously enhanced with rising the TiC volume fraction in the TiCx /Ti–6Al–4V composites. This fact was an indirect confirmation of the continuous increase in the carbon content in the α-Ti solid solution with rising the TiC volume fraction in the composites. According to Table 12.1, enhancing the TiC volume fraction had caused rising residual compressive stresses in the TiCx /Ti–6Al–4V composites. When its content had been less than 10%, this had resulted in a complete dissolution of titanium carbide particles leading to the formation of the TiC eutectic phase, which had not affected the molten pool solidification process. In this case, such compressive stresses were mainly associated with temperature gradients that had risen between sequentially deposited layers (Strantza et al. 2021; Sikan et al. 2021). When the TiC volume fraction had been increased up to 20%, greater compressive stresses were found in

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Fig. 12.7 X-ray diffraction patterns of the TiC1% /Ti–6Al–4V (a), TiC5% /Ti–6Al–4V (b), TiC8% /Ti–6Al–4V (c) and TiC20% /Ti–6Al–4V (d) samples Table 12.1 The lattice parameters of the α-Ti, β-Ti and TiC phases, as well as residual stresses in the TiCx /Ti–6Al–4V composites Sample

Phase

Lattice parameters, nm a

c

TiC1% /Ti–6Al–4V

α-Ti

0.29399

0.46856

TiC5% /Ti–6Al–4V

TiC8% /Ti–6Al–4V

TiC20% /Ti–6Al–4V

c/a

σ, GPa

1.5938

−0.7

β-Ti

0.32307

–

–

TiC

0.42957

–

–

α-Ti

0.29378

0.46840

1.5944

β-Ti

0.32183

–

–

TiC

0.43044

–

–

α-Ti

0.29345

0.46811

1.5952

β-Ti

0.32313

–

–

TiC

0.43001

–

–

α-Ti

0.29331

0.46885

1.5985

β-Ti

0.32296

–

–

TiC

0.42955

–

–

−0.8

−0.9

−1.5

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Fig. 12.8 Dependences of the microhardness levels of the α-Ti matrix (1, black dots) and granular TiC eutectic phase particles (2, red dots) on the TiC volume fraction in the TiCx /Ti–6Al–4V composites

the TiC20% /Ti–6Al–4V sample due to the difference in thermal expansion of the metal matrix and carbide particles (Aghdam and Morsali 2017). Mechanical properties. Rising the TiC volume fraction in the TiCx /Ti–6Al–4V composites caused enhancing the carbon contents in both α-Ti solid solution and TiC eutectic phase. As a result, their microhardness also increased. As the TiC volume fraction rose from 1 up to 8%, the microhardness levels of the matrix and TiC granular particles enhanced from 4600 up to 6400 Mpa and from 6500 up to 8400 Mpa, respectively (Fig. 12.8). It should be noted that the microhardness values of the TiC eutectic phase particles precipitated in the studied composites were significantly lower than those (20–30 Gpa (Wen et al. 2019)) of TiC inclusions in similar TiC/Ti– 6Al–4V metal-matrix samples obtained by conventional hot compression or sintering methods. Lowering sizes of primary β grains, as well as rising both dimensions and distribution density of the TiC eutectic phase particles in the TiCx /Ti–6Al–4V composites, caused increasing their ultimate tensile strength levels but decreasing the elongation values. As follows from Fig. 12.9, these parameters of the Ti–6Al–4V alloy were 890 MPa and 6%, respectively, for samples obtained by the wire-feed EBAM method (Panin et al. 2020). Such low strength properties were due to their specific microstructure, consisting of large primary β-phase grains, which included big martensite packed crystals and interlayers of the relatively soft and plastic residual β-phase along the boundaries of α-phase plates. At the same ductility, these characteristics were significantly higher for the TiC1% /Ti–6Al–4V sample. Its yield point and the ultimate tensile strength were 900 and 1010 MPa, respectively. With enhancing the TiC volume fraction, the strength characteristics of the TiCx /Ti–6Al–4V composites slightly increased but their elongation decreased greatly. These results were in good agreement with the data published in Candel et al. (2010).

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Fig. 12.9 The engineering stress–strain diagrams for the Ti–6Al–4V alloy, fabricated by the wire-feed EBAM method (Panin et al. 2020) (1), as well as the TiC1% /Ti–6Al–4V (2), TiC5% /Ti–6Al–4V (3) and TiC8% /Ti–6Al–4V (4) composites

12.2.4 Conclusions to Sect. 12.2 The implemented EBAM procedure, using the Ti–6Al–4V wire and the TiC powder as a feedstock, enabled to fabricate the TiCx /Ti–6Al–4V composites, characterized by the uniform distribution of TiC globular eutectic particles in the titanium matrix. The segregation of the TiC eutectic phase particles along the boundaries of primary β grains caused reducing their dimensions with rising the TiC volume fraction. In the TiC8% /Ti–6Al–4V and TiC20% /Ti–6Al–4V samples, primary β-phase grain sizes ranged from 30 to 100 μm. Inside them, martensitic α-phase plates were observed in addition to the TiC eutectic phase particles, which distribution density rose as the TiC volume fraction increased. The main phases in the TiCx /Ti–6Al–4V composites were the α-Ti, β-Ti and TiC ones. The β-Ti volume fraction varied within 3–5% regardless of the TiC contents. According to the X-ray diffraction analysis data, rising the TiC volume fraction was accompanied by increasing the lattice parameters of the α-Ti solid solution due to the presence of carbon atoms. At the TiC contents less than 10%, the levels of residual compressive stresses varied in the range from 0.7 to 0.9 GPa, weakly depending on its volume fraction. In the TiC20% /Ti–6Al–4V sample, the value of residual compressive stresses was 1.5 GPa. Enhancing the TiC volume fraction in the TiCx/Ti–6Al–4V composites caused rising the microhardness values of both matrix and eutectic particles. Their maximum levels values (6400 and 8400 MPa, respectively) were found in the TiC20% /Ti–6Al– 4V sample. The TiCx /Ti–6Al–4V composites were also characterized by greater tensile strength values but lower ductility compared to those of the Ti–6Al–4V alloy sample, fabricated by the same EBAM method. The TiC5% /Ti–6Al–4V sample possessed the maximum ultimate tensile strength of 1040 MPa. At the TiC volume fractions of 8%

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and more, the TiCx /Ti–6Al–4V composites experienced almost no plastic strains and brittle fracture occurred when applied stresses exceeded their yield points.

12.3 The Structure and Properties of 3D Printed PEEK-Based Composites for Antifriction Applications 12.3.1 Problem Definition PEEK is a polymer whose structure consists of benzene or arene rings connected by both ether and ketone bridges. Great strength and low flexibility of the bonds of polymeric macromolecules determine its improved mechanical properties and a high melting point of 341 °C. Another important feature of this polymer is its chemical inertness to most substances. Any interactions could occur only with strong oxidizing agents at high temperatures. This pattern makes PEEK bioinert that enables to use it for manufacturing endoprostheses and various medical fasteners (staples, screws, etc.). Modification of PEEK causes a number of undoubted advantages over other prosthetic materials, based on metals and ceramics (Paglia et al. 2022; Fabris et al. 2021). In scientific papers devoted to PEEK-based composites for medical applications, its filling with HA particles is often mentioned. Yu et al. (2005) studied such composites loaded with HA in contents from 10 up to 40%. Both growth and crystallization kinetics were analyzed taking into account the microstructure and morphology of the samples. Similar research results were published on other polymers with resembling compositions. For example, Pratik Roy and R.R.N. Sailaja investigated some properties of polyarietherketone (PAEK) for medical applications (Roy and Sailaja 2015). The research aim was to analyze crystallinity, structure, thermal stability, toxicity, etc. A number of published papers are devoted to the deposition of PEEK coatings on titanium-based prostheses. Lim et al. (2019) fabricated a porous structure from the Ti–6Al–4V alloy by additive manufacturing. Then, it was coated with PEEK for using as a spine link. Vaezi and Yang (2015) published a review of recent research results on the manufacture of porous PEEK-based parts for medical applications. The authors noted that a plug can be formed upon 3D printing at temperatures below 400 °C, which blocks a nozzle and interrupts the process. In addition, much attention was paid to the porosity of the formed structures, the comparison of their properties, and the crystallinity degree. Lee et al. (2019) built a composite by additive manufacturing from a preliminarily prepared feedstock that included a PEEK monomer with epoxy groups and a PEEK polymer. In order to crosslink the epoxy groups, they were heated up to 380 °C. The authors showed that the elastic modulus of the samples decreased to 62% of that for the initial PEEK. Designing composites for medical applications imposes a number of restrictions on the types of loaded fillers. In particular, the use of glass fibers is unreasonable in

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biodegradable polylactide (Varsavas and Kaynak 2018; Akindoyo et al. 2020). On the other hand, HA is an important component, being a stimulator for growth of bone cells. Manufacturing composites from PEEK and HA mixtures is also an urgent task, since it enables to provide their physical and mechanical properties (primarily, the elastic modulus) close to those for bone tissues. However, such characteristics are determined by the composite structures of a bone as a whole (including trabecular and tubular ones). The natural structures vary in their density and composition in different directions. To achieve these uneven properties, an approach should be developed that allows producing architectonics, resembling the real structures of bone tissues. A promising way to solve this issue is 3D printing. At the same time, the selection of an acceptable feedstock composition is one of the key challenges. Such composites should be suitable for additive manufacturing and meet all the requirements for biocompatibility, as well as not be inferior in terms of strength characteristics to the human bone tissues. One of the most relevant areas in implantology is manufacturing of joint prostheses. In addition to improved strength and biocompatibility, their components must have low friction coefficients but high wear resistance. The solution to such a problem can be achieved by designing three- (or more) component composites, when each of the components contributes to achieving individual required properties. Thus, the research task was to design a three-component PEEK-based composite filled with HA and PTFE particles (to provide a solid lubricating effect), which is suitable for manufacturing by the Fused Deposition Modeling (FDM) method. For this purpose, it was necessary to study the structure, as well as both mechanical and tribological characteristics of such samples.

12.3.2 Experimental The matrix material was the “450PF” PEEK powder (Vitrex, UK) with an average particle size of 50 μm. The solid lubricant fillers were the “F-4–PN20” PTFE powder with particle sizes of 6–20 μm (Ruflon LLC, Russia) and HA nanoparticles with diameters of 20–30 nm, obtained by mechanosynthesis at the ISSCM SB RAS. The PEEK-based composites loaded with 10 wt% PTFE, as well as 0.3 and 7 wt% HA nanoparticles were studied. The components were mixed with a “RR/TSMP” test sample injection molding machine (Ray-Ran Test Equipment Ltd, Nuneaton, UK) at a temperature of 400 °C and a pressure of 3 atm (Yang and Zhiwei 2009; Párizs et al. 2022). Then, the extrudate was mechanically milled to obtain granules with a length of 3–4 mm and a diameter of ~2 mm. Blanks in the form of round plates were fabricated in two ways: (a) by hot compression (HC) at a pressure of 15 MPa and a temperature of 400 °C (the cooling rate was 2 °C/min); (b) by the FDM method from granules with the “ArmPrint-2” laboratory (home-made) 3D printer (NR TPU, Tomsk, Russia). The 3D printer was designed for using 2–5 mm granules and equipped with a single-screw micro-extruder. Its nozzle diameter was 0.4 mm. Temperatures of the bed, as well as

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the upper and lower regions for feeding the filament (granules) were 175, 300 and 380 °C, respectively. The thickness of each deposited layer was 0.3 mm. The printer head was moved at a speed of 20 mm/s. Shore D hardness was determined with an “Instron 902” tester according to ASTM D 2240. Mechanical properties were evaluated by tensile tests of “dogbone” shaped specimens using an “Instron 5582” electromechanical testing machine according to ISO 178:2010. Wear tests were carried out using a “CSEM CH-2000 tribometer” (CSEM, Switzerland) in the dry friction mode according to the “pin-on-disk” scheme (ASTM G99) at a load of 10 N and a sliding speed of 0.3 m/s. The diameter of the counterface from the bearing steel and Al2 O3 ceramics was 6 mm. The test distance was 3 km, the rotation trajectory radius was 10 mm; the circular speed was 286 rpm. The wear track surfaces were studied using a “Neophot 2” optical microscope (Carl Zeiss Jenna, Germany). The structures of the composites were observed on the cleavage surfaces of the samples mechanically fractured after exposure to liquid nitrogen. For this purpose, a “LEO EVO 50” scanning electron microscope (Carl Zeiss, Germany) was used at an accelerating voltage of 20 kV.

12.3.3 Results and Discussion Table 12.2 and Fig. 12.10 present the mechanical characteristics of the studied composites. After loading with PTFE particles, the elastic modulus slightly decreased (by ~10%). On the contrary, it increased by ~10% after loading with 0.3 wt% HA and by ~30% at its content of 7 wt%. The addition of 0.3 wt% HA caused negligible rising of both ultimate tensile strength and elongation at break values. The decrease in the elongation at break levels of all investigated specimens indicated that loading with these fillers deteriorated the composite structures (shown below). Table 12.2 The physical and mechanical characteristics of the PEEK-based composites loaded with HA and PTFE Density ρ, g/cm3

No.

Filler content, wt%

Shore D hardness

Elastic modulus E, MPa

Ultimate tensile strength σ UTS , MPa

Elongation at break ε, %

1

–

1.308

80.1 ± 1.2

2840 ± 273

106.9 ± 4.7

25.6 ± 7.2

2

0.3%HA

1.304

78.7 ± 0.4

3076 ± 79

108.5 ± 3.1

18.3 ± 2.7

3

7%HA

1.348

80.5 ± 0.8

3248 ± 144

102.5 ± 2.4

8.1 ± 1.5

4

10%PTFE

1.324

77.3 ± 0.2

2620 ± 158

83.9 ± 2.4

4.4 ± 0.7

5

0.3%HA + 10%PTFE

1.344

77.9 ± 0.2

2688 ± 104

85.2 ± 2.0

9.7 ± 0.9

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Fig. 12.10 The stress–strain diagram for neat PEEK (1) and the “PEEK + 0.3%HA” (2); “PEEK + 7%HA” (3); “PEEK + 10%PTFE” (4); and “PEEK + 0.3%HA + 10%PTFE” (5) composites

SEM-micrographs of the structures of the studied composites are shown in Fig. 12.11. After loading with HA nanoparticles, the size of the characteristic structural elements became smaller than that in neat PEEK. In addition, a more uniform structure was observed (Fig. 12.11c, d), which ensured that the elongation at break value was at a level comparable with neat PEEK. After loading 1 wt% PTFE (Fig. 12.11e, f), pores and microcracks were found at the interface between the matrix and the filler. PTFE particles had poor adhesion to the polymer matrix, lowering the mechanical properties (Table 12.2). In the “PEEK + 0.3%HA + 10%PTFE” composite (Fig. 12.11g), the size of the characteristic structural elements was greater than that of the “PEEK + 10%PTFE” sample. PTFE particles were distributed quasiuniformly along the boundaries of the polymer structural elements (Fig. 12.11g). It was found at high magnification (Fig. 12.11f) that the HA nanofiller segregated and agglomerated in the regions of PTFE inclusions. Dependences of the friction coefficients on the test distance are presented in Fig. 12.12, while their average values for the studied materials are summarized in Table 12.3. The friction coefficient levels decreased significantly only in the case of loading with PTFE. For the “PEEK + 10 wt%PTFE” composite, its value was 0.17 ± 0.02, which was two times less than that for neat PEEK (μ = 0.34 ± 0.03) when tested on the metal counterface. On the ceramic one, the friction coefficient was three times less than for neat PEEK (μ = 0.09 ± 0.01 and 0.27 ± 0.02, respectively). Figure 12.13 shows data on wear rates for neat PEEK and the investigated composites. On the metal counterface, the minimum damage level was observed for the “PEEK + 10%PTFE” sample, which a wear rate value of 0.93 ± 0.07·10–6 mm3 /N·m decreased by 12.5 times compared to that for neat PEEK. Loading with 7 wt% HA reduced the wear rate by about 12 times on the metal counterface. On the ceramic one, this parameter decreased by about 9 times for the composite loaded with 7 wt% HA and 10 wt% PTFE, compared to that for neat PEEK.

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Fig. 12.11 The SEM-micrographs of the structures of neat PEEK (a, b) and the “PEEK + 0.3%HA” (c, d); “PEEK + 10%PTFE” (e, f); and “PEEK + 0.3% HA + 10%PTFE” (g, h) composites; magnification of ×1000 (a, c, e, g) and ×5000 (b, d, f, h)

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1

0.35

Friction coefficient

Fig. 12.12 Dependences of the friction coefficient versus test distance on the metal (a) and ceramic (b) counterfaces for neat PEEK (1) and the “PEEK + 0.3%HA” (2); “PEEK + 7%HA” (3); “PEEK + 10%PTFE” (4); and “PEEK + 0.3%HA + 10%PTFE” (5) composites

293

2

0.30 0.25

3

0.20

4 5

0.15 0.10 0.05

0.0

0.5

1.0

1.5

2.0

Distance, km

2.5

3.0

Friction coefficient

0.35

1

0.30 0.25 0.20

2

0.15

3

0.10

4 5

0.05 0.0

0.5

1.0

1.5

2.0

Distance, km

2.5

3.0

Table 12.3 The tribological properties of the PEEK-based composites with different HA and PTFE contents No.

Filler content, wt%

Friction coefficient μ

Wear rate, 10–6 mm3 /N·m

Metal counterpart

Ceramic counterpart

Metal counterpart

Ceramic counterpart

1

–

0.34 ± 0.03

0.27 ± 0.02

11.67 ± 1.00

3.00 ± 0.33

2

0.3%HA

0.30 ± 0.02

0.19 ± 0.02

7.33 ± 0.33

0.97 ± 0.10

3

7%HA

0.23 ± 0.02

0.12 ± 0.01

0.97 ± 0.07

0.37 ± 0.03

4

10%PTFE

0.17 ± 0.02

0.09 ± 0.01

0.93 ± 0.07

0.47 ± 0.07

5

0.3%HA + 10%PTFE

0.14 ± 0.02

0.08 ± 0.01

0.80 ± 0.07

0.23 ± 0.02

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Fig. 12.13 Wear rates for neat PEEK (1), as well as the “PEEK + 0.3%HA” (2); “PEEK + 7%HA” (3); “PEEK + 10%PTFE” (4); and “PEEK + 0.3%HA + 10%PTFE” (5) composites on the steel and ceramic counterfaces under the dry friction conditions

Figures 12.14 and 12.15 show the topography of the wear track surfaces on the samples and the counterfaces, as well as their profiles. After the tribological test of the “PEEK + 10%PTFE” composite, the metal counterface was worn out to a low extent (Fig. 12.14b). On the composite wear track surface, PTFE particles were quasi-uniformly distributed as fairly large inclusions (Fig. 12.14j). Nevertheless, a transfer film was observed on the metal counterpart surface (Fig. 12.14k), whose width was narrower than that on neat PEEK (Fig. 12.14b). The transfer film indicated the presence of a PTFE layer on the friction surface of the polymer composite, which significantly facilitated the sliding of the counterface. At the same time, the wear track surface roughness of 0.081 μm on the composite decreased by almost a factor of two compared to that of 0.156 μm on neat PEEK. After the tribological tests on the ceramic counterface, all the composites showed almost no wear damages (Fig. 12.15). The wear track surfaces on the composites were quite smooth, as evidenced by their roughness values of less than 0.08 compared to that of 0.131 μm on neat PEEK. In addition, thin transfer films were observed on the ceramic counterface due to the presence of PTFE layers on the friction surfaces of its containing composites. Thus, it was shown that the HA filling degree of 7 wt% was excessively high. The solid lubricating effect was achieved by loading with PTFE particles. The “PEEK + 0.3%HA + 10%PTFE” composite possessed satisfactory physical, mechanical and tribological properties. For this reason, such a composite was further fabricated by 3D-printing to compare the two methods.

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Fig. 12.14 The topography of the wear track surfaces on neat PEEK (a–c), as well as the “PEEK + 0.3%HA” (d–f); “PEEK + 7%HA” (g–i); “PEEK + 10%PTFE” (j–l); and “PEEK + 0.3%HA + 10%PTFE” (m–o) composites after the test distance of 3 km on the steel counterface

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Fig. 12.15 The topography of the wear track surfaces on neat PEEK (a–c), as well as the “PEEK + 0.3%HA” (d–f); “PEEK + 7%HA” (g–i); “PEEK + 10%PTFE” (j–l); and “PEEK + 0.3%HA + 10%PTFE” (m–o) composites after the test distance of 3 km on the ceramic counterface

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12.3.4 Comparison of the Characteristics of the “PEEK + 0.3%HA + 10%PTFE” Composites Fabricated by the HC and HDM Methods Table 12.4 and Fig. 12.16 present the mechanical properties of the “PEEK + 0.3%HA + 10%PTFE” composites fabricated by both HC and FDM methods. For the FDM sample, the ultimate tensile strength and elongation at break values were reduced by 15–20 and 5%, respectively. The mechanical properties of the composites were determined by the formed structure depending on the manufacturing methods (Fig. 12.17). The structure of the HC composite was slightly “loose” (Fig. 12.17a). In addition, no clear signs of Table 12.4 The physical and mechanical characteristics of neat PEEK and the “PEEK + 0.3%HA + 10%PTFE” composites No.

Material

Density ρ, g/cm3

Shore D hardness

Elastic modulus E, MPa

Ultimate tensile strength σ UTS , Mpa

Elongation at break ε, %

1

Neat PEEK

1.31

80.1 ± 1.2

2840 ± 273

106.9 ± 4.7

25.6 ± 7.2

2

PEEK + 0.3%HA + 10%PTFE (HC)

1.34

77.9 ± 0.2

2688 ± 104

85.2 ± 2.0

9.7 ± 0.9

3

PEEK + 0.3%HA + 10%PTFE (FDM)

1.31

75.7 ± 0.5

2460 ± 65

52.4 ± 3.0

4.4 ± 0.6

Fig. 12.16 The stress–strain diagrams of neat PEEK (1) and the “PEEK + 0.3%HA + 10%PTFE” composites fabricated by the HC (2) and FDM (3) methods

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particle cracking were observed. The FDM composite had a more uniform structure compared to the HC one. No pronounced signs of agglomeration of nanofillers were found, but an agglomeration of PTFE particles occurred, which deteriorated all physical and mechanical properties of this sample (Fig. 12.17b). This effect was associated with the FDM process (Challa et al. 2022). Temperature fluctuations upon 3D-printing (between the nozzle and the heating bed, as well as between the first and subsequent layers) resulted in delamination of some layers. Also, loading PEEK with HA nanoparticles and PTFE microparticles enabled to reduce the melt flow rate of the mixture, which in turn further complicated the material flow from the microextruder. Table 12.5 and Figs. 12.18 and 12.19 show the tribological properties of the composites on both metal and ceramic counterfaces under dry friction conditions. The friction coefficient decreased by a factor of three for the metal-polymer tribological contact, regardless of the composite manufacturing method (Fig. 12.18a). In the ceramic-polymer tribological contact, the friction coefficient was reduced by 3.3 times for the HC composite and by 2.7 times for the FDM one (Fig. 12.19b).

Fig. 12.17 The SEM-micrographs of the structures of the “PEEK + 0.3%HA + 10%PTFE” composites fabricated by the HC (a) and FDM (b) methods

Table 12.5 The tribological properties of neat PEEK and the “PEEK + 0.3HA + 10PTFE” composites Material

Friction coefficient μ

Wear rate, 10–6 mm3 /N·m

Metal counterpart

Ceramic counterpart

Metal counterpart

Ceramic counterpart

Neat PEEK

0.34 ± 0.03

0.27 ± 0.02

11.67 ± 1.00

3.00 ± 0.33

PEEK + 0.3%HA + 10%PTFE (HC)

0.14 ± 0.02

0.08 ± 0.01

0.80 ± 0.07

0.23 ± 0.02

PEEK + 0.3%HA + 10%PTFE (FDM)

0.11 ± 0.02

0.10 ± 0.01

1.23 ± 0.05

0.42 ± 0.08

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1

0.35

Friction coefficient

Fig. 12.18 The dependences of friction coefficient versus test distance on the metal (a) and ceramic (b) counterfaces for neat PEEK (1) and the “PEEK + 0.3%HA + 10%PTFE” composites fabricated by the HC (2) and FDM (3) methods

299

0.30 0.25 0.20

2

0.15

3

0.10 0.05

0.0

0.5

1.0

1.5

2.0

Distance, km

2.5

3.0

Friction coefficient

0.35

1

0.30 0.25 0.20 0.15

3

0.10

2

0.05 0.0

0.5

1.0

1.5

2.0

Distance, km

2.5

3.0

In addition, the wear rates on the metal and ceramic counterfaces of the HC sample were reduced by factors of 15 and 10, respectively, compared to those for neat PEEK. The wear rate of the FDM printed composite decreased by a factor of 10 on the steel counterface and by a factor of 7.5 on the ceramic one (Fig. 12.18). Presented in Figs. 12.20 and 12.21 optical images of the topography of the wear track surfaces on both the samples and the counterfaces enabled to visually interpret the data on their wear rates (Fig. 12.19). In contrast to neat PEEK (Fig. 12.14b, c), the counterfaces almost did not wear out during the tests against the composites (Fig. 12.20b, e). On both counterface surfaces, transfer layers were found, the areas of which were smaller than that for neat PEEK (Fig. 12.14a). As expected, there were no microgrooves or other damages on the wear track surfaces of the composites, but PTFE inclusions were observed (Fig. 12.20c, f). On the steel counterpart surface, accumulated debris was found in the form of a continuous film (Fig. 12.20b,

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Fig. 12.19 The wear rate values for neat PEEK (1) and the “PEEK + 0.3%HA + 10%PTFE” composites fabricated by the HC (2) and FDM (3) methods on the metal and ceramic counterfaces under the dry friction conditions

e). Respectively, the transfer film composition on the counterpart surface included degradation products of the polymer matrix in addition to PTFE particles. This could be caused by the lower density after the layer-by-layer deposition as evidenced by the low strength properties according to Fig. 12.16. As shown above, the wear track surface on the ceramic counterface was covered with the thin “iridescent” film in the case of neat PEEK, indicating the development of transfer processes (Fig. 12.14b). Thus, the PTFE transfer layer caused an almost fourfold reduction in the wear rate for the ceramic-polymer tribological contact (Fig. 12.19). A similar pattern was typical for the HC fabricated “PEEK + 0.3%HA + 10%PTFE” composite (Fig. 12.21b). For the FDM one, a slightly higher wear

Fig. 12.20 The topography of the wear track surfaces on the “PEEK + 0.3%HA + 10%PTFE” composites fabricated by the HC (a–c) and FDM (d–f) methods after the test distance of 3 km on the steel counterface

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Fig. 12.21 The topography of the wear track surfaces on the “PEEK + 0.3%HA + 10%PTFE” composites fabricated by the HC (a–c) and FDM (d–f) methods after the test distance of 3 km on the ceramic counterface

rate level (Fig. 12.21b) was caused by both transfer film and debris adherence on the ceramic counterpart surface (Fig. 12.21e). However, the low friction coefficient of about 0.1 indicated that PTFE successfully formed an antifriction layer on the composite friction surface even under structural imperfection conditions.

12.3.5 Conclusions to Sect. 12.3 Compared with neat PEEK, its simultaneous loading with the two studied fillers deteriorated the composite structure to some extent. However, the wear rate level was greatly reduced and microabrasive damages to both steel and ceramic counterparts were eliminated by facilitating the transfer film formation. In addition to the self-lubricating effect of the formed composite structure, another (probable) reason for such protection of the steel counterface was the protective effect of the transfer film from the standpoint of suppressing tribological oxidative processes during its interaction with PEEK. The slight lowering of the physical and mechanical properties of the FDM composite, compared with the HC one, was associated with the specifics of the additive manufacturing process. In this case, the interlayer adhesion was reduced and the complete internal space filling was not provided during the layer-by-layer formation of the FDM composite macrostructure due to the decrease in the melt flow rate after loading PEEK with 0.5 wt% HA nanoparticles.

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12.4 Evolution of the Functional Properties of Composites During the Formation of Intermediate Phases Between the Matrix and Inclusions 12.4.1 General Considerations The problem of predicting functional properties of heterogeneous materials belongs to the classical ones of micromechanics and has more than a century of history. Different approaches and approximations are used to solve it (Kainer 2006; Chawla 2012). All known analytical methods can be divided into two main groups. The first one enables to obtain exact solutions in the approximation of non-interaction of components. It includes variational formulations and solutions for periodic microstructures. The second is based on approximation schemes aimed at taking into account the relationship between inhomogeneities and concentrates on the “placement” of non-interacting inhomogeneities in some effective medium. For predicting the functional properties, a quantitative characterization of the microstructure is needed, which covers its various types and different physical characteristics. Determination of the correct microstructural parameters depends on the analysis of the individual inhomogeneity contribution to the investigated parameter. This is a complex task that involves an understanding of various shape factors, since inhomogeneities in real microstructures can possess both typical and “irregular” shapes. In anisotropic cases of non-random inhomogeneity orientations, the parameters are tensor. If a matrix is anisotropic, this further complicates the analysis. The property contribution tensors are applied within the homogenization method to describe the contribution of a single inhomogeneity to the investigated parameter. These may be elastic compliance or stiffness, thermal or electrical conductivity, as well as thermal expansion coefficient. The contribution tensor has been first introduced for compliance, taking into account the effect of pores and cracks on the functional properties (Nemat-Nasser and Hori 1993). The components of this tensor have been calculated by Kachanov et al. (Kachanov et al. 1994) for two-dimensional pores of various shapes and three-dimensional ellipsoidal pores in an isotropic material. For the general case of elastic inhomogeneities, these tensors have been introduced and calculated for ellipsoidal shapes by Sevostianov and Kachanov (Sevostianov and Kachanov 1999, 2002). Micromechanical approaches are quite applicable to the calculation of the functional properties of polycrystalline materials (Knyazeva et al. 2015), if a polycrystal with grain boundaries is represented as a composite, consisting of a matrix with a high diffusion capacity (grain boundaries and triple junctions of grains) and inhomogeneities with a low diffusion capacity (in the form of a grain volume, including crystal defects such as dislocations). In real composites, an important role is played by the nature of the interfaces between phases, including between inclusions and a matrix, as well as transition layers formed during their synthesis. The presence of active components promotes

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the formation of such transition zones (or even new phases) in the vicinity of inclusions due to the specific solidification conditions near the boundaries, etc. Therefore, the properties of these intermediate layers should be taken into account when calculating integral functional properties of the composites. In their prediction, the idea of replacing an inhomogeneous inclusion (possessing a transition zone) with a homogeneous one has been implemented in most papers on this topic. As noted above, they can be divided into three groups based on the assumptions about effective inclusions (T. Mori, K. Tanaka, etc.), effective matrix (K. Wakashima, M. Otsuka, S. Umekawa; C. Budiansky, etc.), and three phases (A. Riccardi, F. Montheillet). In the papers by Sevostianov and Kachanov (2007), Sevostianov (2007), explicit expressions have been suggested from which it follows that the overall elastic or conductive (thermal or electrical) properties are affected mainly by the transition zone width, normalized to the particle core size. With multiple layers, the idea of radius approximation of variable functional properties (piecewise-continuous variation of constant properties) has been proposed by Garboczi and Bentz (1997), as well as Garboczi and Berryman (2000) for some specific composites. An arbitrary law of changing the functional properties along the radius has been considered. Regardless of this idea, an interesting methodology has been proposed by Shen and Li (2003, 2005), according to which the transition zone width enhances the incremental (differential) form at each stage of the homogenization process. This approach has been modified by Sevostianov and Kachanov (1999), as well as Kachanov et al. (1994). They have obtained a differential equation for inhomogeneous both elastic and thermal properties. In Anisimova et al. (2016), a model has been proposed that explains a nonmonotonic change in the thermal conductivity of an aluminum matrix composite and, at the same time, a monotonic variation of its thermal expansion coefficient at a high volume fraction of diamond particles. The model takes into account the presence of a coating on the particles, which improves the interfacial bond between them and the metal matrix. Another model of an interfacial layer has been developed as a continuous medium with local cohesion and adhesion effects (Lurie et al. 2006). Within the model framework, it has been shown that these effects enable to understand or predict the macroor micromechanics of a material if its boundaries and phase properties are simulated on different scales. The approach has been applied for predicting some mechanical properties of a polymer matrix composite, reinforced with nanosized particles. Zare and Rhee (2019) have calculated some properties of the interfacial layer in a polymer composite and determined the dependence of the Young’s modulus of the transition layer on the distance between the surface of nanoparticles and the matrix. The transition layer is simulated as a multilayer phase in which the Young’s modulus of each layer changes continuously from the particle surface to the matrix. In most papers, transition layers or their phase composition are assumed to be known. However, the dependence of the layer width and composition on technological parameters requires special attention and approach. For example, a glass– ceramic composite is considered, consisting of a lead borate matrix containing

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isolated eucryptite granules (hereinafter referred to as inhomogeneities) with a negative thermal expansion coefficient (Kryukova et al. 2017). The authors have discussed a dependence of the thermal expansion coefficient on the particle size and sintering temperature, as well as the process of lithium diffusion from eucryptite into the glass matrix. The latter causes the formation of interfacial zones between the matrix and inhomogeneities. The simulation has been carried out using various micromechanical schemes. It has been shown that the Maxwell homogenization scheme provides the best agreement for the entire range of eucryptite volume fraction. The dependence of the nature and properties of transition zones on the technological conditions for fabricated composites leads to a discussion of the possibilities of developing predictive models, including the assessment of the properties during the manufacturing process and taking into account the mutual influence of various phenomena.

12.4.2 Research on Fe-Based Composites with Titanium Carbide Inclusions upon Their Laser-Beam Synthesis As an example, a model for the transition zone evolution between a particle and a matrix has been considered under conditions, typical for laser-beam additive manufacturing methods. A layer from a mixture of Fe and TiC powders has been processed by a laser beam moving along a certain preset trajectory. The process of the transition zone formation begins after melting of iron that surrounds refractory particles. The actual composite structure is formed at the cooling stage, when new phases are developed around the particle. Dimensions of a molten pool can be determined from the ratio between the mass concentrations of the substances in the initial powder mixture. Typical temperature changing dynamics, obtained by numerically solving the corresponding heat conduction problem (Knyazeva 2021), is shown in Fig. 12.22. The temperature changing points are presented in Fig. 12.22a.

Fig. 12.22 Selected points on the processed surface (a), thermal cycles at these points (b) and the temperature curves in dimensionless variables (c) at Points B (1) and C (2)

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Under such non-stationarity conditions changing at high rates, the formation of extremely non-equilibrium structures with non-stoichiometric compounds and intermetallics are possible. Upon additive manufacturing of composites on a substrate, similar problems appear under the influence of an electron or laser beam. In the layer-by-layer deposition, repeating thermal cycles only intensify these processes, so research on the phase formation specifics and their controlling is an important simulation stage. The model takes into account that the particle size is small compared to that of the sample (layer). The heat conduction process is characterized by a spatial scale that is much greater than the characteristic diffusion one. Therefore, it is assumed that all particles are in the same conditions at each macropoint of the sample. Temperatures of each particle changes only in time, and the transition zone formation process between the particle and the matrix can be simulated at different scale levels in a suitable coordinate system. A representative cell of a composite, which is equivalent (by a certain set of structural parameters) to the microstructure of a real one, is a spherical region, in the center of which there is a titanium carbide particle. At the initial moment of time, the rest of its volume is occupied by iron. It is assumed that the particle concentration is low. This enables not to consider the interaction of a given representative volume with neighboring ones. If only a single phase and one solution are formed between the particle and the matrix, then the problem of the transition zone formation can be formulated in the spherical coordinate system according to Fig. 12.23 as follows. In each of the areas according to Fig. 12.22, the diffusion equation is represented as: Dk ∂ 2 ∂Ck ∂Ck = 2 r ∂t r ∂r ∂r

(12.1)

where C k is the particle material concentration in different areas; the p, m and f indices refer to the particle and formed phase areas; Dk is the diffusion coefficient; Fig. 12.23 A graphic representation of the problem statement

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t is time. Both particle and matrix material concentrations are constant (C p = 1 and C m = 0, respectively). The boundary and initial conditions have the following form: r =0:

∂C p = 0; ∂r

r = x1 (t) : C p = C0 , C f = C1 ; (0 − C1 )

(12.2) ∂C f d x1 = Df ; dt ∂r

(12.3)

 d x2  ∂C f ∂Cm = Dm − Df ; (12.4) r = x2 (t) : C f = C1 , Cm = C2 ; C1 − C2 dt ∂r ∂r r = Rm :

∂Cm = 0; ∂r

t = 0 : C p = 1, Cm = 0, C f = 0; x1 = x10 < R0 ; x2 = x20 > R0

(12.5) (12.6)

In this case, C 0 , C 1 , C1 , C 2 are the solubility limits of the particle material in the matrix, determined from the state diagrams; x 1 , x 2 are positions of moving boundaries that separate the new phase from the initial ones and are functions of time x(t). Condition (12.2) is the symmetry one. Conditions (12.3) and (12.4) consider the presence of the solubility limits and a discontinuity in diffusion flows associated with the movement of the interfacial boundaries. Condition (12.5) denotes the impermeability of the outer boundary of the selected mesocell of with the Rm radius. Its size is determined by the matrix per particle material ratio with the R0 initial radius. The initial positions of the interfaces are close to R0 . This is a typical problem in the reactive diffusion theory. However, under varying temperature conditions, the C1 , C1 quantities and the diffusion coefficients also change. The solubility limits as a function of temperature follow from the state diagrams. The diffusion coefficients depend on temperature according to the Arrhenius law:   Ek , Dk = Dk0 exp − RT where Dk0 is the pre-exponential factor; E k is the activation energy; k = f , m. If more than one phase is formed, then the number of the coefficients, as well as Conditions (12.3) and (12.4), increases. The model assumes that carbon diffuses from TiC particles into the matrix, first forming a solution. Upon reaching the solubility limit, it reacts with the matrix material and forms iron carbide during the synthesis of the composite from the Fe and TiC powders. These processes take place near an isolated particle. At the same time, a layer of titanium carbide with an excess of titanium (for example, the Ti2 C non-equilibrium phase) is formed inside, since carbon is released from the particle. Accordingly, a third new phase is formed, which grows towards the particle center.

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Such a problem is considered in Anisimova (2021a), which is solved in several steps. Initially, dimensionless variables are introduced, which are convenient for qualitative analysis: θ=

T − T ph t r ; τ= ; ξ= , TM − T ph t∗ R0

phase formation temperwhere T M is the matrix melting point; T ph is the iron   carbide R2

E

ature; R0 is the initial particle size; t∗ = D00f 2 exp RTf M2 is the characteristic diffusion scale at the melting point. In the problem solving process, the temperature dynamics is first determined (for example, by analyzing curves for points B and C, according to Fig. 12.22b, at the cooling stage). In dimensionless variables, they are presented in Fig. 12.22c, where θM =

TM − T ph T ph − T ph and θph = . TM − T ph TM − T ph

The moving boundary value problem for the formation and growth of a transition layer between the matrix and inhomogeneities is solved analytically in the quasi-stationary approximation. When the temperature dynamics is determined, its solution is used to estimate changes in the phase volume concentrations upon the synthesis, and to calculate the functional properties of a representative inhomogeneous inclusion (of a variable size) consisting of a central inhomogeneity and a changing transition layer. At the last stage, changes in the composite functional properties are evaluated by varying phase concentrations during the synthesis. In this case, micromechanical homogenization schemes are used. Figure 12.24 shows the composite synthesis process, beginning from the liquid phase occurrence. At this initial stage, the titanium carbide depletion occurs with the formation of a carbon solution in the matrix (Fig. 12.24a). In this case, the depleted titanium carbide phase grows deep into the particle. In the temperature range from θ M = 1 to θ ph = 0, the cell includes the particle material (TiC), a layer of depleted titanium carbide, a carbon solution in iron, and the matrix (Fe). In the next step, carbon reacts with the matrix material as the temperature decreases. Respectively, iron carbide is formed (Fig. 12.24b). At this stage, a layer of iron carbide is added to the cell. The nature of the movement of the phase boundaries in the composite synthesis process is shown in Fig. 12.25 for a given law of temperature changing (Fig. 12.22c). Repeated heating and cooling lead to a non-uniform phase growth. Due to the scanning beam movement, the temperature at point B has time to drop below the phase formation point, while point C again falls into the zone of the beam thermal influence that slows down the iron carbide creation (dashed line 1 in Fig. 12.25 corresponds to this moment). Reheating (interval between lines 2 and 3) retards the iron carbide phase growth and increases the zone of carbon solution in

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Fig. 12.24 The concentration distributions during the growth of two phases at the initial stage (a) and three phases at temperatures below θ ph = 0 (b); a 1—particle, 2—depleted titanium carbide, 3—solution of carbon in titanium, 4—matrix; b 1—particle, 2—depleted titanium carbide, 3—iron carbide, 4—solution of carbon in titanium, 5—matrix

Fig. 12.25 The ξ i phase boundary positions depending on time for thermal cycles corresponding to Points B (a) and C (b). β = 0.4, σ = 0.2, δf1 = 0.6, εf1 = 4, δf3 = 0.9, εf3 = 2

titanium. As following from Fig. 12.25, the regions occupied by the phases actually do not differ over time in both cases.

12.4.3 The Behavior of the Ti–Al–C Composite Upon Its Synthesis In recent years, composites based on the Ti–Al–C system have become a careful research object due to the unique properties of such ternary (MAX) phases, which combine characteristics of both metals and ceramics. Aluminum-based composites are most often produced by powder metallurgy. For the Ti–Al–C system, various technological methods are used, such as mechanical alloying, reactive hot isostatic

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pressing, hot pressing, spark plasma sintering, pulsed discharge sintering, and combustion synthesis. However, final products include both TiCx carbide and Tix Aly intermetallic phases in most cases. Despite the fact that mechanical and thermal properties play a leading role in the possible applications of materials, few research results on these characteristics have been published to date. These properties of a final product are determined by the microstructure and the phase formation sequence, which, in turn, depend on the composite synthesis method, the initial feedstock composition, as well as processing temperatures and durations. In the combustion synthesis from titanium, aluminum and carbon powders, for example, the ternary Ti2 AlC phase forms in stages. The thermodynamic analysis results (Witusiewicz et al. 2015; Sadeghi et al. 2013) enable to theoretically describe the possibility of a reaction and the phase formation order for the Ti–Al–C system. In Filimonov et al. (2008) and Nekrasov et al. (1980, 1981), both volumetric synthesis and combustion processes have been simulated for the Ti–Al, Ta–C, Hf–B and Ti–C systems with separation of reaction cells. It has been noted in Filimonov et al. (2008) that the main product of the reaction is a solid solution at low both temperatures and synthesis rates, but it is the TiAl intermetallics at high temperatures of 960–1020 °C. Relatively great maximum temperatures and synthesis rates ensure intense reaction diffusion. For the Ti–C system, the formation of solid solutions is not considered due to the negligible solubility of the non-metal in the metal matrix (Nekrasov et al. 1980). In the volumetric synthesis models, transformations are simulated starting from the melting point of the fusible component. Based on the published data (Anisimova 2021b), a model for the composite synthesis from powders of the Ti–Al–C system has been proposed. Since the fusion reactions are high temperature ones in this case (Sadeghi et al. 2013), their acceleration at temperature levels close to the melting points, the early stage transformations can be neglected as a first approximation. The model assumes that first carbon particles melt in the Ti–Al solution at a temperature of 2000 K. Then the material cools down with the formation of new phases. The following possible reactions are considered: Ti + C = TiC, Ti + Al = TiAl, TiC + TiAl = Ti2 AlC. It is believed that carbon particles are evenly distributed in the material and have a spherical shape. Depending on their concentration, it is possible to estimate a molten volume corresponding to one particle. The problem of growing new phases at the interface between a spherical inclusion and the molten matrix is similar to the previous one and includes the diffusion equation and boundary conditions: (1) the symmetry is in the center; (2) the interphase boundaries take into account the solubility limits; (3) diffusion flows are interrupted due to the movement of interphase boundaries. In addition, the impermeability condition is set on the outer boundary of a selected spherical mesocell.

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The material cooling from the temperature of 2000 K is accompanied by the formation of new phases. At the first stage, the Ti–Al melt interacts with carbon at a temperature above the melting point with the TiC formation. With a further decrease in temperature below 1600 K, the TiAl phase is formed in the melt. When the reaction of carbon with titanium is completed, TiC particles react with TiAl ones, resulting in the Ti2 AlC phase occurrence. This follows from the nature of changes in the position of the interfacial boundaries (Fig. 12.26a), while the Ti2 AlC phase is formed at the final stage of the synthesis process (Fig. 12.26b). The titanium carbide phase remains at a volume fraction of 2%. The phase formation dynamics depends on the cooling rate (not shown in the figures). The formation of new phases significantly affects the distribution of the phase fractions in a conventional mesovolume as well as its functional properties and, as a result, the integral composite characteristics. In Fig. 12.27, such curves show sharp Fig. 12.26 The movement of the phase boundaries (a) and the changes in their volume fractions (b) during the synthesis process

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Fig. 12.27 Changes in the cell properties during the formation of new phases: a—the thermal conductivity coefficient, b—the bulk modulus, c—the thermal expansion coefficient

changes in the studied properties at the time points corresponding to the formation of the new phases, which is associated with their noticeable variations.

12.4.4 Mechanical Stresses Around the Diffusion Zone During the Synthesis of a Composite from Titanium and Carbon Powders The formation of a new phase and the movement of its boundary are accompanied by diffusion, leading to a redistribution of concentrations. These processes contribute to the appearance of diffusion (concentration) stresses due to the difference in the mobility of a diffusant in the phases and variations in their properties. This problem has been discussed by many authors in relation to some applications. In Anisimova and Knyazeva (2020), a model has been proposed that enables to study the influence of the composite synthesis conditions on the width of a formed transition layer between a matrix and inclusions, as well as to evaluate resulting stresses in the diffusion zone in dynamics. It should be noted that the formation of a composite is expected during its synthesis from titanium and carbon particles under equilibrium conditions, consisting of a titanium matrix with TiC carbide inclusions (Kobashi et al. 2010; Niyomwas 2011). The heat of a reaction between titanium and carbon is high enough to induce a selfsustaining TiC formation reaction under appropriate processing parameters (both Ti/C ratio and porosity). However, some non-equilibrium Tix Cy phases can be formed (Gusev 2002; Knyazeva and Korosteleva 2020; Krinitcyn et al. 2020), the number of which varies depending on the synthesis conditions. According to published data, the appearance of the TiC2 , Ti3 C2 and Ti2 C phases can be expected (Enyashin and Ivanovskii 2013; Magnuson et al. 2018). However, the Ti3 C2 one is unstable, so it is not taken into account in the model (Anisimova and Knyazeva 2020). Thus, the formation of a multiphase transition layer, including the TiC2 , TiC and Ti2 C phases, is considered. The mathematical formulation of the model is similar to that used in the previous section.

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Figure 12.28 shows the position changes for the boundaries of the obtained nonequilibrium phases (Curves 1–4) in comparison with those for a single-phase transition layer consisting of stoichiometric TiC carbide (Curves 1 and 2 ). The distribution of concentrations is given in Fig. 12.29 for both cases at the same time. With the growth of new phases upon the diffusion zone cooling, both stresses and strains arise, the nature of which is shown in Fig. 12.27 for different points in time. To estimate them, a particle has been considered in the first approximation as an “elastic” ball with a non-uniform distribution of concentrations, symmetrical around its center. Since both diffusion and growth of new phases are irreversible processes, such stresses are assumed to be elastic. In addition, viscous stresses have not been analyzed, since their increments are linearly related to changes in the concentrations. The model includes generalized relations between the components of both stress and strain tensors with concentration. There are no body forces, and the symmetry

Fig. 12.28 Movement of the ξ i multiphase transition layer boundaries (i = 1, …, 4) compared with the (1 ) and (2 ) single-phase ones relative to the R 0 initial particle radius (the dashed line) in time

Fig. 12.29 The concentration distributions and the boundary positions during the transition layer growth process at the τ time of 0.8: a the TiC single-phase transition layer: C 1 = 0.5, C1 = 0.33, b the TiC2 /TiC/Ti2 C multiphase transition layer: C 1 = 0.5, C1 = 0.45, C 2 = 0.44, C2 = 0.39, C 3 = 0.38, C3 = 0.33

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conditions enable to assume that some of the components of the stress and strain tensors are equal to zero, while other ones are functions of only the radial coordinate and time as a parameter. Assuming that the properties do not depend on the coordinate within the phase, a conjugate problem has been formulated in a spherical coordinate system, which has an exact analytical solution (some complex formulas are not given here). According to Figs. 12.30 and 12.31, stresses increase with rising the transition layer width. Strains change insignificantly with the phase growth, their maximum falls on the inner boundary. This enables to state that stresses can cause a local damage during a long synthesis process. Stresses and strains arising in the particle vicinity increase with the new phase area growth. Their maximum values are observed closer to the inner boundary and depend on the difference in the properties of the matrix, particle and transition layer materials. The speed and nature of the movement of the phase boundaries are influenced by the solubility limit values, as well as the number and order of the formation of new phases that can coexist at a given temperature. In turn, this changes the material properties as a result. The composite characteristics have been calculated by the same method as in Sect. 12.4.2. Figure 12.32 shows changes in the particle properties and their overall Fig. 12.30 The ratios of stresses to the particle elastic modulus at different τ times: 1–0.1; 2–0.4; 3–0.8. The boundary positions are shown by the dotted lines

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Fig. 12.31 The strains at different τ times: 1–0.1; 2–0.4; 3–0.8. The boundary positions are shown by the dotted lines

levels for the Ti-C composite over time with different ratios of those for the transition layer and matrix. Curve 1 reflects changes in the properties of a particle with a transition layer containing only TiC. Curve 2 corresponds to those including the TiC2 /TiC/Ti2 C phases. The radii of the particles with and without the transition layers match the positions of the boundaries and are known from the diffusion problem, the solution of which makes it possible to calculate the phase fractions in the composite. It should be noted that stresses caused by both diffusion and phase formation processes are referred to as those of the second kind in materials science. At the scale of particles and thermal stresses, those of the second kind can be taken into account when estimating residual levels under the assumption of additivity of the stress contributions.

12.4.5 Conclusions to Sect. 12.4 The methods of micromechanics and the reactive diffusion theory are applicable for assessing changes in the functional properties of composites during their synthesis, in addition to the results of microstructural studies. For the presented case, this

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Fig. 12.32 The changes in the relative bulk modulus (a), relative thermal conductivity (c) and thermal expansion coefficient (e) of a particle with interfacial layers and the integral composite properties (b, d, f) upon growing of the TiC (1) and TiC2 /TiC/Ti2 C (2) transition layers

is demonstrated by the simplest example of the isolation of conditional reaction mesocells. However, using the ideas about the structure of mesovolumes in powder compacts, which evolve in the composite synthesis, real objects can be proceeded for the investigations described in Sects. 12.4.2 and 12.4.3. The calculation of stresses of the second kind can also be improved by taking into account inelastic effects, changes in the porosity, the appearance of defects, etc. Conclusions Despite the fact that, at first glance, the results collected in this chapter seem to be rather disparate and incompatible, they adequately reflect the specifics of the

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processes, developed during the formation and operation of metal- and polymerbased composites fabricated by additive manufacturing methods. In the metal-matrix case, the special role of the formed structure and phase composition is shown in relation to residual stresses after filling the Ti–6Al–4V alloy with TiC particles. For the FDM PEEK-based composites, the important role of the component compatibility is shown for the dense structure formation and the possibility of the retention of both reinforcing and solid lubricant particles on the friction surface. As a part of the concept development, the model has been proposed, enabling both to study the effect of the composite manufacturing conditions on the width of the formed transition layer between the matrix and inclusions, and to evaluate resulting stresses in dynamics. During the time that has passed since the death of Professor Sevostianov, the authors have continued the research laid down in the idea of the project. Their results are reflected in a number of publications listed below. Of course, working with him, including within the framework of the project mentioned in the Introduction section, would make it possible to develop these ideas at a fundamentally greater level and quality. Acknowledgements The authors thank Yuri Dontsov, an engineer at the School of Engineering and New Manufacturing Technologies at Tomsk Polytechnic University, for fabricating the PEEKbased composites by the FDM method. The work was performed according to the Government research assignment for ISPMS SB RAS, Projects FWRW-2021-0010 (Sects. 12.2 and 12.3), as well as FWRW-2022-0003 (Sect. 12.4). The investigations have been carried out using the equipment of Share Use Centre “Nanotech” of the ISPMS SB RAS. The authors are grateful to Dr. Marina Chaykina (Institute of Solid-State Chemistry and Mechanochemistry, Siberian Branch of the Russian Academy of Sciences) for supplying HA nanoparticles.

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Chapter 13

Analysis of the Periodicity Cell Problems for the Fiber-Reinforced Plate and Applications A. G. Kolpakov and S. I. Rakin

Abstract We discuss the periodicity cells problem of the homogenization theory for the fiber-reinforced plate – elastic and thermoelastic, both linear and nonlinear. A specific feature of the plate periodicity cells is that they have free surfaces, corresponding to the top and bottom surfaces of the plate. We carry out numerical analysis of the plate periodicity cells for linear elasticity, linear thermoelasticity, and nonlinear elasticity problems. As follows from our computations, the boundary layers appear at the top/bottom surfaces of the plates in all the mentioned problems. We investigate the characteristics of the boundary layers for the unidirectional and cross reinforced plates. We introduce the notion of the representative plate and demonstrate that a three-layer plate is the representative plate for a plate with arbitrary number of layers. Furthermore, we use the notion of the representative plate to construct the homogenized strength criterion for the reinforced plates. Keywords Plate · Reinforcing fibers · Matrix · Homogenization method

13.1 Homogenization Method as Applied to Fiber-Reinforced Plates. Linear Problems The problem of the computation of the homogenized (also referred to as overall, macroscopic, effective) characteristics is the oldest in the theory of composite materials (Kachanov and Sevostianov 2018; Dvorak 2013). The homogenization theory provides us with a rigorous mathematical method to investigate the macroscopic and local stress-state state in composite through the solution of the so-called periodicity cell problems (Kolpakov 2004; Caillerie 1984; Kalamkarov and Kolpakov 1997; Bakhvalov and Panasenko 1989) and macroscopic solutions to the problem. In this paper, we apply the homogenization theory to the fiber-reinforced plate. A specific A. G. Kolpakov (B) · S. I. Rakin Siberian Transport University, Novosibirsk 630049, Russia e-mail: [email protected] S. I. Rakin e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_13

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Fig. 13.1 Fiber layers in the “slow” variables x and periodicity cell P of the plate in the “fast” variables y

feature of the plate is that its periodicity cell extends from the top to the bottom of the plate. Note that for a solid composite body, the periodicity cell coincides with the periodicity cell of the structure of the composite, see Fig. 13.1. The top and bottom surfaces of plate periodicity cell are free surfaces (Kolpakov 2004; Caillerie 1984; Kalamkarov and Kolpakov 1997). The presence of free surfaces usually leads to the appearance of boundary layers—thin zones where the local stress–strain state differs from the stress–strain state in the main part of the plate. Earlier, boundary layers at the “free edges” of the plate (also referred to as transversal cut surfaces, see Fig. 13.1) were considered. The “outer edges” boundary layers were intensively discussed in the 1970s-1980s for laminated composites, see, e.g., (Sanchez-Palencia et al. 1987; Andrianov et al. 2011; Pipes and Pagano 1970; Herakovich et al. 1985). Numerical experimental, theoretical, and numerical results were reported on this theme. In this paper, we do not consider the “outer edges” boundary layers. We pay attention to the boundary layers at the top and the bottom surfaces of the plate (see Fig. 13.1). The square of the top and the bottom surfaces of the plate are large. A direct computation in this zone would assume a solution to the specific (elastic, thermoelastic, etc.) problems in the whole domain occupied by the plate. Modern computer power is sufficient for numerical analysis of “outer edges” effects, but not sufficient for direct analysis of the boundary effects at the top and the bottom surfaces of the plate. Denote: ε R radius of fibers, εh—the distance between the fibers in the layers, εδ— the distance between the layers of fiber, Fig. 13.1. It is assumed the characteristic size ε of the fibers and the binder between the fibers is small: ε < 1. . To describe this two-scaled material, the “fast” (microscopic) variables y = x/ε are introduced in addition to the “slow” (macroscopic) variables x (Caillerie 1984; Kalamkarov and Kolpakov 1997). We assume that the plate has periodic structure and the top and the bottom surfaces of the plate are flat. In this case, periodicity cell of plate P = [0, h 1 ]×[0, h 2 ]×[0, h 3 ] in the “fast” variables. This is periodic in the variables y1 , y2 , and extends from the top to the bottom in the variable y3 , Fig. 13.1. We denote P2 = [0, h 1 ] × [0, h 2 ] the two-dimensional periodicity cell corresponding to periodicity cell P. The faces of the periodicity cell P are rectangles i = {y : yi = 0} and i + h i ei .

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We use the homogenization theory as applied to plates developed in Caillerie (1984), Kalamkarov and Kolpakov 1997) and later discussed in numerous publications. We focus on the mechanics and numerical aspects of the problem. We give no detailed explanation of the homogenization theory technique. For a detailed presentation of the homogenization theory as applied to plates, see (Caillerie 1984) (see also the monograph (Kalamkarov and Kolpakov 1997)). Solution to the elasticity and thermoelasticity theory problems for composite thin plate within the homogenization theory as applied to plates is sought in the form (Kalamkarov and Kolpakov 1997) ue = u0 (x1 , x2 ) + εu1 (x1 , x2 , x/ε) + ...

(13.1)

where u0 (x1 , x2 ) = (u 01 , u 02 , u 03 )(x1 , x2 ) is the macroscopic solution (u 01 and u 02 are in-plane displacements, and u 03 is normal deflection). εu1 (x1 , x2 , x/ε) is the corrector. The corrector εu1 (x1 , x2 , x/ε) makes a small contribution to the displacements, but a significant contribution to the local stress–strain state. The corrector has the form   ∂ 2 u 03 ∂u 0α (x1 , x2 )Nαβ0 (x/ε) + (x1 , x2 )Nαβ1 (x/ε) + εN0 (x/ε)T (x1 , x2 ), ε ∂ xβ ∂ xα ∂ xβ where Nαβν (y) and N0 (y) are solutions to the following periodicity cell problems of the homogenization theory as applied to plates (Caillerie 1984; Kalamkarov and Kolpakov 1997): ⎧ αβν ν ⎪ ⎪ (ai jkl (y)Nk,ly + (−1) ai jαβ (y)), j y = 0 in P, ⎨ ⎪ ⎪ ⎩

αβν

(ai jkl (y)Nk,ly + (−1)ν ai jαβ (y))n j = 0 on the top/bottom,

(13.2)

[Nαβν (y)]α = 0

and thermoelastic periodicity cell problem of the homogenization theory as applied to plates (Kalamkarov and Kolpakov 1997) ⎧ 0 (a (y)Nk,ly + bi j (y)), j y = 0 in P, ⎪ ⎨ i jkl 0 (ai jkl (y)Nk,ly + bi j (y))n j = 0 on the top/bottom, ⎪ ⎩ 0 [N (y)]α = 0

(13.3)

In problems (13.2, 13.3),  ai jkl (y) =

aiFjkl in fibers aiMjkl in matrix

 · · · αi j (y) =

αiFj in fibers αiMj in matrix

(13.4)

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where aiFjkl and aiMjkl are the elastic constants of the fibers and the binder, correspondingly; αiFj and αiMj are the coefficients of thermal expansion. In the thermoelastic periodicity cell problem (13.3), bi j (y) = −ai jkl (y)αkl (y). The indices take the following values: ν and μ—0, 1; the other Greek indices—1,2; Latin indices—1,2,3. Notation [ f (y)]α means the jump of the function f (y) on the opposite faces y ∈ α y ∈ α and y ∈ α + h α eα of periodicity cell P ([ f (y)]α = 0 means the periodicity of the function f (y) in yα ). The periodicity cell problems (13.2) and (13.3) have many solutions that differ by rigid body displacements. We eliminate the rigid body displacements by fixing some points of the periodicity cell. The periodicity cell problems (13.2) and (13.3) are problems in the “fast” variables 2 0α y. The macroscopic in-plane strains eαβ = ∂u , curvatures/torsion ραβ = ∂∂xαu∂03xβ , and ∂ xβ temperature T are functions of the “slow” variables x1 , x2 . Periodicity cell problem (13.2) with αβ = 11, 22 corresponds to the macroscopic tension if ν = 0 or bending if ν = 1; with αβ = 23—macroscopic shifts if ν = 0 or torsion if ν = 1. The local stresses in the plate are computed as αβ0

σi j (x1 , x2 , y) = (ai jkl (y)Nk,ly + ai jαβ (y))eαβ (x1 , x2 ) αβ1

+(ai jkl (y)Nk,ly − ai jαβ (y))ραβ (x1 , x2 ) 0 +(ai jkl (y)Nk,ly

(13.5)

− bi j (y))T (x1 , x2 )

The homogenized stiffnesses and the coefficients of the linear thermal expansion of the plate are computed as (Caillerie 1984; Kalamkarov and Kolpakov 1997) ν+μ

Sαβ

=

1 |P2 |



αβν

μ

(ai jkl (y)Nk,ly + (−1)ν ai jαβ (y))y3 dy

(13.6)

P

and μ Bαβ

1 = |P2 |



μ

0 (ai jkl (y)Nk,ly + bi j (y))y3 dy

(13.7)

P

13.2 Elastic Problem Present an analysis of elastic periodicity cell problems. Introduce ν [Nαβν (y) + ξαβν (y)] Zαβν (y) = eαβ

(13.8)

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αβν

where ξαβν (y) is the solution to the equation ai jkl (y)ξk,ly = y3ν ai jαβ (y) (α, β = 1,2, ν = 0,1). The existence of the functions ξαβν (y) was demonstrated in Kolpakov (2004). In (Kolpakov 2004), the explicit formulas for ξαβν (y) are also presented. By using the notation (13.8), we write the periodicity cell problem (13.2) in the form ⎧ αβν ⎪ (a (y)Z k,ly + (−1)ν ai jαβ (y)), j y = 0 in P, ⎪ ⎨ i jkl αβν (ai jkl (y)Z k,ly + (−1)ν ai jαβ (y))n j = 0 on the top/bottom, ⎪ ⎪ ⎩ αβν [Z (y)]α = eαβ [ξαβ0 (y)]α if ν = 0, = ραβ [ξαβ1 (y)]α if ν = 1

(13.9)

The computational processes have some differences for in-plane tensile/shift deformations and bending/torsion of the plate. The difference arises from the difference in the functions ξαβν (y) in periodicity cell problem (13.9) for in-plane deformation and bending/torsion. We have solved periodicity cell problem (13.9) for in-plane deformations (ν = 0) and bending/torsion (ν = 1) for 3-, 4-…, 10-layer periodicity cells for the same macroscopic deformations. In the computations below Young’s modulus and Poisson’s ratio of the fibers and matrix E f = 170GPa, ν f = 0.3, and E b = 2GPa, νb = 0.36, correspondingly. These values correspond to the carbon/epoxy composite. The fiber radius is R = 0.45; the distance between the fibers in the layers is h = 0.1, between the layers of fibers δ = 0.1. Periodicity cell dimensions are h 1 = 1.1, h 2 = 3, h 3 = 1.1, see Fig. 13.1. These geometrical characteristics are given in the non-dimensional “fast” variables y. The corresponding dimensional values are computed by multiplying by the characteristic size ε. For carbon fibers, ε changes from 5 to 20 microns (Agarwal et al. 2017).

13.2.1 Plates with Unidirectional Systems of Fibers We investigate the plates with the unidirectional and cross reinforcements. First, we consider plates with unidirectional reinforcements. Bulk bodies with the system of unidirectional fibers or with a system of parallel channels were investigated by numerous researchers, see (Grigolyuk and Fil’shtinskii 1992; Fy 1966; Grigolyuk et al. 1991). Note that in Grigolyuk and Fil’shtinskii (1992); Fy 1966; Grigolyuk et al. 1991) the problems were considered for the “infinite” bodies of periodic structures. It was the result of the application of a special technique of the double-periodic functions, which was the unique effective computational method in the pre-computer era. We present numerical solutions to periodicity cell problems for the plate with the system of cylindrical inclusions or channels parallel to O x1 -axis. The local deformations of the periodicity cell corresponding to macroscopic inplane tension/ shift are the solution to the problem (13.9) with the index ν = 0. In this case, y3ν = 1. The local deformations of the periodicity cell corresponding to macroscopic bending/torsion are solutions to the problem (13.9) with the index

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Fig. 13.2 Local von mises stresses: in-plane tension (a), bending (b)

ν = 1. The results of numerical computations for the tension and bending of the fiber-reinforced plate are presented in Fig. 13..2. The top/bottom fragments of the periodicity cells are zoomed. In Fig. 13.4, the boundary layer depth is less than the thickness of one structural layer.

13.2.2 Plates with Unidirectional Systems of Channels The analysis of the plates with a system of unidirectional channels provides us with examples of the negative role of the boundary layer. The results of numerical computations for the tension and bending of a plate with channels are presented in Fig. 13.3. It is seen in Fig. 13.3 that the boundary layer increases the local von Mises stress. The numerical analysis demonstrates the similar behavior of 4-, 5-,…, 10-layer plates.

Fig. 13.3 Composite plate with channels: in-plane tension (a), bending (b)

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Fig. 13.4 The local von Mises stress in the periodicity cell (a), the bottom of the periodicity cell (zoomed) (b), and the local von Mises stress in the matrix (c)

13.2.3 Plates with the Cross Reinforcements As above, the in-plane local deformations of the periodicity cell are a solution to the problem (13.9) with the index ν = 0. The deformed shape of the periodicity cells and von Mises stresses are displayed in Fig. 13.4. Figure 4a displays the Local von Mises stress in the periodicity cell. Due to the large difference of the stresses in the fibers and the matrix, the figure in 13.4a is two colored. In Fig. 4c, the stress–strain state in the matrix is displayed. The pictures in Fig. 13.4 are typical for 4-, 5-… 10 layer periodicity cells. All our numerical computational results demonstrate that the solution to the problem is periodic in the “core” part of the periodicity cell and there exists the boundary layer at the top/bottom surfaces of the periodicity cell. The thickness of the boundary layers is less than the thickness of one structural layer S. Looking at Figs. 4a and c, we observe significant differences between the local stress–strain states in the fiber and the matrix. The difference is so large that the fibers and the matrix may be qualified as two different media. Based on our numerical computations, we conclude that the “stiff fibers in the soft matrix” composites demonstrate the multi-continuum behavior predicted for the high-contrast composites theoretically by Panasenko in Panasenko (1991). The in-plane local deformations of the periodicity cell are solutions to the problem (13.9) with the index ν = 1. In this case, the presence of the variable y3 leads to the typical bending/torsion solutions to the problem (13.9). The deformed shape of the periodicity cells and von Mises stresses are displayed in Fig. 13.3. Figure 13.3 displays the deformation of periodicity cell corresponding to the bending in the O yzplane. It is seen that the stress in the high modulus fibers parallel to O y-axis prevails to all other stresses. It corresponds to the general theory of the fiber-reinforced plates. The pictures in Fig. 13.3 are typical for 4-, 5-… 10 layer periodicity cells. Due to the large difference of the stresses in the fibers and the matrix, the matrix in Fig. 13..3 is colored in the same color. In Fig. 13.4, the local stress–strain state in the matrix is displayed. In both unidirectional- and cross-reinforced plates, the boundary layer decreases the von Mises stress in the matrix, see Figs. 13.2 and 13.4. In the core part of the plate, the local stress–strain state looks similar to the local stress–strain state arising in the

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contrast composite plate’s “stiff fibers in the soft matrix” (Kalamkarov and Kolpakov 1997; Kolpakov and Kolpakov 2009; Keller and Flaherty 1973; Kang and Yu 2020; Kolpakov 2007a; Rakin 2014; Trofimov et al. 2017). A similar effect takes place for the cross-reinforced plates. We can conclude that the role of the boundary layer is positive for the strength of the contrast composite plates “stiff fibers in the soft matrix”. An example of such kind of composite structures is the “composite wings” https://www.boeing.com/777x/reveal/state-of-the-art-777x-composite-wing-center -completes-parts/, https://www.compositesworld.com/hashtag/a350 or some sports equipment (Kaufmann 2015). Of course, the composite wings or sports equipment have more complex reinforcement designs and direct application of the results given in this paper is not allowed.

13.3 Thermoelastic Problem Present an analysis of the thermoelastic periodicity cell problem. To the best knowledge of the authors, the thermoelasticity periodicity cell problems (13.3) cannot be written in a compact form similar to the problem (13.9). We have carried out computations for the carbon/epoxy composite with the material and geometrical parameters indicated above. The coefficient of the linear thermal expansion of the fibers and matrix α f = 5 · 10−6 and αb = 60 · 10−6 αb = 60 · 10−6 , correspondingly. The solution was obtained for the temperature T = 100◦ C.

13.3.1 Plates with Unidirectional Systems of Fibers Figure 13.5 displays the solution to the periodicity cell problem for the thermoelastic problem. The computations have been carried out for the carbon/epoxy composite with the material and geometrical parameters indicated above. The solution is periodic in the “core” part of the periodicity cell, and there exists the boundary layer at the top/bottom surfaces of the periodicity cell. The thickness

Fig. 13.5 The local von Mises stresses (a) and top/bottom boundary layers (zoomed) (b, c)

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of the boundary layers is less than the thickness of one structural layer S, like in the elastic periodicity cell problems.

13.3.2 Plates with the Cross Reinforcements The deformed shape of the periodicity cells and von Mises stresses are displayed in Fig. 13.6a. The wrinkling is especially well seen for two adjacent periodicity cells, see Fig. 13.8b. It is seen that solution to the problem is periodic in the “core” part of the periodicity cell and there exist the boundary layer at the top/bottom surfaces of the periodicity cell. The thickness of the boundary layers is less than the thickness of one structural layer S, like in the elastic periodicity cell problems.

13.4 Non-Linear Problems We investigate the influence of the nonlinearity of the material(s) in the cases considered above for the plate with linear components. In the previous sections, we considered the carbon/epoxy composite. The carbon fibers demonstrate the linear elastic behavior in the wide range of strains until the strength limit (Agarwal et al. 2017; Lubin 1982), while the epoxy resin demonstrates the non-linear elastic behavior (Lubin 1982). Similar situations take place for numerous fiber-reinforced composites. For this reason, we consider the nonlinear periodicity cell problem

αβν

g(de f (Nk,ly + (−1)ν xα eβ ))dy → min

P

It is denoted de f (N) = 1/2(Ni, j + N j,i ). Fig. 13.6 The local von mises stresses in the matrix and the top/bottom boundary layers (zoomed)

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The stress–strain curve for epoxy resin may be approximated as follows (detailed information on the deformation of epoxy resin may be found in Lubin (1982)):  σ =

E b e for |a| ≤ e∗ . E 1 (e − e∗ ) + E b e for |a| > e∗

(13.10)

The formula (13.10) is valid both for loading and unloading. Formula (13.10) means that the epoxy resin demonstrates the nonlinear (multilinear) elastic behavior with the proportionally limit e∗ . In computations, e∗ = 0.03 (Lubin 1982) and E 1 = E b /100.

13.4.1 Fiber-Reinforced Plate The results of the numerical computations for the nonlinear problem are presented in Figs. 13.7 and 13.8. Numerical analysis demonstrates both similarities and differences in the linear and nonlinear behavior of the composite plate. The differences are associated with the nonlinearity of the deformation and manifest themselves primarily in the “necks” between adjacent fibers. Note that in the “necks” between the fibers, approximate calculation methods (Kolpakov 2007a; Rakin 2014) can be used. These methods significantly reduce the cost of computing resources. Similarities are manifested primarily in the structure of stress–strain states, it consists of the “skin” boundary layers near the top and bottom surfaces of the plate and periodic stress–strain state in the “core” part of the plate. The wrinkling of top and bottom surfaces of the plate takes place in the nonlinear case (Fig. 13.9).

Fig. 13.7 The local von mises stresses in the plate periodicity cell and the top and the bottom of the periodicity cell (zoomed)

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Fig 13.8. The local von mises stresses in the periodicity cell (a) and the top of the two adjacent periodicity cells (zoomed) (b)

Fig. 13.9 Total von mises stress in the periodicity cell and the boundary layer (zoomed)

13.4.2 Plates with Unidirectional Systems of Channels For a plate with unidirectional systems of channels, the results of the numerical computations for the nonlinear problem are presented in Fig. 13.11. Solutions presented in Fig. 13.11 correspond to the in-plate tension of the plate. Numerical analysis demonstrates both similarities and differences in the linear and nonlinear behavior of the composite plate, compare with Fig. 13.3.

13.5 Applications of the Results of the Numerical Analysis In this section, we present several applications of the results of numerical analysis presented above.

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13.5.1 The Boundary Layers on the Top/bottom Surfaces of Inhomogeneous Plate The boundary layers on the top/bottom surfaces of the plate occur in all the problems considered above: linear elasticity, linear thermoelasticity and nonlinear elasticity. For all the problems, the boundary layer thickness is less than one structural layer thickness.

13.5.2 Wrinkling the Top/bottom Surfaces of Inhomogeneous Plates The above calculations were carried out for a plate with flat upper and lower surfaces. Looking closely at the drawings, one can conclude that, generally, the top/bottom surfaces of the deformed plate do not turn into cylindrical surfaces, but wrinkle. The wrinkling takes place in all the cases considered above: linear elasticity, linear thermoelasticity and nonlinear elasticity. The wrinkling is especially well seen for two adjacent periodicity cells, see Fig. 8b. Both the amplitude and period of the wrinkling have the order of ε. Such kind wrinkling leads to the 1-order change of square (length) of the top and bottom surfaces of the plate. This phenomenon affects, for example, the thermal properties of the surfaces (Sanchez-Palencia 1980). The wrinkling can also affect the interaction of the plate with liquid or gas flow. The wrinkling does not appear in every periodicity cell problem. The known exact solutions to the periodicity cell problem (Kolpakov 2004, 2007b) demonstrate no boundary layers in the periodicity cell problem, corresponding to homogeneous and laminated plates. Consequently, no wrinkling appears in homogeneous and laminated plates. The wrinkling cannot appear in the plates of complex structures. Figure 13.10 displays a solution to periodicity cell corresponding to in-plane shift. Although the boundary layers are clearly seen, no wrinkling is seen at the free surface (the top surfaces is displayed and the bottom one looks similar). Thus, the boundary layer and wrinkling are not identical phenomena (Fig. 13.12).

13.5.3 The Representative Model of Multilayer Plate The concept of the representative element in the mechanics of composite materials is known for a long time (Dvorak 2013). The traditional notion of the representative element (Dvorak 2013; Kolpakov and Rakin 2020) is based on the “similarity of the portion and the whole”. The concept of the representative plate is based on other concepts, which are the following:

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Fig. 13.10 Total von mises total strains (a), and von mises elastic nonlinear strains (b)

Fig. 13.11 Total von mises total stress (a), total von mises total strains (b), and von mises elastic nonlinear strains (c)

Fig. 13.12 Local von mises stresses in the matrix and top of periodicity cell (zoomed)

• The stress-strain state in the representative plate is similar to the stress-strain state in the original plate in some parts; • For any part of the original plate, there exists the corresponding part in the representative plate. The concept of the representative plate is based on the fact that the multilayer plate naturally separates into three zones: Top skin zone, core zone, and bottom skin zone. The top skin zone and the bottom skin zone are boundary layer zones. Since boundary layer thickness is less than one structural layer thickness, then one layer is sufficient

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to represent every skin zone. One layer is sufficient to represent the core zone (it is a direct consequence of the homogenization theory (Caillerie 1984; Kalamkarov and Kolpakov 1997)). Thus, a 3-layer plate periodicity cell would provide us with full information about the local stress–strain state in a multilayer plate. This 3-layer plate is the representative plate. It is seen that the representative plate is not any portion of the original plate. The representative plate method allows accurate computation of the local stress– strain state in all the plate (top and bottom skins and core) with minimal computing resources. The idea of the representative plate would make a possible accurate analysis of the local stress–strain state earlier with less power of computers than now. As far as the authors’ best knowledge, this idea does not arise earlier. In the 3-layer representative plate, we distinguish the informative and not informative layers. In the 3-layer, the stress–strain states in the top and the core layers coincide with stress–strain state in the multilayer plates. But stress–strain state in the bottom layer in the 3-layer plate does not correspond to stress–strain state in the multilayer plate. Then the top and core layers in 3-layer plate are informative, and the bottom one is not informative. Note that the third layer plays an important role in the 3-layer representative model and cannot be removed. The similar situation occurs at the bottom of the plate: The bottom and the core layers are informative and top one is not informative. Constructing the representative plate model above, we use only one property of the boundary layer, namely, that the boundary layer thickness is less than one structural layer thickness. From the boundary layer thickness is less than one structural layer thickness, it follows that the three-layer plate is the representative plate model for the plate formed of arbitrary number of layers. These conclusions were formulated and justified in Kolpakov et al. (2022), Kolpakov (2005) for elastic plates. The results presented in this article show that a three-layer plate is a representative model for a plate consisting of an arbitrary number of layers, both for elastic plates and for thermoelastic plates and non-linear plates.

13.5.4 The Strength of the Composite Plate We call the homogenized strength criterion the strength criterion for constitutive components (fibers and binder) of the plate, written in terms of the macroscopic in-plane strains and curvatures/torsion eεβ b ρεβ or resultant in-plane forces and bending/torsion moments. We assume that the strength criterions of the constitutive elements of the plate loc ) < σ ∗ (y), where the (the fibers and the binder) may be written in the form f (y, σ pq loc ∗ functions f (y, σ pq ) and σ (y) are defined as follows:  loc f (y, σ pq )=

loc ) in fibers f F (σ pq loc f M (σ pq ) in matrix

 . . . σ ∗ (y) =

σ F∗ in fibers σ M∗ in matrix

(13.11)

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In (13.11), σ F∗ is the strength limit of fibers, σ M∗ is the strength limit of matrix. Denote F(eεβ (x1 , x2 ), ρεβ (x1 , x2 ), T (x1 , x2 )) = max y∈P

σi j (x1 , x2 , y) σ ∗ (y)

(13.12)

where σi j (x1 , x2 , y) is given by (13.5). The inequality F(eεβ (x1 , x2 ), ρεβ (x1 , x2 ), T (x1 , x2 )) < 1

(13.13)

ensures no damage to the microscopic elements of the plate in the vicinity of the point (x1 , x2 ). The destruction of the constitutive elements of the plates starts when the condition F(eεβ (x1 , x2 ), ρεβ (x1 , x2 ), T (x1 , x2 )) = 1

(13.14)

is satisfied. The destruction starts at the point(s) y0 ∈ P at which the maximum in (13.12) reaches value 1. Formulas (13.12, 13.13) prove that the HSC for the reinforced plates exists for plates subjected both mechanical and thermal loads. The strength criterion (13.14) is the “first crack” condition and y0 above indicates the „weakest element “ of the plate. The equality (13.14) does not mean that the plate necessarily divides into separate parts or loses its carrying capacity. However, the “first cracks” occur in numerous PCs, thus, the damage to the plate is significant. The mechanical and thermal loads {(eεβ , ρεβ , T ): F(eεβ (x1 , x2 ), ρεβ (x1 , x2 ), T (x1 , x2 )) < 1} satisfying the inequality (13.13) form the “safety zone”.

13.6 Conclusions As follows from our computations, the boundary layers can appear at the free surfaces. Boundary layers do not appear in every periodicity cell problem. The known exact solutions to the periodicity cell problem demonstrate no boundary layers in the periodicity cell problem, corresponding to homogeneous and laminated plates. The top/bottom boundary layers appear in the plates of complex structures. The fiber-reinforced plates are an example of the plates of complex structures. The boundary layer thickness is less than one structural layer thickness (one fiber and the surrounding fragment of binder, see Fig. 13.1). The boundary layer thickness is less than one layer thickness. Our numerical computations demonstrate that the maximum von Mises stress in the fiber-reinforced plate and plate with channels occurs in the top/bottom layers,

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but not exactly at the top/bottom of the plate. The maximum von Mises stress occurs exactly in the top/bottom of the plate for the uniform plates and laminated plates. One of the results of the top/bottom boundary layers in the plate is the “wrinkling” of the originally flat top/bottom surfaces of the fiber-reinforced plate. While the amplitude of the wrinkling is small, it leads to a not small change of square (length) of the top and bottom surfaces of the plate. The boundary layers at the top and the bottom) surfaces of inhomogeneous plate can appear both in the elastic, thermoelastic and non-linear elastic problems. The characteristic thickness of the boundary layers is the same in all these problems. The three-layer plate is a representative model for a plate consisting of an arbitrary number of layers, both for elastic plates and for thermoelastic plates and non-linear plates.

References Agarwal BD, Broutman LJ, Chandrashekhara K (2017) Analysis and performance of fiber composites, 4th edn. Wiley, Hoboken NJ Andrianov IV, Danishevskyy VV, Weichert D (2011) Boundary layers in fibre composite materials. Acta Mech 216(1):3–15 Bakhvalov N, Panasenko G (1989) Averaging processes in periodic media. Math Probl Mech Compos Mater. Kluwer, Dordrecht Caillerie D (1984) Thin elastic and periodic plate. Math Models Meth Appl Sci 6(1):159–191 Dvorak G (2013) Micromechanics of composite materials. Springer, Dordrecht Grigolyuk EI, Fil’shtinskii LA (1992) Periodic Piecewise-Homogeneous elastic structures. Nauka, Moscow [in Russian] Grigolyuk EI, Kovalev YuD, Fil’shtinskii LA (1991) Bending of a layer weakened by through tunnel cuts. Dokl Akad Nauk SSSR 317(1):51–53 Herakovich CT, Post D, Buczek MB, Czarnek R (1985) Free edge strain concentrations in real composite laminates: experimental-theoretical correlation. J Appl Mech 52(4):787–793 https://www.boeing.com/777x/reveal/state-of-the-art-777x-composite-wing-center-completesparts/ https://www.compositesworld.com/hashtag/a350 Kalamkarov AL, Kolpakov AG (1997) Analysis, design and optimization of composite structures. Chichester, Wiley Kachanov M, Sevostianov I (2018) Micromechanics of materials, with applications. Springer, Cham Kalamkarov AL, Kolpakov AG (1997) Analysis. Wiley, Design and optimization of composite structures. Chichester Kang H, Yu S (2020) A proof of the Flaherty-Keller formula on the effective property of densely packed elastic composites. Calc Var 59:22 Kaufmann J (2015) New materials for sports equipment made of anisotropic fiber-reinforced plastics with stiffness related coupling effect. Procedia Engng 112:140–145 Keller JB, Flaherty JE (1973) Elastic behavior of composite media. Comm Pure Appl Math 26:565– 580 Kolpakov AA, Kolpakov AG (2009) Capacity and transport in contrast composite structures: asymptotic analysis and applications. CRC Press, Boca Raton, FL Kolpakov AG (2004) Homogenized models for thin-walled nonhomogeneous structures with initial stresses. Springer, Heidelberg

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Kolpakov AA (2007a) Numerical verification of the existence of the energy-concentration effect in a high-contrast heavy-charged composite material. J Engng Phys Thermoph 80(4):812–819 Kolpakov AA (2007b) Design of a laminated plate possessing the required stiffnesses using the minimum number of materials and layers. J Elast 86(3):245–261 Kolpakov AG, Rakin SI (2020) Homogenized strength criterion for composite reinforced with orthogonal systems of fibers. Mech Mater 148:103489 Kolpakov AA, Kolpakov AG, Rakin SI (2022) “Skin” boundary layers and concept of representative model in inhomogeneous plates. Eur J Mech / A Solids 93:104552 Kolpakov AA, Kolpakov AG, (2005) Solution of the laminated plate design problem: New problems and algorithms Comp & Struct, 83(12–13), 964–975 Lubin G (ed) (1982). Van Noxtrand, New York Panasenko GP (1991) Multicomponent homogenization for processes in essentially nonhomogeneous structures. Math USSR Sb 69(1):143–153 Pipes RB, Pagano NJ (1970) Interlaminar stresses in composite laminates under uniform axial extension. J Comp Mater 4(4):538–548 Rakin SI (2014) Numerical verification of the existence of the elastic energy localization effect for closely spaced rigid disks. J Engng Phys Thermoph 87:246–252 Sanchez-Palencia E (1980) Non-Homogeneous Media and Vibration Theory. Springer, Berlin Sanchez-Palencia E (1987) Boundary layers and edge effects in composites. In: Sanchez-Palencia E, Zaoui A (eds) Homogenization techniques for composite materials. Springer, Berlin, pp 122–193 Trofimov A, Abaimov S, Akhatov I, Sevostianov I (2017) Effect of elastic contrast on the contribution of helical fibers into overall stiffness of a composites Int. J Engng Sci 120:31–50 Vang Fo Fy GA (1966) Elastic constants and thermal expansion of some bodies having unhomogeneous regular structure. Dokl Akad Nauk SSSR, 166(4), 817–820

Chapter 14

Boundary Layers at the Interface of Layers of Unidirectional Fibers in Fibrous Composites Alexander G. Kolpakov, Igor V. Andrianov, and Sergey I. Rakin

Abstract We deal with a layered composite formed from “thick” layers of unidirectional fibers (layers containing many unidirectional fibers). It is demonstrated that the local stress–strain state in such composite consists of local stress–strain states in “thick” layers and local stress–strain states at the layer interfaces. We theoretically predict and numerically confirm the presence of a boundary layer at the junction of adjacent fiber layers. This asymptotic phenomenon, to the best of our knowledge, has not been described earlier.

14.1 Introduction Problems of conjugation of periodic structures often arise in engineering practice. As an example, we mention layered composites. The local stress–strain state due to the presence of conjugation consists of a stress–strain state in the layers of the composite and rapidly decaying boundary layers in the vicinities of conjugation. Both types of stress–strain states are related to each other, and in our approach (Kolpakov and Andrianov 2014; Kolpakov and Gaudiello 2011) they are treated as a single “perturbation”, which can be calculated from the solution of one boundary value problem. Close to the problems we are considering are the problems of multistructures (Zhikov 2002; Bakhvalov and Panasenko 1989; Kozlov et al. 1999; Movchan 2006; Bensoussan et al. 1979; Sanchez-Palencia 1987; Oleinik et al. 1992), where the main attention is paid to the mathematically rigorous formulation of “connection A. G. Kolpakov (B) SusAn and Siberian Transport University, Novosibirsk, Russia e-mail: [email protected] I. V. Andrianov Chair and Institute of General Mechanics, RWTH Aachen University, Eilfschornsteinstraße 18, Aachen D-52062, Germany S. I. Rakin Siberian Transport University, D. Koval’chuk st., 191, Novosibirsk 630049, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_14

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Fig. 14.1 Three types of destruction of “thick” layer composite and the periodicity cell of the composite

conditions” for connecting different parts of a multistructure. It is typical for the present time that either purely mathematical or purely engineering studies are devoted to boundary layers when connecting complex structures. The theory of boundary layer in homogenization is the subject of Allaire and Amar (1999). Examples of the engineering approach are presented in Mishuris and Öchsner (2005). The problem under consideration is illustrated in Fig. 14.1, which displays the basic modes of fracture of laminated composite, traditionally referred to as. A. delamination; B. fiber fracture; C. matrix fracture. The terms above are conditional. The fiber fracture is associated with the tension of the layer along the fibers. The simplest model predicts the local stresses σi j (the fibers are assumed to be parallel to O x1 -axis and subjected to the tension e11 = e along the fibers): σ11 = E f e in the fiber, σ11 = E b e in the binder, σi j = 0 if i j = 1 both in the fiber and in the binder.

(14.1)

The formulas (14.1) are exact if ν f = νb . For ν f = νb , the transversal local deformations appear. Usually, the fibers fail first in the mode “fiber fracture” which is displayed in Fig. 14.1. Generally, the failure is determined by the ratios E f /σ ∗f and E b /σb∗ and binder may fail first. The matrix fracture is associated with the deformation of the layer in the directions perpendicular to the fibers. The proper planar periodic elasticity problem was investigated in Mityushev and Rogozin (2000), Gluzman et al. (2018), Dryga´s et al. (2020); see also references in these books.

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The delamination, from the mathematical point of view, is associated with the boundary layer, which appears on the interface of 0- and 90-“thick” layers. To the best knowledge of the authors, such a boundary layer was not investigated earlier. The local stresses near the interface of 0-layer and 90-layer are asymptotic phenomena.

14.2 Contact and Connection We distinguish between contact and connection. Upon contact, structures directly adjoin one another. When connecting between the structure, there is a certain connecting element (for example, an adhesive layer). Contact is not only a useful mathematical abstraction, but it is also implemented in layered composites. Problems with the contact of periodic structures were studied in Panasenko (1981), Panasenko (2005) by the asymptotic method. When connecting the structure P − and P + , there is some transition zone P0 that differs from the connected structure (Fig. 14.1). We confine ourselves to considering bodies with a periodic structure. Such bodies are characterized by a periodicity cell (εY − and εY + in Fig. 14.1); hereinafter, ε is the characteristic size of the microstructure. We will call the parts P − and P + of the body, sufficiently distant from the connection (the meaning of the term “sufficiently distant” is discussed below), the “main parts” of the bodies. The local perturbation method (Kolpakov and Andrianov 2014; Kolpakov and Gaudiello 2011), as well as the asymptotic expansion method (Bakhvalov and Panasenko 1989; Bensoussan et al. 1979), makes it possible to describe rapidly decaying boundary layers in coupled structures P − and P + . In addition, the local perturbation method makes it possible to describe the stress–strain state also in the connecting node (transition zone). In this case, the calculation of the stress–strain state in the connecting zone is carried out within the framework of the numerical solution of one problem. The authors treat this as the main advantage of the proposed method. Consider the problem of elasticity theory in an inhomogeneous periodic medium occupying an area P with a typical periodicity cell(s) size ε −1, j = 1, 2, 3,

(15.3)

where functions ψ j (x), j = 1, 2, 3, satisfy the Hölder condition in [−1, 1]. Substituting (15.3) in system (15.1) and analyzing the behavior of the system equations in the vicinity of the ends of the interval [−1, 1] using the Muskhelishvili results (Muskhelishvili 1946) concerning the behavior of singular integrals, we find that α1 = − 21 − arctan



λ(1−2ν1 ) 2(1−ν1 )



, β1 = − 21 + arctan

α2 = α3 = β2 = β3 =



λ(1−2ν1 ) 2(1−ν1 )



, (15.4)

− 21 .

Using the method of mechanical quadrature (Sahakyan and Amirjanyan 2018) sufficiently smooth functions ψ j (x), j = 1, 2, 3, and pc (x) can be replaced by interpolation polynomials, i.e., ψ1 (x) can be replaced by interpolating Jacobi polynomials (α ,β ) Pn 1 1 (x), ψ2 (xc ), and ψ3 (xc ) can be replaced by interpolating Chebyshev polynomials Tn (xc ) (Abramowitz and Stegun 1972). The first of the equations in (15.1) is (−α ,−β ) satisfied at the roots of Jacobi polynomials Pn−1 1 1 (x) while the second and the third of the equations in (15.1) are satisfied at the roots of the Chebyshev polynomials Un−1 (xc ) of the second order (Abramowitz and Stegun 1972). That reduces solution of equations (15.1) to solution of the system of linear algebraic equations. However, condition (15.2) makes this system nonlinear. Nonetheless, the discretized nonlinear systems (15.1) and (15.2) can be solved iteratively as a system of linear algebraic equations (discretized equations (15.1)) by fixing the distribution of the crack contact stress pc (xc ) and updating conditions (15.2) after each iteration until the process converges.

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After this system is solved, all components of the problem solution can be√deterE2 l can mined. Specifically, the stress intensity factors at the crack tips scaled by 1−ν 2 2 be calculated according to the following formulas: k10 (∓1) = ± 4√1 2 ψ2 (∓1), k20 (∓1) = ± 4√1 2 ψ3 (∓1).

(15.5)

15.3 Some Results for an Interface Crack The current analysis is focused on the case of cracks small compared to the contact region size and located far from the contact region compared to the crack size. In this case, the influence of the interface crack on the contact pressure is negligibly small. The numerical results obtained for the stress intensity factors at the crack tips by the described above procedure for the case when both coatings are made of the same material as the substrate were practically identical to the results obtained analytically by a perturbation method for small subsurface cracks in (Kudish and Covitch 2010). For the further analysis, we will fix the half-thickness of the interface coating as h 2 = 0.05 and crack half-length as l = 0.01, as well a = 1 and P0 = 1. This crack half-length is chosen in such a way that k10 (−1) and k10 (1) as well as k20 (−1) and k20 (1) practically coincide outside of a narrow region of c values for which the interface crack is just partially open/close, respectively. The thickness of the upper coating we will consider much greater than the one of the interface layer. Therefore, we will consider the following range for 0.2 ≤ h 1 ≤ 3. Under these assumptions, the typical behavior of the stress intensity factors k10 and k20 as functions of the distance between the crack and the contact region c is shown in Figs. 15.2 and 15.3. Therefore, as the punch moves from right to left, the crack behaves as follows: when the punch is to the right of the crack, above it or a little to the left of it the crack is completely closed, as the punch continues to move to the left the crack quickly goes through the stage when it opens up partially and then with further punch movement to the left, the crack opens up completely and opens wider and wider, then the crack opening reaches its maximum and starts to diminish as the punch continues to move to the left. In Figs. 15.2 and 15.3, the curves of k10 and k20 are calculated for different values of the upper coating thickness h 1 and some specific values of the elastic characteristic of the materials of the coatings and the substrate. It is important to note that for a fixed thickness h 1 the variation in elastic characteristics mostly causes the change in values of k10 and k20 while the change in the location of the curves along the c-axis is insignificant. Also, it should be noted that the curves of k20 may behave monotonically with respect to c as c moves to the left of the minimum of k20 or to the right of the maximum of k20 (see Fig. 15.2) or first cross the c-axis and only after that diminish and vanish. There is a very important question concerning the path of crack propagation caused by application of cyclic loading as it determines the fatigue life of such a coated solid.

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Fig. 15.2 Typical graphs of the normal stress intensity factor k10 at the tips of a small interface crack versus its position c relative to the contact region

Fig. 15.3 Typical graphs of the tangential stress intensity factor k20 at the tips of a small interface crack versus its position c relative to the contact region

We will consider only the initial change of the direction of the crack propagation at the crack tips. According to (Kudish and Covitch 2010; Panasyuk et al. 1976), the change of the angle of crack propagation θ is determined as follows: θ = 2 arctan

k10 −

 2 2 k10 + 8k20 4k20

.

(15.6)

From this formula, it is clear that the crack kinks if θ = 0 which occurs only when k20 = 0 and the crack continues to grow along its initial path when θ = 0 and k20 = 0.

(15.7)

Now, we can raise a question for the given elastic characteristics of the coatings and substrate to find such a distance c between the punch and the crack for which k10 will have its maximum value while k20 = 0. If it is possible then this situation depends on the upper coating thickness h 1 . Obviously, in this case under the action of cyclic loading, the crack will not kink and continues to grow in the initial direction of the interface. Otherwise, right away the initial interface crack will kink and grow across the interface layer toward the boundary with the upper coating.

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(a) E1 /E2 = 2.2, E3 /E2 = 1.5, ν1 = 0.15, and ν3 = 0.36.

I. I. Kudish et al.

(b) E1 /E2 = 4.2, E3 /E2 = 3, ν1 = 0.15, and ν3 = 0.4.

max (solid line) and k 0 (dashed line) at Fig. 15.4 Graphs of the normal stress intensity factors k10 10 the tips of a small interface crack versus the thickness h 1 of the upper coating for different interface material layer Poisson’s ratios ν2 = 0.2 and ν2 = 0.3

Now let us try to solve the following problem: find such a combination of elastic characteristics of the coatings and substrate for which the condition when the maximum of k10 is reached where k20 = 0 is satisfied for almost any thickness h 1 of the upper coating. Let us assume that (a) Young’s modulus of the upper coating is higher than Young’s modulus of the substrate and (b) Young’s modulus of the upper coating and substrate are greater than Young’s modulus of the interface material layer, i.e., E 1 /E 3 > 1, E 1 /E 2 > 1, and E 3 /E 2 > 1. After extensive numerical calculations, certain combinations of elastic characteristics of the coatings and substrate were found which represent an approximate solution to the formulated problem. The max , and the curves of curves of the dependence of the maximum of k10 , called k10 0 the value of k10 where k20 = 0, called k10 , versus thickness h 1 of the upper coating for different elastic characteristics are presented in Fig. 15.4a, b where solid lines max 0 while dashed lines to k10 . correspond to curves of k10 It follows from Fig. 15.4a, b that the influence of Poisson’s ratio ν2 of the interface max 0 and k10 occurs mostly for the material layer on the proximity of the curves of k10 small values of the upper coating thickness h 1 for which the interface crack may kink while even a small variation of one of the parameters shown in captions causes a noticeable divergence of the above curves and leads to the interface crack kinking. The numerical results presented in this section show that there are sets of elastic and geometric parameters of our system for which the system being subjected to cyclic loading will lead to the interface crack propagation along its initial direction, i.e., horizontally.

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15.4 Statistical Model of Coating Delamination The process of coating delamination along its interface with the substrate is statistical in nature. In our model, the whole lower coating will be playing the role of the interface as its thickness 2h 2 is much smaller than the thickness h 1 of the upper coating. We will consider the statistical description of the delamination process. We will assume that coating delamination can develop only at a location of some interface imperfection such as crack, void, and inclusion. Moreover, we will assume that the interface imperfections that eventually caused coating delamination quickly become straight cracks along the middle line of the lower coating of the lengths approximately equal to the lengths of the original interface imperfections. In other words, we will assume that the crack initiation process is much shorter than the crack propagation process. Within the bounds of the above model by the coating delamination process, we will call the process of interface crack propagation along its original orientation (i.e., along the middle line of the lower coating) until it reaches its critical size, becomes unstable, and coating delamination occurs. It was shown above that under certain conditions, the assumption that the crack fatigue growth occurs along its initial direction is correct, i.e., fatigue cracks propagate parallel to the interface boundary. For other conditions, it is possible to expect cracks to kink and grow toward the interface boundary between the two coatings. Here we will consider only the former case of cracks growing parallel to the interface, i.e., the case of K I -mode growth. Here we need to consider two different levels of interface defect “infestation”, i.e., different level of knowledge about initial interface crack distribution. In case of low level of the initial knowledge of interface defect “infestation” (Case B), we will assume that we know the initial crack distribution f (0, l0 ) versus crack half-length l0 in the whole set of M coated samples but we do not know any specifics about this distribution in each particular sample from the set. In this case, our goal is to develop a statistical model which would predict the survival probability p(N ) of this set of M samples subjected to N repeating identical loading cycles. In case of advanced level of the initial knowledge of interface defect “infestation” (Case A), we will assume that we know the initial crack distribution f (0, x, l0 ) versus crack half-length l0 along the interface in each coated sample. In this case, our goal is to develop a model capable of determining the survival probability p(N , x) of each coated sample at point −∞ < x < ∞. Obviously, the latter model is more detailed and can be easily used to determine the survival probability P(N ) of the entire set of M coated samples. Specifically, assuming that the first fatigue failure occurs in just one coated sample, it can be shown that Kudish and Covitch (2010) P(N ) = min x p(N , x),

(15.8)

where the minimization is done over all values of the x-coordinate, i.e., −∞ < x < ∞.

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First, let us consider Case A. We will assume that in a small neighborhood of every point x on the interface (which further is called the middle line of the lower coating) compared to the contact half-width a, there are a number of very small imperfections which can be replaced by small cracks with various half-lengths l0 which are far from each other compared to their own sizes. By collapsing this neighborhood to the point x and assigning these small cracks to that point x, we can come up with the distribution of these cracks f (0, x, l0 ) versus crack half-length l0 for every point x such that ∞

f (0, x, l0 )dl0 = n(0, x),

(15.9)

0

where n(0, x) is the linear density of cracks at point x at the initial time moment N = 0. We will assume that we have a not necessarily uniform distribution f (0, x, l0 ) along the interface, i.e., versus x. The number of small interface cracks at x practically does not change while the sample is subjected to N loading cycles. This statement is based on three assertions: (a) the small cracks in the small neighborhood of x are far from each other compared to their own sizes means that they practically do not interact with each other and can be considered as single cracks at x; (b) during most of their life these small cracks grow very slowly (see below) and, therefore, they practically never join one another; and (c) even if some new cracks will be developed in the process of cycling loading these new cracks will be much smaller than the previously existed ones and they will grow much slower than the previously existed ones. Therefore, neglecting the newly developed cracks, we can conclude that the number of cracks at x after N loading cycles over d N loading cycles does not change, i.e., f (N , x, l)dl = f (0, x, l0 )dl0 ,

(15.10)

where dl is the crack half-length growth over d N loading cycles. Calculation of f (N , x, l) can be done from (15.10) as follows: f (N , x, l) = f (0, x, l0 )/ dldl0 ,

(15.11)

where dl/dl0 is determined by the crack growth process. Integrating both sides of (15.10) over (0, ∞), one obtains that the linear density of cracks n(N , x) after N loading cycles is equal to the initial one, i.e., n(N , x) = n(0, x).

(15.12)

To determine dldl0 we will use the simplest experimentally established Paris law (Kudish and Covitch 2010) for fatigue crack growth. We can simplify this calculation by taking into account that for the most of their lives, interface cracks remain small. It means that for any crack at its position, the stress intensity factors √ at both crack tips are practically equal to each other, i.e., k1 (±1)(N , x) = k10 (x) l + . . . , l = l(N ).

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Then the initial value problem for Paris equation can be represented as follows: dl dN

E2 n n/2 = g0 K 10 l , K 10 = maxx k10 (x) 1−ν 2 , l(0) = l 0 , 2

(15.13)

where g0 is the coefficient of proportionality, g0 > 0; n is the power, n > 2; and x varies between −∞ and ∞. Both constants g0 and n are determined experimentally. It is clear from (15.13) that smaller cracks grow slower than the larger ones. The solution of problem (15.13) has the form 2  − n−2   1−n/2 n l(N ) = l0 + 1 − n2 g0 K 10 N .

(15.14)

It is clear from (15.14) that for initially small cracks (i.e., for small l0 ) the cracks remain small (i.e., l(N ) small) as long as the number of loading cycles N satisfies the inequality 1−n/2 2 1 K −n l . N n−2 (15.15) g0 10 0 For n > 2 and small l0 , the right-hand side in (15.15) may be quite large which substantiates the claim that small cracks remain small during large periods of loading cycling. Using (15.14), one obtains dl dl0

−n/2

= l0

n  − n−2   1−n/2 n l0 + 1 − n2 g0 K 10 N .

(15.16)

Now, using (15.11), (15.14), and (15.16), one can actually calculate the crack distribution f (N , x, l) after any number of loading cycles N and for any initial crack half-length l0 . Moreover, it is clear that with increased number of loading cycles N , the crack distribution f (N , x, l) stretches in the direction of larger values of l. As it was mentioned before, the interface cracks may grow until they reach the critical size for which their stress intensity factor reaches the value of fatigue toughness K f , i.e., the critical crack half-length lc at the point where k10 (x) reaches its maximum K 10 is equal to  2 K (15.17) lc = K f . 10

Based on that we can propose the definition of the survival probability p(N , x) of the sample material at point x after N loading cycles as the number of cracks below the critical length relative to the whole number of cracks at x. Therefore, we will get p(N , x) =

1 n(N ,x)

lc

f (N , x, l)dl i f f (N , x, l) = 0,

0

p(N , x) = 1 i f f (N , x, l) ≡ 0.

(15.18)

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Let us introduce the initial crack half-length l0c (N ) which after application of N loading cycles would grow to the critical crack half-length lc at the point where k10 (x) reaches its maximum K 10 . Then using (15.14), one will obtain that 2  − n−2  − n−2  n l0c (N ) = lc 2 + n2 − 1 g0 K 10 N .

(15.19)

After that using (15.10), (15.12), and converting integration in l into integration in l0 from (15.18), one gets a simplified expression for the survival probability p(N , x) =

1 n(0,x)

l0c (N  ,x)

f (0, x, l0 )dl0 i f f (0, x, l0 ) = 0,

0

(15.20)

p(N , x) = 1 i f f (0, x, l0 ) ≡ 0. Assuming that the first coating delamination event occurs at just one point x, it can be shown that the survival probability of the sample as a whole can be determined from (15.8) (Kudish and Covitch 2010). Formula (15.20) clearly shows the dependence of the survival probability p(N , x) on the initial crack distribution f (0, x, l0 ) and crack growth mechanism as well as the critical crack size through (15.19). In Case B, function f is independent of x. Therefore, in Case B, we get the same formula for the survival probability P(N ) of the whole set of M coated samples as it is described by formula (15.20) in which the dependence on x is gone and in the left-hand side we would have P(N ), i.e., P(N ) =

1 n(0)

l0c(N )

f (0, l0 )dl0 i f f (0, l0 ) = 0,

0

(15.21)

P(N ) = 1 i f f (0, l0 ) ≡ 0. To be able to make specific calculations, let us assume that the initial crack distribution is described by a log-normal one, i.e., f (0, x, l0 ) = 0 i f l0 < 0, f (0, x, l0 ) =

√ n(0,x) 2πσln (x)l0

 2 ln (x) i f l0 > 0, exp − 21 ln(l0σ)−μ ln (x)

(15.22)

where μln and σln are the mean and standard deviation of the initial interface crack distribution at x determined as follows (Kudish and Covitch 2010):  μln = ln √

μ2

μ2 +σ 2

, σln =

ln[1 +

 2 σ μ

],

(15.23)

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where μ and σ are regular mean and standard deviation of the initial interface crack distribution. Using (15.20) and (15.22), one obtains p(N , x) =

1 2

   ln (x) 1 + er f ln(l0c√(N2σ))−μ , (x)

(15.24)

ln

where er f (x) is the error integral (Abramowitz and Stegun 1972). In Case B in (15.24) in the left-hand side we will have P(N ) while x is gone from the right-hand side. Therefore, for the given crack mean μln (x) and standard deviation σln (x), the probability of survival p(N , x) can be determined at any point x after N loading cycles. Using (15.8), one can find the survival probability P(N ) from the formula P(N ) =

1 2

   ln (x) 1 + min x er f ln(l0c√(N2σ))−μ , (x)

(15.25)

ln

where minimum is taken over the interval (−∞, ∞). To determine the sample delamination life N with survival probability P∗ , one has to solve the equation P(N ) = P∗ .

(15.26)

The derivation that follows is valid for Case B as well. Let us assume that μln and σln are constants. Then the operation of min can be dropped and the solution of (15.26) can be represented in the form n −1 N = [( n2 − 1)g0 K 10 ] {exp[(1 − n2 )(μln 2−n √ + 2σln er f −1 (2P∗ − 1))] − lc 2 },

(15.27)

where er f −1 (x) is the inverse function of the error integral er f (x). Discounting the tail of the initial crack distribution with high crack lengths, we assume that the half-length of all initial cracks is smaller than lc . Due to the fact that most of the delamination life N cracks remain small and only over the last very short period of time grow exponentially fast (Kudish and Covitch 2010), we can conclude that over loading cycles cracks remain much shorter than their critical half-length lc . Therefore, if we neglect the short period of crack explosive growth than the delamination life N from (15.27) if practically independent of fracture toughness K f and critical half-length lc . Neglecting the last term in (15.27), we get (Kudish and Covitch 2010) N=

1

n ( n2 −1)g0 K 10

 exp

  √  1 − n2 (μln + 2σln er f −1 (2P∗ − 1)) .

(15.28)

Taking into account formulas (15.23), (15.28) and the fact that K 10 is proportional to the characteristic Hertzian pressure p H and, also, depends on the friction coefficient λ, interface position h 1 + h 2 , elastic properties of the coating, substrate, and the interface layer, formula (15.28) can be rewritten in the form

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Fig. 15.5 Graphs of the normal stress intensity max (solid line) and factors k10 0 k10 (dashed line) at the tips of a small interface crack versus the thickness h 1 of the upper coating for the basic set of values of the input parameters

N=

Cd (n−2)g0 pnH

(

√

μ2 +σ 2 n −1 ) 2 {exp[(1 μ2

− n2 )(ln √ μ2

2

μ +σ 2

 + 2 ln[1 + ( σμ )2 ]er f −1 (2P∗ − 1))]},

(15.29)

where Cd is a constant which depends on λ, interface position h 1 + h 2 , elastic properties of the coating, substrate, and the interface layer. If we assume that σ μ then (15.29) leads to N=

Cd {exp[(1 n (n−2)g0 pnH μ 2 −1

− n2 )

√ 2σ er f −1 (2P∗ μ

− 1)]}.

(15.30)

A simple analysis of formula (15.30) shows that for n > 2 the delamination life N decreases with increase in coefficient g0 from crack propagation equation, characteristic Hertzian pressure p H , mean μ, and standard deviation σ. Let us consider some specific examples of the coating delamination life calculation. As it was mentioned earlier, we will be concerned only with the case when the interface cracks continue to propagate along their original direction, i.e., parallel to the boundary of the coatings and substrate. That occurs when the maximum of max 0 = k10 . That k10 (x) is reached at the point x where k20 (x) = 0 in which case k10 condition imposes certain limitations on the selection of the system elastic characteristic and geometry as well as the coefficient of friction. Due to the fact that this condition takes place for certain combinations of the above problem input parameters as a base combination we will take one of these combinations. Specifically, we will assume the following basic set of values for the input parameters: applied normal load per unit length P = 1.45 × 103 MPa · m; contact half-width a = 50 µm; friction coefficient λ = 0.002; Young’s modulus E 1 = 5.7 × 103 MPa, E 2 = 3.8 × 103 MPa, E 3 = 7.6 × 103 MPa; and Poisson’s ratios ν1 = 0.13, ν2 = 0.27, ν3 = 0.4 of the coatings and substrate, respectively; interface material layer fracture toughness K f = 15 MPa · m1/2 ; coefficient of proportionality in Paris equation g0 = 15 × 10−7 MPa−n · m1−n/2 · cycle−1 ; power in Paris’ equation n = 7; mean of the interface crack

15 Contact Problem for a Coating/Substrate Interface Crack Under Action …

(a) h1 = 0.2 - curve 1, h1 = 0.5 - curve 2, h1 = 1 - curve 3 and h1 = 2.5 - curve 4

365

(b) λ = 0.001 - curve 1, λ = 0.002 curve 2 and λ = 0.0025 - curve 3

Fig. 15.6 Graphs of the delamination life probability 1 − P(N ) versus number of loading cycles N for different values of the upper coating thickness h 1 and friction coefficient λ

(a) σ = 2μm curve 1, σ = 4μm curve 2 and σ = 6μm curve 3

(b) μ = 5μm curve 1, μ = 10μm curve 2 and μ = 15μm curve 3

Fig. 15.7 Graphs of the delamination life probability 1 − P(N ) versus number of loading cycles N for different values of the mean μ and standard deviation σ of the initial interface crack distribution

initial half-lengths µ = 10 µm; and standard deviation of the interface crack initial half-lengths σ = 5 µm. In the dimensionless variables for h 1 = 0.2, the maximum of max−∞ 0, p(x1 , x2 ) = 0 ⇒ ∀(x1 , x2 ) ∈ / Ac

where p(x, y) is the contact pressure and h(x, y) stands for the gap between two surfaces of contact bodies, for the element centered at (x, y); Ωc denotes the contact region. For elastic contact of the heterogeneous material employing the inclusion theory, gap h(x 1 , x 2 ) has a form as b,p

h(x1 , x2 ) = h 0 (x1 , x2 ) + u 3 (x1 , x2 ) + u b,s 3 (x 1 , x 2 ) m,p m,s + u 3 (x1 , x2 ) + u 3 (x1 , x2 ) + u ∗3 (x1 , x2 ) − δ

(18.3) b,p

in which, h0 (x, y) is initial gap between two surfaces of contact bodies; u 3 (x, y) and u b,s 3 (x, y) stand for surface normal displacement of the spherical indenter induced m,p by normal pressure p(x 1 , x 2 ) and shear traction s(x 1 , x 2 ); u 3 (x, y) and u m,s 3 (x, y) denote surface normal displacement of the homogeneous half space produced by normal pressure p(x 1 , x 2 ) and shear traction s(x 1 , x 2 ); u ∗3 represents normal displacement resulting from eigenstrain within the inclusion; δ denotes the relative rigid approach between the spherical indenter and the heterogeneous half space with embedded inhomogeneities. Gap calculation is a necessary step for contact analyses, where displacement involved in Eq. (18.3) needs to be obtained.

18.2.1 Equivalent Inclusion Method in Contact Problems Eshelby’s celebrated equivalent inclusion method (EIM) (Eshelby 1959; Eshelby and Peierls 1957) is proposed innovatively by modeling inhomogeneity as a corresponding homogeneous inclusion with appropriately selected eigenstrains. By employing the EIM, the solution of contact problems involving an inhomogeneitycontaining half-space consists of three portions: (1) the contact solution (cf. Sect. 18.2.2), (2) the inclusion solution to the elastic fields due to eigenstrains (cf. Sect. 18.2.3), and (3) the determination of the unknown eigenstrains. The consistency condition of EIM for eigenstrain determination is enforced at the inhomogeneity domain:  C   C  ∗ = Ci∗jkl εkl + εkl − εkl + εkl in  Ci jkl εkl

(18.4)

C is the strain due to the contact load, εkl is the strain disturbance caused by where εkl ∗ is the unknown eigenstrain in the equivalent inclusion. the inhomogeneity and εkl After manipulation, Eq. (18.4) may be recast into the following form:

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 C  ∗ Ci jkl εkl = Hi jkl εkl + εkl in 

(18.5)

where Hi jkl = −Ci−1 jmn C mnkl , and C mnkl is the difference between the inhomogeneity and matrix stiffnesses. Equation (18.5) can also be expressed in terms of compliance modulus Di jkl = Ci−1 jkl as   εi∗j = Di jkl σklC + σkl in 

(18.6)

where σklC is the stress induced by the contact load, σkl is the disturbed stress caused by the inhomogeneity, Di jkl is the compliance difference of the two materials, i.e., Di jkl = Di∗jkl − Di jkl , and Di∗jkl , Di jkl are the compliance moduli for the inhomogeneity and the matrix, respectively. Either Eqs. (18.5) or (18.6) is sufficient for the determination of eigenstrain εi∗j and the EIMs with respect to Hi jkl and Di jkl are appropriate for problems with different material combinations. In order to simplify the implementation of the EIM, Dundurs’ parameters (Dundurs 1968) are introduced. For ease of numerical implement, those tensorial equations may be represented in matrix forms (Zhou et al. 2016a): ⎤ 0 0⎥ ⎥ ⎥ 0⎥ Hi jkl ⎥ 0⎥ ⎥ 0⎦ η3 κ1 + 1 k= 2[(α − 2β − 1)(κ1 − 1) − 4β][(α − β)κ1 − β − 1] η1 = (κ1 − 1)(α − β)(α − 4β + 1) − β(6α − 8β − 2) η2 = −(ακ1 − βκ1 − β − α)(1 + α) ⎡

η1 ⎢η ⎢ 2 ⎢ ⎢η =k·⎢ 2 ⎢0 ⎢ ⎣0 0

η2 η1 η2 0 0 0

η2 η2 η1 0 0 0

0 0 0 η3 0 0

0 0 0 0 η3 0

η3 = [(α − 2β − 1)(κ1 − 1) − 4β](1 + α)

(18.7)

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In a similar manner, Di jkl could also be written as ⎡

Di jkl

γ1 ⎢γ ⎢ 2 κ1 + 1 ⎢ ⎢ γ2 =− ⎢ 4μ1 (1 + α) ⎢ 0 ⎢ ⎣0 0

γ2 γ1 γ2 0 0 0

γ2 γ2 γ1 0 0 0

0 0 0 γ3 0 0

0 0 0 0 γ3 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ γ3

γ1 = α γ2 = 2β − α γ3 = 4(α − β)

(18.8)

where α and β are Dundurs parameters defined as follows (Dundurs 1968): α=

μ2 (κ1 − 1) − μ1 (κ2 − 1) μ2 (κ1 + 1) − μ1 (κ2 + 1) ,β= μ2 (κ1 + 1) + μ1 (κ2 + 1) μ2 (κ1 + 1) + μ1 (κ2 + 1)

(18.9)

Note that in Eq. (18.9), μ1 and μ2 are shear moduli of the matrix and inhomogeneity, respectively, and κ1 , κ2 are Kolosov’s constants for the two materials. Inspired by the works by Hutchinson (1987) and Shi and Li (2006), an iterative process is proposed for implementing the EIM. As shown in Fig. 18.3, the present method is a combination of the two schemes. For the cases of stiff matrix with much compliant inhomogeneities, the compliance-difference-based scheme using Eq. (18.6) may generally encounter a convergence problem; and on the other hand, the same obstacle occurs when the stiffness-difference-based scheme (cf. Eq. (18.5)) is utilized to deal with the problems involving stiff inhomogeneity surrounded by much compliant matrix. Therefore, the proper computational scheme should be selected according to the comparison between α and β, and the proposed iterative method can be capable of handling almost all different material combinations (Zhou et al. 2016a, 2015).

18.2.2 The Elementary Solution of the Elastic Field Regarding the displacement field (Fig. 18.4) for a homogeneous half space, the elementary solutions for rectangular contact was first studied by Love and later revisited by Jin et al. (2016b) for the complete elastic fields. Following their presentation (Jin et al. 2016b), the displacements can be expressed as ⎤ ⎡ ⎤⎡ ⎤ Φ11 Φ12 Φ13 p10 u1 ⎣ u 2 ⎦ = ⎣ Φ21 Φ22 Φ23 ⎦⎣ p20 ⎦ u3 Φ31 Φ32 Φ33 p30 ⎡

(18.10)

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Fig. 18.3 Flowchart for the proposed iterative method for a single inhomogeneity

where u i , (i = 1, 2, 3) are the components of the elastic displacement at any field point(x1 , x2 , x3 ); p j0 , ( j = 1, 2, 3), is the magnitude of the uniform load along the x j direction applied over a rectangular patch of dimension1 ×2 ; and Φi j is termed as the elementary influence coefficient. Each Φi j is related to a primitive function,ϕi j , and is represented by Fig. 18.4 Schematic of uniformly distributed loads applied over a rectangular patch (|x 1 |≤1 /2, |x 2 |≤2 /2) on the surface of the half-space, where p10 , p20 and p30 stand for constant traction loadings along the x 1 , x 2 , and x 3 directions, respectively

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Φi j =

 1 · ϕi j (x1 , x2 , x3 )[1 , 2 ], (i, j = 1, 2, 3) 4π μ

407

(18.11)

where μ is shear modulus of material. Note that in Eq. (18.11), the following short hand notation is invoked: ϕi j (x1 , x2 , x3 )|[x1 , x2 ] =

2 2

  (−1)α+β ϕi j x1α , x2β , x3 ,

α=1 β=1

⎧ x1 ⎪ ⎨ x1α = x1 + (−1)α 2 , with ⎪ ⎩ x = x + (−1)β x2 2β 2 2

(18.12) (α, β = 1, 2)

The notation on the left hand side of Eq. (18.12) contains all the necessary information to evaluate the response at the field point x(x1 , x2 , x3 ), due to a constant excitation distributed over a rectangular patch of dimensions1 × 2 , centered on   , x20 , 0), then Eq. (18.11) can the origin. If the rectangular patch is centered at (x10 be further written as Φi j =

   1  · ϕi j (x1 , x2 , x3 ) x10 , x20 ; 1 , 2 , 4π μ

(i, j = 1, 2, 3)

(18.13)

With Voigt notation (Jin et al. 2016b), the elementary solution of the stress field at any field point inside the half-space is expressed as follows: ⎤ ⎡ 11 σ1 ⎢σ ⎥ ⎢ ⎢ 2 ⎥ ⎢ 21 ⎢ ⎥ ⎢ ⎢ σ3 ⎥ ⎢ 31 ⎢ ⎥=⎢ ⎢ σ4 ⎥ ⎢ 41 ⎢ ⎥ ⎢ ⎣ σ5 ⎦ ⎣ 51 σ6 61 ⎡

12 22 32 42 52 62

⎤ 13 23 ⎥ ⎥⎡ p ⎤ ⎥ 33 ⎥⎣ 10 ⎦ ⎥ p 43 ⎥ 20 ⎥ p 53 ⎦ 30 63

(18.14)

where the elementary influence coefficient i j is associated with primitive function λi j and is represented as: i j =

 1 · λi j (x1 , x2 , x3 )[1 , 2 ], 2π

(i = 1, 2, ..., 6; j = 1, 2, 3)

(18.15)

For convenience of references, the detailed expressions for the primitive functions in Eqs. (18.11) and (18.15) are given in Appendix 1.

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18.2.3 Elementary Solution for Inclusion in Half-Space For the purpose of handling the contact problems involving arbitrarily shaped inhomogeneities, the region enclosing all the influential inhomogeneities is generally meshed into a group of N x × N y × N z cuboidal elements. The eigenstrain distributions within individual equivalent inclusions are then calculated through a numerical EIM (NEIM) process, which will be explained in detail later (cf. Sect. 18.3.1). Before the discussion of the NEIM, it is necessary to introduce the cuboidal elementary solution first, since the inclusion solution could be one of the core components for implementing the NEIM with the iterative method shown in Fig. 18.3. The method of images proposed by (Chiu 1978) has been widely used (Zhang et al. 2022; Zhang et al. 2020) to solve the elastic fields due to inclusions beneath a surface through a superposition approach. However, the method of images is indirect methods, with which surface domain truncations is complex and leads to inevitable truncation errors. Based on the Galerkin vector, (Liu et al. 2012) derived the direct analytical solution method, which is utilized in this section. For a cuboidal inclusion within a half-space, as shown in Fig. 18.5, the stress disturbance is expressed as: ⎤ ⎡ γ11 σ1 ⎢σ ⎥ ⎢γ ⎢ 2 ⎥ ⎢ 21 ⎢ ⎥ ⎢ ⎢ σ3 ⎥ ⎢ γ31 ⎢ ⎥=⎢ ⎢ σ4 ⎥ ⎢ γ41 ⎢ ⎥ ⎢ ⎣ σ5 ⎦ ⎣ γ51 σ6 γ61 ⎡

γ12 γ22 γ32 γ42 γ52 γ62

γ13 γ23 γ33 γ43 γ53 γ63

γ14 γ24 γ34 γ44 γ54 γ64

γ15 γ25 γ35 γ45 γ55 γ65

⎤⎡ ⎤ ∗ ε10 γ16 ⎢ ∗ ⎥ γ26 ⎥ ⎥⎢ ε20 ⎥ ⎥⎢ ∗ ⎥ γ36 ⎥⎢ ε30 ⎥ ⎥⎢ ∗ ⎥ ⎥ γ46 ⎥⎢ ε40 ⎥⎢ ∗ ⎥ ⎦ γ56 ⎦⎣ ε50 ∗ γ66 ε60

(18.16)

where the Voigt notation (Jin et al. 2016b) is adopted, and thus the stress components and eigenstrains are represented as  ∗ ∗ ∗ ∗ ∗ ∗ T σ = [σ1 , σ2 , σ3 , σ4 , σ5 , σ6 ]T , ε ∗0 = ε10 , ε20 , ε30 , ε40 , ε50 , ε60

(18.17)

Note that the elementary influence coefficient γi j is associated with primitive function Ti j and could be represented using an effective notation proposed by (Jin Fig. 18.5 Schematic of a cuboidal inclusion with uniformly distributed eigenstrain in an elastic half-space

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et al. 2016b): γi j =

   −μ (1) (2) 2 (3)     + T + x T + x T · Ti(0) 3 3 i j  x 10 , x 20 , x 30 ; 1 , 2 , 3 j ij ij 4π(1 − ν) (18.18)

where μ is the shear modulus and ν is Poisson’s ratio of the material. The detailed (1) (2) (3) expressions of the primitive function, i.e., Ti(0) j , Ti j , Ti j and Ti j , are given in Appendix 2. On the other hand, the elementary solution of the displacement field may be presented in the matrix form as follows: ⎡

⎤ ⎡ N11 N12 N13 N14 N15 u1 ⎣ u 2 ⎦ = ⎣ N21 N22 N23 N24 N25 u3 N31 N32 N33 N34 N35 ⎡

⎤ ∗ ε10 ∗ ⎥ ⎤⎢ ε20 ⎥ N16 ⎢ ⎢ ∗ ⎥ ε ⎢ ⎥ N26 ⎦⎢ 30 ∗ ⎥ ⎢ ε40 ⎥ N36 ⎢ ∗ ⎥ ⎣ ε50 ⎦ ∗ ε60

(18.19)

where each Ni j is associated with a primitive function, Ui j , and similar to Eq. (18.18), could be represented as Ni j =

   −1 (1) (2) 2 (3)     · Ui(0) j + Ui j + x 3 Ui j + x 3 Ui j  x 10 , x 20 , x 30 ; 1 , 2 , 3 8π(1 − ν) (i = 1, 2, 3; j = 1, 2, ..., 6) (18.20)

(1) (2) (3) The detailed expressions of the primitive function, i.e., Ui(0) j , Ui j , Ui j and Ui j , are available in Appendix 2.

18.3 Numerical Solutions for Contact of Heterogeneous Materials 18.3.1 Numerical EIM for Multiple Inhomogeneities During the enforcement of the NEIM, the computational domain of interest is discretized into a total number of N x × N y × N z cuboidal elements, as shown in Fig. 18.6. Each elementary cuboid has the same size of 1 , 2 and 3 in the x1 , x2 and x3 directions, respectively. When the consistency condition of EIM is enforced at ∗ , inside each element is assumed approxeach individual element, the eigenstrain, εkl imately uniform. Further, according to the superposition principle, the elastic field

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Fig. 18.6 Discretization of the computational domain into cuboidal elements

could be obtained by summation of contributions from all the inclusion elements, which have been discussed in Sect. 18.2.

18.3.2 The Conjugate Gradient Method Assume that one is attempting to solve the following system of linear equations ⎡

a11 ⎢ a21 ⎢ ⎢ . ⎣ .. an1

⎤⎡ ⎤ ⎡ ⎤ x1 b1 a12 · · · a1n ⎢ ⎢ ⎥ ⎥ a22 · · · a2n ⎥⎢ x2 ⎥ ⎢ b2 ⎥ ⎥ .. . . .. ⎥⎢ .. ⎥ = ⎢ .. ⎥ . . ⎦⎣ . ⎦ ⎣ . ⎦ . an2 · · · ann xn bn

(18.21)

For simplicity, the above Eq. (18.21) might alternatively be written as AX = B

(18.22)

where the coefficients of Eq. (18.21) make up the real nonsingular n × n square matrix A. Let AT denote the transpose of A, if A = AT , matrix A is symmetric. Additionally, a matrix A is positive-definite if, for every nonzero vector X, XT AX > 0

(18.23)

The influence coefficient matrix that appears in the numerical solution of a contact problem is a classic illustration of a symmetric positive-definite (SPD) matrix. It is possible to pre-multiply both sides of Eq. (18.22) by the transposition of A if matrix A in Eq. (18.22) is neither symmetric nor positive-definite. Therefore,

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Eq. (18.22) is changed into the following form: 

ˆ = Bˆ AX ˆ = AT A and Bˆ = AT B where A

(18.24)

One can easily demonstrate that A is symmetric and positive-definite using elementary matrix operations (SPD). Without losing generality, the coefficient matrix A is assumed to be SPD in the following. Two non-zero vectors Di and D j are referred to as conjugate (with respect to A) or A-orthogonal, if DiT AD j = 0, for i = j

(18.25)

Assume that {Di , i = 0, 1, 2, ..., n −1} is a set of n mutually conjugate directions, which forms the basis forRn . Let X represent the true solution of Eq. (18.22), i.e., X = A−1 B

(18.26)

In this context, expanding X produces X=

n−1

αi Di

(18.27)

i=0

The coefficients are identified as follows from Eqs. (18.22) and (18.25) αi =

DiT B DiT ADi

(18.28)

From a different perspective, resolving Eq. (18.22) is similar to minimizing the quadratic functional φ(X), which is defined by φ(X) =

1 T X AX − XT B 2

(18.29)

To demonstrate this, consider vector E as an error term, after some manipulation (Shewchuk 1994), 1 φ(X + E) = φ(X) + ET AE 2

(18.30)

The final component on the right side of Eq. (18.30) is nonnegative since A is SPD, Since A is SPD, and the minimum value of φ is −BT A−1 B/2, obtained by setting X = X = A−1 B.

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An iterative solution begins with an initial estimate X(0) of the true solution X, and then one decides successively new estimates X(1) , X(2) , · · · of X until one is satisfactorily close to the solution X, the residual vector is defined as the following at step k + 1 of the procedure R(k+1) = B − AX(k+1)

(18.31)

Moreover, there may be a connection between the new estimate X (k+1) and the preceding X (k) as X(k+1) = X(k) + αk P(k) , k = 0, 1, 2, · · ·

(18.32)

where P (k) is the direction of the search, and the scalar αk represents the step length along this direction. Equation (18.32) is substituted into (18.29) to produce 1 T T AP(k) − αk P(k) R(k) φ(X(k+1) ) = φ(X(k) ) + αk2 P(k) 2

(18.33)

Consequently, the function φ varies quadratically with αk , and the local minimum is obtained by setting ∂φ/∂αk = 0, which gives αk =

T R(k) P(k) T P(k) AP(k)

(18.34)

Equations (18.27) and (18.32) differ in that Eq. (18.32) has an initial guess term X(0) . If P(k) are A-orthogonal and X (0) ≡ 0 holds, it may be easily proven that equations Eqs. (18.34) and (18.28) are identical. The method of steepest descent is one way for minimizing φ. The function φ decrements most rapidly in the direction of the negative gradient at a current point X(k) . The direction P(k) is given by Eqs. (18.29) and (18.31) P(k) = −∇φ(X(k) ) = B − AX(k) = R(k)

(18.35)

Equation (18.34) thus becomes αk =

T R(k) R(k) T R(k) AR(k)

In conclusion, the steepest descent algorithm is:

(18.36)

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⎧ R(i) = B − AX(i) ⎪ ⎪ ⎪ ⎪ T ⎨ R(i) R(i) αi = T ⎪ R(i) AR(i) ⎪ ⎪ ⎪ ⎩ X(i+1) = X(i) + αi R(i)

413

(18.37)

Unfortunately, the above iterative scheme’s convergence may be unreasonably slow, given the steepest descent, we are compelled to travel back and forth across the valley (the minimal) rather than down the valley (Gene 1996). The search direction is optimized as a benefit of the conjugate gradient method over the steepest descent. With the right set of A-orthogonal search directions and the appropriate step lengths along each route, as demonstrated in Eqs. (18.27) and (18.28), we can arrive at the true solution in n steps. Step i allows for the construction of the current A-orthogonal search directions D(i) using the residual vectors R(i) and search directions created in the preceding iterative steps { D( j) , j = 0, 1, 2, ..., i − 1} through the conjugate Gram-Schmidt process (Shewchuk 1994; Strang 1988). One of the intriguing aspects of the CG method is that with the exception of D(i−1) , R(i) is already A-orthogonal to all of the prior search directions. The i-th search direction is determined D(0) = R(0) D(i) = R(i) + βi−1 D(i−1) , i = 1, 2, ... T T where βi−1 = −D(i−1) AR(i) /D(i−1) AD(i−1)

(18.38)

By induction, Hestenes and Stiefel (1952) demonstrated that the residual vectors are mutually orthogonal, i.e., T R( j) = 0, for i = j R(i)

(18.39)

This property allows for the equivalent expression of the scalar coefficients αk in Eq. (18.34) and βi−1 in Eq. (18.38) by αk =

T R(k) R(k) T P(k) AP(k)

βi−1 =

(18.40)

T T R(i) R(i) /R(i−1) R(i−1)

Finally, the conjugate gradient method algorithm is depicted in the flowchart (Fig. 18.7). The procedure begins with inputs for the SPD matrix A and the vector B. Any initial guess X(0) , the maximum number of iterations i max , and the error tolerance ε < 1 can all be specified. When i has exceeded the maximum number of iterations i max or R(i)  ≤ εR(0)  is satisfied, the algorithm ends.

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Fig. 18.7 Flowchart of the conjugate gradient method

18.3.3 Numerical Solutions for Distributed Inhomogeneities Under a Rough Surface For the rough surface contact of inhomogeneous materials, the external loads will cause elastic deformation in the surface region of pressure distribution. Contact may cause the inhomogeneities to interact with the matrix, leading to concentrated localized stress. Furthermore, the surface may deform further due to the disturbed elastic field in the subsurface, which further affects the distribution of contact pressure on

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the surface. For contact problems involving heterogeneous materials, this interaction between the inhomogeneities and the contact pressure must be thoroughly taken into account. Figure 18.8 illustrates the proposed unified numerical solution scheme for rough surface contact involving materials with distributed inhomogeneities. Before launching the iterative procedure, the initial geometry is input. The surface contact pressure is then calculated using the rough-surface contact solver. A numerical EIM process is then used to calculate the eigenstrain distributions within specific equivalent elementary inclusions. Following the determination of the eigenstrain, the resulting surface deformation due to the eigenstrain and contact loads is solved to update the surface geometry until the eigenstrain-surface deformation interaction process converges to a satisfactory degree.

Fig. 18.8 The numerical solution scheme for heterogeneous rough surface contact

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18.4 Numerical Studies on Heterogeneous Contact Inhomogeneity produces disturbance on the elastic field of the matrix subjected to contact load, further imposing effect on contact behaviors. Heterogeneous contact characteristics tend to be more complex if roughness of contact interfaces is taken into account. This section provides some specific cases, in terms of different inhomogeneity shapes, numbers, distributions and rough surfaces, to demonstrate contact behaviors of heterogeneous material with the NEIM. In the following discussions, spherical indenter with a radius of 100r (r means contact radius for homogeneous material under given load) is far stiffer than the heterogeneous material, hence it is treated as rigid; elastic modulus and Poisson ratio of the matrix are set as 210 GPa and 0.3; the whole calculated domain is selected as 6r × 6r × 3r and meshed into a total number of 144 × 144 × 72 cuboidal elements of identical size.

18.4.1 Smooth Surface In order to clearly state effects of inhomogeneity on contact behaviors and mechanical responses of heterogeneous material, the contact interface is assumed as smooth to eliminate contributions from the rough surface. Therefore, influences of inhomogeneity shapes, numbers and distributions on heterogeneous contact characteristics are discussed in the condition of smooth contact interface for two bodies.

18.4.1.1

Inhomogeneity Shapes

Particles or fibers are reinforcement of composites, which are usually regarded as spheres or ellipsoids in simulation. On the other hand, various defects in the material have different shapes. Effects of three different inhomogeneities of spherical, cuboidal and ellipsoidal shape centered at (0, 0, r) with identical volume on contact behaviors and mechanical responses are investigated. The spherical inhomogeneity has a diameter of r, and three semi-axes of the ellipsoidal inhomogeneity are 0.65r, 0.3r and 0.3r, respectively. The inhomogeneity with higher elastic modulus than the matrix, namely, E i = 2E m is defined as stiff inhomogeneity, or else it is a compliant  inhomogeneity (E i = 0.5E m ). Figure 18.9 is dimensionless contact pressure p ph for heterogeneous contact and surface displacement induced by the inhomogeneity u ∗3 . In the current cases, there are negligible differences in contact pressure distribution and normal surface displacement induced by the inhomogeneity for the spherical and cuboidal inhomogeneity cases, while appreciable discrepancy is identified between the ellipsoidal inhomogeneity and the other two. The reason for this phenomenon may be that the spherical and cuboidal inhomogeneities share a spatial position with minor differences, while the ellipsoidal has a notably different spatial position from the others, thus leading to distinct disturbance on the elastic field

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shown in Fig. 18.9b–d. It can be seen that contact pressure for the heterogeneous material containing stiff inhomogeneity is more concentrated than that for compliant inhomogeneity. Note that inwards normal surface displacement is defined positive in the present investigation. Surface displacement induced by the inhomogeneity illustrated in Fig. 18.9c demonstrates that the stiff inhomogeneity produces negative surface displacement to counteract positive surface displacement arising from contact pressure, thus forming a smaller contact region and further resulting in more concentrated contact pressure than the compliant inhomogeneity. Figure 18.10a–b portrays surface and subsurface von Mises stress of heterogeneous material containing spherical, cuboidal and ellipsoidal inhomogeneity. The stiff inhomogeneity produces greater von Mises stress than the compliant inhomogeneity no matter what shape the inhomogeneity has, which may indicate that the stiff inhomogeneity brings about stress concentration more easily than the compliant inhomogeneity here. The region near the contact center has a bigger von Mises stress for

(a)

(b)

(c)

(d)

Fig. 18.9 Surface contact pressure and displacement induced by inhomogeneity of heterogeneous material involving inhomogeneity of different shapes. a Surface contact pressure, E i = 2E m ; b surface contact pressure, E i = 0.5E m ; c surface displacement induced by inhomogeneity, E i = 2E m ; d surface displacement induced by inhomogeneity, E i = 0.5E m

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the stiff inhomogeneity but smaller for the compliant inhomogeneity. Stress concentration occurs in the vicinity of the upper border between the inhomogeneity and the matrix for the stiff inhomogeneity, while it occurs around the border shoulder for the compliant inhomogeneity. Contact pressure and elastic field caused by the ellipsoidal inhomogeneity have remarkable differences compared with that by the spherical/cuboidal inhomogeneity, which is attributed to their positions. Hence effect of angle of the ellipsoidal inhomogeneity on contact behaviors and mechanical responses is to be revealed by setting three different angles. (α, β, γ ) are angles to describe orientation of an ellipsoidal inhomogeneity, and here angles α and β are fixed at zero, γ set as 0°, 45° and 90°. The results in Fig. 18.11 demonstrate that as the γ increases, contact pressure around the contact center grows greater for the stiff inhomogeneity but becomes smaller for the compliant inhomogeneity. Displacement induced by the inhomogeneity increases with the angle γ . Note that inclined inhomogeneity produces asymmetric surface displacement as depicted in Fig. 18.11c, d, so does the surface von Mises stress shown in Fig. 18.12a, b. It can be seen in Fig. 18.12c, d that the maximum von Mises stress increases with the angle γ , which may be ascribed to the fact that the ellipsoidal inhomogeneity with a greater γ approaches the contact center thus bringing stronger interactions.

18.4.1.2

Inhomogeneity Interactions

Inhomogeneity embedded in materials is usually numerous and interactions among them should be considered. For the sake of a better understanding of interactions among multiple inhomogeneities, only two spherical inhomogeneities with a diameter of r centered at (−d/2, 0, r) and (d/2, 0, r) are considered, where d is distance between two inhomogeneity centers. Three different distances are set as 1.2r, 1.5r and 2r, and simulation results are exhibited in Fig. 18.13. An augment in d leads to a decline in contact pressure around the contact center for the stiff inhomogeneity but an increase for the compliant inhomogeneity. Displacement induced by the inhomogeneity decreases with the inhomogeneity center distance no matter what the inhomogeneity elastic modulus is, indicating that the interactions between the two inhomogeneities becomes weaker. Similar conclusion can be made from Fig. 18.14 that a closer distance means stronger interactions, thus resulting in a greater von Mises stress. Another phenomenon observed is that the major stress concentration region occurs in the matrix beneath the contact center for a greater d.

18.4.1.3

Inhomogeneity Distributions

Inhomogeneities are usually randomly distributed in the engineering material with different volume fractions. In the present investigation, the stiff spherical inhomogeneities are  assumed  tofollow normal   distribution, which is characterized by  x ∼ N x, dx2 , y ∼ N y, d y2 , z ∼ N z, dz2 . N represents normal distribution; x, y,

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Fig. 18.10 Surface and subsurface von Mises stress of heterogeneous material involving inhomogeneity of different shapes. a Surface von Mises stress along x aixs, E i = 2E m ; b surface von Mises stress along x aixs,E i = 0.5E m ; c subsurface von Mises stress, spherical inhomogeneity; d subsurface von Mises stress, cuboidal inhomogeneity; e subsurface von Mises stress, ellipsoidal inhomogeneity

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(a)

(b)

(c)

(d)

Fig. 18.11 Surface contact pressure and displacement induced by inhomogeneity of heterogeneous material involving ellipsoidal inhomogeneities of different angles. a Surface contact pressure, E i = 2E m ; b surface contact pressure, E i = 0.5E m ; c surface displacement induced by inhomogeneity, E i = 2E m ; d surface displacement induced by inhomogeneity, E i = 0.5E m

z and dx2 , d y2 , dz2 are mean values and standard deviations of the inhomogeneity center coordinates x, y, z. The spherical inhomogeneities have an identical diameter of 0.4r, and their locations are set as(x, y) ∼ N (0, r ), z ∼ N (r, r ). Heterogeneous materials involving normally distributed spherical inhomogeneities with volume fraction of 5%, 10% and 15% are shown in Fig. 18.15a, and the corresponding surface contact pressure, von Mises stress distribution are illustrated in Fig. 18.15b–d. The greater the volume fraction of the inhomogeneities, the stronger interactions among them, the higher the contact pressure and von Mises stress thus produced. In addition, contact pressure and von Mises stress distribution are severely disturbed by normally distributed inhomogeneities, thus leading to stress concentration more easily.

18.4.2 Rough Contact Machined surface of engineering material is not ideal smooth but rough due to limitation of cutting technology. Surface asperities are commonly simplified to sinusoidal

18 Heterogeneous Contact Modelling and Analysis via Numerical …

421

Fig. 18.12 Surface and subsurface von Mises stress of heterogeneous material involving ellipsoidal inhomogeneities of different angles. a Surface von Mises stress along x axis, E i = 2E m ; b surface von Mises stress along x axis, E i = 0.5E m ; c subsurface von Mises stress, E i = 2E m ; d Subsurface von Mises stress, E i = 0.5E m

distribution in numerical studies of contact behaviors for material with rough surface. Real rough surface can also be obtained by using a surface topography apparatus without difficulties, so it is not a challenge for modeling contact between bodies with real rough surface.

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(a)

(b)

(c)

(d)

Fig. 18.13 Surface contact pressure and displacement induced by inclusion of heterogeneous material involving two spherical inhomogeneities of different distances. a Surface contact pressure, E i = 2E m ; b surface contact pressure, E i = 0.5E m ; c surface displacement induced by inclusion, E i = 2E m ; d surface displacement induced by inclusion, E i = 0.5E m

18.4.2.1

Sinusoidal Rough Surface

The sinusoidal rough surface, featured by asperity amplitude Arms and wave length lw , is governed by z rms (x1 , x2 ) = Arms · cos(2π x1 /lw ) · cos(2π x2 /lw ). Effect of sinusoidal rough surface with different Arms is first discussed. Wave length is set as 0.2 r, and asperity amplitude Arms is set to be (0.5, 1 and 1.5) × 10−3 r. Parameters for distributed inhomogeneities are identical to that in Sect. 18.4.1.3 but only the case with 10% inhomogeneities is selected. Portrayed in Fig. 18.16 are sinusoidal rough surfaces of different asperity amplitudes, contact pressure and von Mises stress distribution of heterogeneous material with normally distributed stiff/compliant inhomogeneities. As the asperity amplitude increases, the contact pressure and subsurface von Mises stress grows higher. The stiff inhomogeneity cases result in more severe contact pressure and stress concentration than the compliant ones. Note that larger contact pressure occurs at asperities around the contact center symmetrically if the material with sinusoidal rough surface is homogeneous, while it may appear randomly if the material is heterogeneous and embedded with distributed

18 Heterogeneous Contact Modelling and Analysis via Numerical …

423

Fig. 18.14 Surface and subsurface von Mises stress of heterogeneous material involving two spherical inhomogeneities of different distances. a von Mises stress along the line through two inhomogeneity centers, E i = 2E m ; b von Mises stress along the line through two inhomogeneity centers, E i = 0.5E m ; c subsurface von Mises stress, E i = 2E m ; d subsurface von Mises stress, E i = 0.5E m

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Fig. 18.15 Elastic contact responses of heterogeneous material involving normally distributed stiff inhomogeneities with different volume fractions of 5%, 10% and 15%. a Heterogeneous material containing inhomogeneities of normal distribution; b surface contact pressure distribution of heterogeneous material; c surface von Mises stress contours of heterogeneous material; d subsurface von Mises stress contours of heterogeneous material

inhomogeneities. Furthermore, as the asperity amplitude becomes bigger, major stress concentration region approaches the surface gradually, indicating that the asperity may dominate stress concentration of heterogeneous when its amplitude exceeds a certain value. Besides asperity amplitude, wave length of sinusoidal rough surface will also affect contact behaviors and mechanical responses of heterogeneous material

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425

Fig. 18.16 Contact pressure and subsurface von Mises stress of heterogeneous material involving 10% normally distributed inhomogeneities with sinusoidal rough surface of different amplitudes

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containing normally distributed inhomogeneities. Asperity amplitude Arms of sinusoidal rough surface is fixed at 10−3 r, while wave length lw is set as 0.05r, 0.3r and 0.5r. Other simulation parameters are the same as those in the precedent for different asperity amplitudes. Figure 18.17 represents the sinusoidal rough surface with different wave lengths, resulting contact pressure and subsurface von Mises stress distribution of heterogeneous material involving normally distributed stiff/compliant inhomogeneities. The greater the wave length, the less the contact asperities, but the larger the contact region for each asperity. Although the maximum contact pressure does not vary monotonously, the maximum von Mises stress decreases, as the wave length increases, for both the stiff and inhomogeneities.

18.4.2.2

Real Rough Surface

Sinusoidal rough surface mentioned above is an approximate version of a real rough surface. Here real machined rough surface is abstracted by a surface topography apparatus, whose related information and real topography are given in Fig. 18.18. Other simulation parameters are identical to those in Sect. 18.4.2.1. It is obvious that banded peaks and valleys formed on the material surface during the cutting or polishing process, reflecting the processing direction. Similar distribution with respect to the real rough surface can also be observed on the contact pressure and surface von Mises stress distribution in the case of normally distributed stiff/compliant inhomogeneities, as depicted in Fig. 18.19a, b. Contact pressure distribution is severely affected by elastic modulus difference of inhomogeneities close to the surface. In addition, the maximum contact pressure and surface von Mises stress occur more randomly due to random rough surface and distributed inhomogeneities.

18.5 Extended Application for Heterogeneous Contact According to definition of the inhomogeneity that a subdomain with different material properties from the matrix, coated material and joined quarter spaces are also within its scope. Therefore, the NEIM can be extended to deal with elastic contact between a spherical indenter and a layered material/joined quarter spaces, as demonstrated in Fig. 18.20.

18.5.1 Coated Material As shown in Fig. 18.20a, the coating, a solid thin film, is assumed to be perfectly bonded on the matrix. Two kinds of coatings, sharing an identical thickness of r, are designed, i.e., E i = 2E m for stiff coating and E i = 0.5E m for compliant coating, respectively. Contact pressure, gap distribution and subsurface von Mises stress are

18 Heterogeneous Contact Modelling and Analysis via Numerical …

427

Fig. 18.17 Contact pressure and subsurface von Mises stress of heterogeneous material involving 10% normally distributed inhomogeneities with sinusoidal rough surface of different wave lengths

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(a)

(b)

Fig. 18.18 Real rough surface abstracted from machined material by a surface topography apparatus. a Basic information for rough surface; b 3D real rough surface

Fig. 18.19 Contact pressure and von Mises stress of heterogeneous material with real rough surface abstracted by a surface topography apparatus. a Surface contact pressure distribution; b surface von Mises stress distribution; c subsurface von Mises stress

explored using the proposed method. It can be seen in Fig. 18.21a that the stiff coating leads to a greater contact pressure, while the compliant coating generates more moderate contact pressure, compared to the homogeneous material. The stiffer the coating, the less the contact radius (see Fig. 18.21b). Exhibited in Fig. 18.22 are subsurface von Mises stress contours and their values along the z axis of the coated material. The stiff (compliant) coating produces greater (less) von Mises in and near the coating than the homogeneous material. Stress jumping occurs at the interface

18 Heterogeneous Contact Modelling and Analysis via Numerical …

(a)

429

(b)

Fig. 18.20 Elastic contact between a a spherical indenter and a layered material; b a spherical indenter and joined quarter spaces

(a)

(b)

Fig. 18.21 Contact pressure and gap distribution for layered material of different coating elastic moduli. a Surface contact pressure distribution; b gap distribution

of the coated material due to elastic modulus difference between the coating and the matrix. In the present cases, the major stress concentration region lies in the coating, indicating that the coating may provide protection for the matrix.

18.5.2 Two Joined Quarter Spaces As illustrated in Fig. 18.20b, two quarter spaces are perfectly joined together, whose interface paralleling to x2 -x3 plane has a distance of 0.5r from the interface to x2 x3 plane across the origin. The left quarter space is treated as the matrix, the right

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Fig. 18.22 Subsurface von Mises stress of layered material of different coating elastic moduli. a Subsurface von Mises stress along z axis; b subsurface von Mises stress distribution

one as the inhomogeneity. The elastic modulus of the right quarter space is set to be E i = 2E m and E i = 0.5E m . The simulation results of contact pressure, gap distribution and subsurface von Mises stress are demonstrated in Fig. 18.23. Notable difference of the elastic modulus between the two joined quarter spaces leads to different pressure distribution, and the contact pressure increases for the stiff right quarter space but decreases for the compliant right quarter space sharply at the interface. Such phenomenon cannot be observed from the gap distribution in Fig. 18.23b but it can be observed from the von Mises stress distribution in Fig. 18.24. Similar to the effect of the inhomogeneity, the stiff one produces greater von Mises stress.

18.6 Summary This chapter briefly presents the theory of contact modeling of heterogeneous materials and implementation of the numerical equivalent inclusion method (NEIM). The necessary ingredients and the companion numerical recipes for programming a powerful contact solver for problems involving material heterogeneity are discussed. With the NEIM, effects of inhomogeneity parameters and contact interface conditions on contact behavior and mechanical response of the heterogeneous material are

18 Heterogeneous Contact Modelling and Analysis via Numerical … 0.60

0.030

Ei=2Em

0.48

Ei=2Em

0.024

Ei=0.5Em

Ei=0.5Em

Ei=Em

0.36

Ei=Em

g/r

p / ph

0.018

0.24

0.012

0.12 0.00 -2

431

0.006

-1

0

x1 / r

(a)

1

2

0.000 -2

Undeformed spherical indenter

-1

0

x1 / r

1

2

(b)

Fig. 18.23 Contact pressure and gap distribution for two joined quarter spaces of different elastic moduli. a Surface contact pressure distribution; b gap distribution

Fig. 18.24 Subsurface von Mises stress of two joined quarter spaces of different elastic moduli. a Surface von Mises stress along x axis; b subsurface von Mises stress distribution

further elaborated. The present study shows that not only the inhomogeneity’s shape but also its orientation has a significant impact on the distributions of contact pressure and subsurface von Mises stress. The closer the inhomogeneity distances, the stronger the interaction among inhomogeneities, and stress field is severely disturbed

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by randomly distributed inhomogeneities. It is also demonstrated that stress concentration occurs around the asperities for rough surface contact of the heterogeneous material, and it may appear around the inhomogeneity as well. Finally, two examples of the coated material and joined quarter spaces are employed to demonstrate the powerful ability and potential of the NEIM. It is believed that there must be more applicable scenarios for the NEIM to exploit. Acknowledgements This research is financially supported by the Natural Science Foundation of China (Grant No. 52275205, 51875059 and 52205192), Graduate Research and Innovation Foundation of Chongqing, China (Grant No. CYS21011) and Chongqing City Science and Technology Program (Grant No. cstc2020jcyj-msxmX0850).

Appendix 1: Response Primitive Functions of Half Space Contact Problem The primitive functions, ϕi j , corresponding to the displacement, cf. Equation (18.11), are detailed as: ϕ11 = 2(1 − ν)ξ1 ln(r + ξ2 ) + 2ξ2 ln(r + ξ1 ) + 4ξ3 X 3 + 2(1 − 2ν)ξ3 X 1 (18.41) ϕ21 = (2ν − 1)ξ3 ln(r + ξ3 ) − 2νr ϕ31 = (1 − 2ν)ξ2 ln(r + ξ3 ) − 2νξ3 ln(r + ξ2 ) − 2(1 − 2ν)ξ1 X 1

(18.42) (18.43)

ϕ12 (ξ1 , ξ2 , ξ3 ) = ϕ21 (ξ2 , ξ1 , ξ3 )

(18.44)

ϕ22 (ξ1 , ξ2 , ξ3 ) = ϕ11 (ξ2 , ξ1 , ξ3 )

(18.45)

ϕ32 (ξ1 , ξ2 , ξ3 ) = ϕ31 (ξ2 , ξ1 , ξ3 )

(18.46)

ϕ13 = (2ν − 1)ξ2 ln(r + ξ3 ) + 2(1 − 2ν)ξ1 X 1 − 2(1 − ν)ξ3 ln(r + ξ2 )

(18.47)

ϕ23 = (2ν − 1)ξ1 ln(r + ξ3 ) + 2(1 − 2ν)ξ2 X 2 − 2(1 − ν)ξ3 ln(r + ξ1 )

(18.48)

ϕ33 = 2(1 − ν)ξ1 ln(r + ξ2 ) + 2(1 − ν)ξ2 ln(r + ξ1 ) + 2(1 − 2ν)ξ3 X 3

(18.49)

where the auxiliary functions are defined as

18 Heterogeneous Contact Modelling and Analysis via Numerical …

ξ1 = x1 − x1 , ξ2 = x2 − x2 , ξ3 = x3 − x3 ; r = X 1 = arctan

ξ1 ; r + ξ2 + ξ3

X 2 = arctan

433

 ξ12 + ξ22 + ξ32

ξ2 ; r + ξ1 + ξ3

X 3 = arctan

r + ξ1 + ξ 2 ; ξ3 (18.50)

The primitive functions, λi j , for the stress, cf. Equation (18.15), are expressed as: λ11 = 2 ln(r + ξ2 ) − (1 − 2ν)

ξ32 ξ2 ξ2 + + r + ξ3 r r (r + ξ2 )

λ21 = 2ν ln(r + ξ2 ) + (1 − 2ν) λ31 = −

ξ2 ξ2 − r + ξ3 r

(18.51) (18.52)

ξ32 r (r + ξ2 )

(18.53)

ξ3 r

(18.54)

λ41 = − λ51 = 2X 3 −

ξ3 ξ1 r (r + ξ2 )

λ61 = ln(r + ξ1 ) + (1 − 2ν)

ξ1 ξ1 − r + ξ3 r

(18.55) (18.56)

λ12 (ξ1 , ξ2 , ξ3 ) = λ21 (ξ2 , ξ1 , ξ3 )

(18.57)

λ22 (ξ1 , ξ2 , ξ3 ) = λ11 (ξ2 , ξ1 , ξ3 )

(18.58)

λ32 (ξ1 , ξ2 , ξ3 ) = λ31 (ξ2 , ξ1 , ξ3 )

(18.59)

λ42 (ξ1 , ξ2 , ξ3 ) = λ51 (ξ2 , ξ1 , ξ3 )

(18.60)

λ52 (ξ1 , ξ2 , ξ3 ) = λ41 (ξ2 , ξ1 , ξ3 )

(18.61)

λ62 (ξ1 , ξ2 , ξ3 ) = λ61 (ξ2 , ξ1 , ξ3 )

(18.62)

λ13 = 4ν X 3 + 2(1 − 2ν)X 1 −

ξ3 ξ1 r (r + ξ2 )

(18.63)

λ23 = 4ν X 3 + 2(1 − 2ν)X 2 −

ξ3 ξ2 r (r + ξ1 )

(18.64)

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 λ33 = 2X 3 + ξ3

ξ1 ξ2 + r (r + ξ2 ) r (r + ξ1 )

 (18.65)

λ43 = −

ξ32 r (r + ξ1 )

(18.66)

λ53 = −

ξ32 r (r + ξ2 )

(18.67)

λ63 = −(1 − 2ν) ln(r + ξ3 ) −

ξ3 r

(18.68)

Appendix 2: The Expression of the Primitive Functions in the Elementary Solution In this appendix, introducing the auxiliary functions by the formula 2r + ξγ ξα ξβ 1 ), Vγ = , Wγ =  2 , r ξγ r (r + ξγ ) r 3 r + ξγ ξγ ), Yγ = ln(r + ξγ ), (α, β, γ = 1, 2, 3) X γ = arctan( r + ξα + ξβ

Uγ = arctan(

(18.69)

where  ξ1 = x 1 −

x1 ,ξ2

= x2 −

x2 ,ξ3

=

(0) x3 − x3 , for Ti(0) j and Ui j , r =  x3 + x3 , for other cases

 ξ12 + ξ22 + ξ32 (18.70)

The Explicit Expression of Tij (1) Ti(0) j

(0) T11 = 2U2 + 2U3 + ξ1 ξ3 V2 + ξ1 ξ2 V3

(18.71)

(0) (0) T12 = T21 = 2νU3 − ξ1 ξ2 V3

(18.72)

18 Heterogeneous Contact Modelling and Analysis via Numerical … (0) (0) T13 = T31 = 2νU2 − ξ1 ξ3 V2

435

(18.73)

ξ1 r

(18.74)

(0) (0) = T51 = Y2 − ξ12 V2 T15

(18.75)

(0) (0) T16 = T61 = Y3 − ξ12 V3

(18.76)

(0) T22 = 2U1 + 2U3 + ξ2 ξ3 V1 + ξ1 ξ2 V3

(18.77)

(0) (0) T23 = T32 = 2νU1 − ξ2 ξ3 V1

(18.78)

(0) (0) T24 = T42 = Y1 − ξ22 V1

(18.79)

ξ2 r

(18.80)

(0) (0) = T62 = Y3 − ξ22 V3 T26

(18.81)

(0) T33 = 2U1 + 2U2 + ξ2 ξ3 V1 + ξ1 ξ3 V2

(18.82)

(0) (0) T34 = T43 = Y1 − ξ32 V1

(18.83)

(0) (0) T35 = T53 = Y2 − ξ32 V2

(18.84)

ξ3 r

(18.85)

(0) (0) = T41 = 2νY1 − T14

(0) (0) = T52 = 2νY2 − T25

(0) (0) = T63 = 2νY3 − T36

(0) T44 = (1 − ν)U1 − ξ2 ξ3 V1

(18.86)

(0) (0) = T54 = (1 − ν)Y3 − T45

ξ3 r

(18.87)

(0) (0) T46 = T64 = (1 − ν)Y2 −

ξ2 r

(18.88)

(0) T55 = (1 − ν)U2 − ξ1 ξ3 V2

(18.89)

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ξ1 r

(18.90)

(0) T66 = (1 − ν)U3 − ξ1 ξ2 V3

(18.91)

(1) T11 = −2U1 − 8v 2 U2 + 3ξ1 ξ3 V2 + r ξ1 ξ2 V3 A

(18.92)

(1) T12 = −4vU1 + 2vU3 + 4vξ2 ξ3 V1 − r ξ1 ξ2 V3 A

(18.93)

(1) T13 = −4U1 − 6vU2 − 4vξ2 ξ3 V1 − 3ξ1 ξ3 V2

(18.94)

(2) Ti(1) j

(1) = 2vY1 − 4vξ32 V1 + (3 − 4v) T14

ξ1 r

(1) T15 = Y2 + (3 − 4v)ξ1 ξ3 V2 − 4vξ1 ξ3 V2

(18.95) (18.96)

  ξ3 (1) = 1 − 2v − 4v 2 Y3 + 4v − r ξ1 ξ2 V3 A + 2(1 − v)(1 − 2v)r ξ3 V3 (18.97) T16 r (1) T21 = 2vU3 − 4vU2 + 4vξ1 ξ3 V2 − r ξ1 ξ2 V3 A

(18.98)

(1) T22 = −8v 2 U1 − 2U2 + 3ξ2 ξ3 V1 + r ξ1 ξ2 V3 A

(18.99)

(1) T23 = −6vU1 − 4U2 − 3ξ2 ξ3 V1 − 4vξ1 ξ3 V2

(18.100)

(1) T23 = −6vU1 − 4U2 − 3ξ2 ξ3 V1 − 4vξ1 ξ3 V2

(18.101)

(1) T24 = Y1 + (3 − 4v)ξ22 V1 − 4vξ32 V1

(18.102)

ξ2 − 4vξ32 V2 r

(18.103)

(1) = 2vY2 + (3 − 4v) T25

  ξ3 (1) = 1 − 2v − 4v 2 Y3 + 4v − r ξ22 V3 A + 2(1 − v)(1 − 2v)r ξ3 V3 (18.104) T26 r (1) T31 = −2vU2 + ξ1 ξ3 V2

(18.105)

18 Heterogeneous Contact Modelling and Analysis via Numerical …

437

(1) T32 = −2vU1 + ξ2 ξ3 V1

(18.106)

(1) T33 = 2U3 − ξ2 ξ3 V1 − ξ1 ξ3 V2

(18.107)

(1) T34 = Y1 − ξ32 V1

(18.108)

(1) T35 = Y2 − ξ32 V2

(18.109)

(1) = −2vY3 + T36

ξ3 r

(18.110)

ξ1 r

(18.111)

(1) = Y1 − ξ22 V1 T42

(18.112)

(1) T43 = Y1 − ξ32 V1

(18.113)

(1) T44 = −(1 − v)U1 + ξ2 ξ3 V1

(18.114)

(1) T41 = 2vY1 −

(1) = −(1 − v)Y3 + T45 (1) T46 = (1 − v)Y2 −

ξ3 r

ξ2 r

(18.115) (18.116)

(1) T51 = Y2 − ξ12 V2

(18.117)

ξ2 r

(18.118)

(1) T53 = Y2 − ξ32 V2

(18.119)

(1) = 2vY2 − T52

(1) T54 = −(1 − v)Y3 +

ξ3 r

(1) T55 = −(1 − v)U2 + ξ1 ξ3 V2 (1) T56 = (1 − v)Y1 −

ξ1 r

(18.120) (18.121) (18.122)

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  (1) T61 = 1 + 2v − 4v 2 Y3 − r ξ12 V3 A + 2(1 − v)(1 − 2v)r ξ3 V3

(18.123)

  (1) T62 = 1 + 2v − 4v 2 Y3 − r ξ22 V3 A + 2(1 − v)(1 − 2v)r ξ3 V3

(18.124)

ξ3 r

(18.125)

(1) T64 = −(1 − v)Y2 + (3 − 4v)

ξ2 r

(18.126)

(1) T65 = −(1 − v)Y1 + (3 − 4v)

ξ1 r

(18.127)

(1) T63 = (4 − 6v)Y3 − (3 − 4v)

(1) T66 = (1 − v)U3 − r ξ1 ξ2 V3 A

(18.128)

where A is the auxiliary function expressed as A=

3 − 4v 2(1 − v)(1 − 2v) − r r + ξ3

(18.129)

(2) (3) Note that ξ3 = x3 + x  for Ti(1) j , Ti j and Ti j .

(3) Ti(2) j

(2) T11 = −6ξ1 V2 + 2ξ13 W2 (2) T12 = 4v(ξ1 V2 − ξ2 V1 ) +

(18.130)

2ξ1 ξ2 r3

(2) T13 = 4vξ2 V1 + 6ξ1 V2 + 2ξ1 ξ32 W2 (2) T14 = 4vξ3 V1 − 4(1 − v)ξ1 V3 −

2ξ3 ξ1 r3

(2) T15 = 4(1 − v)ξ2 V3 + 6ξ3 V2 − 2ξ12 ξ3 W2

(2) T16

  2 ξ22 + ξ32 4v =− − r r3

(2) T21 = 4v(ξ2 V1 − ξ1 V2 ) +

2ξ1 ξ2 r3

(18.131) (18.132) (18.133) (18.134)

(18.135) (18.136)

18 Heterogeneous Contact Modelling and Analysis via Numerical …

439

(2) T22 = −6ξ2 V1 + ξ23 2W1

(18.137)

(2) T23 = 4vξ1 V2 + 6ξ2 V1 + 2ξ2 ξ32 W1

(18.138)

(2) T24 = 4(1 − v)ξ1 V3 + 6ξ3 V1 − 2ξ22 ξ3 W1

(18.139)

(2) T25 = 4vξ3 V2 − 4(1 − v)ξ2 V3 −

(2) T26

2ξ3 ξ2 r3

  2 ξ12 + ξ32 4v =− − r r3

(18.140)

(18.141)

(2) T31 = −4vξ2 V1 − 2ξ1 V2 + 2ξ1 ξ32 W2

(18.142)

(2) T32 = −4vξ1 V2 − 2ξ2 V1 + 2ξ2 ξ32 W1

(18.143)

  (2) T33 = −2(ξ1 V2 + ξ2 V1 ) − 2 ξ1 ξ32 W2 + ξ2 ξ32 W1

(18.144)

(2) T34 = 2ξ3 V1 − 2ξ33 W1

(18.145)

(2) T35 = 2ξ3 V2 − 2ξ33 W2

(18.146)

(2) T36

  2 ξ12 + ξ22 4v − = r r3

(2) T41 = 4v(ξ3 V1 + ξ1 V3 ) +

2ξ1 ξ3 r3

(18.147) (18.148)

(2) T42 = −4vξ1 V3 − 2ξ3 V1 + 2ξ22 ξ3 W1

(18.149)

(2) T43 = 2ξ3 V1 + 2ξ33 W1

(18.150)

(2) T44 = −2ξ2 V1 − 2ξ2 ξ32 W1

(18.151)

(2) T45 =

  2 ξ12 + ξ22 4 − r3 r

(2) T46 = 4vξ2 V3 +

2ξ3 ξ2 r3

(18.152) (18.153)

440

W. Yang et al. (2) T51 = −4vξ2 V3 − 2ξ3 V2 + 2ξ12 ξ3 W2 (2) T52 = 4v(ξ3 V2 + ξ2 V3 ) +

2ξ2 ξ3 r3

(2) T53 = 2ξ3 V2 + 2ξ33 W2

(2) T54

  2 ξ12 + ξ22 4 = − r3 r

(2) = −2ξ1 V2 − 2ξ1 ξ32 W2 T55

2ξ3 ξ1 r3   2 ξ22 + ξ32 4v (2) − T61 = r r3   2 ξ12 + ξ32 4v (2) − T62 = r r3   2 4(2 − v) 2 ξ1 + ξ22 (2) T63 = − r r3 (2) T56 = 4vξ1 V3 +

(18.154) (18.155) (18.156)

(18.157) (18.158) (18.159)

(18.160)

(18.161)

(18.162)

(2) = −4(1 − v)ξ2 V3 − T64

2ξ3 ξ2 r3

(18.163)

(2) T65 = −4(1 − v)ξ1 V3 −

2ξ3 ξ1 r3

(18.164)

(2) T66 =

2ξ2 ξ1 r3

(18.165)

(4) Ti(3) j

(3) T11 = 2ξ1 ξ3 W2 + 2ξ1 ξ2 W3

(18.166)

(3) (3) T12 = T21 = −2ξ1 ξ2 W3

(18.167)

(3) (3) T13 = T31 = −2ξ1 ξ3 W2

(18.168)

18 Heterogeneous Contact Modelling and Analysis via Numerical … (3) (3) T14 = −T41 =

ξ1 r3

441

(18.169)

(3) (3) T15 = −T51 = −V2 + ξ12 W2

(18.170)

(3) (3) T16 = T61 = V3 − ξ12 W3

(18.171)

(3) T22 = 2ξ1 ξ2 W3 + 2ξ2 ξ3 W1

(18.172)

(3) (3) T23 = T32 = −2ξ2 ξ3 W1

(18.173)

(3) (3) T24 = −T42 = −2V1 + 2ξ22 W1

(18.174)

(3) (3) = −T52 =2 T25

ξ2 r3

(18.175)

(3) (3) T26 = T62 = 2V3 − 2ξ22 W3

(18.176)

(3) T33 = 2ξ1 ξ3 W2 + 2ξ2 ξ3 W1

(18.177)

(3) (3) T34 = −T43 = −2V1 + 2ξ32 W1

(18.178)

(3) (3) T35 = −T53 = −2V2 + 2ξ32 W2

(18.179)

(3) (3) = T63 = −2 T36

ξ3 r3

(3) T44 = 2ξ2 ξ3 W1 (3) (3) = T54 =2 T45

(18.181)

ξ3 r3

(3) (3) = −T64 = −2 T46

ξ2 r3

(3) T55 = 2ξ1 ξ3 W2 (3) (3) = −T65 = −2 T56 (3) T66 = −2ξ1 ξ2 W3

(18.180)

(18.182) (18.183) (18.184)

ξ1 r3

(18.185) (18.186)

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W. Yang et al.

The Explicit Expression of Uij (1) Ui(0) j

(0) U11 = (3 − 2ν)ξ2 Y3 + (3 − 2ν)ξ3 Y2 − 2(1 − ν)ξ1 U1

(18.187)

(0) U12 = −(1 − 2ν)ξ3 Y2 + 2νξ2 Y3 − 2νξ1 U1

(18.188)

(0) U13 = −(1 − 2ν)ξ2 Y3 + 2νξ3 Y2 − 2νξ1 U1

(18.189)

(0) U14 = −r

(18.190)

(0) U15 = 2(1 − ν)ξ1 Y2 + (1 − 2ν)ξ2 Y1 − 2(1 − ν)ξ3 U3

(18.191)

(0) U16 = 2(1 − ν)ξ1 Y3 + (1 − 2ν)ξ3 Y1 − 2(1 − ν)ξ2 U2

(18.192)

(0) U21 = −(1 − 2ν)ξ3 Y1 + 2νξ1 Y3 − 2νξ2 U2 (0) U22 = (3 − 2ν)ξ1 Y3 + (3 − 2ν)ξ3 Y1 − 2(1 − ν)ξ2 U2 (0) U23 = −(1 − 2ν)ξ1 Y3 + 2νξ3 Y1 − 2νξ2 U2 (0) U24 = 2(1 − ν)ξ2 Y1 + (1 − 2ν)ξ1 Y2 − 2(1 − ν)ξ3 U3 (0) U25 = −r (0) U26 = 2(1 − ν)ξ2 Y3 + (1 − 2ν)ξ3 Y2 − 2(1 − ν)ξ1 U1

(18.193) (18.194) (18.195) (18.196) (18.197) (18.198)

(0) U31 = −(1 − 2ν)ξ2 Y1 + 2νξ1 Y2 − 2νξ3 U3

(18.199)

(0) U32 = −(1 − 2ν)ξ1 Y2 + 2νξ2 Y1 − 2νξ3 U3

(18.200)

(0) U33 = (3 − 2ν)ξ1 Y2 + (3 − 2ν)ξ2 Y1 − 2(1 − ν)ξ3 U3

(18.201)

(0) U34 = 2(1 − ν)ξ3 Y1 + (1 − 2ν)ξ1 Y3 − 2(1 − ν)ξ2 U2

(18.202)

18 Heterogeneous Contact Modelling and Analysis via Numerical … (0) U35 = 2(1 − ν)ξ3 Y2 + (1 − 2ν)ξ2 Y3 − 2(1 − ν)ξ1 U1 (0) U36 = −r

443

(18.203) (18.204)

(2) Ui(1) j   (1) U11 = (5 − 6v)ξ3 Y2 + 3 − 4v 2 ξ2 Y3 − 2(1 − v)ξ1 U1 − 2(1 − v)(1 − 2v)r ξ2 ξ3 V3 (18.205) (1)

U12 = 2v(3 − 4v)ξ3 Y2 − (1 − 2v)2 ξ2 Y3 − 2v(3 − 4v)ξ1 U1 + 2(1 − v)(1 − 2v)r ξ2 ξ3 V3

(18.206) (1) U13 = (1 − 2v)ξ3 Y2 + (4 − 6v)ξ2 Y3 − (4 − 6v)ξ1 U1 (1) U14 = (3 − 4v)r (1) U15 = −2(1 − ν)ξ1 Y2 + (1 − 2ν)ξ2 Y1 + 2(1 − ν)ξ3 U3

(18.207) (18.208) (18.209)

(1) U16 = 2(1 − v)ξ3 Y1 + (1 − 2v)2 ξ1 Y3 − 2(1 − v)ξ2 U2 + 2(1 − v)(1 − 2v)r ξ1 ξ3 V3 (18.210)

(1)

U21 = 2v(3 − 4v)ξ3 Y1 − (1 − 2v)2 ξ1 Y3 − 2v(3 − 4v)ξ2 U2 + 2(1 − v)(1 − 2v)r ξ1 ξ3 V3

(18.211)   (1) U22 = (5 − 6v)ξ3 Y1 + 3 − 4v 2 ξ1 Y3 − 2(1 − v)ξ2 U2 − 2(1 − v)(1 − 2v)r ξ1 ξ3 V3 (18.212) (1) U23 = (1 − 2v)ξ3 Y1 + (4 − 6v)ξ1 Y3 − (4 − 6v)ξ2 U2

(18.213)

(1) U24 = −2(1 − ν)ξ2 Y1 + (1 − 2ν)ξ1 Y2 + 2(1 − ν)ξ3 U3

(18.214)

(1) U25 = (3 − 4v)r

(18.215)

(1) U26 = 2(1 − v)ξ3 Y2 + (1 − 2v)2 ξ2 Y3 − 2(1 − v)ξ1 U1 + 2(1 − v)(1 − 2v)r ξ2 ξ3 V3 (18.216)

(1)

U31 = −v(6 − 8v)ξ2 Y1 − (1 − 2v)ξ1 Y2 + 2v(3 − 4v)ξ3 U3 − 8(1 − v)(1 − 2v)ξ3 X 1

(18.217) (1) U32 = −(1 − 2v)ξ2 Y1 − v(6 − 8v)ξ1 Y2 + 2v(3 − 4v)ξ3 U3 − 8(1 − v)(1 − 2v)ξ3 X 2

(18.218)

444

W. Yang et al. (1) U33 = −(5 − 6v)(ξ2 Y1 + ξ1 Y2 ) + 2(1 − v)ξ3 U3

(18.219)

(1) U34 = −2(1 − ν)ξ3 Y1 + (1 − 2ν)ξ1 Y3 + 2(1 − ν)ξ2 U2

(18.220)

(1) U35 = −2(1 − ν)ξ3 Y2 + (1 − 2ν)ξ2 Y3 + 2(1 − ν)ξ1 U1

(18.221)

  (1) U36 = 4(1 − v)(1 − 2v)ξ3 Y3 − 1 − 8v + 8v 2 r

(18.222)

(3) Ui(2) j

(2) U11 = −2(1 − 2v)Y2 − 2ξ12 V2 (2) U12 = 4vY2 −

2ξ2 r

(18.224)

(2) = 2(3 − 2v)Y2 − 2ξ32 V2 U13 (2) U14 = −4(1 − v)Y3 +

2ξ3 r

(2) = 4(1 − v)U1 + 2ξ1 ξ3 V2 U15 (2) =− U16

2ξ1 r

(2) U21 = 4vY1 −

(18.223)

(18.225) (18.226) (18.227) (18.228)

2ξ1 r

(18.229)

(2) U22 = −2(1 − 2v)Y1 − 2ξ22 V1

(18.230)

(2) U23 = 2(3 − 2v)Y1 − 2ξ32 V1

(18.231)

(2) U24 = 4(1 − v)U2 + 2ξ2 ξ3 V1

(18.232)

(2) U25 = −4(1 − v)Y3 + (2) U26 =−

2ξ2 r

2ξ3 r

(18.233) (18.234)

18 Heterogeneous Contact Modelling and Analysis via Numerical …

445

(2) U31 = 4(1 − 2v)U1 − 4vU3 − 2ξ1 ξ3 V2

(18.235)

(2) U32 = 4(1 − 2v)U2 − 4vU3 − 2ξ2 ξ3 V1

(18.236)

(2) U33 = −4vU3 + 2ξ2 ξ3 V1 + 2ξ1 ξ3 V2

(18.237)

(2) U34 = 2(1 − 2v)Y1 + 2ξ32 V1

(18.238)

(2) U35 = 2(1 − 2v)Y2 + 2ξ32 V2

(18.239)

(3) U11 = −2ξ3 V2 − 2ξ2 V3

(18.240)

(3) U12 = 2ξ2 V3

(18.241)

(3) U13 = 2ξ3 V2

(18.242)

(4) Ui(3) j

(3) U14 =−

2 r

(18.243)

(3) U15 = −2ξ1 V2

(18.244)

(3) U16 = 2ξ1 V3

(18.245)

(3) U21 = 2ξ1 V3

(18.246)

(3) U22 = −2ξ1 V3 − 2ξ3 V1

(18.247)

(3) U23 = 2ξ3 V1

(18.248)

(3) U24 = −2ξ2 V1

(18.249)

(3) U25 =−

2 r

(18.250)

446

W. Yang et al. (3) U26 = 2ξ2 V3

(18.251)

(3) U31 = 2ξ1 V2

(18.252)

(3) U32 = 2ξ2 V1

(18.253)

(3) U33 = −2ξ1 V2 − 2ξ2 V1

(18.254)

(3) U34 = −2ξ3 V1

(18.255)

(3) U35 = −2ξ3 V2

(18.256)

(3) U36 =

2 r

(18.257)

References Benaicha Y, Perret-Liaudet J, Beley JD, Rigaud E, Thouverez F (2022) On a flexible multibody modelling approach using FE-based contact formulation for describing gear transmission error. Mech Mach Theory 167 Chen WW, Zhou K, Keer LM, Wang QJ (2010) Modeling elasto-plastic indentation on layered materials using the equivalent inclusion method. Int J Solid Struct 47:2841–2854 Chiu YP (1977) On the stress field due to initial strains in a cuboid surrounded by an infinite elastic space. J Appl Mech 44:587–590 Chiu YP (1978) On the stress field and surface deformation in a half space with a cuboidal zone in which initial strains are uniform. J Appl Mech-T ASME 45:302–306 Dundurs J (1969) Discussion: “Edge-Bonded Dissimilar Orthogonal Elastic Wedges Under Normal and Shear Loading” (Bogy DB, (1968) ASME J Appl Mech 35:460–466). J Appl Mech 36:650– 652 Dong Q, Zhou K (2017) Numerical modeling of elastohydrodynamic lubrication in point or line contact for heterogeneous elasto-plastic materials. Mech Adv Mater Struc 24:1300–1308 Dong Q, Zhou K, Chen WW, Fan Q (2016) Partial slip contact modeling of heterogeneous elastoplastic materials. Int J Mech Sci 114:98–110 Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. P Roy Soc A-Math Phy 241:376–396 Eshelby JD (1959) The elastic field outside an ellipsoidal inclusion. P Roy Soc A-Math Phy 252:561– 569 Eshelby JD, Peierls RE (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. P Roy Soc A-Math Phy 241:376–396 Fu Q, Zhang L, Guo X, Yang H (2022) A piecewise recursive semi-analytical method for effective evaluation of heterogeneous aging viscoelastic materials. Mech Mater 174 Golub GH, Loan CFV (1996) Matrix computations. Johns Hopkins University Press, Baltimore Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49:28

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Huang Y, Wang Z, Zhou Q (2016) Numerical studies on the surface effects caused by inhomogeneities on torsional fretting. Tribol Int 96:202–216 Hutchinson JW (1987) Crack tip shielding by micro-cracking in brittle solids. Acta Metall 35:1605– 1619 Jin X, Keer LM, Wang Q (2009) New Green’s function for stress field and a note of its application in quantum-wire structures. Int J Solids Struct 46:3788–3798 Jin X, Keer LM, Wang Q (2011) A closed-form solution for the Eshelby tensor and the elastic field outside an elliptic cylindrical inclusion. J Appl Mech 78:031009 Jin X, Wang Z, Zhou Q, Keer LM, Wang Q (2014) On the solution of an elliptical inhomogeneity in plane elasticity by the equivalent inclusion method. J Elast 114:1–18 Jin X, Lyu D, Zhang X, Zhou Q, Wang Q, Keer LM (2016a) Explicit analytical solutions for a complete set of the Eshelby tensors of an ellipsoidal inclusion. ASME J Appl Mech 83:121010 Jin X, Niu F, Zhang X, Zhou Q, Lyu D, Keer LM et al (2016b) Love’s rectangular contact problem revisited: a complete solution. Tribol Int 103:331–342 Jin X, Zhang X, Li P, Xu Z, Hu Y, Keer LM (2017) On the displacement of a two-dimensional Eshelby inclusion of elliptic cylindrical shape. J Appl Mech 84:074501 Li G, Yin BB, Zhang LW, Liew KM (2021) A framework for phase-field modeling of interfacial debonding and frictional slipping in heterogeneous composites. Comput Method Appl M 382 Liu S, Wang Q (2005) Elastic fields due to eigenstrains in a half-space. J Appl Mech 72:871–878 Liu S, Jin X, Wang Z, Keer LM, Wang Q (2012) Analytical solution for elastic fields caused by eigenstrains in a half-space and numerical implementation based on FFT. Int J Plasticity. 35:135–154 Lyu D, Zhang X, Li P, Luo D, Hu Y, Jin X et al (2018) Explicit analytical solutions for the complete elastic field produced by an ellipsoidal thermal inclusion in a semi-infinite space. J Appl Mech 85:051005 Mura T (2013) Micromechanics of defects in solids. Springer Science & Business Media, Dordrecht Shewchuk J (1994) An introduction to the conjugate gradient method without the agonizing pain Shi J, Li Z (2006) An approximate solution of the interaction between an edge dislocation and an inclusion of arbitrary shape. Mech Res Commun 33:804–810 Spilker K, Nguyen VD, Adam L, Wu L, Noels L (2022) Piecewise-uniform homogenization of heterogeneous composites using a spatial decomposition based on inelastic micromechanics. Compos Struct 295:26 Strang G (1988) Linear algebra and its applications. Harcourt, Brace, Jovanovich, Publishers, San Diego Wang Z, Jin X, Zhou Q, Ai X, Keer LM, Wang Q (2013) An efficient numerical method with a parallel computational strategy for solving arbitrarily shaped inclusions in elastoplastic contact problems. J Tribol 135:031401 Zhang M, Zhao N, Wang Z, Wang Q (2018) Efficient numerical method with a dual-grid scheme for contact of inhomogeneous materials and its applications. Comput Mech 62:991–1007 Zhang M, Wang Q, Wang Z, Zhao N, Peng Y (2019) Multiscale computational scheme for semianalytical modeling of the point contact of inhomogeneous materials. Int J Solid Struct. 168:90– 108 Zhang MQ, Yan ZQ (2022) Effects of near-surface composites on frictional rolling contact solved by a semi-analytical model. J Tribol-T Asme 144 Zhang B, Boffy H, Venner CH (2020) Multigrid solution of 2D and 3D stress fields in contact mechanics of anisotropic inhomogeneous materials. Tribol Int 149 Zhang B, Minov B, Morales-Espejel GE, Venner CH (2022) Effect of steel anisotropy on contact pressure and stress distribution in dry and lubricated point contacts: A case study with measured material properties. Tribol Int 175 Zhou K, Hoh HJ, Wang X, Keer LM, Pang JHL, Song B et al (2013) A review of recent works on inclusions. Mech Mater 60:144–158

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Zhou Q, Jin X, Wang Z, Wang J, Keer LM, Wang Q (2014) An efficient approximate numerical method for modeling contact of materials with distributed inhomogeneities. Int J Solids Struct 51:3410–3421 Zhou Q, Jin X, Wang Z, Wang J, Keer LM, Wang Q (2015) Numerical implementation of the equivalent inclusion method for 2D arbitrarily shaped inhomogeneities. J Elast 118:39–61 Zhou Q, Jin X, Wang Z, Wang J, Keer LM, Wang Q (2016a) Numerical EIM with 3D FFT for the contact with a smooth or rough surface involving complicated and distributed inhomogeneities. Tribol Int 93:91–103 Zhou Q, Xie L, Wang X, Jin X, Wang Z, Wang J et al (2016b) Modeling rolling contact fatigue lives of composite materials based on the dual beam FIB/SEM technique. Int J Fatigue 83:201–208 Zhou Q, Jin X, Wang Z, Yang Y, Wang J, Keer LM et al (2016c) A mesh differential refinement scheme for solving elastic fields of half-space inclusion problems. Tribol Int 93:124–136 Zhou Q, Wang J, Wan Q, Jin F, Yang W, Miao Q et al (2017) Numerical analysis of the influence of distributed inhomogeneities on tangential fretting. P I Mech Eng J-J Eng 231:1350–1370 Zhou Q, Xie L, Jin X, Wang Z, Wang J, Keer LM et al (2015) Numerical modeling of distributed inhomogeneities and their effect on rolling-contact fatigue life. J Tribol 137

Chapter 19

Effective Engineering Constants for Micropolar Composites with Imperfect Contact Conditions R. Rodríguez-Ramos, V. Yanes, Y. Espinosa-Almeyda, C. F. Sánchez-Valdés, J. A. Otero, F. Lebon, R. Rizzoni, M. Serpilli, S. Dumont, and F. J. Sabina Abstract In this work, the homogenization theory is applied within the framework of three-dimensional linear micropolar media. The fundamental results derived by the asymptotic homogenization method to compute the effective engineering moduli for a laminated micropolar elastic composite with centro-symmetric constituents are summarized, in which the interface between the layer phases is considered imperfect spring type. The layers are considered with isotropic symmetry. Non-uniform and, as a particular case, uniform imperfections are assumed, where different imperfection parameters and cell lengths in the y3 -direction are assigned for the analysis. The analytical expressions of the engineering constants related to the stiffness and torque are given as functions of the imperfection parameters. The behavior of the engineering coefficients depending on the imperfection is studied. The influence of the imperfection and the cell length in the direction of the imperfection is observed. The present study allows validating other models and experimental results, as well as the investigation of fracture prediction in laminated composite materials.

R. Rodríguez-Ramos (B) Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, Vedado, La Habana, CP 10400, Cuba PPG-MCCT, Universidade Federal Fluminense, Av. dos Trabalhadores 420, Vila Sta. Cecília, CP 27255-125 Volta Redonda, RJ, Brazil e-mail: [email protected]; [email protected] V. Yanes Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Pza. Cardenal Cisneros 3,, Madrid 28040 Madrid, Spain e-mail: [email protected] Y. Espinosa-Almeyda · C. F. Sánchez-Valdés Instituto de Ingeniería y Tecnología, Universidad Autónoma de Ciudad Juárez, Av. Del Charro 450 Norte Cd. Juárez, Chihuahua, CP 32310, Mexico e-mail: [email protected] C. F. Sánchez-Valdés e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_19

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19.1 Introduction Several investigations in biomechanics have shown that models related to Cosserattype media better capture the actual response of biological tissues (Eringen 1968; Cowin 1970; Yang and Lakes 1981, 1982; Lakes et al. 1990). The micromechanical study in Cosserat’s media has had an impact on the mechanics of bones (Park and Lakes 1987; Lakes 1993; Tanaka and Adachi 1999; Fatemi et al. 2002, 2003; Goda et al. 1990; Jasiuk 2018), cardiac tissues (Sack et al. 2016; Hussan et al. 2012), etc. The applicability of laminated structures in various branches of industry is well known. The investigation of their properties is important to improve and design new materials. There are micromechanical methods based on multiscale homogenization schemes that provide information about the properties of heterogeneous laminated micropolar or Cosserat media, for example: Properties of micropolar multi-layered media have been calculated using the finite element technique (Adhikary and Dyskin 1997; Riahi and Curran 2009; Lebée and Sab 2010). In these works, the potentiality of the Cosserat continuum model to predict the mechanical behavior of layered structures is analyzed. Moreover, the Cosserat continuums with 2D and 3D layered-like microstructure are analyzed by a finite element scheme in Riahi and Curran (2009, 2010). On the other hand, multiscale homogenization approaches applied to micropolar heterogeneous structures have been carried out by Nika (2022); Bigoni and Drugan (2007); Forest and Sab (1998); Forest et al. (2001); Forest and Trinh (2011); Gorbachev and Emel’yanov (2014, 2021), among others. In these approaches, the generalized stress and strain are linked to the displacements, strains, and stresses defined in the representative volume element. Different works address the imperfect interface effects on multi-laminated media through the linear spring interface with zero thickness and the interphase models, (Bövik 1994; Ensan et al. 2003; Duong et al. 2011; Sertse and Yu 2017; Khoroshun 2019; Brito-Santana et al. 2019), among others. In the framework of heterogeneous J. A. Otero Escuela de Ingeniería y Ciencias, Tecnológico de Monterrey, Carr. al Lago de Guadalupe Km. 3.5, Estado de México 52926, Mexico e-mail: [email protected] F. Lebon Laboratoire de Mecanique et d’Acoustique, Universite Aix-Marseille, CNRS, Centrale Marseille,, CS 40006 13453 Marseille Cedex 13, France e-mail: [email protected] R. Rizzoni Department of Engineering, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy M. Serpilli Department of Civil and Building Engineering, and Architecture, Universitá Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy S. Dumont University of Nîmes, Place Gabriel Péri, 30000 Nîmes, France e-mail: [email protected]

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micropolar or Cosserat elastic media, the problem of the existence of an imperfect interface between two contiguous phases has been considered. For example, the imperfect interface model applied to elastic composites (Achenbach and Zhu 1989; Hashin 2002) is generalized to micropolar media assuming that the couple tractions are continuous across the interface and proportional to the jumps of the out-of-plane microrotation (Videla and Atroshchenko 2017). In addition, the boundary element method is used to simulate microstructured Cosserat media with both perfect and uniform imperfect interfaces. The asymptotic analysis (see, for example, Ciarlet 1997) has proven to be a powerful mathematical tool to derive simplified models for thin films and structures (Geymonat et al. 2014; Serpilli et al. 2013). This technique has also been extensively used to study the mechanical behavior of layered composites, constituted by two solids bonded together by a thin interphase, considering different continuum theories with microstructure, such as micropolar elasticity (Serpilli 2018), poroelasticity (Serpilli 2019), and flexoelectricity (Serpilli et al. 2022). Recently, the effective behavior of multi-laminated micropolar composites is studied using the asymptotic homogenization method (Yanes et al. 2022; Rodríguez-Ramos et al. 2022). In both works, centro-symmetric cubic or isotropic constituents and perfect interface conditions are assumed. Other previous works dealing with the problem of imperfection in micropolar structures can be found in Rubin and Benveniste (2004); Dong et al. (2014, 2015); Kumari et al. (2022). Therefore, further analyses are required on this topic. In the present work, based on the methodology presented in Yanes et al. (2022); Rodríguez-Ramos et al. (2022); Espinosa-Almeyda et al. (2022), the main results derived by the asymptotic homogenization method (AHM) to compute the effective engineering moduli for a laminated micropolar centro-symmetric composite are summarized, in which the interface between the layer phases is considered imperfect spring type. The layers are considered with isotropic symmetry. The imperfection is considered non-uniform and as a particular case uniform, controlled by different imperfection parameters and the cell length in the y3 -direction. The analytical expressions of the effective engineering moduli associated with the stiffness and torque are given as a function of the non-uniform imperfect parameters. An analysis of the behavior of the effective engineering coefficients depending on the imperfection is performed. The influence of the imperfection and the cell length in the direction of the imperfection is observed.

F. J. Sabina Instituto de Investigaciones en matemáticas Aplicadas y Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-126, Alcaldía Álvaro Obregón, 01000 CDMX, Mexico e-mail: [email protected]

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19.2 Heterogeneous Problem Statement and Fundamental Equations A periodic centro-symmetric linear elastic  micropolar continuum  at the Cartesian coordinate system x = x1 , x2 , x3 ⊂ R3 is defined by two independent sets of degrees of freedom given by the displacement u m (x) [m] and the microrotation ωs (x) fields associated with each material point (Eringen 1999). For the static case, it is formulated by the linear and angular balance equations   Ci jmn (x) enm (x) , j + f i (x) = 0,     Di jmn (x) ψnm (x) , j + i jk Ck jmn (x) enm (x) + gi (x) = 0,

(19.1)

where Ci jmn (x) [N/m2 ] is the stiffness tensor, Di jmn (x) [N] is the torque tensor, f i (x) [N/m3 ] are the body forces, and gi (x) [N/m2 ] are the body couples functions, with i, j, k, m, n, s = 1, 2, 3. The micropolar strain emn (x) and the torsion-curvature ψmn (x) [m−1 ] tensors are given by enm (x) = u m,n (x) + mns ωs (x), ψnm (x) = ωm,n (x),

(19.2)

where mns is the Levi-Civita tensor, u m is the displacement vector, and ωm is the microrotation vector, independent of the displacement. The notation f , m ≡ ∂ f /∂ xm and the square brackets contain the physical units of measure for the variable. In Eqs. (19.1) and (19.2), the symmetric part of emn (x) corresponds to the classical strain tensor, whereas its skew-symmetric part accounts for the local reorientation of the microstructure. Also, the symmetry conditions Ci jmn (x) = Cmni j (x) and Di jmn (x) = Dmni j (x) are satisfied. The system, Eq. (19.1), together with the boundary conditions on ∂   Ci jmn (x) enm (x) n j |∂2 = Fi (x),   ωm (x) |∂3 = 0, Di jmn (x) ψnm (x) n j |∂4 = G i (x), u m (x) |∂1 = 0,

(19.3)

where Fi (x) and G i (x) are the surface body forces and moments, representing the static boundary value problem associated with the linear theory of micropolar elasticity whose coefficients are rapidly oscillating. In Eq.  (19.3), n j is the unit outer ∂ j = ∅ (disjoint sets) and normal vector to ∂ and the subsets ∂i satisfy ∂i ∂ =

4 

i= j

∂i .

i=1

In addition to the problem statement (Eqs. ( 19.1)–(19.3)), we deal with the spring model described above considering imperfect contact conditions at the interface , such as

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453

     Ci jmn (x)enm (x) n j = K i j u j , Ci jmn (x)enm (x) n j = 0, on       (19.4) Di jmn (x)ψnm (x) n j = Q i j ω j , Di jmn (x)ψnm (x) n j = 0, on  where [[ p]] = p (1) − p (2) means the jump of the function p across the interface and microtational imperfec. K i j [N/m3 ] and Q i j [N/m] are ⎞ ⎞ ⎛ the extensional ⎛ Kt 0 0 Qt 0 0 tion parameters, such that K i j = ⎝ 0 K s 0 ⎠ and Q i j = ⎝ 0 Q s 0 ⎠. Here, 0 0 Kn 0 0 Qn K t , K s , K n , Q t , Q s , and Q n are the interface parameters in the normal and tangential directions, which are considered equals for the sake of simplicity as follows: K t = K s = K n and Q t = Q s = Q n . An equivalent form of the imperfect contact conditions (19.4) has been derived for soft micropolar interfaces in Serpilli (2018), by means of the asymptotic analysis.

19.3 Asymptotic Homogenization Method and Effective Engineering Moduli for Periodic Laminated Micropolar Media From now on, let us consider that the three-dimensional heterogeneous centrosymmetric linear elastic micropolar continuum  is described by a parallelepiped of dimensions li (i = 1, 2, 3) generated by repetitions of a periodic cell Y, whose layered direction is along the y3 -axis. At the microscale, the transversal cross-section of Y is characterized by a bi-laminated composite in the plane O y2 y3 , see Fig. 19.1, where the constituent material phases are denoted by Sγ (γ = 1, 2) with volume Vγ , such as Y = S1 ∪ S2 , S1 ∩ S2 = ∅, and V1 + V2 = 1. Imperfect contact conditions

y3 ∂Ωε

l3

Ωε

1

Y

1

Γ

2

Y

2

Γ

N

Y

N

Γ

Y Layer 2 (V2 ) Interface Γ Layer 1 (V1 )

0

x3 x1

x2

L

(a)

Θ1 l2 Θ2 l2

...

l 2 y2 ΘN l 2

(b)

Fig. 19.1 a Heterogeneous Cosserat composite; b Cross-section of a periodic bi-laminated structure Y at the plane O y2 y3 with non-uniform imperfect interface  partitioned in N disjoint sub-interfaces r  (r = 1, 2, . . . , N )

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(uniform or non-uniform) are assumed at the interface region  between the layers following Eq. (19.4). The non-uniform imperfect interface is defined by partitioning  along the y2 direction, where θr 2 is the length of the r-partition (denoted by r ) with imperfection length fraction θr (r = 1, . . . , N ), 2 is the characteristic length of Y along the y2 N  r direction, and N is the number of partitions, such as  = . In this context, K i j r =1

and Q i j are considered piecewise linear functions in each unit cell partition r Y (with r  −1  θs 2 < y2 < rs=1 θs 2 , θ0 fixed), such as r Y = y ∈ R3 : 0 < yi < i , and rs=0 ⎧1 1 ⎪ ⎨ f in Y N  r , where f might be replaced by K i j and = 0} and Y = Y. Also, r f = ... ⎪ r =1 ⎩N N f in Y Q i j or any function defined in r Y. On the other hand, as a particular case, a uniform interface is taken into account when the values of the imperfection parameters in each cell partition r Y are equal. In this framework, the applied methodology based on the AHM for centrosymmetric micropolar composites with perfect contact conditions (Yanes et al. 2022; Rodríguez-Ramos et al. 2022) is implemented in the case of an imperfect interface. The AHM provides averaged expressions for the rapidly oscillating elasticity tensors of the original problem and proposes a homogeneous equivalent medium with the same behavior. Its main assumptions are that all fields are considered as power series of the small and positive definite dimensionless parameter ε whose coefficients are dependent on the macro (x) and micro ( y) scales; see, for instance, (Sanchez-Palencia 1980; Pobedrya 1984; Bakhvalov and Panasenko 1989). Both scales are related as y = x/ε, where ε = /L 1 is defined by the ratio between the characteristic size of the periodicity cell ( ) and the diameter of the body (L). The AHM starts from the substitution of the expansions for the displacements u εm (x) and the microrotations ωmε (x) u εm (x) =

∞  α=0

ε εα u (α) m (x, y), ωm (x) =

∞ 

εα ωm(α) (x, y),

(19.5)

α=0

into the problem (Eqs. 19.1–19.4), and following algebraic operations and differx , y ) and ωm(i) (xx , y ) (i = 0, 1, 2, . . . ) are infinitely differentiation rules. Here, u (i) m (x entiable and Y-periodic functions with respect to y . Thus, a sequence of problems given by partial differential equations is obtained in relation to the power of the ε parameter. From them, the formulation of local problems on Y, the effective moduli, and the equivalent homogenized problem with its asymptotic solution are obtained. Details about the AHM methodology related to micropolar laminated composites are shown in Forest et al. (2001), Gorbachev and Emel’yanov (2014), Yanes et al. (2022), Rodríguez-Ramos et al. (2022) and are omitted here. The mathematical statement of the pq r L1 and pq r L2 (with p, q = 1, 2, 3) local problems over each partition r Y are given by

19 Effective Engineering Constants for Micropolar …

455

    r r  Ci3 pq + Ci3m3r pq N m = 0, in Yr Ci3 pq + Ci3m3pq Nm n 3 = r 1  r0, in  r pq L  r r C n + C N = K ⎪ i3m3 pq m i j pq N j , in  ⎪ 3 ⎩ i3 pq r N = 0, pq m r Y

(19.6)

    r r  Di3 pq + Di3m3r pq M m = 0, in Yr Di3 pq + Di3m3pq Mm n 3 = r 2  r0, in  r pq L  r r D n + D M = Q ⎪ i3m3 pq m i j pq M j , in  ⎪ 3 ⎩ i3 pq r pq Mm r Y = 0,

(19.7)

⎧ ⎪ ⎪ ⎨

⎧ ⎪ ⎪ ⎨

where (•) = d(•)/dy3 . In Eqs. (19.6) and (19.7), pqr N m and pqr M m are the local pqdisplacements and pq-microrotations defined in the r-partition of the cell Y, respectively. The periodicity conditions pq Nm (0) = pq Nm (li ) and pq Mm (0) = pq Mm (li ) are satisfied and the unknown functions pqr N m and pqr M m only depend on y3 as well. N  p (i) Vi with The symbol p denotes Voigt’s average of the property p, i.e. p = N the number of phases in Y and (1)

N 

i=1

Vi = 1. In case of a bi-laminated composite,

i=1

(2)

p = p V1 + p V2 where V1 = γ / 3 and V2 = 1 − γ / 3 are the volume fractions per unit length occupied by layers 1 and 2, respectively, such as V1 + V2 = 1. γ is the y3 coordinate of the constituent contact. Once the unknown functions pqr N m and pqr N m are determined, the corresponding effective properties in terms of the r -interface-partition formulation can be found as follows: Ci∗j pq =

N  r =0

Di∗j pq =

N  r =0

  θr Ci j pq + Ci jm3 rpq N m r ,

(19.8)

  θr Di j pq + Di jm3 rpq M m r .

(19.9)

Y

Y

The local functions pqr N m and pqr M m can be determined as it is shown in Yanes et al. (2022), Rodríguez-Ramos et al. (2022) and after their replacement into Eq. (19.8), the corresponding stiffness and torque effective properties are obtained as functions of the constituent’s properties, the imperfection parameters, and the constituent’s volume fractions   −1 Ca3 pq + Ci∗j pq = Ci j pq − Ci jm3 Cm3a3    −1    −1  N −1 r −1 −1 Ca3b3 + −1 Cb3c3 Cc3 pq , K ab 3 r =1 θr C i jm3 C m3a3   −1 Di∗j pq = Di j pq − Di jm3 Dm3a3 Da3 pq +    −1    −1  N −1 r −1 −1 Da3b3 + −1 Db3c3 Dc3 pq . Q ab 3 r =1 θr Di jm3 Dm3a3

(19.10)

(19.11)

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Since both local problems (Eqs. 19.6 and 19.7) and the effective properties (Eqs. 19.10 and 19.11) have the same structure, only the analytical expressions for the stiffness are shown. The analytical expressions for effective torque moduli can be found replacing D for C, and Q for K .

19.3.1 Effective Engineering Moduli Assuming that the constituents are centro-symmetric isotropic materials, these are characterized by 6 independent constants C1122 , C1212 , C1221 , D1122 , D1212 , and D1221 , see Hassanpour and Heppler (2017), through the relations Ci jmn = C1122 δi j δmn + C1212 δim δ jn + C1221 δin δ jm , Di jmn = D1122 δi j δmn + D1212 δm δ jn + D1221 δin δ jm ,

(19.12) (19.13)

where δi j is the Kronecker delta tensor. The global symmetry after the homogenization process is orthotropic, defined by 18 non-zero effective moduli, as it is pointed out in Yanes et al. (2022), Rodríguez∗ = Ramos et al. (2022). The nine non-zero stiffness effective properties are C1111 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , C2222 , C3333 , C1122 , C1133 = C2233 , C1313 = C2323 , C3232 = C3131 , C1331 = C2332 ∗ ∗ ∗ = C2121 , and C1221 . Similarly, the other nine torque properties can be derived. C1212 Following the strain-stress relationships for a centro-symmetric micropolar media according to Eq. (19.1), and applying the effective relations reported in Eqs. (64)–(66) of (Rodríguez-Ramos et al. 2022) for the corresponding stiffness effective properties (see, Eq. 19.10), the independent effective engineering moduli written as functions of the stiffness matrix components and the imperfection parameters are given as follows: Effective Young’s moduli:    2  −1 C1111 − C1122 C1111 + C1122 − 2 C1122 C1111 ∗ ∗  2  , S E1 = S E2 = −1 C1111 − C1122 C1111   2  −1 C1111 + C1122 − 2 C1122 B1 ( r K 33 ) C1111 ∗ E = . S 3  2    −1 −1 2 C1111 + C1122 − 2 C1122 + 2 C1122 C1111 C1111 B1 ( r K 33 ) (19.14) Effective shear moduli:

19 Effective Engineering Constants for Micropolar …

C1212 2 − C1221 2 , C1212    2  −1 C1212 − C1221 B2 ( r K 22 ) C1212 = ,  2    −1 −1 2 C1212 − C1221 + C1221 C1212 C1212 B2 ( r K 22 )  2  −1 ∗ ∗ S G 32 = S G 31 = C 1212 − C 1221 C 1212 . ∗ S G 12

∗ S G 13

= S G ∗23

457

= S G ∗21 =

(19.15)

Effective Poisson’s ratios: ∗ S ν21

  2 −1 − C1122 C1122 C1111  = 2 , −1 C1122 C1111 − C1111

  −1 B1 ( r K 33 ) C1122 C1111 = = .  2    −1 −1 2 C1111 + C1122 − 2 C1122 C1111 + 2 C1122 C1111 B1 ( r K 33 ) (19.16) Effective shear-strain ratios:

∗ S ν32

∗ S ν31

∗ S ζ2112

=

C1221 , C1212

∗ S ζ3223

  −1 , = C1221 C1212

(19.17)

where the following parameters B1 ( r K 33 ) and B2 ( r K 22 ) are introduced for better presentation of the formulae B1 ( r K 33 ) =

N  −1   1 −1 −1 + r K 33 θr C1111 ,

3 r =1

N  −1   1 −1 −1 B2 ( K 22 ) = + r K 22 θr C1212 .

3 r =1

(19.18)

r

The effective engineering constants for torque moduli can be written in the analogous form, and they are denoted by a subscript T , for example: the torsional Young’s moduli T E i∗ , the torsional shear moduli T G ∗12 , T G ∗13 and T G ∗32 , the twist Poisson’s ∗ ∗ ∗ ∗ and T ν32 , and the twist shear-strain ratios S ζ2112 and S ζ3223 . coefficient T ν21

19.4 Numerical Results In this section, Eqs. (19.14)–(19.18) are implemented to analyze the effect of a non-uniform or uniform imperfect interface  on the effective engineering moduli of a centro-symmetric bi-laminated Cosserat composite (layer 1/layer 2 = SyF/PUF) with isotropic constituents. The values of the Cosserat elastic parameters listed in Table 19.1 are used for computations through the relations C1122 ≡ λ, (C1212 + C1221 )/2 ≡ μ, (C1212 − C1221 )/2 ≡ α, D1122 ≡ β, (D1212 + D1221 )/2 ≡

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R. Rodríguez-Ramos et al.

Table 19.1 Constituent material properties. a Syntactic foam—hollow glass spheres in epoxy resin, and b Dense polyurethane foam Material λ (MPa) μ (MPa) α (MPa) β (N) γ (N)  (N) properties SyFa PUFb

2097.0 762.7

1033.0 104.0

114.8 4.333

−2.91 −26.65

4.364 39.98

−0.133 4.504

γ , and (D1212 − D1221 )/2 ≡ , where λ and μ are the Lamé parameters, α is the micropolar couple modulus, and the properties β, γ , and  represent the additional micropolar elastic constants introduced in micropolar theory, according to the following constitutive law for a micropolar isotropic centro-symmetric material: σi j = (μ + κ)ei j + (μ − κ)e ji + λekk δi j , χi j = (γ + β)ψi j + (γ − β)ψ ji + αψkk δi j ,

(19.19)

where σi j and χi j represent the stress and couple-stress tensor components, respectively. These material data are taken from Hassanpour and Heppler (2017). For the micropolar constants, the same notation of (Hassanpour and Heppler 2017) is used, where α is κ, β is α, γ remains γ , and  is β.

19.4.1 Non-uniform Imperfect Interface Here, the non-uniform imperfect interface  is defined by a partition of N disjoint sub-interfaces r  characterized by an imperfection length fraction r θ and by two sets of imperfection parameters (r K i j and r Q i j ) with a considerably large gap between their values for each partition; see Sect. 19.3.  In Table 19.3, the effective engineering moduli related to the stiffness S E 3∗ ,  ∗ ∗ ∗ ∗ ∗ S G 13 , S ν31 and torques T E 3 , T G 13 , T ν31 affected by the imperfection are shown for four SyF volume fractions (V1 ) equal to 0.2, 0.4, 0.6, and 0.8. Two different partitions of  are analyzed, one with N = 2 partitions and another with N = 4

Table 19.2 Sets of values for the r K i j and r Q i j imperfection parameters considered in each partition of r  N =2 N =4 1K 2K 1Q 2Q 1K 2K 3K 4K 1Q 2Q 3Q 4Q Set ij ij ij ij ij ij ij ij ij ij ij ij S1 S2 S3

103 105 107

104 106 108

101 103 105

102 104 106

103 105 107

104 106 108

105 107 109

106 108 1010

101 103 105

102 104 106

103 105 107

104 106 108

19 Effective Engineering Constants for Micropolar …

459

Table 19.3 Variation of the effective engineering moduli related to non-uniform imperfect interface for four SyF volume fractions (V1 ). The moduli S E 3∗ , S G ∗13 are measured in [MPa]; T E 3∗ , T G ∗13 in ∗ , ν ∗ are dimensionless [N]; S ν31 T 31 Moduli

V1

N =2 S1

∗ S E3

∗ S G 13

∗ S ν32

∗ T E3

∗ T G 13

∗ T ν32

∗

N =4 S2

S3

S1

Perfect S2

S3

0.2

0.0055∗

0.5493

48.8405

0.2774

25.1987

307.7864

581.9802

0.4

0.0055∗

0.5495

50.4403

0.2775

25.7829

392.7140

853.5091

0.6

0.0055∗

0.5496

51.5785

0.2776

26.2485

489.5963

1194.0030

0.8

0.0055∗

0.5497

52.5544

0.2776

26.6766

626.0288

1719.9066

0.2

0.0055∗

0.5437

25.9370

0.2752

14.7569

47.3517

62.7962

0.4

0.0055∗

0.5457

31.2829

0.2759

17.0148

67.8537

97.8025

0.6

0.0055∗

0.5470

35.9172

0.2765

19.1634

92.8556

142.0619

0.8

0.0055∗

0.5481

41.0084

0.2770

21.6470

135.7580

218.5210

0.2

3.2 × 10−6

3.2 × 10−4

0.0283

2 × 10−4

0.0146

0.1785

0.3376

0.4

1.9 × 10−6

1.9 × 10−4

0.0171

9 × 10−5

0.0088

0.1335

0.2901

0.6

1.3 × 10−6

1.3 × 10−4

0.0118

6 × 10−5

0.0060

0.1117

0.2724

0.8

0.9 × 10−6

0.9 × 10−5

0.0086

5 × 10−5

0.0044

0.1024

0.2813

0.2

6 × 10−5∗

0.0044

0.0219

0.0025

0.0210

0.0227

0.0228

0.4

6 × 10−5∗

0.0041

0.0151

0.0024

0.0147

0.0155

0.0156

0.6

6 × 10−5∗

0.0033

0.0083

0.0021

0.0081

0.0084

0.0084

0.8

5 × 10−5∗

0.0010

0.0012

0.0008

0.0012

0.0012

0.0012

0.2

6 × 10−5∗

0.0055∗

0.5043

0.0028∗

0.2585

3.9585

8.2248

0.4

6 × 10−5∗

0.0055∗

0.4800

0.0028∗

0.2477

2.7316

5.1540

0.6

6 × 10−5∗

0.0055∗

0.4512

0.0028∗

0.2360

1.9728

3.3367

0.8

6 × 10−5∗

0.0055∗

0.3998

0.0028∗

0.2179

1.2555

1.7927

0.2

−0.0024

−0.1943

−0.9585

−0.1085

−0.9211

−0.9964

−0.9990

0.4

−0.0035

−0.2605

−0.9704

−0.1511

−0.9432

−0.9965

−0.9987

0.6

−0.0065

−0.3955

−0.9833

−0.2483

−0.9677

−0.9975

−0.9989

0.8

−0.0439

−0.8207

−0.9973

−0.6981

−0.9949

−0.9994

−0.9996

The values with more significant digits are given in Appendix A

partitions. In the case of N = 2, θ1 = θ2 = 0.5 and the corresponding imperfection parameters are defined by r K i j and r Q i j (r = 1, 2), whereas, for N = 4, θ1 = θ2 = θ3 = θ4 = 0.25 and the imperfection parameters are r K i j and r Q i j (r = 1, . . . , 4) with i j = 22, 33. For both partitions, three different sets of imperfection parameters (S1 , S2 and S3 ) are considered for r K i j and r Q i j ; see Table 19.2. For example, when N = 2, S1 is the set of values 1 K i j = 10−1 , 2 K i j = 100 , 1 Q i j = 10−1 , and 2 Q i j = 100 . The remaining sets can be understood in a similar form. The characteristic lengths of Y along the x2 and x3 directions are 3 = 10−6 m and 2 = 1, respectively. In addition, the effective values associate with the perfect contact case are reported for the same volume fractions. From Table 19.3, it can be observed that the influence of the non-uniform imperfect interface is remarkable in the effective engineering properties, regardless of the V1 volume fraction, and even the microstructure of the imperfection region determined by the partition N affects the behavior of the properties. A non-uniform interface with values for r K i j and r Q i j as in S1 or lower implies the delamination of the

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material, and, hence, a loss of the effective properties. However, as the values of the imperfection parameters increase as in S2 and S3 , an approach to the existence of a perfect interface is appreciated, and then the engineering moduli have an increment. The perfect contact is reached when the values of r K i j and r Q i j parameters are 1014 in each r . The highest values of the engineering constants are achieved in this perfect case. Furthermore, it can be seen that the effect of the is  imperfection ∗ ∗ ∗ E , G , ν more noticeable in the engineering moduli related to compliance S S S 3 13 31   ∗ , and is even more pronounced for than those related to torque T E 3∗ , T G ∗13 , T ν31 high SyF volume fraction. As N increases, the microstructure of the imperfection becomes finer, and its effect on the constant engineering behaviors is evident. On the other hand, in Table 19.4, the remaining effective engineering moduli, which are independent of the imperfection effect, are reported for four SyF volume fractions (V1 = 0.2, 0.4, 0.6, and 0.8). As can be seen in Eqs. (19.14)–(19.18), these effective moduli only depend on the material constituents and their volume fractions, therefore, their behaviors are related to the hardness or softness of the SyF material properties. According to Table 19.1, we can see that SyF is harder than PUF. Thus, for the perfect case, as V1 volume fraction increases, the effective engineering constants ∗ ∗ ∗ ∗ ∗ ∗ ∗ S E 1 , S E 3 , S G 13 , S G 12 , and S G 31 for compliance and the other ones T ζ2112 and T ζ3223 for torques increase too. The opposite happens for the remaining Cosserat elastic parameters, which are softer for SyF and thus for the composite as V1 increases. The effective engineering constants are stiffer in this case.

19.4.2 Uniform Imperfect Interface Now, the effect of a uniform imperfect interface on the effective engineering moduli is analyzed. The uniform imperfect interface is defined as a particular case of the previously described non-uniform imperfect ones assuming that the values of r K i j and r Q i j imperfection parameters are the same along , such as K ≡ r K i j and Q ≡ r Qi j . The numerical simulations are conducted for different grades of imperfection, such that the values for K are 106 , 5 × 106 , 107 , 3 × 107 , 5 × 107 , 108 , and the latest 1010 (perfect contact); and for Q they are 105 , 2 × 105 , 3 × 105 , 5 × 105 , 106 , 5 × 106 , and finally 107 (perfect contact), respectively. Also, the characteristic lengths 3 = 10−4 and 2 = 1. In Figs. 19.2 and 19.3, only the behaviors of the effective engineering moduli affected by the imperfection are illustrated for a bi-laminated Cosserat composites (SyF/PUF) versus SyF volume fraction considering different imperfect parameters. We remark that these effective engineering moduli are sensitive to the imperfection, that is, they get weaker and only reach their highest values in the case of perfect ∗ are more sensitive to the imperfection K contact. Notice that S E 3∗ , S G ∗13 , and S ν32 ∗ ∗ have the same performance for Q when when V1 increases, whereas T G 13 and T ν32 low values of V1 are attached. However, T E 3∗ undergoes slight changes caused by the effect of the Q imperfection. In this sense, a zoom illustrates the slight weakening

19 Effective Engineering Constants for Micropolar …

461

Table 19.4 Effective engineering moduli calculated for four SyF volume fractions V1 . The moduli ∗ ∗ ∗ ∗ ∗ ∗ ∗ S E 1 , S G 12 , and S G 31 are measured in [MPa]; T E 1 , T G 12 , and T G 31 in [N]; and the others S ν21 , ∗ ∗ ∗ ∗ ∗ S ζ2112 , S ζ3223 , T ν21 , T ζ2112 , and T ζ3223 are dimensionless V1

∗ S E1

∗ S G 12

∗ S G 31

∗ S ν21

∗ S ζ2112

∗ S ζ3223

0.2 0.4 0.6 0.8 V1 0.2 0.4 0.6 0.8

793.4487 1285.0272 1776.1720 2267.1612 ∗ T E1 0.0228 0.0156 0.0084 0.0012

96.8720 176.1125 255.2062 334.2504 ∗ T G 12 12.9020 9.6077 6.3040 2.9721

95.9654 175.2921 254.6187 333.9454 ∗ T G 31 12.8438 9.4956 6.1475 2.7994

0.3690 0.3510 0.3427 0.3380 ∗ T ν21 −0.9997 −0.9997 −0.9998 −0.9999

0.8329 0.8149 0.8071 0.8027 ∗ T ζ2112 0.8037 0.8133 0.8306 0.8706

0.8960 0.8720 0.8480 0.8240 ∗ T ζ3223 0.8506 0.9036 0.9567 1.0098

of the property when V1 is close to 0.4. Despite the imperfection effect, the S E 3∗ and ∗ ∗ ∗ S G 13 become stronger as V1 increase, whereas the opposite occurs for T E 3 and T G 13 . ∗ On the other hand, the behavior of the effective Poisson S ν32 and twist Poisson ∗ ∗ T ν32 moduli is remarkable. The module S ν32 has a concave upward behavior, whereas ∗ r r T ν32 is concave downward for all K i j and Q i j imperfection parameters in the whole ∗ ∗ V1 interval. Also, S ν32 is positive and T ν32 is negative. These behaviors are similar to the one reported by Dunn and Ledbetter (1995) for an elastic solid weakened by porosity and microcracks. Paraphrasing his statement 4 from the conclusions (Dunn and Ledbetter 1995), the Poisson and twist Poisson moduli can increase, decrease, or remain unchanged depending on the imperfection parameters and the SyF volume fractions. The trend of the pictures is reversed by passing from Figs. 19.2 and 19.3, they are mirror-like. This can be understood by looking at the values in Table 19.1; the elastic coefficients are larger for SyF, but the opposite happens for the micropolar constants—they are larger for PUF. ∗ in V1 = 0.8333087 Notice the existence of a change correlation point for T ν32  −1 2 D1111 =0 (Fig. 19.3). This point is a consequence of D1111 + D1122 − 2 D1122   −1 −1 ∗ ∗ in Eq. (19.16) for the torque. Thus, T ν32 = T ν31 = 0.5 D1122 D1111 ≡ H (V1 ). All the curves are intercepted in this correlation point and H (V1 ) shows the indepen∗ with respect to r Q i j . Moreover, it is worth mentioning that the comdence of T ν32 ∗ in Fig. 19.3 are comparable with those obtained experputed values for S E 3∗ and S ν32 imentally in Hassanpour and Heppler (2017). Indeed, the experimental values for the torsional micropolar Young’s modulus and twist Poisson’s ratio are, respectively, equal to 0 and -1. The slight difference, highlighted in the present plots, is likely due to the presence of the interface and numerical approximations.

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  ∗ related to stiffness versus V1 Fig. 19.2 Effective engineering moduli S E 3∗ , S G ∗13 , and S ν32 volume fraction of a bi-laminated Cosserat composite with uniform imperfect contact conditions

  ∗ related to torque versus V volume Fig. 19.3 Effective engineering moduli T E 3∗ , T G ∗13 , and T ν32 1 fraction of a bi-laminated Cosserat composite with uniform imperfect contact conditions

19 Effective Engineering Constants for Micropolar …

463

19.5 Conclusions In this work, the asymptotic homogenization method is applied to heterogeneous micropolar media. In particular, effective engineering expressions with isotropic symmetry layers are provided for multi-laminated Cosserat media under non-uniform imperfect contact conditions. The effective engineering properties for centrosymmetric laminated Cosserat composites are derived as a function of the material properties, the imperfection parameters, the cell length in the y3 direction, and the constituent’s volume fractions. The typical length scales of the periodic cell and the microstructure imperfection play an important role in the macroscopic behavior of the laminate structures. The homogenized Cosserat engineering constants are characterized by two effective Young’s moduli, three effective shear moduli, two effective Poisson’s ratios, and two effective shear-strain ratios. Actually, only the transverse properties perpendicular to the layer distribution, i.e. along the x3 , depend on the imperfection parameter. Finally, numerical results are we conclude that  discussed. In general,   ∗ ∗ and torque T E 3∗ , T G ∗13 , T ν31 effective engi(i) The stiffness S E 3∗ , S G ∗13 , S ν31 neering constants transverse to the distribution of the laminae are sensible to the imperfection effects; (ii) The effective engineering constants related to stiffness and torque, i.e. Young’s ∗ , and moduli E 1∗ = E 2∗ , shear moduli G ∗12 = G ∗21 , G ∗32 = G ∗31 , Poisson’s coefficient ν21 ∗ ∗ shear-strain ratios ζ2112 and ζ3223 are independent of the imperfection parameters and the cell length; (iii) The volume fraction has an influence on the behavior of the stiffness and torque effective engineering moduli when the imperfect contact is considered; and (iv) The cell length changes the effective engineering constants when imperfect contact conditions are assumed. Acknowledgements YEA gratefully acknowledges the CONACYT for the postdoctoral scholarship “Estancias Postdoctorales por México para la Formación y Consolidación de Investigadores por México” at IIT, UACJ, 2022-2024, and the financial support of Grant A1-S-37066 during the postdoctoral stay at IIT, UACJ, 2021-2022. CFSV is grateful for the support of the CONACYT Basic Science Grant A1-S-37066. VY is grateful for the support and funding of the XS-Meta project during the course of his PhD. FJS and RRR acknowledge the funding of PAPIIT-DGAPA-UNAM IN101822, 2022-2023. This work was partially written during the visit of RRR at Aix-Marseille University, Centrale Marseille, the LMA-CNRS, and the University of Ferrara 2022. RRR thanks co-funding of the Departmental Strategic Plan, Department of Engineering, University of Ferrara. RRR would also like to thank to EDITAL UFF PROPPI No. 05/2022 and PPG-MCCT of Universidade Federal Fluminense, Brazil. This work is devoted to Igor Sevostianov, who gave significant contributions in the micro-mechanic area.

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Appendix The corresponding full approximation values with more significant digits of the effective engineering moduli S E 3∗ , S G ∗13 , T E 3∗ , and T G ∗13 labeled with the symbol “*” in Table 19.5 are given. Table 19.5 The values with more significant digits of the effective engineering moduli. The moduli ∗ ∗ ∗ ∗ S E 3 and S G 13 are measured in [MPa]; T E 3 , and T G 13 in [N] Moduli

V1

N =2 S1

∗ S E3

∗ S G 13

∗ T E3

∗ T G 13

N =4 S2

S1

0.2

0.00549993

–

–

0.4

0.00549995

–

–

0.6

0.00549996

–

–

0.8

0.00549997

–

–

0.2

0.00549937

–

–

0.4

0.00549957

–

–

0.6

0.00549970

–

–

0.8

0.00549981

–

–

0.2

5.48674 × 10−5

–

–

0.4

5.48063 × 10−5

–

–

0.6

5.46415 × 10−5

–

–

0.8

5.25861 × 10−5

–

0.2

5.49995 × 10−5

0.00549501

0.00277542

0.4

5.49992 × 10−5

0.00549196

0.00277411

0.6

5.49988 × 10−5

0.00548791

0.00277255

0.8

5.49979 × 10−5

.00547929

0.00276982

–

References Achenbach JD, Zhu H (1989) Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites. J Mech Phys Solids 37(3):381–393 Adhikary DP, Dyskin DV (1997) A Cosserat continuum model for layered materials. Comput Geotech 20(1):1545 Bakhvalov N, Panasenko G (1989) Homogenization: averaging process in periodic media. In: Hills L (ed) In mathematics and its applications (Soviet Series), 1st edn. Moscow Bigoni D, Drugan W (2007) Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J Appl Mech 74(4):741–753 Bövik P (1994) On the modelling of thin interface layers in elastic and acoustic scattering problems. Q J Mech Appl Math 47(1):17–42 Brito-Santana H, Christoff BG, Mendes Ferreira AJ, Lebon F, Rodríguez-Ramos R, Tita V (2019) Delamination influence on elastic properties of laminated composites. Acta Mech 230:821–837 Ciarlet PG (1997) Mathematical elasticity, Theory of Plates, North-Holland, Amsterdam, II

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Cowin SC (1970) An incorrect inequality in micropolar elasticity theory. J Appl Math Phys 21:494– 497 Dong H, Wang J, Rubin M (2014) Cosserat interphase models for elasticity with application to the interphase bonding a spherical inclusion to an infinite matrix. Int J Solids Struct 51(2):462–477 Dong H, Wang J, Rubin M (2015) A nonlinear cosserat interphase model for residual stresses in an inclusion and the interphase that bonds it to an infinite matrix. Int J Solids Struct 62:186–206 Dunn M, Ledbetter H (1995) Poisson’s ratio of porous and microcracked solids: theory and application to oxide superconductors. J Mater Res 10(11):2715–2722 Duong VA, Diaz Diaz A, Chataigner S, Caronn J-F (2011) A layerwise finite element for multilayers with imperfect interfaces. Compos Struct 93:3262–3271 Ensan MN, Shahrour I (2003) A macroscopic constitutive law for elasto-plastic multilayered materials with imperfect interfaces: application to reinforced soils. Comput Geotech 30:339–345 Eringen AC (1968) Theory of micropolar elasticity. In: Liebowitz H (ed) Fracture. Academic Press, New York, pp 621–729 Eringen AC (1999) Microcontinuum field theories I: Foundations and solids. Springer, New York Espinosa-Almeyda Y, Yanes V, Rodríguez-Ramos R, Sabina FJ, Lebon, F., Sánchez-Valdés CF, Camacho-Montes H (2022) Chapter 6: overall properties for elastic micropolar heterogeneous laminated composites with centro-symmetric constituents. In: Altenbach H, Prikazchikov D, Nobili A (eds) Book series advanced structured materials 8611, Mechanics of high-contrast elastic solids: contributions from euromech colloquium 626. Springer Fatemi J, Van Keulen F, Onck PR (2002) Generalized continuum theories: application to stress analysis in bone. Meccanica 37:385–396 Fatemi J, Onck PR, Poort G, Van Keulen F (2003) Cosserat moduli of anisotropic cancellous bone: a micromechanical analysis. J Phys IV France 105:273–280 Forest S, Sab K (1998) Cosserat overall modeling of heterogeneous media. Mech Res Commun 25(4):449–454 Forest S, Trinh D (2011) Generalized continua and non-homogeneous boundary conditions in homogenisation methods. J Appl Math Mech ZAMM 91(2):90–109 Forest S, Padel F, Sab K (2001) Asymptotic analysis of heterogeneous Cosserat media. Int J Solids Struct 38:4585–4608 Geymonat G, Hendili S, Krasucki F, Serpilli M, Vidrascu M (2014) Asymptotic expansions and domain decomposition. In: Erhel J, Gander M, Halpern L, Pichot G, Sassi T, Widlund O (eds) Domain decomposition methods in science and engineering XXI. Lecture notes in computational science and engineering, vol 98. Springer, Cham Goda I, Assidi M, Belouettar S, Ganghoffer JF (1990) Fracture mechanics of bone with short cracks. J Biomech 23:967–975 Gorbachev VI, Emel’yanov AN (2014) Homogenization of the Equations of the Cosserat Theory of Elasticity of Inhomogeneous Bodies. Mech Solids 49(1):73–82 Gorbachev V, Emel’yanov A (2021) Homogenization of problems of Cosserat theory of elasticity of composites. Additional materials. In: International scientific symposium in problems of mechanics of deformable solids dedicated to A.A. I1’Yushin on the occasion of his 100th birthday 49:81–88, [in Russian] Hashin Z (2002) Thin interphase/imperfect interface in elasticity with application to coated fiber composites. J Mech Phys Solids 50(12):2509–2537 Hassanpour S, Heppler GR (2017) Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. Math Mech Solids 22:224–242 Hussan JR, Trew ML, Hunter PJ (2012) A mean-field model of ventricular muscle tissue. J Biomech Eng 134(7):071003 Jasiuk I (2018) Micromechanics of bone modeled as a composite material. In: Meguid S, Weng G (eds) Micromechanics and nanomechanics of composite solids. Springer, Cham Khoroshun LP (2019) Effective elastic properties of laminated composite materials with interfacial defects. Int Appl Mech 55:187–198

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Kumari R, Singh AK, Chaki MS (2022) Influence of abrupt thickening on the shear wave propagation on reduced cosserat media with imperfect interface. Int J Geomech 22(4):04022018 Lakes R (1993) Materials with structural hierarchy. Nature 361:511–515 Lakes R, Nakamura S, Behiri J, Bonfield W (1990) Fracture mechanics of bone with short cracks. J Biomech 23:967–975 Lebée A, Sab K (2010) A Cosserat multiparticle model for periodically layered materials. Mech Res Commun 37:293–297 Nika G, On a hierarchy of effective models for the biomechanics of human compact bone tissue. HAL Id: hal-03629864 Park HC, Lakes RS (1987) Fracture mechanics of bone with short cracks. J Biomech 23:967–975 Pobedrya B (1984) Mechanics of composite materials, 1st edn. Izd-vo MGU, Moscow (in Russian) Riahi A, Curran JH (2009) Full 3D finite element Cosserat formulation with application in layered structures. Appl Math Model 33:3450–3464 Riahi A, Curran JH (2009) Full 3d finite element Cosserat formulation with application in layered structures. Appl Math Modell 33:3450–3464 Riahi A, Curran JH (2010) Comparison of the Cosserat continuum approach with finite element interface models in a simulation of layered materials. Trans A: Civ Eng 17(1):39–52 Rodríguez-Ramos R, Yanes V, Espinosa-Almeyda Y, Otero JA, Sabina FJ, Sánchez-Valdés CF, Lebon F (2022) Micro-macro asymptotic approach applied to heterogeneous elastic micropolar media. Analysis of some examples. Int J Solids Struct 111444:239–240 Rubin MB, Benveniste Y (2004) A Cosserat shell model for interphases in elastic media. J Mech Phys Solids 52(5):1023–1052 Sack KL, Skatulla S, Sansour C (2016) Biological tissue mechanics with fibres modelled as onedimensional Cosserat continua. Applications to cardiac tissue. Int J Solids Struct 81:84–94 Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer, Berlin, Heidelberg Serpilli M (2018) On modeling interfaces in linear micropolar composites. Math Mech Solids 23(4):667–685 Serpilli M (2019) Classical and higher order interface conditions in poroelasticity. Ann Solid Struct Mech 11:1–10 Serpilli M, Krasucki F, Geymonat G (2013) An asymptotic strain gradient Reissner–Mindlin plate model. Meccanica 48(8):2007–2018 Serpilli M, Rizzoni R, Rodríguez-Ramos R, Lebon F, Dumont S (2022) A novel form of imperfect contact laws in flexoelectricity. Comp Struct 300:116059 Sertse H, Yu W (2017) Three-dimensional effective properties of layered composites with imperfect interfaces. Adv Aircr Spacecr Sci 4(6):639–650 Tanaka M, Adachi T (1999) Lattice continuum model for bone remodeling considering microstructural optimality of trabecular architecture. In: Pedersen P, Bendsoe MP (eds) Proceedings of the IUTAM symposium on synthesis in bio solid mechanics. Kluwer Academic Publishers, The Netherlands, pp 43–54 Videla J, Atroshchenko E (2017) Analytical study of a circular inhomogeneity with homogeneously imperfect interface in plane micropolar elasticity. Z Angew Math Mech 97(3):322–339 Yanes V, Sabina FJ, Espinosa-Almeyda Y, Otero JA, Rodríguez-Ramos R (2022) Asymptotic homogenization approach applied to Cosserat heterogeneous media. In: Andrianov I, Gluzman S, Mityushev V (2022) Mechanics and physics of structured media. Academic Press, Elsevier, USA Yang JFC, Lakes RS (1981) Transient study of couple stress in compact bone: torsion. J Biomech Eng 103:275–279 Yang JFC, Lakes RS (1982) Experimental study of micropolar and couple-stress elasticity in bone in bending. J Biomech 15:91–98

Chapter 20

The Mixed Problems of Poroelasticity for Rectangular Domains Natalya Vaysfeld and Zinaida Zhuravlova

Abstract In this paper, the exact solution of the poroelasticity problem for finite and semi-infinite rectangular regions in terms of Biot’s model is derived. To solve the problem, the analytical method based on the formulation of a mathematical physics’ boundary problem with mixed boundary conditions is used. It allowed to construct the exact solutions of the problems for permeable and impermeable boundary surfaces in an explicit form. The formulas for displacements, stress and pore pressure are used not only for quantitative, but also for qualitative analysis: the influence of different poroelastic constants, loading types and boundary conditions on the stress state of rectangular domains was investigated. The obtained formulas can serve as an etalon for the verification of effective numeric solving methods in poroelasticity problems. Keywords A poroelastic rectangle · A poroelastic semi-strip · Exact solution · An integral transform · Matrix differential calculation

20.1 Introduction The investigation of poroelastic materials characteristics is a relevant scientific challenge over the past decades. It is explained by the important properties of these materials for its use in medicine and engineering. During this time different models of poroelasticity were developed, among which the fundamental ones are given at well-known papers of Terzaghi (1925) and Biot (1941). Continuum-type mechanics of porous media having a generally anisotropic, product-like fractal geometry was considered in Li and Ostoja-Starzewski (2020).

N. Vaysfeld (B) King’s College, Strand Building, London S2.35, UK e-mail: [email protected] Z. Zhuravlova Faculty of Mathematics, Physics and Information Technologies, Odessa I.I. Mechnikov National University, Str. Dvoryanskaya, 2, Odessa 65082, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_20

467

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The methods worked out in papers of Igor Sevostianov made it possible to solve a new class of problems in the theory of poroelasticity. The methods he developed can be used to solve the problem of poroelasticity in a more complex statement, such as with regard of tortuosity of the pore spare (Sevostianov 2022), with consideration of time (Liu et al. 2021) and with the influence of thermal or electrical sources (Łyd˙zba et al. 2021). Among the methods of investigation two main directions can be selected: purely numerical investigations and investigations using different analytical mathematical methods. Between various numerical approaches for the solving of poroelasticity problems for rectangular areas the following should be noted: the Galerkin method and the boundary element method. The Galerkin method was used for solving equations of motion of an air saturated rectangular porous plate under the terms of the model based on a mixed displacement–pressure formulation of the Biot–Allard theory (Etchessahar et al. 2001) and in the non-linear form for the impact analysis of the poroelastic plates (Shariyat et al. 2020). The poroelastic plane strain BEM formulation was used for the study of dynamic response of embedded strip and rectangular shallow foundations subjected to time-harmonic vertical excitations (Chopra and Dargush 1997). The numerical and experimental investigation for sound attenuation of a silencer consisting of a lamella network made with melamine foam inserted in rectangular duct was done in Li et al. (2021). Semi-analytical approaches are also widely used for the solving of such problems. The dynamic problem of an infinite plate resting on a poroelastic layered half-space soil medium with imperfect interfaces to a moving load in terms of Kirchhoff thin plate theory and Biot’s fully dynamic poroelastic theory was solved semi-analytically with the help of Fourier transform and vector functions in Zheng et al. (2022). The dynamic response of a rectangular, simply supported, thin, fluid-saturated poroelastic plate to harmonic load was derived analytically-numerically in Theodorakopoulos and Beskos (1994). The quantity of papers dedicated to the application of analytical methods for solving poroelasticity problems for rectangular areas is significantly limited. Three dimensional wave propagation in poroelastic plate immersed in an inviscid elastic fluid employing Biot’s theory with the help of dilatational and vector potentials was analyzed in Shah and Tajuddin (2011). An analytical solution for plane-strain poroelasticity induced by surface-normal loading within a finite rectangular fluidsaturated domain was presented in Li et al. (2016) for special boundary conditions. The Laplace transform was inverted numerically there, which, as it is known, is an incorrect problem of mathematical physics and which can cause problems with calculations. The analytical solution for rectangular domain configuration with special boundary conditions was provided in Vaughan et al. (2013) by direct integration. The poroelastic problem for a semi-plane was solved in Vaysfeld and Zhuravlova (2022) using the integral transforms method and matrix differential calculation apparatus. The analytical investigation of the axisymmetric and steady flow of incompressible couple stress fluid through a rigid sphere embedded in a porous medium was done in Madasu and Sarkar (2022). A closed-form analytical expression for

20 The Mixed Problems of Poroelasticity for Rectangular Domains

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hydrodynamic drag on the bounded spheroidal particle was determined in Bucha and Prasad (2021). The fully developed laminar flow of a viscous incompressible fluid in a long composite cylindrical channel was considered in Singh and Verma (2020), and analytical expressions for velocity profiles, rate of volume flow and shear stress on the boundaries surface were obtained. As can be seen from the review, the number of analytical methods developed for solving poroelasticity problems is significantly less than solving with numerical approaches. So, the problem of working out new analytical approaches of such problems is relevant, especially with regard of that the proposed method allows to derive the exact solutions for some boundary problems of poroelasticity.

20.2 The Statement of the Poroelasticity Problems for Domains of Rectangular Shapes Typical model poroelasticity problems are boundary problems that consider the stress state of the rectangular domains. In the present paper, the purpose of investigation was to derive a classical solution for the problem about the stress state of the rectangular area and to find out under which parameters of the problem the influence of boundary conditions become so insignificant that instead of a solution for the finite rectangular domain, the solution for a semi-infinite poroelastic strip can be used. The poroelastic rectangle domain, which is described in dimensionless coordinates by formulas 0 < x < d, 0 < y < 1, is considered in the terms of Biot’s model (Biot 1941): ⎧ 2 ∂ u κ−1 ∂ 2 u 2 ∂2v κ−1 ∂ p ⎪ ⎪ ⎨ ∂ x 2 2+ κ+1 ∂ y 2 2+ κ+1 ∂ x∂ y2 − α κ+1 ∂ x = 0, ∂ v ∂ v + κ+1 + 2 ∂ u − α ∂∂ py = 0, (20.1) ∂x2 κ−1 ∂ y 2  κ−1 ∂ x∂ y ⎪ ⎪ ⎩ ∂ 2 p2 + ∂ 2 p2 − α ∂u + ∂v − S P p = 0 ∂x

∂y

∂x

K

∂y

K

u (x,y)

, v(x, y) = y h are dimensionless displacements of the Here u(x, y) = u x (x,y) h ˜ solid skeleton, p(x, y) = p(x,y) is dimensionless pore pressure, κ = 3 − 4μ is G Muskhelishvili’s constant, μ is Poisson ratio, G is shear modulus, α is Biot’s coefh2 , S P = SG are dimensionless values, S is storativity of the pore ficient, K = Gk space, k is permeability, d = a/h, h is the width of the rectangle, a is the length of the rectangle. The boundary x = 0 is loaded by dimensionless mechanical loadings L(y), T (y) and fluid pressure P(y) σx |x=0 = −L(y) − α P(y), τr z |x=0 = T (y), p|x=0 = P(y), 0 < y < 1 Here σr (x, y) = effective stress.

∼

σ r (x,y) , τr z (x, G

y) =

(20.2)

∼

τ r z (x,y) G

are dimensionless normal and shear

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The boundary x = d is fixed, and different conditions of surface permeability could be stated: if the surface is supposed permeable, the boundary conditions of the problem are formulated in the form u|x=d = 0, v|x=d = 0, p|x=d = 0

(20.3)

If the surface x = d is supposed impermeable, then boundary conditions (3) are changed u|x=d = 0, v|x=d

 ∂ p  = 0, =0 ∂ x x=d

(20.4)

At the boundaries y = 0, y = 1 the conditions of ideal contact for stress and undrained conditions for pore pressure are fulfilled   ∂ p  v| y=0 = 0, τx y  y=0 = 0, =0 ∂ y  y=0   ∂ p   v| y=1 = 0, τx y y=1 = 0, =0 ∂ y  y=1

(20.5) (20.6)

The exact solution in the explicit form of the stated boundary-valued problem (1)– (6) must be found for different pore pressure conditions (3)–(4) on the rectangular surface.

20.3 Deriving the Exact Solution of the Stated Problem for the Poroelastic Finite Rectangular Domain The boundary-valued problem for the rectangle is reduced to a one-dimensional problem with the help of finite sin-, cos- Fourier transform applied with regard to variable y ⎤⎡ ⎡ ⎤ ⎤

1 u(x, y) cosβy u β (x) ⎣ v(x, y) ⎦⎣ sinβy ⎦dy, βn = π n, n = 0, 1, 2, ... ⎣ vβ (x) ⎦ = 0 p(x, y) pβ (x) cosβy ⎡

(20.7)

The one-dimensional problem in the transform domain has the following form ⎧ 2β  κ−1   2 κ−1 ⎪ ⎨ u β (x) + κ+1 v β (x) − α κ+1 p β (x) − β κ+1 u β (x) = 0, 2β  u β (x) − β 2 κ+1 v + αβpβ (x) = 0, v  β (x) − κ−1 κ−1 β (x) ⎪ ⎩ p  (x) − α u  (x) − α v (x) − β 2 + S P  p (x) = 0, β β β β K K K

(20.8)

20 The Mixed Problems of Poroelasticity for Rectangular Domains

(1 − μ)u  β (0) + μβvβ (0) =

471

 1 − 2μ  α Pβ − L β , v  β (0) − βu β (0) = Tβ , pβ (0) = Pβ , 2

(20.9)

In the case of conditions (3) of permeable surface x = d the transformed conditions have the form u β (d) = 0, vβ (d) = 0, pβ (d) = 0

(20.10)

In the case of impermeable surface condition formula (20.9) should be changed by u β (d) = 0, vβ (d) = 0, p  β (d) = 0

(20.11)

The idea of solving the indicated boundary value problem (8)-(11) is to formulate the vector boundary problem in the transform domain (Vaysfeld and Zhuravlova 2023) and to construct its exact solution with the help of the matrix differential calculation method. With this aim the differential operator of the second order L 2 → → → → is introduced. It has the form L 2 − y β (x) = I − y  (x) − Pβ − y β (x), y  β (x) − Q β − ⎛ β ⎞ ⎛ ⎞ 2β 0 − κ+1 α κ−1 u β (x) κ+1 ⎜ 2β ⎟ → where I is the unit matrix, − y β (x) = ⎝ vβ (x) ⎠, Q β = ⎝ κ−1 0 0 ⎠, α pβ (x) 0 0 K ⎞ ⎛ 2 κ−1 0 0 β ⎟ ⎜ κ+1 Pβ = ⎝ 0 β 2 κ+1 −αβ ⎠. κ−1 αβ SP 2 β + 0 K K The boundary conditions are formulated in the form of boundary matrix functionals with the help of matrices of 3 × 3 order Ai,β , Bi,β , i = 0, 1, where ⎛ ⎛ ⎞ ⎞ 1−μ 0 0 0 βμ 0 A0,β = ⎝ 0 1 0 ⎠, B0,β = ⎝ −β 0 0 ⎠. The matrices A1,β , B1,β take the 0 00 0 0 1 ⎛ ⎛ ⎞ ⎞ 000 100 form A1,β = ⎝ 0 0 0 ⎠, B1,β = ⎝ 0 1 0 ⎠ for the permeable boundary condi000 001 ⎛ ⎛ ⎞ ⎞ 000 100 tions (3), and A1,β = ⎝ 0 0 0 ⎠, B1,β = ⎝ 0 1 0 ⎠ for the impermeable boundary 001 000 conditions (4). Then the boundary problem (8)-(11) is rewritten in the following form ⎧ − → ⎨ L 2 y β (x) = 0, 0 < x < d, → → → y  β (0) + B0,β − y β (0) = − A − g β, ⎩ 0,β − → − →  A1,β y β (d) + B1,β y β (d) = 0

(20.12)

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   T → Here − g β = (1 − 2μ) α Pβ − L β /(2G); Tβ /G; Pβ , L β , Tβ , Pβ are transforms of the known given loading functions. To solve the vector Eq. (20.12), according to the method (Popov 2013) one has to first solve the corresponding matrix equation L 2 Yβ (x) = 0, 0 < x < d

(20.13)

Here Yβ (x) is the matrix 3 × 3 order. For this the formula (Grantmakher 1967) is used, according to which the solution of the matrix equation is constructed in the form Yβ (x) =

1 2πi



eξ x M −1 (ξ )dξ,

(20.14)

C

Here M −1 (ξ ) is the inverse matrix to the matrix M(ξ ) = ⎞ ⎛ 2β κ−1 2 ξ 2 − κ−1 β ξ −α ξ κ+1 κ+1 κ+1 ⎟ ⎜ 2β β2 αβ ⎠. The closed contour C covers all ⎝ − κ−1 ξ ξ 2 − κ+1 κ−1 SP 2 2 − αβ ξ − β − − Kα ξ K K singularity points of the matrix M −1 (ξ ). To derive the matrix M(ξ ) the following procedure should be done. The matrix Yβ (x) is chosen in the form Yβ (x) = eξ x I , and it is substituted in Eq. (20.13). It leads to the equality L 2 eξ x I = M(ξ )eξ x . The determinant of the matrix M(ξ ) has 2 multiple  poles of the second

(κ−1) ,ξ = order ξ = β, ξ = −β and 2 simple poles ξ = β 2 + SKP + αK (κ+1)  2 (κ−1) − β 2 + SKP + αK (κ+1) . According to the formula (20.14) after the use of the residual theorem the explicit expression for the matrix Yβ (x) is derived for 4 indicated poles. To construct the solution of vector boundary problem (20.11) we can use these matrix solutions by multiplying them with the vectors of unknown constants 2

⎞ ⎛ ⎞ c1 c4 − → y β (x) = (Y1 (x) + Y3 (x))⎝ c2 ⎠ + (Y2 (x) + Y4 (x))⎝ c5 ⎠ c3 c6 ⎛

(20.15)

where unknown constants ci , i = 1, 4 are found from the boundary conditions in (20.9)–(20.11). The solution of the vector boundary-valued problem (20.8)–(20.11) in transform domain is defined by the formulae (20.15). To derive the exact solution of the stated problem (20.1)–(20.6) one must apply the inverse Fourier transform by the scheme ⎤ ⎡ ⎡ ⎤ ⎤ ⎤⎡ u 0 (x) u βn (x) cosβn y u(x, y) ∞ ⎣ vβn (x) ⎦⎣ sinβn y ⎦, βn = π n ⎣ v(x, y) ⎦ = ⎣ v0 (x) ⎦ + 2 n=1 p(x, y) p0 (x) pβn (x) cosβn y ⎡

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473

Here v0 (r ) ≡ 0, u 0 (r ), p0 (r ) are found earlier, and u βn (r ), wβn (r ), pβn (r ) are defined by the formula (20.15). Thus, the exact solution of the problem of poroelasticity for a rectangular domain in the case of a permeable and impermeable surface is constructed in the explicit form. If we consider the rectangular area, the width of which is essentially longer than its length, the indicated problem can be modelled by the problem for a poroelastic semistrip. It is useful to find out under which ratio of the sizes of the rectangular area, the semi-strip’s model, is quite adequate. In order to find this ratio of the sizes the exact solution of the problem for a poroelastic semi-infinite rectangular domain should be derived in explicit form.

20.4 Deriving the Exact Solution of the Stated Problem for the Poroelastic Semi-infinite Rectangular Domain The poroelastic semi-infinite rectangular domain in dimensionless coordinates 0 < x < d, 0 < y < ∞ is considered. The boundary conditions (20.2) are fulfilled when x = 0. At the boundary x = d the displacements are supposed to be equal to zero and conditions for pore pressure for the drained case (20.3) or undrained case (20.4) are executed. At the boundary y = 0 the conditions of ideal contact with undrained conditions for pore pressure (20.5) are given. The stress field and pore pressure satisfying boundary problem (20.1)–(20.5) inside the semi-infinite rectangular domain should be found. Here the scheme of deriving the solution repeats the construction of the solution for the finite rectangular area with the difference that instead of the Fourier transform on the finite interval (20.7) the semi-infinite sin-, cos- Fourier transform is applied regarding variable y with the parameter β ⎤

u β (x) ⎣ vβ (x) ⎦ = pβ (x) ⎡

∞ 0

⎤⎡ ⎤ cosβy u(x, y) ⎣ v(x, y) ⎦⎣ sinβy ⎦dy p(x, y) cosβy ⎡

Then the construction of the solution in the transform domain needs the formulation of the one-dimensional vector boundary problem (12) with its further solving and finding the constants from the corresponding boundary conditions (20.9)–(20.11). The finalizing of the solving is done by the application of semi-infinite inverse Fourier transform (Tikhonov and Samarskii 2011), which gives the possibility to write the expressions for the stress, displacements and pore pressure in explicit form.

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20.5 The Behavior Features of Mechanical Characteristics and Pore Pressure in Rectangular Domains The authors conducted a study for 3 different types of dimensionless load applied to the rectangular domains: (1) concentrated normal mechanical load L(y) = δ(y − 1/2), T (y) = 0, P(y) = 0; (2) distributed normal mechanical load L(y) = sin(π y), T (y) = 0, P(y) = 0; (3) distributed fluid pressure L(y) = 0, T (y) = 0, P(y) = sin(π y). Three different poroelastic materials (Cheng 2016) were investigated. Characteristics of poroelastic materials are presented in Table 20.1, and were used in the dimensionless form for numerical calculations.

20.5.1 The Behavior Features of Mechanical Characteristics and Pore Pressure in Finite Rectangular Domain The extreme values of stress and pore pressure for the finite rectangular domain were investigated regarding its sizes. It was supposed that the load is applied to the lateral vertical boundary x = 0, that is the conditions (2) are fulfilled. Numerous numerical calculations show that the largest absolute values of stress and pore pressure were observed for all loads in the case when the boundary, by which the load is applied, of the length h is essentially longer than the upper and bottom horizontal boundaries of rectangular area of the length a, which corresponds to the value d < 1, d = a/ h. Some features of the stress distribution and the pore pressure values inside the area are demonstrated further for 3 types of loading in the case of permeable and impermeable lateral surface x = d.

20.5.1.1

The Case of the Concentrated Normal Mechanical Load

The case with a concentrated normal mechanical load is presented at Figs. 20.1, 20.2, 20.3, 20.4. Since the concentrated load is applied at the center point y = 1/2 normal stress and pore pressure are symmetric regarding the line y = 1/2. The largest absolute values of normal stress and pore pressure are reached at the point y = 1/2. For the case with a permeable surface x = 1 (Fig. 20.1) stretching normal Table 20.1 The characteristics of poroelastic materials Material

G, N /m 2

μ

α

k, m 4 /N s

S p , m 2 /N

Charcoal granite

1.87 × 1010

0.27

0.242

1 × 10−16

1.377 × 10−11

Westerly granite

1.5 × 1010

0.25

0.449

4 × 10−16

1.412 × 10−11

Ruhr sandstone

1.33 × 1010

0.12

0.637

2 × 10−13

2.604 × 10−11

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stress are observed only when d < 1 near the boundaries y = 0 and y = 1, but for the case with an impermeable surface x = d (Fig. 20.2) stretching normal stress are observed for all values of d, and the biggest values are seen for d < 1. The pore pressure for the case with a permeable surface x = d is positive for all values of d. This is caused by the drainage, which starts at the boundary x = 0, and which produces a tendency for shrinkage of the rectangle’s boundary. However, for the case with an impermeable surface x = d negative values of pore pressure are observed near the boundaries y = 0 and y = 1. The absolute values of normal stress and pore pressure are bigger for the case with a permeable surface x = d. The features of the stress state of the poroelastic area were also investigated depending on the different types of poroelastic materials (Table 20.1). The highest absolute values of normal stress and pore pressure are inherent in Ruhr sandstone with the largest Biot’s constant among the observed poroelastic materials for the case of the permeable surface x = d (Fig. 20.3) and in Charcoal granite with the least Biot’s constant for the case of the impermeable surface x = d

Fig. 20.1 The distributions of dimensionless effective stress and pore pressure inside the finite rectangular domain (Ruhr sandstone) with the permeable surface x = d

Fig. 20.2 The distributions of dimensionless effective stress and pore pressure inside the finite rectangular domain (Ruhr sandstone) with the impermeable surface x = d

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Fig. 20.3 The distributions of dimensionless effective stress and pore pressure inside the finite rectangular domain depending on type of poroelastic material with the permeable surface x = d when d = 1

Fig. 20.4 The distributions of dimensionless effective stress and pore pressure inside the finite rectangular domain depending on the type of poroelastic material with the impermeable surface x = d when d = 1

(Fig. 20.4). Stretching normal stress and negative pore pressure for the case of a permeable surface x = d is observed near the boundaries y = 0 and y = 1 for Charcoal granite.

20.5.1.2

The Case of the Distributed Normal Mechanical Load

The case of the distributed normal mechanical load is shown in Figs. 20.5, 20.6, 20.7, 20.8. As it is seen from the graphs the absolute values of normal stress and pore pressure are less here than the absolute values of normal stress and pore pressure in the case of concentrated normal mechanical load. The load is supposed to be symmetric regarding the line. The general behavior of normal stress and pore pressure is analogical to the previous case of concentrated loading regarding the change of the rectangular domain’s proportions.

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Fig. 20.5 The distributions of dimensionless effective stress and pore pressure inside the finite rectangular domain (Ruhr sandstone) with the permeable surface x = d

Fig. 20.6 The distributions of dimensionless effective stress and pore pressure inside the finite rectangular domain (Ruhr sandstone) with the impermeable surface x = d

Fig. 20.7 The distributions of dimensionless effective stress and pore pressure inside the finite rectangular domain (Ruhr sandstone) with the permeable surface x = d

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Fig. 20.8 The distributions of dimensionless effective stress and pore pressure inside the finite rectangular domain (Ruhr sandstone) with the impermeable surface x = d

The behavior of stress and pore pressure depending on the type of poroelastic material presumes the same tendencies that in the case of the concentrated load. So these graphics are not given here.

20.5.1.3

The Case of the Distributed Fluid Pressure Load

Figures 20.7, 20.8 present the case with a distributed fluid pressure load. Pore pressure is positive for the case with a permeable surface x = d (Fig. 20.7), while negative pore pressure is observed near the rectangle’s boundaries for the case with an impermeable surface x = d (Fig. 20.8). Stretching normal stress is seen near the boundaries y = 0 and y = 1 for the case with an impermeable surface x = d for all values of d. For the poroelastic finite rectangular domain the investigation results show that the values of all mechanical characteristics and pore pressure essentially depend on the ratio of the sizes: the longer the vertical boundary, by which the load is applied, than the horizontal boundaries, the higher the absolute values of all characteristics inside the area. The assumption of the permeability of the boundary x = d has the essential influence on the stress distribution. For all types of materials and for all loads it was established that in the case of impermeable surface the absolute values of all mechanical characteristics and pore pressure are higher than in the case of the permeable surface. Also the investigation of the influence of Biot’s constant was done. It was found out that increasing of the Biot’s constant in the case when the boundary x = d is permeable leads to the increase of absolute values of normal stress and pore pressure. If the boundary x = d is impermeable then the maximal absolute values of characteristics are reached for the smallest values of the Biot’s constant. This fact is noted for all types of the applied load.

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20.5.2 The Behavior Features of Mechanical Characteristics and Pore Pressure in Semi-infinite Rectangular Domain For the semi-infinite rectangular domain the investigations analogical to the ones that were conducted in 5.1 were done. The influence of the applied load characteristics on the distribution of stress and pore pressure was investigated, and also the influence on the extreme values of characteristics of the size of zone of the applied on the lateral side load was considered. The calculations showed that the general character of the relations remained the same as in the previous case: the largest absolute values of stress and pore pressure are observed under the impact of the concentrated load, and the increasing of the length of the zone of applied load leads to the essential increasing of the characteristics inside the area. Separately, the influence of permeability of the lateral surface x = d on the stress state of the poroelastic domain was studied. Here it was noticed that inw the finite rectangular area the change of permeability does not influence essentially on the domain’s stress state. But the change of Biot’s constant has a much larger impact on the characteristics. With the increasing of Biot’s constant the decreasing of pore pressure and stress is observed. In the other cases the tendencies and character of change of normal stress and pore pressure remain the same as in the case for the finite rectangular domain. It was important to establish the ratio of the sizes of the finite rectangular area under which the influence of the boundary could be neglected. These numerical investigations were conducted for different loading and different material types. The analysis carried out allowed to conclude that the influence of the conditions of permeability/impermeability of the boundary has a significant impact on the ability to neglect the influence of the boundary. For the case when the right side of the finite area is impermeable, for the d ≤ 0.1 the stress and pore pressure become almost identical. However, this situation does not take place for the case of the permeable boundary x = d. Here even for the values d ≤ 0.001 the significant difference in derived results is observed. So, the simplified model of poroelastic rectangular area can be considered only in the case of impermeable conditions on its boundary.

20.6 Conclusions A new analytical approach for the investigation of poroelasticity rectangular domains is proposed in the article. This analytical approach is based on using the apparatus of boundary mixed problems of mathematical physics. The poroelasticity problem is formulated by the system of differential equations in partial derivatives in terms of Biot’s model with corresponding boundary conditions. The stated problem is solved exactly by the methods of integral transforms and matrix differential calculations. With the help of the proposed approach the exact solution for the poroelasticity problems for rectangular domains was constructed in an explicit form. The derived

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explicit solution for stress and pore pressure of the poroelastic area allowed to investigate the dependencies of these characteristics on the ratios of area’s sizes, load types and poroelasticity properties of material. The indicated method can be expanded for the solving of poroelasticity problems in the presence of defects inside the area. Acknowledgements The research is supported by Ukrainian Department of Science and Education under Project No 0121U111664. Zinaida Zhuravlova is very grateful to the University of L’Aquila and personally to Prof. Bruno Rubino for the support and opportunity to be visiting researcher of the University of L’Aquila during war time in Ukraine. Gratitude to Simon Dyke for the editing of the article’s text.

References Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:155–164 Bucha T, Prasad MK (2021) Slow flow past a weakly permeable spheroidal particle in a hypothetical cell. Arch Mech Eng 68(2):119–146. https://doi.org/10.24425/ame.2021.137044 Cheng AH-D (2016) Poroelasticity, theory and applications of transport in porous media. Springer, p 27 Chopra MB, Dargush GF (1997) Dynamic response of embedded strip and rectangular foundations using a poroelastic BEM. Trans Modelling Simul 18 Etchessahar M, Sahraoui S, Brouard B (2001) Bending vibrations of a rectangular poroelastic plate. C. R. Acad. Sci. Paris 329:615–620 Grantmakher FR (1967) Theory of matrices. Nauka, Moscow (in Russian) Li J, Ostoja-Starzewski M (2020) Thermo-poromechanics of fractal media. Philos Trans Roy Soc A Math Phys Eng Sci 378(2172):20190288. https://doi.org/10.1098/rsta.2019.0288 Li P, Wang K, Lu D (2016) Analytical solution of plane-strain poroelasticity due to surface loading within a finite rectangular domain. Int J Geomech. https://doi.org/10.1061/(ASCE)GM.19435622.0000776 Li K, Nennig B, Perrey-Debain E, Dauchez N (2021) Poroelastic lamellar metamaterial for sound attenuation in a rectangular duct. Appl Acoust 176:107862 Liu J, Cui Z, Sevostianov I (2021) Effect of stresses on wave propagation in fluid-saturated porous media. Int J Eng Sci. https://doi.org/10.1016/j.ijengsci.2021.103519 Łyd˙zba D, Ró˙za´nski A, Sevostianov I, Stefaniuk D (2021) A new methodology for evaluation of thermal or electrical conductivity of the skeleton of a porous material. Int J Eng Sci. https://doi. org/10.1016/j.ijengsci.2020.103397 Madasu KP, Sarkar P (2022) Couple stress fluid past a sphere embedded in a porous medium. Arch Mech Eng 69(1):5–19. https://doi.org/10.24425/ame.2021.139314 Popov GY (2013) Exact solutions of some boundary problems of deformable solid mechanic. Astroprint, Odessa (in Russian) Sevostianov I (2022) Characterization of physical properties of a porous material in terms of tortuosity of the porous space: a review. Adv Struct Mater 155:399–427 Shah SA, Tajuddin M (2011) Three dimensional vibration analysis of an infinite poroelastic plate immersed in an inviscid elastic fluid. Int J Eng Sci Technol 3(2):1–11 Shariyat M, Jahangiri M Asgari M (2020) Numerical low-velocity impact and structural damping analysis of a rectangular poroelastic plate. J Mech Eng 49(89):115–124 Singh SK, Verma VK (2020) Exact solution of flow in a composite porous channel. Arch Mech Eng 67(1):97–110. https://doi.org/10.24425/ame.2020.131685 Terzaghi K (1925) (1925) Erdbaumechanik auf bodenphysikalischer Grundlage. Deuticke, Wien

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Theodorakopoulos DD, Beskos DE (1994) Flexural vibrations of poroelastic plates. Acta Mech 103:191–203. https://doi.org/10.1007/BF01180226 Tikhonov AN, Samarskii AA (2011) Equations of mathematical physics. Dover Publications. Reprint edition Vaughan BL Jr, Galie PA, Stegemann JP, Grotberg JB (2013) A poroelastic model describing nutrient transport and cell stresses within a cyclically strained collagen hydrogel. Biophys J 105:2188–2198 Vaysfeld N, Zhuravlova Z (2022) The plane problem of poroelasticity for a semi-plane. In: Abdel Wahab M (eds) Proceedings of the 4th international conference on numerical modelling in engineering. NME 2021. Lecture notes in mechanical engineering. Springer, Singapore. https:// doi.org/10.1007/978-981-16-8806-5_10 Vaysfeld N, Zhuravlova Z (2023) Exact solution of the axisymmetric problem for poroelastic finite cylinder. In: Altenbach H, Mkhitaryan SM, Hakobyan V, Sahakyan AV (eds) Solid mechanics, theory of elasticity and creep. Copyright: 2023, Published: 17 Dec 2022 Zheng Y, Luo C, Liu H, Du C (2022) The dynamic responses of an infinite plate resting on a poroelastic layered half-space soil medium with imperfect interface to a moving load. Z Angew Math Mech. https://doi.org/10.1002/zamm.202100242

Chapter 21

Pore-Fluid Filtration by Squeezing a Fluid-Saturated Poroelastic Medium Vladimir B. Zelentsov and Polina A. Lapina

Abstract Filtration of fluid from a fluid-saturated poroelastic medium by pressing is widely used in the processing industries of agriculture, medical, chemical, oil, gas and many other industries. The need for mathematical modeling of such processes is associated primarily with the selection of power pressing modes, the selection of materials for a power filtration unit, etc. Mathematical modeling of the pressing process consists in solving the quasi-static contact problem of upsetting a rigid indenter with a flat base shape into a fluid-saturated poroelastic medium. The base of the indenter is transparent to the pore fluid, which, when the base of the indenter settles, can accumulate both inside and outside the indenter. With the help of integral transformations, the posed contact problem is reduced to solving a two-dimensional integral equation of the first kind with a two-dimensional kernel depending both on the difference in coordinate variables and on the difference in time variables. Solving a two-dimensional integral equation using the Laplace transform reduces to solving the corresponding one-dimensional integral equation. The solution of the two-dimensional integral equation is obtained by inverting the solution of the onedimensional integral equation using the inverse Laplace transform. The resulting solution makes it possible to obtain formulas for the velocities of the pore fluid, both at the contact and outside the contact, as well as the fluid flow rate through the base of the indenter. Keywords Contact problem · Fluid-saturated porous medium · Biot medium · Consolidation · Contact stress · Pore fluid pressure

V. B. Zelentsov · P. A. Lapina (B) Research and Education Center “Materials”, Don State Technical University, Rostov-On-Don 344000, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_21

483

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21.1 Introduction To model porous media, describing the mechanical behavior of mixtures that do not mix with each other, obtained by the superposition method, since the middle of the nineteenth century, the theory of porous fluid-saturated media has been developed. The theory has been widely applied in various fields: in materials science, in the oil and gas industry, in soil mechanics. The famous scientists contributed to its creation: R.Woltman (1757–1837), H.Darcy (1803–1858), P.Fillunger (1883–1937), K.Von Terzaghi (1883–1963). In 1935, a Belgian physicist-mathematician M. Biot (1905–1985) developed the theory of consolidation (Biot 1939, 1941, 1955, 1956) of poroelastic fluid-saturated media, including anisotropic materials. Within the framework of the anisotropic linear theory of elasticity, the Terzaghi principle is valid only for the relative deformation of pores, that is shown in the work (Carroll and Katsube 1983). In (Dell’Isola et al. 2000), a variational approach was used to obtain differential equilibrium equations for a fluid-saturated poroelastic medium. In works (Ehlers and Kubik 1994; Lovera 1987; Mikeli´c and Wheeler 2012), problems were solved using the developed modified methods. In (Cieszko and Kubik 1999), various formulations of problems were studied, where a fluid-saturated poroelastic body is in contact with another elastic medium. In (Bo 1999), an analytical method was developed for solving a contact problem in which a vertically vibrating circular plate lies on a fluid-saturated poroelastic half-space. In (Kaczmarek et al. 2000; Kubik et al. 2000; Johnson et al. 1987), the mechanical parameters, micro and macrostructures of fluid-saturated poroelastic materials (biomaterials) were studied based on the response to ultrasonic vibrations. In (Nagy and Blaho 1994), an experiment was carried out to determine the quasi-static stiffness of fluid-saturated poroelastic materials. In (Santos et al. 1990), an experimental method for determining the elastic constants for an isotropic fluid-saturated poroelastic material was presented.

21.2 Statement of the Problem of Squeezing a Fluid-Saturated Poroelastic Material Let a fluid-saturated poroelastic medium be represented by a half-space, and in the section let it be presented by a half-plane in the Cartesian coordinate system x, z. The motion of the medium is described by a system of differential equations, where the forces of inertia and body forces are absent (Biot 1939, 1941, 1955, 1956) ∂σ µ ∂ −α =0 1 − 2ν ∂ x ∂x µ ∂ ∂σ µw + −α =0 1 − 2ν ∂z ∂z ∂ = c ∂t

µu +

(21.1)

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485

where  – is volumetric deformation of the elastic skeleton of a fluid-saturated poroelastic medium (Biot 1939, 1941, 1955, 1956) =

∂w 2(1 + ν) µ ∂u + , α= ∂x ∂z 3(1 − 2ν) H

− is Laplace operator, u, w− are horizontal and vertical displacements of the medium, σ − is fluid pore pressure, c− is Darcy coefficient, μ, ν— are shear modulus and Poisson’s ratio of the elastic skeleton of the medium, the coefficient H1 is a measure of the compressibility of the soil for a change in water pressure (Biot 1939, 1941, 1955, 1956). Dividing each of the first two equations by μ and introducing the notation σ  = αµ−1 σ

(21.2)

we obtain a simplified form of system (1) (the sign ’ is omitted) ∂σ ∂ − =0 ∂x ∂x ∂ ∂σ w + (2η − 1) − =0 ∂z ∂z ∂ = c ∂t

u + (2η − 1)

(21.3)

where η is determined by the formula η=

1−ν 1 − 2ν

Let a non-deformable indenter with a flat base shape be introduced on the surface z = 0 into a fluid-saturated poroelastic medium in the form of a half-plane. The introduction of the indenter is carried out without taking into account the friction force between the base of the indenter and the elastic skeleton of a fluid-saturated medium. The law of indentation or settlement of the indenter under the action of the force P(t) is known and denoted by ε(t). Further, it is assumed that the pore pressure on the surface z = 0 is equal to zero, including the one under the base of the indenter. The latter circumstance practically guarantees the flow of pore fluid through the flat base of the indenter when it settles into a fluid-saturated medium (Fig. 21.1). In addition, it is assumed that the surface z = 0 of the elastic skeleton of a fluid-saturated medium outside the contact area is also free from normal σzz and tangential σx z stresses. The boundary conditions of the formulated contact problem on the displacement of an indenter with a flat base shape have the form z = 0 w(x, 0, t) = −ε(t)

−a < x 0 becomes zero Vx (x, 0, t) = 0, |x| < ∞, t > 0

(21.65)

The vertical component Vz (x, z, t) according to (21.63) is determined by the formula η 1 Vz (x, z, t) = −k α 2πi



1 e dp 2π

∞ ϕ L F (u, p)

pt



−∞

∂N (u, z, p)e−iux du ∂z

(21.66)

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 ∂N p (u, z, p) = ∂z c

√

u 2 + cp z

− |u|e|u|z  u 2 + η cp − |u| u 2 + cp

u2

+

p e c

(21.67)

On the surface at z = 0 the vertical component Vz (x, 0, t) of the pore fluid velocity Vz (x, z, t) is different from zero and is calculated by the formula (21.66), where 

u 2 + cp − |u| ∂N p  (u, 0, p) = ∂z c u 2 + η p − |u| u 2 + c

(21.68)

p c

Having made the change of the integration variable u in (21.66) at ∂∂zN from (21.68) according to the formulas  u=

p  u , du = c



p  du c

(21.69)

and given that a ϕ

LF

ϕ L (x, p)eiux d x

(u, p) =

(21.70)

−a

we get 1 k η Vz (x, 0, t) = − α 2η − 1 2πi

 

p pt 1 e dp c 2π

a

∞ ϕ (ξ, p)dξ

R(u)eiu

L

−a

√p c

|ξ −x|

du

−∞

(21.71) √ u 2 + η + |u| u 2 + 1 η2

, θ2 = R(u) =   √ 2η − 1 u2 + θ 2 u 2 + 1 + |u|

(21.72)

To simplify formula (21.71), we approximate the function R(u) from (21.72) at |u| < ∞ by the formula √ u 2 + η + |u| u 2 + 1 η 1

≈√ R(u) =  (21.73) , where|A| =  √ 2 2 2η −1 u +A u2 + θ 2 u 2 + 1 + |u| After substituting (21.73) into (21.71), the internal integral is calculated (Bateman and Erdelyi 1954) according to the formula

21 Pore-Fluid Filtration by Squeezing a Fluid-Saturated Poroelastic Medium

∞ 0

      p p |ξ − x| du = K 0 A |ξ − x| cos u √ c c u 2 + A2 1

497

(21.74)

where K 0 (u) is the Macdonald function. Rearranging the order of integration in (21.71) we obtain k η 1 Vz (x, 0, t) = − α 2η − 1 π

a dξ −a

1 2πi

 

   pt p L p |ξ − x| e dp ϕ (ξ, p)K 0 A c c (21.75)

Using the convolution theorem in the inner integral in (21.75), and making the change in the outer integral.ξ = aξ  , dξ = adξ  , x = ax  we obtain the formula for the vertical component of the velocity k ηa 1 ∂ Vz (x, 0, t) = − cα 2η − 1 2π ∂t

1 −1

dτ t −τ

1 −1

 A2 a 2 (ξ − x)2 dξ, ϕ(ξ, τ )ex p − 4c(t − τ ) 

|x| < ∞, t > 0

(21.76)

where ϕ(x, t) is from (21.56). Formula (21.76) was obtained using the formula (Bateman and Erdelyi 1954) 1 2πi

 

   pt   A2 (ξ − x)2 p 1 |ξ − x| e dp = ex p − K0 A c 2t 4ct

To calculate the volume of squeezing through the contact area in time t it is enough to use the formula 1 Vz (x, 0, t)d x, t > 0

a −1

21.6 Conclusion To study the pressing process, the problem of upsetting a rigid indenter with a flat shape of the base of the indenter transparent to the fluid is considered. The problem is reduced to solving a two-dimensional integral equation, the asymptotic solution of which is constructed for a short time. The solution obtained makes it possible

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to study the dynamics of a pore fluid in a medium subject to the penetration of an indenter with a flat base that is transparent to the fluid. Acknowledgements Zelentsov V. B. acknowledges the support of the grant of the RSF (grant number 22-19-00732). Lapina P. A. acknowledges the support of the grant of the RSF (grant number 22-49-08014).

References Biot MA (1939) Nonlinear theory of elasticity and the linearized case for a body under initial stress. Philosophical Magazine, 27(7), pp 468–489 Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164 Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Lowfrequency range. J Acoust Soc Am, 28, pp 168–178. https://doi.org/10.1121/1.1908239. Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26(2):182–185 Carroll MM, Katsube N (1983) The role of Terzaghi effective stress in linearly elastic deformation. J Energy Resour Technol 105(4):509–511 Bateman H, Erdelyi A (1954) Tables of integral transforms. McGraw-Hill, New York, p 391 Bo J (1999) The vertical vibration of an elastic circular plate on a fluid-saturated porous half space. Int J Eng Sci 37(3):379–393 Cieszko M, Kubik J (1999) Derivation of matching conditions at the contact surface between fluid-saturated porous solid and bulk fluid. Transp Porous Media 34(1–3):319–336 Dell’Isola F, Guarascio M, Hutter KA (2000) Variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. Archive of Applied Mechanics. 70(5). pp 323–337 Ehlers W, Kubik J (1994) On finite dynamic equations for fluid-saturated porous media. Acta Mech 105(1–4):101–117 Johnson DL, Koplik J, Dashen R (1987) Theory of dynamic permeability and tortuosity in fluidsaturated porous media. Journal of fluid mechanics. 176. pp 379–402 Kaczmarek M, Pakuła M, Kubik J (2000) Multiphase nature and structure of biomaterials studied by ultrasounds. Ultrasonics 38(1–8):703–707 Kubik J, Kaczmarek M, Pakula M (2000) Study of structure of porous biomaterials. Application of ultrasonic method. ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik. 80(S1) pp 157–160 Lovera OM (1987) Boundary conditions for a fluid-saturated porous solid. Geophysics 52(2):174– 178 Mikeli´c A, Wheeler MF (2012) Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot system. J Math Phys, 53(12). article no. 123702 Nagy PB, Blaho G (1994) Experimental measurements of surface stiffness on water-saturated porous solids. J Acoust Soc Am 95(2):828–835 Noble B (1958) Methods based on the wiener-hopf technique for the solution of partial differential equations. Pergamon Press, London, p 255 Santos JE, Corberó JM, Douglas Jr.J (1990) Static and dynamic behavior of a porous solid saturated by a two phase fluid. J Acoust Soc Am. 87(4). pp 1428–1438 Vorovich II, Alexandrov VM, Babeshko VA (1974) Neklassicheskiye smeshannyye zadachi teorii uprugosti [Non-classical mixed problems of the theory of elasticity]. Nauka Publication, Moscow. p 456 (in Russian)

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Zelentsov VB (1999a) Ob odnom asimptoticheskom metode resheniya nestatsionarnykh dinamicheskikh kontaktnykh zadach [On one asymptotic method for solving non-stationary dynamic contact problems]. Prikladnaya Matematika i Mekhanika. 63(2):303–311 (in Russian) Zelentsov VB (1996b) Nestatsionarnaya kontaktnaya zadacha o vnedrenii zhestkogo shtampa v upruguyu poluploskost’ [Non-stationary contact problem on the introduction of a rigid stamp into an elastic half-plane] Izvestiya RAN. Mekhanika tverdogo tela. 3. pp 34–44 (in Russian)

Chapter 22

2D Asymptotic Analysis of a Thin Elastic Beam with Density-Dependent Generalized Young’s Modulus Barı¸s Erba¸s, Julius Kaplunov, and Kumbakonam R. Rajagopal

Abstract The elastic equilibrium of a thin elastic beam is studied using asymptotic analysis starting from a 2D formulation within the context of plane elasticity. The aim of the paper is to elucidate the influence of density and hence small volume strain of Young’s modulus on the response of the beam. The adopted scaling at leading order supports the classical 1D Euler-Bernoulli beam approximation. The effect of strain-dependent Young’s modulus arises at the next order resulting in weak coupling between bending and extension deformations due to the asymmetry of the problem. The refined bending equation of interest is reduced to the form involving an explicit asymptotic correction to the prescribed transverse loading, similar to the previous considerations on the subject.

22.1 Introduction One of the early investigators1 of the response of beams to external stimuli was Leonardo da Vinci (Parsons 1939). Later, Galileo discusses the bending of a cantilever 1 As has been borne out time and time again, it is dangerous and foolhardy to assert that an important

idea was first expounded by a particular individual, an idea close enough would have been expounded earlier by someone else. B. Erba¸s Department of Mathematics, Eski¸sehir Technical University, Yunus Emre Campus, Eski¸sehir, Türkiye e-mail: [email protected] J. Kaplunov (B) School of Computing and Mathematics, Keele University, Keele, Staffordshire ST5 5BG, UK Faculty of Industrial Engineering Novo mesto, Šegova ulica 112, SI–8000 Novo Mesto, Slovenia e-mail: [email protected] K. R. Rajagopal Department of Mechanical Engineering, Texas A&M University, 3123 TAMU, College Station, TX 77843-3123, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Mechanics of Heterogeneous Materials, Advanced Structured Materials 195, https://doi.org/10.1007/978-3-031-28744-2_22

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beam with the aim of determining the load required to break the beam in his celebrated work (Arato 1992). These early studies on beams were followed by Mariotte and Leibniz in the seventeenth century, by Jacob (James) Bernoulli, Coulomb, Euler, Daniel Bernoulli, and Riccati among others in the eighteenth century2 and by Peter Barlow, Hodgkinson, Clayperon, W. H. Barlow, Rankine, and Mohr in the nineteenth century. The celebrated Euler-Bernoulli beam theory is the product of the efforts of Euler (1728) and Bernoulli (1728) (see the discussion in Truesdell (1960)) and the basic tenets for the theory are that the body in question is a linearized elastic solid (obeys Hooke’s law) and that plane sections of the beam remain plane and that the neutral axis of the beam is normal to the cross-section. Transverse shear was integrated into beam theory by Rankine (1881), and rotational inertial effects were fused independently by Bresse (1859) and Rayleigh (1877–1878). Shear and rotational inertial effects were taken into consideration by Timoshenko, and this theory is usually referred to as the Timoshenko beam theory, e.g., see Timoshenko (1916), though there were others that had independently incorporated the effects of shear and rotary inertia. All the above studies, except those by Leonardo and Galileo, are based on the assumption that the body in question obeys the constitutive relation propounded by Hooke (Gunther 1931), which is popularly referred to as Hooke’s law. The constitutive relation developed by Hooke describes elastic solids, provided the displacement gradients are small. A 3D constitutive relation for the response of elastic bodies, from a molecular perspective, was first developed by Navier (1827) assuming a specific force between the molecules. Cauchy introduced the concept of stress and developed the proper basis for nonlinear elasticity that we use (see Cauchy (1827)). If we can associate the notion of stored energy in a sub-class of Cauchy elastic bodies from which the stress can be derived, such bodies are called Green elastic (see Green (1848)) or hyperelastic bodies. If one starts with the general constitutive theory for a Cauchy elastic body and linearizes under the assumption that the norm of the displacement gradient is small, then one arrives at the constitutive relation that is referred to as Hooke’s law. The constitutive relation obtained by linearizing Cauchy elastic bodies by requiring that the norm of the displacement gradient be small, for isotropic bodies, leads to an expression of the stress in terms of the linearized strain wherein we have two material parameters referred to as the Lame constants. This in turn implies that Young’s modulus and Poisson’s ratio are constant. However, there is copious experimental evidence that shows that Young’s modulus depends on the density (see Pauw (1960), Nguyen et al. (2014), Lydon and Balendran (1986), and Munro (2004) for experiments on concrete; Zhang et al. (2014) and Luo and Stevens (1999) for experiments on ceramics; Manoylov et al. (2013), Kováˇcik (1999), and Hirose et al. (2004) for experiments in powder metallurgy; Helgason et al. (2008) and Vanleene et al. (2008) for experiments on bone; and Cristescu (2012) for experiments on rocks). 2

A detailed discussion of early research into various problems concerning beams as well as other structural members can be found in the papers by Truesdell (1960), and Fordham (1938). An exhaustive review of the history of structures can be found in Benvenuto (1991).

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Such a possibility is precluded by Cauchy elasticity for such a starting point would lead unerringly to the classical linearized theory of elasticity wherein the material parameters are constant. The question then is whether we can generalize the class of elastic bodies so that when we restrict ourselves to deformation associated with small displacement gradients, then one can obtain an approximation wherein the linearized strain is a function of the stress, with generalizations for Young’s modulus and Poisson’s ratio being dependent on the density. The answer to that question is an emphatic yes. If by an elastic body we mean a body that is incapable of dissipation, then the class of such elastic bodies is much larger than the class of Cauchy elastic bodies (see Rajagopal 2003, 2007). Many of these bodies are given by implicit constitutive relations between stress and the deformation gradient. Included in this large class of elastic bodies are those that are strain limiting or stress limiting. Moreover, but most importantly, linearizing such elastic bodies leads to a constitutive relation wherein the stress and the linearized strain appear linearly, and the material moduli which can be viewed as the generalized Young’s modulus and the generalized Poisson’s ratio are dependent on the density. While the development of a general class of implicit constitutive relations by Rajagopal (2021) appeals to representation theory to obtain the constitutive relations, Rajagopal and Saccomandi (2022) obtained a class of constitutive relations wherein the material moduli depend on density using a special generalization of the Blatz-Ko model based on physical considerations. Using the constitutive relation for the strain given in terms of the stress, whose material moduli depend on the density developed by Rajagopal (2021), several boundary problems have been solved (see Murru and Rajagopal, 2021a, b; Vajipeyajula et al. 2022) to determine the stress concentration due to circular and elliptic holes for different types of loadings. Itou et al. investigated the problem of a body whose material moduli depend on density with a crack subjected to non-penetration conditions between the opposite crack faces. They treated well-posedness of the problem by using a thresholding technique. There have been several generalizations of the Euler-Bernoulli beam equations, within the context of classical elasticity as well as other theories. Pr˘uša et al. (2022) studied the stresses and strains due to the bending of a prismatic beam of an elastic solid wherein the material parameters depend on the density. This study is devoted to a generalization of the Euler-Bernoulli theory within the context of the new implicit theory for elastic bodies with density-dependent material moduli undergoing small displacement gradients using asymptotic analysis, wherein the dimensionless wavelength is the perturbation parameter to obtain higher order terms in a manner similar to the study by Kaplunov et al. (2022). Such an asymptotic analysis has been successfully applied by Kaplunov and co-workers for a variety of problems concerning thin elastic bodies within the framework of linear isotropic elasticity (Nolde et al. (2018), Erba¸s et al. (2022), and Ege et al. (2022), to mention a few).

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22.2 Statement of the Problem First, we turn to a discussion of the constitutive relation for the material of a beam studied below. The constitutive relation stems from the class of implicit relations to describe the elastic response of the form (see Rajagopal (2003)) f(ρ, T, B) = 0,

(22.1)

where ρ is density, T is the Cauchy stress, and B is the Cauchy-Green tensor. On using the standard representation theorem (see Spencer (1971)), we obtain α0 I + α1 T + α2 B+α3 B2 + α4 T2 + α5 (TB + BT) + α6 (TB2 + B2 T)

(22.2)

+ α7 (T B + BT ) + α8 (B T + T B ) = 0, 2

2

2

2

2

2

where αi , i = 0, . . . , 8 depend on ρ, tr T, tr B, tr T2 , tr B2 , tr T3 , tr B3 , tr TB, tr T2 B, tr B2 T, and tr T2 B2 . Next, we recall that by the conservation of mass ρ R = ρ(det F),

(22.3)

where ρ R is the density of the reference configuration, ρ, the density in the deformed current configuration and F is the deformation gradient. When the displacement gradient is small, the above reduces to ρ R ≈ ρ(1 + tr ε).

(22.4)

Also, when the displacement gradient is small, B ≈ 1 + 2ε.

(22.5)

It then follows that a special subclass of (22.2), under the assumption of a small displacement gradient, is given as ε = λ1 (1 − β tr ε) (tr T) I + λ2 (1 − β tr ε) T,

(22.6)

where β, λ1 , and λ2 are constants. Assuming, in addition, that β tr ε  1, we can approximate the latter as ε=

λ1 λ2 T. (tr T) I + 1 + β tr ε 1 + β tr ε

(22.7)

Associating λ1 and λ2 with −ν/E 0 and (1 + ν)/E 0 , respectively, we can rewrite the above as ν 1+ν T− (22.8) ε= (tr T) I. E 0 (1 + β tr ε) E 0 (1 + β tr ε)

22 2D Asymptotic Analysis of a Thin Elastic Beam with Density …

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α3 α2

2h

α1

Fig. 22.1 Geometry of the problem

We can now interpret E 0 (1 + β tr ε) as a strain-dependent Young’s modulus and ν is Poisson’s ratio. Now, consider an elastic beam with an elongated cross-section of thickness 2h, see Fig. 22.1, for which the plane strain approximation appears to be an adequate original setup, e.g., see Nolde et al. (2018) for more details. Then, equilibrium of the beam in Cartesian coordinates α1 , α3 is given by ∂σ31 ∂σ11 + =0 ∂α1 ∂α3 ∂σ31 ∂σ33 + =0 ∂α1 ∂α3

(22.9) (22.10)

where σi j denote stress components. The stress-displacement relations are taken in the form, e.g., see Kaplunov et al. (2022), E ∂v1 ν σ33 + 2 1 − ν ∂α1 1−ν Eν ∂v1 ν = + σ33 1 − ν 2 ∂α1 1−ν

σ11 =

(22.11)

σ22

(22.12)

∂v3 = σ33 − ν(σ11 + σ22 ) ∂α3 ∂v1 ∂v3 E = −E + 2(1 + ν)σ31 ∂α3 ∂α1 E

(22.13) (22.14)

where E is Young’s modulus, ν is Poisson’s ratio while v1 and v3 stand for the horizontal and vertical displacements, respectively. The boundary conditions along the faces α3 = ±h are assumed in the form σ31 = 0

and

σ33 = P ± ,

(22.15)

where P ± are the vertical load applied, respectively, to the upper and lower faces. Let us define Young’s modulus according to (22.8) as E = E 0 (1 + βE)

(22.16)

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where E 0 is a constant, β  1 is a large parameter, and the volume strain E = tr ε is given by E=

∂v3 ∂v1 1 ∂v1 + = + (σ33 − ν (σ11 + σ22 )) . ∂α1 ∂α3 ∂α1 E

(22.17)

The aim of this paper is to investigate the effect of the strain- dependent Young’s modulus on the long-wave beam behaviour for which a typical wavelength L is much greater than the thickness, i.e., L  h. For the sake of simplicity, in what follows, we assume β ∼ L/ h  1. (22.18) In addition, we restrict ourselves to a weakly nonlinear behaviour of Young’s modulus in (22.16) dealing with small strains for which E ∼ 1/β 2 .

(22.19)

Below, we adopt the asymptotic procedure widely used in the mechanics of thin structures; see Nolde et al. (2018), Kaplunov et al. (1998), and references therein.

22.3 Asymptotic Scaling First, we scale the original coordinates as α1 = Lξ,

α3 = hζ

(22.20)

and introduce the dimensionless displacements and stresses by v1 = hβ −1 v1∗ ,

v3 = hv3∗

(22.21)

and ∗ ∗ , σ33 = E 0 β −4 σ33 , σii = E 0 β −2 σii∗ , σ31 = E 0 β −3 σ31

In addition, we set

together with

i = 1, 2.

(22.22)

E = β −2 E ∗

(22.23)

P ± = P0 P∗±

(22.24)

where P0 = E 0 β −4 . Here and below, all the starred quantities are assumed to be of order unity. The last two formulae are essential for adapting the same asymptotic behaviour as for a homogeneous structure in Kaplunov et al. (1998).

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The relations in the previous section rewritten in scaled coordinates and dimensionless functions become ∗ ∗ ∂σ11 ∂σ31 + =0 ∂ξ ∂ζ ∗ ∗ ∂σ33 ∂σ31 + =0 ∂ξ ∂ζ

(22.25) (22.26)

with ν 1 + β −1 E ∗ ∂v1∗ + β −2 σ∗ 2 1−ν ∂ξ 1 − ν 33  ∗ ν ν  −1 ∗ ∂v1 + β −2 σ∗ 1 + β = E 1 − ν2 ∂ξ 1 − ν 33   −2 ∗ β −2 ∗ ∗ = + σ22 ) β σ33 − ν(σ11 1 + β −1 E ∗ ∂v∗ 2(1 + ν) ∗ = − 3 + β −2 σ ∂ξ 1 + β −1 E ∗ 31

∗ σ11 =

(22.27)

∗ σ22

(22.28)

∂v3∗ ∂ζ ∂v1∗ ∂ζ and E∗ =

 −2 ∗   ∗ 1 ∂v1∗ ∗ + β σ33 − ν σ11 . + σ22 −1 ∗ ∂ξ 1+β E

(22.29) (22.30)

(22.31)

The boundary conditions (22.15) at ζ = ±1 take the form ∗ σ31 =0

and

∗ σ31 = P∗± .

(22.32)

The starred quantities may now be expanded in an asymptotic series of the form f ∗ = f (0) + β −1 f (1) + β −2 f (2) + · · · .

(22.33)

22.4 Two-Term Approximation First, formula (22.29), at leading order, gives v3(0) = V3(0) ,

(22.34)

with V3(0) = V3(0) (ξ) being the sought for transverse displacement depending on the longitudinal coordinate ξ only. Inserting, then, (22.34) into (22.30) and integrating in ζ, ∂V (0) (22.35) v1(0) = −ζ 3 + V1(0) , ∂ξ

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where V1(0) = V1(0) (ξ). The leading order components of main two normal stresses may now be obtained by substituting (22.35) into (22.27) and (22.28), resulting in the formulae (0) =− σ11

ζ ∂ 2 V3(0) 1 ∂V1(0) + 1 − ν 2 ∂ξ 2 1 − ν 2 ∂ξ

(22.36)

(0) σ22 =−

νζ ∂ 2 V3(0) ν ∂V1(0) + . 1 − ν 2 ∂ξ 2 1 − ν 2 ∂ξ

(22.37)

Then, we insert (22.36) into (22.25) and integrate along the thickness. Taking into account the first of the boundary conditions (22.32), we arrive at ∂ 2 V1(0) = 0. ∂ξ 2

(22.38)

Equation (22.25), together with (22.38), also leads to the expression (0) σ31 =

ζ 2 − 1 ∂ 3 V3(0) . 2(1 − ν 2 ) ∂ξ 3

(22.39)

Now, we may deduce from (22.26) and the inhomogeneous boundary condition (22.32) the classical beam bending equation ∂ 4 V3(0) 2 = P∗+ − P∗− . 3(1 − ν 2 ) ∂ξ 4

(22.40)

The homogeneous equation (22.38) above governing beam extension is usually of less practical interest. We also have, on account of (22.32), (0) =− σ33

(ζ + 1)2 (ζ − 2) ∂ 4 V3(0) + P∗− . 6(1 − ν 2 ) ∂ξ 4

(22.41)

At next order, we have from formula (22.29) v3(1) = V3(1) .

(22.42)

Substituting the latter into (22.30) and integrating along the thickness, we obtain v1(1) = −ζ

∂V3(1) + V1(1) . ∂ξ

In the formula above, V1(1) = V1(1) (ξ) and V3(1) = V3(1) (ξ). Next, we have from (22.27)

(22.43)

22 2D Asymptotic Analysis of a Thin Elastic Beam with Density … (1) σ11

1 = 1 − ν2



∂v1(1) ∂v(0) + E (0) 1 ∂ξ ∂ξ

509

 (22.44)

where, as it follows from (22.31), E

(0)

  1 − 2ν ∂v(0) (0) (0) =− = 1 − ν σ11 + σ22 ∂ξ 1−ν



∂V1(0) ∂ 2 V3(0) − ζ ∂ξ 2 ∂ξ

 .

(22.45)

Finally, using (22.43) and (22.45), we have (1) σ11

  2  1 − 2ν ∂V1(0) 1 ∂V1(1) ∂ 2 V3(1) ∂ 2 V3(0) = − − − ζ ζ . (1 − ν)(1 − ν 2 ) ∂ξ 2 ∂ξ 1 − ν2 ∂ξ 2 ∂ξ (22.46)

Now, inserting the last expression in (22.25) and integrating along the thickness, we arrive at 2  1 − 2ν ∂ ∂ 2 V3(0) ∂ 2 V1(1) =− . (22.47) ∂ξ 2 3(1 − ν) ∂ξ ∂ξ 2 In addition, we obtain (1) σ31

⎛ ⎞ 2  (0) 3 2 (0) 3 (0) ζ ∂ + 1 V V ∂V 1 − 2ν ∂ ∂ 3 3 1 ⎠ ⎝ =− − (ζ 2 − 1) (1 − ν)(1 − ν 2 ) 3 ∂ξ ∂ξ 2 ∂ξ 3 ∂ξ (22.48) +

ζ 2(1

2

− 1 ∂ V3(1) − ν 2 ) ∂ξ 3 3

−

ζ + 1 ∂ V1(1) . 1 − ν 2 ∂ξ 2 2

Then, substituting (22.48) into (22.26) and, again, integrating in ζ over the thickness, we arrive at the governing equation 2 ∂ 4 V3(1) 2(1 − 2ν) ∂V1(0) + (P∗ − P∗− ). = − 3(1 − ν 2 ) ∂ξ 4 1−ν ∂ξ

(22.49)

The last derivation also made use of the relations (22.40) and (22.47).

22.5 Refined 1D Equations First, combining (22.40) and (22.49) and using the notation     u = hβ −1 V1(0) + β −1 V1(1) , w = h V3(0) + β −1 V3(1) ,

(22.50)

we write down the sought for refined bending equation in the original variables

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 2E 0 h 3 ∂ 4 w 2β(1 − 2ν) ∂u  + P − P− . = 1− 3(1 − ν 2 ) ∂α14 1 − ν ∂α1 In addition, ∂2u 1 − 2ν ∂ + βh 2 2 3(1 − ν) ∂α1 ∂α1



∂2w ∂α12

(22.51)

2 = 0.

(22.52)

Thus, we observe a weak coupling of bending and extension deformation caused by the effect of small volume strain on Young’s modulus. In fact, we get from equation (22.52) at leading order ∂u/∂α1 = β −2 c where c is a constant of order unity, see asymptotic behaviour (22.23), which should be determined from boundary conditions. In this case, equation (22.51) becomes

 2c(1 − 2ν)  + 2E 0 h 3 ∂ 4 w P − P− = 1− 4 2 3(1 − ν ) ∂α1 β(1 − ν)

(22.53)

involving O(β −1 ) correction to the prescribed external loading. We also introduce two-term expansions for the stress components of interest through     (0) (1) (0) (1) , s31 = E 0 β −3 σ31 . + β −1 σ11 + β −1 σ31 s11 = E 0 β −2 σ11

(22.54)

Then, using (22.36) and (22.46), we have the following equations in the original variables:

∂2w E0 ∂u −1 1 − 2ν 1−β , (22.55) α3 2 − s11 = − 1 − ν2 1−ν ∂α1 ∂α1

3 ∂ w 2(1 − 2ν) ∂2w E 0 (α32 − h 2 ) ∂u s31 = 1 − β 2α − 3 . (22.56) 3 2 2 2(1 − ν ) 3(1 − ν) ∂α1 ∂α1 ∂α13 It is now a straightforward matter to produce the transverse shear resultant N , the bending stress couple G, and longitudinal stress resultant T by the formulae h N=

h σ31 dα3 , G =

−h

h σ11 α3 dα3 , T =

−h

σ11 dα3 .

(22.57)

−h

On substituting (22.55) and (22.56) into the latter, we obtain

2(1 − 2ν) ∂u ∂ 3 w 2E 0 h 3 1 + β , 3(1 − ν 2 ) 1 − ν ∂α1 ∂α13

2 ∂ w 2E 0 h 3 −1 1 − 2ν 1−β G=− , 2 3(1 − ν ) 1 − ν ∂α12 N =−

(22.58) (22.59)

22 2D Asymptotic Analysis of a Thin Elastic Beam with Density …

T =

2E 0 h 1 − ν2



1 − 2ν ∂u 1 − β −1 . 1 − ν ∂α1

511

(22.60)

Finally, we mention that the correction in the formula (22.58) is of order O(β −1 ) due to the relation ∂u/∂x = β −2 c above. For the same reason and on the same grounds, the correction in formula (22.59) is outside the assumed accuracy.

22.6 Concluding Remarks The effect of Young’s modulus depending on small volume strain according to (22.16), (22.19) on the equilibrium of a thin beam is studied starting from the 2D plane problem in isotropic elasticity. The asymptotic scaling adopted in Sect. 22.3 results at leading order in the classical Euler-Bernoulli theory for beam bending. The considered inhomogeneity of Young’s modulus appears at the next order supporting a weak non-linear coupling between bending and extension deformations, as follows from refined equations (22.51) and (22.52). In this case, the first of these equations can be reduced to the form (22.53) involving just an asymptotically small correction of the prescribed transverse loading, similar to other higher order theories for thin elastic structures, e.g., see Kaplunov et al. (1998). The asymptotic behaviour adapted in the paper is restricted to a scenario, in which the effect of the volume strain on Young’s modulus is negligible at leading order but at the same time is in order of magnitude greater than that associated with transverse shear deformation; see Elishakoff et al. (2015), Erba¸s et al. (2022), and Nolde et al. (2018) for more detail. Obviously, other setups assume a special asymptotic treatment. In addition, the same problem for a strain- dependent Poisson ratio may also be considered. Further possible extensions may also include analysis of dynamic behaviour and a similar perturbation of the conventional equations for thin elastic plates.

References Arato F (1992) “Galileo Galilei”, Discorsi e dimostrazioni matematiche intorno a due nuove scienze attinenti alla mecanica ed i movimenti locali", a cura di ENRICO GIUSTI. Giornale Storico della Letteratura Italiana 169(545):449 Benedikt H, Egon P, Enrico S, Fulvia T, Sigurður B, Marco V (2008) Mathematical relationships between bone density and mechanical properties: a literature review. Clin Biomech 23(2):135– 146 Benvenuto E (1991) An introduction to the history of structural mechanics, part: I: statics and resistance of solids. Springer, New York Bernoulli D (1728) Methodus universalis determinandae curvaturae filt a potentiis quamcunque legem inter se observantibus extensi, una cum solutione problematum quorundam novorum eo pertinentium. Commun Acad Sci Petrop 3:62–69 (1732)

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