Mechanics and physics of solids at micro- and nano-scales 9781786305312, 1786305313

Citation preview

Mechanics and Physics of Solids at Micro- and Nano-Scales

Series Editor Gilles Pijaudier-Cabot

Mechanics and Physics of Solids at Microand Nano-Scales

Edited by

Ioan R. Ionescu Sylvain Queyreau Catalin R. Picu Oguz Umut Salman

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019950358 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-531-2

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Part 1. Plastic Deformation of Crystalline Materials . . . . . . .

1

Chapter 1. Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity . . . . . . . . . . . . . . . . . Oguz Umut SALMAN and Roberta BAGGIO

3

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . 1.2. The model . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Linear stability analysis . . . . . . . . . . . 1.3. Numerical implementation . . . . . . . . . . . 1.4. Simulation results . . . . . . . . . . . . . . . . . 1.4.1. Stress field of a single-edge dislocation . 1.4.2. Dislocation annihilation . . . . . . . . . . . 1.4.3. Homogeneous nucleation . . . . . . . . . . 1.5. Conclusion . . . . . . . . . . . . . . . . . . . . . 1.6. References. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

3 6 9 11 12 12 13 14 18 18

Chapter 2. Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefanos PAPANIKOLAOU and Michail TZIMAS

25

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Effects of loading rates and protocols in crystal plasticity . 2.4. Size effects in microcrystal plasticity . . . . . . . . . . . . . . 2.5. Unveiling the crystalline prior deformation history using unsupervised machine learning approaches . . . . . . . . . . . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . .

25 27 29 36

...

38

vi

Mechanics and Physics of Solids at Micro- and Nano-Scales

2.6. Predicting the mechanical response of crystalline materials using supervised machine learning . . . . . . . . . . . . . . . . . . . 2.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

Chapter 3. Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions . . . . . . . . . . . . . . . Sylvie AUBRY, Sylvain QUEYREAU and Athanasios ARSENLIS 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Review of existing approaches . . . . . . . . . . . . . . . . . . . 3.2.1. Modeling discrete precipitates with DD simulations . . . 3.2.2. Investigation of precipitation strengthening and some related effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Dislocation dynamics modeling of dislocation interactions with Eshelby inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Stress field and forces at dislocation lines . . . . . . . . . 3.3.2. Stress at a point induced by an inclusion . . . . . . . . . . 3.3.3. Force on a dislocation coming from an inclusion . . . . . 3.3.4. Far field interactions induced by an Eshelby inclusion . 3.3.5. Parallel implementation . . . . . . . . . . . . . . . . . . . . . 3.4. DD simulations of the interaction with Eshelby inclusions . 3.4.1. Eshelby force for a single dislocation and a single inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Simulations of bulk crystal plasticity . . . . . . . . . . . . 3.5. Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . 3.6. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Appendix: derivation of the Eshelby force . . . . . . . . . . . 3.8. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

55

.. .. ..

55 57 57

..

61

. . . . . . .

. . . . . . .

63 63 64 64 68 68 69

. . . . . .

. . . . . .

69 70 77 79 80 82

Chapter 4. Scale Transition in Finite Element Simulations of Hydrogen–Plasticity Interactions . . . . . . . . . . . . . . . . . . Yann CHARLES, Hung Tuan NGUYEN, Kevin ARDON and Monique GASPERINI 4.1. Introduction . . . . . . . . . . . . . . . . . . . . 4.2. Modeling assumptions . . . . . . . . . . . . . 4.2.1. Crystal plasticity mechanical behavior 4.2.2. Hydrogen transport equation . . . . . . . 4.2.3. Implementation . . . . . . . . . . . . . . . 4.2.4. Mechanical parameters . . . . . . . . . .

43 48 49 49

. . . . . .

. . . . . .

87

87 92 92 93 95 96

Contents

4.3. Identification of a trap density function at the crystal scale . 4.3.1. Geometry, mesh, and boundary conditions applied on the polycrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Adaptation of the Dadfarnia’s model at the crystal scale. . . 4.4.1. Formulation at the polycrystal scale . . . . . . . . . . . . . 4.4.2. Application to single crystals . . . . . . . . . . . . . . . . . 4.4.3. Boundary and initial conditions . . . . . . . . . . . . . . . . 4.4.4. Crystal orientations. . . . . . . . . . . . . . . . . . . . . . . . 4.4.5. Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6. Consequences on hydrogen transport through a polycrystalline bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Appendix: Numbering of the slip systems in the UMAT . . 4.7. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

vii

97

. . . . . . . .

. . . . . . . .

98 100 104 104 106 107 108 108

. . . .

. . . .

113 118 118 119

Part 2. Mechanics and Physics of Soft Solids . . . . . . . . . . .

131

Chapter 5. Compression of Fiber Networks Modeled as a Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prashant K. PUROHIT

133

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Experimental observations in compressed fibrin clots and CNT forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Compression of platelet-poor plasma clots and platelet-rich plasma clots . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Compression of CNT forests coated with alumina . . . . 5.3. Theoretical model based on continuum theory of phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Compression of PPP and PRP clots . . . . . . . . . . . . . 5.3.2. Phase transition theory . . . . . . . . . . . . . . . . . . . . . 5.3.3. Effect of liquid pumping . . . . . . . . . . . . . . . . . . . . 5.3.4. Application of phase transition model to PPP and PRP clots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5. Predictive capability of our model . . . . . . . . . . . . . . 5.3.6. Application of phase transition model to CNT networks 5.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

133

..

134

.. ..

134 138

. . . .

. . . .

141 141 143 145

. . . . .

. . . . .

146 148 148 151 153

viii

Mechanics and Physics of Solids at Micro- and Nano-Scales

Chapter 6. Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion . . . . . . . . . . . . . . . . . Catalin R. PICU and Vineet NEGI 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Mechanics in the presence of adhesion . . . . . . . . . 6.2.1. The adhesive interaction of two fibers . . . . . . . 6.2.2. Triangle of fiber bundles . . . . . . . . . . . . . . . 6.3. Structure of non-crosslinked networks with inter-fiber adhesion . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Tensile behavior of non-crosslinked networks with inter-fiber adhesion . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Structure of networks with inter-fiber adhesion and crosslinks . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Tensile behavior of crosslinked networks with inter-fiber adhesion . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8. References. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

157 160 160 163

.......

165

.......

169

.......

171

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

173 179 180

Chapter 7. Surface Effects on Elastic Structures . . . . . . . . . Hadrien BENSE, Benoit ROMAN and José BICO

185

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Liquid surface energy. . . . . . . . . . . . . . . . . . . . . . . 7.2.1. Can a liquid deform a solid? . . . . . . . . . . . . . . . 7.2.2. Slender structures . . . . . . . . . . . . . . . . . . . . . . . 7.2.3. Wrapping a cylinder . . . . . . . . . . . . . . . . . . . . . 7.2.4. Capillary origamis . . . . . . . . . . . . . . . . . . . . . . 7.3. Dielectric elastomers: a surface effect? . . . . . . . . . . . 7.3.1. Introduction: electrostatic energy of a capacitor as a surface energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Mechanics of dielectric elastomers . . . . . . . . . . . . 7.3.3. Buckling experiments . . . . . . . . . . . . . . . . . . . . 7.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

157

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

185 186 186 187 188 190 192

. . . . .

. . . . .

. . . . .

. . . . .

192 194 202 209 210

Chapter 8. Stress-driven Kirigami: From Planar Shapes to 3D Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexandre DANESCU, Philippe REGRENY, Pierre CRÉMILIEU and Jean-Louis LECLERCQ 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Bilayer plates with pre-stress . . . . . . . . . . . . . . . . . . . . . .

215

215 216

Contents

8.3. Constant curvature ribbons and geodesic curvature . 8.3.1. Experimental evidence . . . . . . . . . . . . . . . . 8.3.2. Geodesic objects . . . . . . . . . . . . . . . . . . . . 8.4. Directional bending of large surfaces . . . . . . . . . . 8.4.1. Photonic crystals tubes . . . . . . . . . . . . . . . . 8.4.2. Control the directional bending . . . . . . . . . . . 8.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. References. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

Chapter 9. Modeling the Mechanics of Amorphous Polymer in the Glass Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hélène MONTES, Aude BELGUISE, Sabine CANTOURNET and François LEQUEUX 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Modeling the mechanics of amorphous . . . . . . . . . . . . . . 9.2.1. Input physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2. Temperature dependence of the intrinsic relaxation times 9.2.3. Length scales in the model . . . . . . . . . . . . . . . . . . . . 9.2.4. Numerical implementation . . . . . . . . . . . . . . . . . . . . 9.3. Linear regime in bulk geometry . . . . . . . . . . . . . . . . . . . 9.3.1. Stress relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Numerical predictions versus experiments in the linear regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3. Role of elastic coupling between domains . . . . . . . . . . 9.4. Linear regime in confined geometries . . . . . . . . . . . . . . . 9.4.1. Apparent linear viscoelasticity in various geometries . . . 9.4.2. Comparison of the results of our model with the observation of Tg shift in filled elastomers . . . . . . . . . . . . . . 9.4.3. Role of mechanical coupling in confined geometry . . . . 9.4.4. Conclusion on the effects of confinement . . . . . . . . . . 9.5. Nonlinear mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1. Input of nonlinearities . . . . . . . . . . . . . . . . . . . . . . . 9.5.2. Results of the model . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3. Role of elastic coupling in the nonlinear regime . . . . . . 9.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

219 220 222 223 224 225 227 227 231

. . . . . . . .

231 233 233 235 236 237 239 239

. . . .

240 241 244 244

. . . . . . . . . .

247 250 252 253 254 255 256 257 258 259

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267

 

Introduction

Structural and technological systems are as “good” as the materials that compose them. Understanding and predicting the mechanical behavior of materials, in particular, is, therefore, essential to the improvement of the performance of current materials, to the control of their durability and to the design of new materials. Substantial effort has been dedicated to these matters over the last century. Motivated by an ongoing need for advanced materials, with either enhanced strength, reduced weight, or smaller size to name a few, this book brings together a selection of current investigations in this field. The book is organized in two parts, the first set of chapters deals with crystalline “hard” materials, reflecting the continued interest in their plastic deformation, and a second set with “soft” materials that are becoming more and more studied for their mechanical properties. Among crystalline materials, metals certainly play a central role due to their countless applications. Large – mostly irreversible – plastic deformations are applied during forming, machining, or during the service life of the components. Plastic flow is known to be controlled by dislocation motion and interactions for “bulk” systems, while dislocation nucleation is also important in smallscale systems. Second phase particles in alloys or impurities and punctual defects when materials are submitted to harsh environments can lead to additional interactions with dislocations. On a small scale, which could be labeled as “nano-scale,” the elementary mechanics of interaction among individual dislocations are rather well-understood, from atomistic                                         Introduction written by Ioan R. IONESCU, Sylvain QUEYREAU, Catalin R. PICU and Oguz Umut SALMAN.

xii

Mechanics and Physics of Solids at Micro- and Nano-Scales

simulations or classical approaches based on the theory of dislocations. Similarly, the description of plastic deformation at the macroscopic and continuous scale is also well-established, but remains mostly phenomenological. However, the linkage between the nano-scale and microscale physical pictures of the plastic behavior remains poorly described and understood. This missing link requires the capacity of describing the collective behavior of large populations of dislocations and represents the physical basis for macroscale descriptions of plasticity. The first part of the book addresses these issues, with three chapters devoted to “mesoscale modeling,” i.e. the representation of plastic deformation at a scale between nano- and macro-scales. Mesoscale simulations typically use an explicit description of crystalline defects while using continuous expressions for forces or energies. Chapter 1, by Salman and Baggio, proposes a new formalism based on a generalized Landau theory and crystal symmetry to describe the plastic transformation in a “virtual” lattice. At micrometer scale and below, plastic flow is heterogeneous and occurs through intermittent bursts of dislocation motion or avalanches. Papanikolaou and Tzimas (Chapter 2) propose a review of this critical nature of plastic flow by means of 2D dislocation dynamics investigation in small systems where these effects are easily observable. Then, Chapter 3, from Aubry et al., presents a model of Eshelby particle in 3D dislocation dynamics along with some massive simulations to investigate precipitation strengthening in alloys. Finally, Chapter 4 by Charles et al. addresses the problem of plastic deformation occurring simultaneously with the diffusion of punctual defects in polycrystals, which is of particular importance in the context of hydrogen embrittlement in steels. All these chapters cover various perspectives, from mesoscale to macro-scale approaches, and focus on various aspects of plastic deformation. Without claiming to be exhaustive, this section of the book offers the reader a broad picture of mesoscale plasticity. The second part of the book is dedicated to the mechanics of soft materials. These topics have gained more and more interest over the last few decades, due to their biomedical, bioengineering, and biomimetics applications. “Soft materials” is a generic term, referring to polymers and materials made from fibers or molecular networks, such as gels, rubber, and non-wovens. The mechanics of fibrous materials is discussed in Chapters 5 and 6, by Purohit and by Picu and Negi, respectively; the first addresses the compressive response of fibrin networks, while the second discusses the mechanical behavior of fibrous structures in which fibers interact adhesively.

Introduction

xiii

Chapter 7, by Bense et al., considers the interaction of liquids and soft structures such as films and thin fibers, while Chapter 8, by Danescu et al., presents the designs and associated methods required to produce 3D structures from pre-stressed 2D thin films. Chapter 9, by Montes et al., features an analysis of the homogenized time-dependent behavior of stochastic composites in terms of the local behavior of material subdomains. This model is used to explain specific behaviors of polymers in the vicinity of the glass transition temperature. With a broad selection of state-of-the-art scientific and technological topics, the book offers a representative perspective on the current research in mechanics of materials. It brings together selected articles from the invited lectures presented at the 11th US–France symposium “Mechanics and physics of solids at micro- and nano-scales” held in Paris on 19th–21st of June 2018, under the auspices of the International Center for Applied Computational Mechanics (ICACM).

 

Part 1 Plastic Deformation of Crystalline Materials

Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

 

 

1 Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

1.1. Introduction Crystalline solids exhibit plasticity when macroscopic stresses exceed certain thresholds. It is well-known that plastic deformation of crystals is originated from the generation and motion of interacting dislocations, which evolve collectively in a complex energy landscape driven by the applied loading and long-range mutual interactions (Wilson 1954). Controlling crystal plasticity is needed in a variety of applications from metal hardening (Cottrell 2002) and fatigue failure (Irastorza-Landa et al. 2016) to nano-scale forming (Chen et al. 2010) and micro-pillar optimization (Pan et al. 2019; Zhang et al. 2017). The current trend of manufacturing small-scale metallic crystalline materials calls for a deeper understanding of their mechanical behavior at micro- and nano-scales. At small scales, a smooth description of plastic flow breaks down, plastic response exhibits strong intermittency (Csikor et al. 2007; Devincre et al. 2008; Ispánovity et al. 2014), and mechanical properties of materials depend strongly on size, initial microstructure, quenched disorder, and prior deformation [ see, e . g . Zhang et al. (2016, 2017)]. These properties render inadequate phenomenological continuum plasticity theory that adopts the smooth description of crystal plasticity (Forest 1998; Franciosi and Zaoui 1991) although it has been very successful in reproducing the most important plasticity phenomenology such as yield, hardening, and shakedown (Lubliner 2008). This inadequacy led to the use or                                         Chapter written by Oguz Umut SALMAN and Roberta BAGGIO.

4

Mechanics and Physics of Solids at Micro- and Nano-Scales

development of other approaches going beyond the phenomenological continuum theory. Molecular dynamics simulations that rely minimally on phenomenology are widely used to study crystal plasticity; the main advantage is that it is being formulated without any auxiliary hypotheses beyond the choice of interatomic potential (Bulatov et al. 1998; Moretti et al. 2011; Zepeda-Ruiz et al. 2017). However, the major drawback in molecular theories is the limitation on the accessible time and length scales, typically 104–106 atoms (which is equivalent to a few nanometers) and a time span of a few nanoseconds (Baruffi et al. 2019). An intermediate discrete dislocation dynamics approach describes the dynamics of elastically interacting defects (dislocations) but it requires phenomenological rules describing short-range interactions, annihilation, and nucleation (Devincre et al. 2008; Queyreau et al. 2010; Shilkrot et al. 2004). A bridge between the discrete dislocation dynamics theory and molecular dynamics has recently emerged in the form of phase-field crystal theory, but such a detailed description still remains prohibitively expensive when one deals with a large number of dislocations (Elder et al. 2002; Salvalaglio et al. 2019; Skaugen et al. 2018); quasicontinuum numerical approaches that attempt to match phenomenological continuum theory with microscopic molecular dynamics at selective points face the same problem (Kochmann and Amelang 2016). Dynamics of many defects can be also described by an evolving continuum dislocation density; on the other hand, despite many interesting recent advances, a rigorous coarse-graining in a strongly interacting system of many dislocations still remains a major challenge (Acharya and Roy 2006; Chen et al. 2013; ElAzab 2000; Groma 2019; LeSar 2014; Sandfeld et al. 2011; Valdenaire et al. 2016; Xia and El-Azab 2015). A very powerful meso-scale approach is the phase-field method based on the Ginzburg–Landau theory and it has been successful in modeling dislocation dynamics by both employing the continuum microelasticity theory to describe the elastic interactions (Khachaturyan 1967) and incorporating the γ-surface into the crystalline energy to describe the core structures (Finel and Rodney 2000; Hunter et al. 2011; Rodney et al. 2003; Ruffini et al. 2017; Shen and Wang 2004; Wang et al. 2001; Zheng et al. 2018). In this work, we use a recently developed complimentary mesoscopic approach (Baggio et al. 2019) that provides a nonlinear elasticity perspective on crystal plasticity and can be viewed as a far-reaching generalization of the Frenkel–Kontorova theory (Frenkel and Kontorova 1939). The approach is meso-scale in the sense it deals with macroscopic quantities such as stresses

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

5

and strains, and at the same time, it accounts properly for the exact symmetry of the underlying crystal structure. The aim is bridging fully atomistic descriptions and macroscopic theory based on continuum mechanics. The approach exploits the global invariance of the energy in the space of metric tensors compatible with geometrically nonlinear kinematics of crystal lattices. It takes the form of Landau theory, in which geometrically nonlinear metric tensor measuring the local deformation is the order parameter, with an infinite number of equivalent energy wells whose position is governed by the infinite symmetry group and corresponds to lattice-invariant shears. Therefore, plastic deformation can be described by an escape from the reference well when the crystal is loaded and dislocations appear as domain boundaries. This approach can be traced back to a few classic papers by J. L. Ericksen (1977, 1980, 1983) and follows subsequent development from the work of Conti and Zanzotto (2004). Similar approaches with periodic energies based on geometrically linear kinematics have also been used to study many aspects of crystal plasticity including, but not limited, to the description of dislocation cores and dislocation nucleation and intermittent nature of plastic flows (see Bonilla et al. 2007; Carpio and Bonilla 2003, 2005; Geslin et al. 2014; Kovalev et al. 1993; Landau 1994; Lomdahl and Srolovitz 1986; Minami and Onuki 2007; Onuki 2003; Plans et al. 2007; Salman and Truskinovsky 2011, 2012; Srolovitz and Lomdahl 1986). In this work, we use the model to study dislocation nucleation in a homogeneously sheared 2D square crystal. We consider athermal dynamics that reduces to parametric minimization of our elastic energy function with infinite periodicity. Our results suggest that the crystal does not necessarily follow the imposed deformation path such that a remarkable collective dislocation nucleation scenario takes place. The motivation for this particular study is due to the fact that, at small scales, it is experimentally possible to manufacture crystals with very low heterogeneity sources such as grain boundaries, precipitates, voids, cracks, and so on. Similarly, in nanocrystalline materials or ceramics with very fine grains, classical nucleation sources, e.g. Frank–Read, are not effective and nucleation occurs not only heterogeneously at pre-existent grain boundaries, but also homogeneously in grain interiors (Gutkin and Ovid’ko 2008). These peculiarities make relevant to develop a detailed mathematical modeling for a better understanding of homogeneous dislocation nucleation that remains a challenge in materials science. Most of the previous works rely on molecular theories. For instance, molecular dynamics simulations have been used to investigate nucleation in an initially defect-free crystal during nano-indentation (Miller and Rodney

6

Mechanics and Physics of Solids at Micro- and Nano-Scales

2008; Zhu et al. 2004; Zimmerman et al. 2001) or also during compression/tension simulations to explain the resulting asymmetrical material behavior through non-Schmid effects (Tschopp et al. 2007). Simple shear molecular dynamics simulations have also been performed to examine the dependence of plastic flow on crystal orientation in single-crystal nickel (Horstemeyer et al. 2002). These studies aimed to develop nucleation criteria that can be used in higher scale models such as meso-scale discrete dislocation dynamics or continuum crystal plasticity models. On a more fundamental level, the minimum shear deformation required to induce plastic deformation (although its measurement does not seem feasible in experiments) has been studied using ab initio calculations (Ogata et al. 2002) to investigate the higher shear strength of aluminum than that of copper. 1.2. The model We consider the deformation of a continuum body , where y is the current configuration and x is the reference state. Due to Euclidean invariance, the strain-energy density of an elastic solid must depend on deformation gradient through Cauchy–Green tensor . It is straightforward to show that the strain-energy C density possesses rotational invariance for ∈ [see Bhattacharya (1993), Salman (2009), Finel et al. (2010), and Salman et al. (2019)]. Second, in order to account for all deformations that map a Bravais lattice into itself, we must require that the strain energy density must also satisfy ,

[1.1]

where m belongs to in GL(2, Z) that denotes the group of 2 2 invertible matrices with entries in Z and det m = ±1 (Pitteri and Zanzotto and 2003). This is the suitable group since two sets of lattice vectors = L( ) if and only if they are related as generate the same lattice. L (Conti and Zanzotto 2004; Pitteri and Zanzotto 2003): with

∈

2, Z .

[1.2]

In the presence of such symmetry, the space of metric tensors C partitions into periodicity domains, each one containing an energy well equivalent to the reference one. The metric tensor C characterizing local deformation plays the role of an order parameter in the theory. To illustrate the implied periodicity, we focus on 2D crystal lattices in what follows and,

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

7

we consider linear transformations F that maps the original lattice ea into the deformed one fa. The Born hypothesis that states the positions of the atoms within the crystal lattice follow the overall strain of the medium (Ericksen 2008) reads [1.3]

After choosing the set of lattice vectors of a square lattice 1, 0 and 0, 1 as the reference state, and using equation [1.2] together with the Born hypothesis given by equation [1.3], one can observe that there are infinitely many homogeneous deformations, the so-called lattice-invariant deformations (Ericksen 2008) that solve the following equation: [1.4]

such that ⊗ and det F = 1. Thus, these deformations are naturally called lattice-invariant shears since ⊗

1

⊗

[1.5]

with ⋅ 0, which is a form associated with shearing deformations (Ericksen 2008). For instance, the point S1 in Figure 1.1 describes the reference square lattice. Another square lattice S2 in Figure 1.1 can be reached by a simple shear 1 ⊗ , where is perpendicular to vector , whereas the square lattice S3 can be reached by the shear 1 ⊗ . Similarly, the point T1 corresponds to a triangular lattice with basis vectors being 1, 0 and 1/2, √3/2 , where 4/3 / , and possesses hexagonal symmetry. The other equivalent triangular lattices can be reached by performing shears of the form 1 ⊗ . We note that the choice of GL(2, Z) infinite group symmetry as the appropriate material symmetry in equation [1.1] enforces that the set of lattice vectors generating the same lattice that must have the same energy and the strain energy possesses an infinite number of stable phases, or minimizers, in three-dimensional metric space. The task of constructing a strain energy for a given deformation state satisfying the above invariance is rather complex. However, it can be achieved by noticing that there is a subdomain in the infinite tensor space, the so-called fundamental domain, to which any metric can be reduced in order to obtain its crystallographically (Ericksen 1983) equivalent reduced form (metrics are said to have the “reduced form of Lagrange”) by the so-called iterative procedure Lagrange

8

Mechanics and Physics of Solids at Micro- and Nano-Scales

reduction (Conti and Zanzotto 2004; Engel 1986). The boundaries of the fundamental domain can be analytically found in 2D in terms of the components of the lattice metric (Conti and Zanzotto 2004; Engel 1986). The colored region in Figure 1.1 corresponds to the fundamental domain and the black dot at the origin corresponds to the metric 1 of the reference square lattice. The Lagrange reduction allows one to find the reduced form of any metric associated with an arbitrary shear deformation when it lies outside of the fundamental domain, and hence, we conclude that it is enough to construct a strain energy inside the fundamental domain: ≡

,

[1.6]

is only defined inside the fundamental domain. In this where work, we adopt the polynomial energy developed by Conti and Zanzotto (2004) for the study of reconstructive martensitic phase transformations that satisfies the continuity of elastic moduli on the boundaries of the fundamental domain. Also, see the work of Folkins (1991) for the general non-polynomial representation. Modular forms can also be used to construct infinitely periodic potentials [see Baggio et al. (2019)]. In this work, we choose a strain energy density , which det and an isochoric / det / decouples into a volumetric parts. Since det is invariant under GL(2, Z), the symmetry constraints concern only the isochoric part. The minimal potential used in the following is given in the form (Conti and Zanzotto 2004): [1.7] where 4 /11 8 have the structure: 1 3

41 /99 7 /66 /1056, and /11 17 /528. The hexagonal invariants here

,

1 4

1 12

4

,

, 4 4 . The and choice 1/4 enforces the square symmetry on the reference state while choosing 4, we bias the reference state toward hexagonal symmetry; the energy landscape for which the square crystal is the reference state, as

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

9

illustrated in Figure 1.6(a). The volumetric energy density is chosen in the 1 , where the coefficient µ plays the role of a bulk form modulus. The choice of a polynomial is typical in the Landau type of theories, but our work remains qualitative in this sense. Quantitatively, adequate Landau potentials with correct periodicity can be indeed extracted from ab initio (say Density Functional Theory) calculations for affine configurations with subsequent application of the Cauchy–Born rule [see, e.g. Liu et al. (2010) Tadmor et al. (1996)].

Figure 1.1. Partition of the section det C = 1 of the space C. Points Si represent the same square lattice and points Ti represent the same triangular lattice. The colored region is the fundamental domain. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

1.2.1. Linear stability analysis We first start by investigating the onset of plasticity by studying the loss of ellipticity condition that is related to the convexity of the function that describes the strain-energy density of the crystal. The mechanical equilibrium condition states that the divergence of the first Piola–Kirchhoff ≡

stress tensor ∙

0→

,

vanishes in the absence of body forces: 0,

[1.8]

where the subscript after comma denotes the differentiation with respect to relevant coordinate and lower-case and upper-case indices refer to the deformed and reference configurations, respectively. Mechanical equilibrium condition, equation [1.8], can be written compactly in the form ,

0,

[1.9]

10

Mechanics and Physics of Solids at Micro- and Nano-Scales

where A is the fourth-order moduli tensor and x is the position vector at the deformed state and ≡

.

[1.10]

After linearizing equation [1.9] for a small incremental deformation superimposed on an applied homogeneous deformation FA, we cast it into deformed coordinates, i.e. Eulerian coordinates, such that the incremental displacement vector u is now a function of the deformed position u(y) (Ogden 1984): 0,

,

[1.11]

where the tensor A is related to A as ,

[1.12]

where A is the spatial fourth-order incremental moduli tensor. Following Ogden (1984) (see also Kumar and Parks 2015; Merodio and Ogden 2002), ⋅ for incremental deformations in the form , where is the amplitude vector, k is the wave number, and is taken to be a unit vector cos , sin , equation [1.11] can be written as ⋅

0,

[1.13]

A where is the Eulerian acoustic tensor. Now, we can express strong ellipticity condition as A 0, for all arbitrary unit vectors and . For a given deformed state, the loss of strong ellipticity first occurs when the following equation has solution such that det

0.

[1.14]

The unit vector and the corresponding zero eigenvectors , for which the loss of ellipticity occurs, provide information on the nature of ensuing instability. The unit vector corresponds to the normal to the surface of the discontinuity of the deformation, whereas describes the type of the deformation. We stress here that we do not enforce the incompressibility condition det F = 1 in our model, and hence, we do not strictly have the orthogonality condition ⋅ 0. However, due to the penalization of

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

11

volumetric deformations in the strain-energy density, we find that the orthogonality condition is almost satisfied at the onset of the instability in our numerical calculations. 1.3. Numerical implementation As we have already mentioned, the obtained infinitely periodic energy densities are necessarily non-convex. Due to this non-convexity, the ground (relaxed) state for such materials must be of a fluid type (Fonseca 1987). To recover the rigidity of a solid, the model needs to be regularized, and to this end, we introduce a finite cut-off size [see also Conti and Zanzotto (2004)]. More specifically, we divide the reference space into triangular finite elements and reduce the space of admissible deformations to compatible piece-wise-affine mappings on such a finite mesh. We are, therefore, assuming that our elements are (i) large enough, i.e. contain enough number of atoms, that the homogeneous deformation of such an element exhibits a sufficient level of periodicity dictated by GL(N, Z) group symmetry and (ii) small enough for the piece-wise-affine approximation to be sufficient to resolve the phenomena of interest. Essentially, instead of a continuum medium, we consider a deformable 2D network whose discrete nodes x have integer-valued coordinates. With each node, we associated a deformed cell defined by the basis vectors , where 1, 2. The strain energy density of a cell associated with node x is assumed to be a function of the metric tensor C(x) with ⋅ . Since the basis vectors , where the components matrix m has integer entries and det 1, define the same cell, and therefore, the function φ must have the symmetry of the infinite group GL(2, Z). Therefore, even at the meso-scale, there must be a footprint of the discrete atomic lattice in the form of the information about the infinity of mappings that leave the energy density invariant. The reference body Ω is discretized into a number of standard finite that lead to elements with linear triangular shape functions ,  

[1.15]

12

Mechanics and Physics of Solids at Micro- and Nano-Scales

denote the values of displacement at node ij. The discrete where deformation gradient is then given by 1

⊗

.

[1.16]

, we are using a variant of To minimize the energy functional Ω conjugate gradient optimization, the so-called L-BFGS algorithm (King 2009), which essentially selects solutions of the equilibrium equations: /

Ω

0,

[1.17]

where / , reachable through overdamped dynamics. We use the hard device boundary condition on each boundary, i.e. the positions of . surface nodes are given by the applied shear deformation such that In our numerical calculations, we will study the homogeneous nucleation for a simple shear given by 1 ⊗ , where 1, 0 and 0, 1 and α is the amount of the applied shear. 1.4. Simulation results 1.4.1. Stress field of a single-edge dislocation In order to test the model, we first begin by calculating the stress field around an edge dislocation. To do so, we introduce a dislocation in the square crystal by using the classical isotropic displacement solution of an infinite edge dislocation with line direction along the {0, 0, 1} axis and Burgers vector along the {1, 0, 0} axis (Po et al. 2018) as the initial condition for the displacement field on the nodes of the finite elements. We then relaxed this configuration by conjugate gradient minimization (King 2009), of the total strain energy of the system with open boundary conditions, i.e. P N = 0, where N is the unit outward normal Ω in the reference configuration. After relaxation, we obtain a single-edge dislocation trapped in the middle of the domain together with a step on the right free , , and surface. Figures 1.2(a)–1.2(c) show the three components of the Cauchy stress of the dislocation in the computational domain. We observe a qualitatively good agreement of contour shapes with those of analytical solution (Lothe and Hirth 2017). Figures 1.2(d)–1.2(f) show the corresponding stress profiles along the glide plane that match the classical

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

13

continuum 𝑟 decay. Note also that the model resolves the core, which is regularized at a scale of the mesh. Finally, Figure 1.3 shows the core structure of a single dislocation in the configurational space of metric tensors, where blue dots are associated with the metric tensor of among finite elements. We observe that the dislocation core appears as a domain boundary between the two equivalent phases S1 and S2.

Figure 1.2. Edge dislocation in a square lattice: (a)–(c) finite-element nodes with color indicating the level of different components of Cauchy stress σxx, σyy, and σxy; (d)–(f) stress profiles along the glide plane. System size 500 × 500. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

1.4.2. Dislocation annihilation As previously mentioned in section 1.1, our model deals with dislocation reactions such as nucleation and annihilation without ad hoc rules in contrast with discrete dislocation dynamics modeling that needs phenomenological treatment. We illustrate here this behavior in the case of annihilation by introducing a dislocation dipole in the square crystal by superimposing the classical isotropic displacement solution of two infinite edge dislocations with

14

Mechanics and Physics of Solids at Micro- and Nano-Scales

different signs as the initial condition for the displacement field 𝐮 and further relaxing the associated configuration. We present the elementary mechanism of dislocation annihilation in the physical space in Figures 1.4(a)–1.4(d), where we observe two dislocations of opposite sign meet and then they effectively cancel out leaving the crystal dislocation free. The configurational space representation shows two domain boundaries between phases S1 and S2, each associated with the dislocations of different signs, see Figures 1.4(e)–1.4(h). After the annihilation event takes place, the crystal returns to its reference state, and all the metrics that are associated with finite elements are now localized in the bottom of the first energy well S1, see Figure 1.4(h).

Figure 1.3. The image of the edge dislocation in the configurational space. Blue dots are associated with the metric tensor among finite elements. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

1.4.3. Homogeneous nucleation We now return to the problem of homogeneous dislocation nucleation. In order to predict the dislocation nucleation in a dislocation free perfect square lattice, we calculate the contour plot of the determinant of the Eulerian acoustic tensor using equation [1.14] as a function of the amount of shear α and the angle ξ for a simple shear, as shown in Figure 1.5. We observe that up to a critical value of the amount of shear α = αc, the determinant of Eulerian acoustic tensor det 𝐐 𝜻 is positive, and thus, the crystal is stable. At the critical value αc, we observe an almost degenerate bifurcation with two instability angles 𝜉 0.11rad such that the unit vector 𝜻 cos 𝜉 , sin 𝜉 is almost perpendicular to the deformed e2, whereas the second instability angle 𝜉 1.55 rad leads to a destabilized direction almost perpendicular with the deformed e1. The almost degeneracy of the bifurcation suggests that two slips can be activated despite the fact that the

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

15

crystal is driven toward the phase S2 by the loading device. Note also that, interestingly, the former instability direction that points toward the phase S3 occurs for a slightly smaller α.

Figure 1.4. Dislocation annihilation in the physical space (a)–(d) and in the configurational space (e)–(h) where blue dots are associated with the metric tensor of different finite elements. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

We compare the strain–stress relation of the crystal for the homogeneous state with one of the numerical solutions shown in Figure 1.7. The latter follows the homogeneous solution almost perfectly up to αc predicted by the linear stability analysis, for which the loss of strong ellipticity occurs, leading to a large stress drop associated with the collective dislocation nucleation. Following the loss of stability, the perfect sheared crystal shown in Figure 1.8 evolves during the different stages of the first nucleation event, and Figures 1.8(b)–(e) show the non-equilibrium configurations during the minimization of the strain energy. The collective dislocation pattern that emerged after the stress drop (avalanche) is shown in Figure 1.8(f), where we observe the formation of dislocations on perpendicular slip planes. Recall here that our stability analysis predicted two almost simultaneous modes aligned with the slip directions in the deformed state, and the second one almost perpendicular with the deformed e1 is reached for a slightly smaller value of α. This instability mode grows faster, as shown in Figure 1.8(b), although it points toward the well S3 not favored by the loading. Indeed, this

16

Mechanics and Physics of Solids at Micro- and Nano-Scales

indicates an early dominance of the secondary slip mechanism. Note that the final spatial dislocation distribution is quasi-regular with pile-ups at the rigid boundaries together with the formation of characteristic junctions between dislocations on two slip planes blocking each other.

Figure 1.5. Contour plot of the determinant of Eulerian acoustic tensor as a function of amount of shear α and orientation angle ζ for a simple shear. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Figure 1.6. (a) Energy landscape corresponding to potential 1.7 with β = 0.25. The energy level is blue – low and red – high. Black lines are the limit of linear stability for the homogeneous deformations. Green line is the loading path enforced by the loading device for a simple shear. (b) Level sets of the strain energy density around the point T1, where we use the parametrization C11 = 1/Y, C22 = X2 + Y 2/Y, and C12 = X/Y. For a color version of this figure, see www.iste.co.uk/ionescu/ mechatronics.zip

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

17

Figure 1.7. Strain–stress relation for the square crystal: black line is the Cauchy stress for the homogeneous state, whereas brown line corresponds to the numerical solution. Notice the large stress drop when α = αc for which the loss of strong ellipticity occurs. For a color version of this figure, see www.iste.co.uk/ ionescu/mechatronics.zip

Figure 1.8. Collective dislocation nucleation in the perfect crystal (a) following the loss of stability as it evolves during the different stages (b)–(e) of the nucleation. The final mechanical equilibrium configuration (f). For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

To better characterize the simultaneous formation of dislocations on two perpendicular slip planes, we plot the level sets of the strain energy density on the surface det C = 1 in the vicinity of the triangular critical point T1 located on the boundary of the fundamental domain. The strain

18

Mechanics and Physics of Solids at Micro- and Nano-Scales

energy density φ at the point T1 has a shallow maximum surrounded by the three non-degenerate saddles R1, R2, and R3 describing rhombic lattices, see Figure 1.7. The flow of configurational points pointing initially toward such an unstable state (say, T1) will necessarily split into three streams directed toward the stable states (say, S1, S2, and S3). Note that the implied coupling of the plastic mechanisms would have to be postulated in the phenomenological plasticity theory (Ask et al. 2018; Forest 1998; Franciosi and Zaoui 1991); however, it can also be reconstructed from ab initio calculations [see Dezerald et al. (2014)]. 1.5. Conclusion To conclude, we have shown that nonlinear elasticity can be used to model crystal plasticity if the global invariance of the energy is taken into account. In such an approach, the complex geometry of the strongly deformed lattice is represented adequately with both physical and geometrical nonlinearities, shaping the infinitely periodic Landau energy. A thermal evolution in the regularized theory of this type can lead to temporal and spatial complexities. In particular, our study highlights the crucial role played in plastic deformation by the degenerate saddle points of the Landau potential, representing seemingly irrelevant, unstable crystallographic phases. Immediate development of this framework would be accounted for crystal orientations and the extension of the theory to the three-dimensional case by considering the group GL(3, Z). This will effectively extend the phase-field description to the case of multi-slip without extending the number of order parameters. Such a model will be able to capture the differences in flow behavior reported in crystals with hexagonal close-packed (HCP), facecentred cubic (FCC), and body-centred cubic (BCC) symmetries. Besides phenomena related to crystal plasticity, the proposed generalization of the Landau theory can also be used to describe mechanically- or thermally-driven irreversible reconstructive transitions, with their associated phenomena of compatibility-sensitive microstructure evolution. 1.6. References Acharya, A. and Roy, A. (2006). Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: part I. Journal of the Mechanics and Physics of Solids, 54, 1687–1710.

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

19

Ask, A., Forest, S., Appolaire, B., Ammar, K., and Salman, O. U. (2018). A cosserat crystal plasticity and phase field theory for grain boundary migration. Journal of the Mechanics and Physics of Solids, 115, 167–194. Baggio, R., Arbib, E., Biscari, P., Conti, S., Truskinovsky, L., Zanzotto, G., and Salman, O. U. (2019). Landau theory of crystal plasticity, available at: https://arxiv.org/abs/1904.03429. Baruffi, C., Finel, A., Le Bouar, Y., Bacroix, B., and Salman, O. U. (2019). Overdamped langevin dynamics simulations of grain boundary motion. Materials Theory, 3(1), 4. Bhattacharya, K. (1993). Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Continuum Mechanics and Thermodynamics, 5(3), 205–242. Bonilla, L. L., Carpio, A., and Plans, I. (2007). Dislocations in cubic crystals described by discrete models. Physica A: Statistical Mechanics and its Applications, 376, 361–377. Bulatov, V., Abraham, F. F., Kubin, L., Devincre, B., and Yip, S. (1998). Connecting atomistic and mesoscale simulations of crystal plasticity. Nature, 391(6668), 669–672. Carpio, A. and Bonilla, L. L. (2003). Edge dislocations in crystal structures considered as traveling waves in discrete models. Physical Review Letters, 90(13), 135502. Carpio, A. and Bonilla, L. L. (2005). Discrete models of dislocations and their motion in cubic crystals. Physical Review B Condensed Matter, 71(13), 134105. Chen, Y. S., Choi, W., Papanikolaou, S., Bierbaum, M., and Sethna, J. P. (2013). Scaling theory of continuum dislocation dynamics in three dimensions: selforganized fractal pattern formation. International Journal of Plasticity, 46, 94–129. Chen, Y. S., Choi, W., Papanikolaou, S., and Sethna, J. P. (2010). Bending crystals: emergence of fractal dislocation structures. Physical Review Letters, 105(10), 105501. Conti, S. and Zanzotto, G. (2004). A variational model for reconstructive phase transformations in crystals, and their relation to dislocations and plasticity. Archive for Rational Mechanics and Analysis, 173(1), 69–88. Cottrell, A. H. (2002). Commentary: a brief view of work hardening. In Dislocations in Solids, vol. 11, Nabarro, F. R. N. and Duesbery, M. S. (eds.). Elsevier, Amsterdam, pp. vii–xvii. Csikor, F. F., Motz, C., Weygand, D., Zaiser, M., and Zapperi, S. (2007). Dislocation avalanches, strain bursts, and the problem of plastic forming at the micrometer scale. Science, 318(5848), 251–254. Devincre, B., Hoc, T., and Kubin, L. (2008). Dislocation mean free paths and strain hardening of crystals. Science, 320(5884), 1745–1748.

20

Mechanics and Physics of Solids at Micro- and Nano-Scales

Dezerald, L., Ventelon, L., Clouet, E., Denoual, C., Rodney, D., and Willaime, F. (2014). Ab initio modeling of the two-dimensional energy landscape of screw dislocations in bcc transition metals. Physical Review B Condensed Matter, 89(2), 024104. El-Azab, A. (2000). Statistical mechanics treatment of the evolution of dislocation distributions in single crystals. Physical Review B Condensed Matter, 61(18), 11956–11966. Elder, K. R., Katakowski, M., Haataja, M., and Grant, M. (2002). Modeling elasticity in crystal growth. Physical Review Letters, 88(24), 245701. Engel, P. (1986). Geometric Crystallography: An Axiomatic Introduction to Crystallography. Springer, The Netherlands. Ericksen, J. L. (1977). Special topics in elastostatics. In Advances in Applied Mechanics, vol. 17. Elsevier, New York, pp. 189–244. Ericksen, J. L. (1980). Some phase transitions in crystals. Archive of Rational Mechanics and Analysis, 73(2), 99–124. Ericksen, J. L. (1983). The Cauchy and born hypothesis for crystals. MRC Technical Summary Report #2591. Ericksen, J. L. (2008). On the Cauchy—born rule. Mathematics and Mechanics of Solids, 13(3–4), 199–220. Finel, A., Le Bouar, Y., Gaubert, A., and Salman, O. U. (2010). Phase field methods: microstructures, mechanical properties and complexity. Comptes Rendus Physique, 11(3–4), 245–256. Finel, A. and Rodney, D. (2000). Phase field methods and dislocations. MRS Fall Meeting, Boston, MA. Folkins, I. (1991). Functions of two-dimensional Bravais lattices. Journal of Mathematical Physics, 32(7), 1965–1969. Fonseca, I. (1987). Variational methods for elastic crystals. Archive for Rational Mechanics and Analysis, 97(3), 189–220. Forest, S. (1998). Modeling slip, kink and shear banding in classical and generalized single crystal plasticity. Acta Materialia, 46(9), 3265–3281. Franciosi, P. and Zaoui, A. (1991). Crystal hardening and the issue of uniqueness. International Journal of Plasticity, 7, 295–311. Frenkel, J. and Kontorova, T. (1939). On the theory of plastic deformation and twinning. Izvestiya Akademii Nauk SSR, Seriya Fizicheskaya, 1, 137–149. Geslin, P. A., Appolaire, B., and Finel, A. (2014). Investigation of coherency loss by prismatic punching with a nonlinear elastic model. Acta Materialia, 71, 80–88.

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

21

Groma, I. (2019). Statistical theory of dislocation. In Mesoscale Models: From Micro-Physics to Macro-Interpretation, Mesarovic, S., Forest, S., and Zbib, H. (eds.). Springer International Publishing, Cham, pp. 87–139. Gutkin, M. Y. and Ovid’ko, I. A. (2008). Homogeneous nucleation of dislocation loops in nanocrystalline metals and ceramics. Acta Materialia, 56(7), 1642–1649. Horstemeyer, M. F., Baskes, M. I., Godfrey, A., and Hughes, D. A. (2002). A large deformation atomistic study examining crystal orientation effects on the stress– strain relationship. International Journal of Plasticity, 18(2), 203–229. Hunter, A., Beyerlein, I. J., Germann, T. C., and Koslowski, M. (2011). Influence of the stacking fault energy surface on partial dislocations in fcc metals with a threedimensional phase field dislocations dynamics model. Physical Review B Condensed Matter, 84(14), 144108. Irastorza-Landa, A., Van Swygenhoven, H., Van Petegem, S., Grilli, N., Bollhalder, A., Brandstetter, S., and Grolimund, D. (2016). Following dislocation patterning during fatigue. Acta Materialia, 112, 184–193. Ispánovity, P. D., Laurson, L., Zaiser, M., Groma, I., Zapperi, S., and Alava, M. J. (2014). Avalanches in 2D dislocation systems: plastic yielding is not depinning. Physical Review Letters, 112(23), 235501. Khachaturyan, A. G. (1967). Some questions concerning the theory of phase transformations in solids. Soviet Physics–Solid State, 8(9), 2163–2168. King, D. E. (2009). Dlib-ml: a machine learning toolkit. Journal of Machine Learning Research, 10, 1755–1758. Kochmann, D. M. and Amelang, J. S. (2016). The quasicontinuum method: theory and applications. In Multiscale Materials Modeling for Nanomechanics, Weinberger, C. R. and Tucker, G. J. (eds.). Springer International Publishing, Cham, pp. 159–193. Kovalev, A. S., Kondratyuk, A. D., Kosevich, A. M., and Landau, A. I. (1993). Theoretical description of the crowdion in an anisotropic crystal based on the Frenkel-Kontorova model including and elastic three-dimensional medium. Physics State Solid (B), 177(1), 117–127. Kumar, S. and Parks, D. M. (2015). On the hyperelastic softening and elastic instabilities in graphene. Proceeding of Royal Society A, 471(2173), 20140567. Landau, A. I. (1994). Application of a model of interacting atomic chains for the description of edge dislocations. Physics State Solid (B), 183(2), 407–417. LeSar, R. (2014). Simulations of dislocation structure and response. Annual Review of Condensed Matter Physics, 5(1), 375–407. Liu, X., Gu, J., Shen, Y., Li, J., and Chen, C. (2010). Lattice dynamical finiteelement method. Acta Materialia, 58(2), 510–523. Lomdahl, P. and Srolovitz, D. (1986). Dislocation generation in the two-dimensional Frenkel-Kontorova model at high stresses. Physical Review Letters, 57(21), 2702.

22

Mechanics and Physics of Solids at Micro- and Nano-Scales

Lothe, J. and Hirth, J. (2017). Theory of Dislocations. Cambridge University Press, Cambridge. Lubliner, J. (2008). Plasticity Theory. Courier Corporation, North Chelmsford, MA. Merodio, J. and Ogden, R. (2002). Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation. Archives of Mechanics, 54(5), 525–552. Miller, R. E. and Rodney, D. (2008). On the nonlocal nature of dislocation nucleation during nanoindentation. Journal of the Mechanics and Physics of Solids, 56(4), 1203–1223. Minami, A. and Onuki, A. (2007). Nonlinear elasticity theory of dislocation formation and composition change in binary alloys in three dimensions. Acta Materialia, 55(7), 2375–2384. Moretti, P., Cerruti, B., and Miguel, M. C. (2011). Yielding and irreversible deformation below the microscale: surface effects and non-mean-field plastic avalanches. PLoS ONE, 6(6), e20418. Ogata, S., Li, J., and Yip, S. (2002). Ideal pure shear strength of aluminum and copper. Science, 298(5594), 807–811. Ogden, R. (1984). Non-Linear Elastic Deformations. John Wiley and Sons, Chichester. Onuki, A. (2003). Plastic flow in two-dimensional solids. Physical Review E, Statistical, Nonlinear, and Soft Matter Physics, 68(6), 061502. Pan, Y., Wu, H., Wang, X., Sun, Q., Xiao, L., Ding, X., Sun, J., and Salje, E. K. H. (2019). Rotatable precipitates change the scale-free to scale dependent statistics in compressed Ti nano-pillars. Scientific Reports, 9(1), 3778. Pitteri, M. and Zanzotto, G. (2003). Continuum Models for Phase Transitions and Twinnining. Chapman & Hall, London. Plans, I., Carpio, A., and Bonilla, L. L. (2007). Homogeneous nucleation of dislocations as bifurcations in a periodized discrete elasticity model. Europhysics Letter, 81(3), 36001. Po, G., Lazar, M., Admal, N. C., and Ghoniem, N. (2018). A non-singular theory of dislocations in anisotropic crystals. International Journal of Plasticity, 103, 1–22. Queyreau, S., Monnet, G., and Devincre, B. (2010). Orowan strengthening and forest hardening superposition examined by dislocation dynamics simulations. Acta Materialia, 58(17), 5586–5595. Rodney, D., Le Bouar, Y., and Finel, A. (2003). Phase field methods and dislocations. Acta Materialia, 51(1), 17–30.

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

23

Ruffini, A., Le Bouar, Y., and Finel, A. (2017). Three-dimensional phase-field model of dislocations for a heterogeneous face-centered cubic crystal. Journal of the Mechanics and Physics of Solids, 105, 95–115. Salman, O. U. (2009). Modeling of spatio-temporal dynamics and patterning mechanisms of martensites by phase-field and Lagrangian methods. PhD Thesis, Université Paris 6, Paris. Salman, O. U., Muite, B., and Finel, A. (2019). Origin of stabilization of macrotwin boundaries in martensites. European Physical Journal B, 92(1), 20. Salman, O. U. and Truskinovsky, L. (2011). Minimal integer automaton behind crystal plasticity. Physical Review Letters, 106(17), 175503. Salman, O. U. and Truskinovsky, L. (2012). On the critical nature of plastic flow: one and two dimensional models. International Journal of Engineering Science, 59, 219–254. Salvalaglio, M., Voigt, A., and Elder, K. R. (2019). Closing the gap between atomicscale lattice deformations and continuum elasticity. npj Computational Materials, 5(1), 48. Sandfeld, S., Hochrainer, T., Zaiser, M., and Gumbsch, P. (2011). Continuum modeling of dislocation plasticity: theory, numerical implementation, and validation by discrete dislocation simulations. Journal of Materials Research and Technology, 26(5), 623–632. Shen, C. and Wang, Y. (2004). Incorporation of γ-surface to phase field model of dislocations: simulating dislocation dissociation in fcc crystals. Acta Materialia, 52(3), 683–691. Shilkrot, L. E., Miller, R. E., and Curtin, W. A. (2004). Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. Journal of the Mechanics and Physics of Solids, 52(4), 755–787. Skaugen, A., Angheluta, L., and Viñals, J. (2018). Separation of elastic and plastic timescales in a phase field crystal model. Physical Review Letters, 121(25), 255501. Srolovitz, D. and Lomdahl, P. (1986). Dislocation dynamics in the 2-D FrenkelKontorova model. Physica D: Nonlinear Phenomena, 23(1–3), 402–412. Tadmor, E. B., Ortiz, M., and Phillips, R. (1996). Quasicontinuum analysis of defects in solids. Philosophical. Magazine A, 73(6), 1529–1563. Tschopp, M. A., Spearot, D. E., and McDowell, D. L. (2007). Atomistic simulations of homogeneous dislocation nucleation in single crystal copper. Modelling and Simulation in Materials Science and Engineering, 15(7), 693. Valdenaire, P. L., Le Bouar, Y., Appolaire, B., and Finel, A. (2016). Density-based crystal plasticity: from the discrete to the continuum. Physical Review B Condensed Matter, 93(21), 214111.

24

Mechanics and Physics of Solids at Micro- and Nano-Scales

Wang, Y. U., Jin, Y. M., Cuitiño, A. M., and Khachaturyan, A. G. (2001). Phase field microelasticity theory and modeling of multiple dislocation dynamics. Applied Physics Letter, 78(16), 2324–2326. Wilson, A. J. C. (1954). Dislocations and plastic flow in crystals by A. H. Cottrell. Acta Crystallographica, 7(4), 384. Xia, S. and El-Azab, A. (2015). Computational modelling of mesoscale dislocation patterning and plastic deformation of single crystals. Modelling and Simulation in Materials Science and Engineering, 23(5), 055009. Zepeda-Ruiz, L. A., Stukowski, A., Oppelstrup, T., and Bulatov, V. V. (2017). Probing the limits of metal plasticity with molecular dynamics simulations. Nature, 550(7677), 492–495. Zhang, H., Tersoff, J., Xu, S., Chen, H., Zhang, Q., Zhang, K., Yang, Y., Lee, C. S., Tu, K. N., Li, J., and Lu, Y. (2016). Approaching the ideal elastic strain limit in silicon nanowires. Science Advances, 2(8), e1501382. Zhang, P., Salman, O. U., Zhang, J. Y., Liu, G., Weiss, J., Truskinovsky, L., and Sun, J. (2017). Taming intermittent plasticity at small scales. Acta Materialia, 128, 351–364. Zheng, S., Zheng, D., Ni, Y., and He, L. (2018). Improved phase field model of dislocation intersections. npj Computational Materials, 4(1), 20. Zhu, T., Li, J., Van Vliet, K. J., Ogata, S., Yip, S., and Suresh, S. (2004). Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper. Journal of the Mechanics and Physics of Solids, 52(3), 691–724. Zimmerman, J. A., Kelchner, C. L., Klein, P. A., Hamilton, J. C., and Foiles, S. M. (2001). Surface step effects on nanoindentation. Physical Review Letters, 87(16), 165507.

2 Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

2.1. Introduction In macro-scale mechanics, it is natural to define constitutive laws for the mechanical response of a particular material class and/or geometry. A characteristically simple example is the deformation of a crystalline sample, described by Hooke’s law for the elastic regime and a yield surface with a plastic flow rule for the inelastic one. In the extreme limit of small finite volumes, while Hooke’s law persists, the concepts of a yield surface and smooth plastic flow are controversial, manifesting into mechanical properties’ strong dependence on size, rate, and prior deformation. Characterizing the extent of failure of traditional inelastic constitutive laws, in relation to phenomena at sub-micron length scales, has been a consistent focus of material science over the last two decades. The key aspect has been the understanding of the effects of strain gradients, intrinsic or not (Hutchinson 2000), on plastic deformation of crystals in various geometries, the most prominent of which has been nanoindentation (Oliver and Pharr 2010). In small finite volumes (Uchic et al. 2002), the focus mainly has been the investigation of uniaxial compression in micro- and nano-pillars (Greer and De Hosson 2011; Uchic et al. 2009): Size dependence has been evident in the material strength due to intrinsic defect-induced strain gradients (Uchic et al. 2003), while rate dependence has been strongly suspected due to the

Chapter written by Stefanos PAPANIKOLAOU and Michail T ZIMAS. Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

26

Mechanics and Physics of Solids at Micro- and Nano-Scales

fact that plastic response displays strong intermittent features (Papanikolaou et al. 2012), typically labeled as avalanches (Zaiser 2006). Moreover, the well-accepted phenomenon of “mechanical annealing” (Shan et al. 2008), namely, the drastic increase of a pillar’s strength through prior compression, has unraveled a well-suspected but elusive, strong connection between small finite volume’s initial conditions and prior deformation history of micro-sized specimens (Nov´ak et al. 1984). For the theoretical investigation and explanation of crystal plasticity in the uniaxial compression of nano-sized specimens, modeling efforts have spanned the whole multiscale modeling spectrum. The investigation of the combined effects of all possible material and geometry details led to atomistic and molecular simulations (Rabkin et al. 2007; Yamakov et al. 2004) that are limited at ultra-high strain rates and tiny loading volumes. These studies have unveiled various delicate features of relevant dislocation mechanisms, such as surface dislocation multiplication. For collective dislocation behaviors, three-dimensional dislocation dynamics simulations have been utilized for the relevant mechanisms behind observed size effects (Greer and De Hosson 2011; Kraft et al. 2010; Uchic et al. 2009), rate effects (Maass and Derlet 2017; Papanikolaou et al. 2012), and also various statistical aspects such as avalanche size distributions (Cui et al. 2016). However, the demanding nature of the simulation of realistic micropillar dislocation densities [1014 /m2 (Shan et al. 2008)] in sub-micron volumes ( τnuc ) for a sufficiently long time tnuc . The model considers only gliding of dislocations, so the dislocation

28

Mechanics and Physics of Solids at Micro- and Nano-Scales

motion is solely controlled by the component of the Peach–Koehler force in the slip direction. Point obstacles are randomly distributed on active slip planes with a constant density that corresponds, on average, eight randomly-distributed −2 obstacles per each bulk dislocation source (i.e. ρbulk obs = 480 µm ). In this way, the source and obstacle densities remain statistically similar as finite volume dimensions change. Obstacles account for precipitates and forest dislocations on out-of-plane slip systems. Our simple obstacle model is that a dislocation stays effectively pinned until its Peach–Koehler force exceeds the obstacle-dependent value τobs b. The strength of the obstacles τobs is taken to be 300 MPa with 20% standard deviation, to account for large variability in realistic scenarios of dislocation pinning. A model volume is shown in Figure 2.1, where slip planes (lines) span the sample, equally spaced at d = 10b. Planes close to corners are deactivated to maintain a smooth loading boundary. Initially, samples are stress free and mobile-dislocation free, and the aspect ratio of height h over width w is maintained constant for all samples, a = h/w = 4. Dislocations can either exit the sample through the traction-free sides, annihilate with a dislocation of opposite sign when their mutual distance is less than 6b, or become effectively pinned at an obstacle. The simulation is carried out incrementally, using a time step that is a factor of 20 smaller than the nucleation time tnuc = 10 ns. At the beginning of every time increment, nucleation, annihilation, and pinning at and release from obstacle sites are evaluated. After updating the dislocation structure, the new stress field in the sample is determined, using the finite-element method to solve for image fields (Van der Giessen and Needleman 1995). Overall, the model is based on the singular theory of dislocations, but dislocations may never overlap into dislocation junctions, instead they follow the rules presented for dislocation annihilation. Similarly, dislocation nucleation is performed at a length scale where the dislocation dipole is stable, and the complexity induced by dislocation singularities disappears. In this way, no singularities are never encountered during simulation. A phenomenological comparison to experiments using single crystals was performed by Papanikolaou et al. (2017b) and displays qualitative agreement that involves not only strengthening effects, but also noise observations. No comparisons to 2.5 DDD simulations have been performed. However,

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

29

it is worth noting that the model agrees in various ways with previous 2D simulations, which were compared to experiments such as those by Nicola et al. (2006).

Figure 2.1. Simulation of uniaxial compression of thin films: (a) The 2D discrete dislocation plasticity model of uniaxial compression of thin films. Slip planes (lines), surface and bulk dislocation sources (red dots), and forest obstacles (blue dots) are shown. (b) Strain profile of sample of w = 2 µm, upon reloading at 0.1% strain. Initial loading at 10% strain. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

2.3. Effects of loading rates and protocols in crystal plasticity The differences between strain-controlled loading (SC) and displacementcontrolled loading (DC) rates have been known to be absent at small loading rates in crystal plasticity (Asaro and Lubarda 2006). However, in small finite volumes, due to the very existence of abrupt avalanche phenomena, there have been evidence and suspicion (Papanikolaou et al. 2012) for significant but statistical rate-dependent effects. The detailed rate effects that originate in the distinct loading protocols have been studied recently for uniaxial compression of micro-pillars (Maass et al. 2015; Sparks and Maass 2018) where several rate-dependent scaling behaviors were identified for rates higher than 102 s−1 . At the macro-scale, crystals are known to display strong rate effects due to viscoplastic dislocation drag effects, as the strain rate surpasses

30

Mechanics and Physics of Solids at Micro- and Nano-Scales

∼5000s−1 (Armstrong and Walley 2008; Clifton 2000; Murphy et al. 2010; Tong et al. 1992). This increase in flow stress has also been shown in DDD simulations (Agnihotri and der Giessen 2015; Hu et al. 2017) and stems from a natural competition between two timescales in dislocation dynamics. The first timescale refers to the dissipative motion of a dislocation inside the crystal (dislocation drag). The second timescale refers to the dislocation nucleation process from a randomly placed source. Nucleation of dislocations is particularly important for small-scale plasticity. These two timescales minimally represent two natural and distinct possibilities in the complex landscape of possible dislocation processes. The competition of these two timescales should extend in small finite volumes, providing a transition regime around 103 s−1 loading rates; thus, a statistically reliable study in loading strain rates ϵ˙ from 10s−1 to 105 s−1 would suffice (Song et al. 2019a). In the case of pure elasticity, SC and DC loading modes can be compared by using σ˙ = E ∗ ϵ, ˙ where σ˙ is the stress rate and ϵ˙ is the strain rate. Typical simulation parameters are listed in Table 2.1. Slip planes d = 10b θ = 30◦

Sources ρnuc = 60 µm−2 τ¯nuc = 50 MPa δτnuc = 5 MPa

Obstacles ρobs = 480 MPa τ¯obs = 150 MPa δτobs = 20 MPa

Table 2.1. Model parameters for the study of rate effects in uniaxial compression: slip plane spacing d, slip plane orientation θ, source density ρnuc , average source strength τ¯nuc , nucleation time tnuc , obstacle density ρobs , and average obstacle strength τ¯obs (Song et al. 2019a)

Timescale competitions are generic in most non-equilibrium systems (Sahni et al. 1983) and one may devise simple nonlinear dynamical models to explain the basic effects. For example, one may consider a minimal model for the strain evolution due to a dislocation segment that may or may not be trapped into a dislocation source, dϵ/dt = σ + µϵ − ϵ3 , where ϵ and σ are the scalars resembling strain and stress variables, respectively, and µ is a mobility parameter. The mobility parameter should have a different sign dependent on the dislocation trapping status. In the absence of dislocation interactions, on a slip plane with a single mobile dislocation, the mobility parameter is µ = µdrift < 0, and the time for stress σ relaxation inside the volume in every incremental timestep is δtdrift = |µdrift |−1 . By contrast, if there

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

31

exists a dislocation source, but not any mobile dislocations on the slip plane, then the mobility parameter becomes µ = µnuc > 0 and the corresponding timescale is δtnuc = µ−1 nuc . In most cases, the association between these timescales is δtnuc ≫ δtdrift , so stress increments are accommodated by nucleation events. However, if a system contains multiple dislocation sources, dislocation interactions may frustrate the system due to the disparity of relaxation time and cause a complexity in the evolution dynamics. In the aforementioned model (Song et al. 2019a), dislocations have mobility µd driven by local stress-induced forces. Gliding of dislocation occurs in a single slip system [slip planes oriented at 30◦ , see Figure 2.2(a)]. Figure 2.2(b) shows the stress-strain curves of SC for low, 102 (blue line), and high 105 (green line) stress rates. Correspondingly, the strain patterns at the same final strain (5%) are shown in Figure 2.2(c) and 2.2(d) where the plasticity is localized at low stress rates and is uniform at higher loading rates. Figure 2.3(a) shows that ϵ˙ = 104 s−1 in DC and correspondingly σ˙ = ∗ 104 s−1 . One may note the onset of expected work hardening in SC conditions, while in DC conditions, one observes softening, with the difference becoming more pronounced as the system width decreases. The model also displays consistent size effects (Papanikolaou et al. 2017a; Papanikolaou et al. 2017b) (σY ∼ w−0.4−0.6 ) for both loading protocols [cf. Figure 2.3(b)] for average flow stresses (at 0.2% engineering strain) of 50 realizations.

E∗

Figure 2.3(c) shows that a flow stress rate dependence is observed in both DC and SC loading modes, even though DC shows a weaker dependence. Upon closer examination of Figure 2.3(c) (Song et al. 2019a), one finds that low SC rates statistically resemble larger DC rates. The origin of this strainrate crossover is hidden in the amount of strain that nucleation events can accommodate, with ϵ˙ > 103 s−1 forcing dislocation drag to take over in the dynamics of dislocations instead of dislocation nucleation. This is consistent with metallurgy phenomenology (Clifton 1990; Follansbee and Kocks 1988; Tong et al. 1992). While both DC and SC display a flow stress rate effect, their statistical noise behavior is very different; evidence arises from the study of the (SC)∑ strain jump statistics in Figure 2.3(d): in SC, the event size is defined as S = i ∈ {δϵi >ϵthreshold } δϵi . By contrast, in DC, an event is characterized by stress drops δσ that lead to temporary displacement overshoots – thus, in order to compare ∑the two loading conditions, a DC strain burst event size is defined as S = i ∈ {−δσi >σthreshold } δϵi (Cui et al. 2016).

32

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 2.2. Rate effects on thin films. (a) The pillar under compression (single slip system). (b) Sample stress strain curves of compression at high (105 /s) and low (102 /s) stress rates σ. ˙ (c) Stain pattern for low σ. ˙ (d) Strain pattern for high σ˙ (Song et al. 2019a). For a color version of this figure, see www.iste.co.uk/ionescu/ mechatronics.zip

The model has two intrinsic timescales (Agnihotri and der Giessen 2015): the dislocation nucleation timescale δtnuc = 10 ns, which can be associated with the dislocation multiplication timescale in other models of plasticity, and the “drag” timescale, which may be defined via the ratio between dislocation mobility and material Young’s modulus B/E. In this model, the drag timescale is 10−6 ns, consistent with single-crystal thin-film experiments for the moduli and dislocation mobility (Nicola et al. 2006; Xiang and Vlassak 2006). As shown in Figure 2.3(d), plastic events’ statistics can be estimated through the analysis of the stress strain curves shown in Figure 2.3 (a); histograms of sizes have different τ exponents with consistent power-law behavior: τ is close to 3.5 for DC and 1.5 for SC. Another interesting fact

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

33

is that this exponent difference decreases as the stress loading rate increases: in Figure 2.4(a), we see statistics for different stress rates varying from σ˙ = E ∗ ∗ 10/s to σ˙ = E ∗ ∗ 104 /s. Power-law events distribution appear for all stress rates, yet with different exponent which changes from 3.5 for σ˙ = E ∗ ∗ 10/s to 1.5 for σ˙ = E ∗ ∗ 104 /s. This dependence of exponents on the stress rate indicates a non-trivial connection between the event statistics and the transition from nucleation-dominated to drag-dominated dislocation dynamics. To verify such a connection, one may increase the dislocation mobility B for the same stress rate (σ˙ = E ∗ ∗ 102 /s).

Figure 2.3. Effect of loading protocol: stress-controlled (SC) versus displacementcontrolled (DC). Blue curves are DC and red curves SC. (a) Stress–strain curves of different w using two different loading protocols. Strain bursts are shown; (b) Size effect of flow stress at 2% strain. (c) Dependence of flow stress (for w = 1µm) on rate. (d) Events (strain jumps) statistics for different loading protocols (Song et al. 2019a). For a color version of this figure, see www.iste.co.uk/ionescu/ mechatronics.zip

Figure 2.4(b) shows an enhanced drag effect (red curve) due to the increase of B and a subsequent exponent change from 2.5 to 2.2. The drag effect may also be magnified when other dislocation mechanisms come into play, such

34

Mechanics and Physics of Solids at Micro- and Nano-Scales

as cross-slip. This can be seen in Figure 2.4(b) blue curve, where a lower dislocation source density leads to a change of the τ exponent from 2.5 to 2.1.

Figure 2.4. SC rate effect crossover. (a) Event statistics for different σ˙ using SC. (b) Effect of dislocation source density ρnuc and mobility B on power-law exponent. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

The aforementioned exponent crossover is associated to an onset of inhomogeneity along the boundaries of the finite volume, signifying spatiotemporally correlated plastic activity. At first sight, this is not unexpected since crystal plasticity is known to be unstable to strain localization, thus adding an inhomogeneity component to avalanche dynamics (Asaro and Lubarda 2006). However, the combination of the exponent crossover with the onset of inhomogeneity in randomly evolving systems is uncommon. Figure 2.5(a) and 2.5(b) show the spatial distribution of events along all slip planes n for the loading process. Figure 2.5(a) shows the event spatial distribution for a loading rate of σ˙ = E ∗ ∗ 102 . Events are localized around certain slip planes and, furthermore, do not always happen at the same slip planes. For a higher loading rate of σ˙ = E ∗ ∗ 104 , the event distribution is more uniform among slip planes, as shown in Figure 2.5(b). The event size with increasing strain in Figure 2.5(c) unveils an oscillatory-like behavior at small stress rate, which disappears at higher stress rates. The observed behavior is akin to a mean-field integrated behavior (Papanikolaou et al. 2017a), labeled as the onset of an avalanche oscillator (Papanikolaou et al. 2012) as the strain rate decreases. The novel terminology is required to distinguish typical integrated depinning behaviors taking place at large loading rates in various systems (Fisher 1998). In this model, at low

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

35

strain rates, critical exponents τ and α are higher than the mean field, but the spectral density x (Papanikolaou et al. 2011) remains at the mean-field limit at low rates while x’ implies the integrated mean-field behavior (Papanikolaou et al. 2011). This novel behavior might explain large exponents in crystal plasticity of small grains in polycrystals (Lebedkina et al. 2018) or crystalline pillar experiments (Sparks and Maass 2018).

Figure 2.5. Spatial and temporal event distributions in SC. Event distribution on all slip planes during the loading up to 10% strain (a) for σ˙ = E ∗ ∗ 102 and (b) for σ˙ = E ∗ ∗ 104 . The color changes from dark purple to yellow with increasing loading strain. (c) Average avalanche size for σ˙ = E ∗ ∗ 102 in a sample. (d) Average avalanche size for σ˙ = E ∗ ∗ 104 in a sample. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

It is interesting to compare the statistical behavior of this model with meanfield plasticity avalanche behavior (Papanikolaou 2016; Uhl et al. 2015), as given in Table 2.2. In comparison, the presented model has free nanoscale boundaries and a timescale competition between dislocation nucleation and drag: these are model characteristics that are not typically included in mean-field avalanche models. Overall, it is found that these differences

36

Mechanics and Physics of Solids at Micro- and Nano-Scales

lead to an integrated behavior that is driven by quasi-periodic avalanche bursts (Papanikolaou et al. 2012). Finally, it is worth noting that the model is limited to small deformations, does not include other possible three-dimensional dislocation motions and does not include boundary roughness stress effects or thermal effects on obstacles/sources (Papanikolaou et al. 2019; Song et al. 2019b). Exponent τ α x x′

Mean-field theory 3/2 2 2 2

Avalanche oscillator Rate-Dependent > 3/2 Rate-Dependent > 2

2 1

Table 2.2. Universality and exponents. Basic mean-field avalanche exponents characterize power-law behaviors in avalanche sizes P (S) ∼ S −τ , durations P (T ) ∼ T −α , spectral response S(ω) ∼ ω −x , and average size–duration ′ relationship ⟨S⟩ ∼ T x

2.4. Size effects in microcrystal plasticity Experiments of uniaxial tension and compression in nanopillars have shown apparent material strengthening with decreasing pillar width w, with the yield strength varying as σY ∼ w−n with n ∈ (0.4, 0.8) (Greer and De Hosson 2011; Uchic et al. 2009). The basic overall explanation behind size effects has been the gradual exhaustion of dislocation multiplication mechanisms, as the finite volume becomes smaller. A variety of possible mechanisms can explain most of the existing experimental phenomenology on strength size effects. However, the non-smooth post-yielding plasticity behaviors have been known to display size effects as well. Analysis of the statistics of abrupt plastic events has revealed that nanopillar events appear to follow power-law distributions for strain steps with a large event cutoff that depends on specimen width (Miguel et al. 2001a, 2001b, Weiss et al. 2000; Weiss and Marsan 2003). These findings have evaded a unified model explanation until recently (Papanikolaou et al. 2017b). In this study (Papanikolaou et al. 2017b), 2D-DDD simulations of uniaxial compression for varying pillar widths w ranged from 0.0625 to 1 µm.

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

37

Figure 2.6(a) shows the typical stress–strain curves, with strengthening and large flow stress fluctuations as w decreases. Due to strain-controlled loading in simulations, avalanches are captured as stress drops. The total number of observed avalanches is not controlled, however, the total simulated strain is. Typically, these model simulations are performed up to 10% strain for any dislocation density. For example, for large dislocation densities (ρ = 1014 /m2 ), a single sample volume may respond to uniaxial compression through 103 avalanches during strain-controlled loading.

Figure 2.6. Axial stress–strain curves, σzz versus ϵzz . Strengthening and large stress drops emerge as w decreases, with the width shown in the legend, in µm. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

As shown in Figure 2.7, the simulations identify clear size effects in the yield strength. Figure 2.7(a) shows that the yield strength σY decreases with increasing w. For a pillar aspect ratio α = 4 (black line), we see a clear powerlaw dependence σY ∼ w−0.45 , which is similar to experimental observations. Morever, the sample strength depends on the aspect ratio α, as also identified in experiments (Senger et al. 2011). According to Figure 2.7(b), the yield strength decreases strongly with a power law σY ∼ α−0.36 for small widths, while in larger samples, this dependence is virtually absent.

38

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 2.7. (a) The width dependence of the yield stress. Different aspect ratios α are indicated by colored numbers. The fit to α = 4 is shown in bold black line, (b) the dependence of the yield stress on the aspect ratio α. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Avalanche behavior in the model of (Papanikolaou et al. 2017b) is shown through power-law tails of the event probability distributions P (S) ∼ S −τ P(S/S0 ). The onset of power-law behavior at decreasing w is shown in Figure 2.8(a) with an exponent τ = 1.2 ± 0.2 while S0 ∼ w−1 . The existence of power-law behavior in the asymptotically small width limit becomes apparent in samples with low aspect ratio, as shown in the inset of Figure 2.8(a), where the average event size Sav ∼ 1/w line is shown as a guide to the eye. Figure 2.8(b) shows the avalanche behavior statistics P (S) for three widths (0.0625, 0.25, and 1 µm) and two aspect ratios (4 and 32). Power-law behavior for varying aspect ratio is shown for the smallest system size w = 0.0625 µm. For larger systems, the distribution displays larger event sizes as the aspect ratio increases. This tendency is also seen in the behavior of Sav [inset of Figure 2.8(b)], where the aspect ratio independence is observed for small widths (0.0625 and 0.125 µm), while for large widths, there is a trend Sav ∼ α1 (shown as a guide to the eye).

2.5. Unveiling the crystalline prior deformation history using unsupervised machine learning approaches Elements of prior deformation history in crystals are needed for any prediction of mechanical properties in plasticity. The most common example

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

39

is the accumulated dislocation density, which is typically used for the prediction of flow stress. It is natural to expect that a wealth of additional mechanical property predictions can be made through the use of multidimensional deformation information, possibly originating in in situ strain maps. However, the efficient and systematic development of such mechanical property predictions requires data-intensive dimensional reduction and classification that has been common in machine learning (ML) methods.

Figure 2.8. Histograms of abrupt events and cutoff dependence. (a) Width dependence of P (S), demonstrating a power-law distribution as w decreases for α = 4 (symbol size reflects w). In the inset, the average event size is shown as a function of w for different aspect ratios α. (b) Dependence of abrupt event statistics on pillar aspect ratio α. Three different widths ( • : w = 0.0625µm,  : w = 0.25µm, and I: w = 1µm) are shown for two different aspect ratios α = 4 and 32 (the symbol sizes follow the aspect ratio’s magnitude for clarity). For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

ML methods have been recently used in science and engineering (DeCost et al. 2017; Mueller et al. 2016; Pilania et al. 2013; Ramprasad et al. 2017) and may predict microstructural properties (Pilania et al. 2013), optimize material design (Liu et al. 2015), and infer deformation history (Papanikolaou et al. 2019). The usage of ML in mechanical deformation studies started from analyzing nanoindentation responses toward the prediction of material properties (Huhn et al. 2017; Iskakov et al. 2018; Khosravani et al. 2017; Meng et al. 2015, 2017). In a new direction on this topic, a recent work (Papanikolaou et al. 2019) showed that the analysis of small-deformation strain correlation images may unveil the prior deformation history of materials. This process, which can be built on any version of DIC (Schreier et al. 2009),

40

Mechanics and Physics of Solids at Micro- and Nano-Scales

shows that the use of unsupervised ML methods on strain correlations may establish an equation-free approach for the recognition of prior deformation history for a large sample width w. The reason for the method’s effectiveness is the fact that the primary features of crystal plasticity, such as spatial strain gradients in the microstructure, may also be reflected in spatially resolved strain correlations (Chaikin et al. 1995; Papanikolaou et al. 2013, 2007; Raman et al. 2008). In three dimensions, it is expected that multiple cross-sections’ strain information would be required for analogous method effectiveness.

Figure 2.9. w = 1µm – prior deformation history of samples (see Papanikolaou et al. 2019): (a) Red • are samples with 0.1% prior strain, blue N samples with 1% prior strain, and green  are samples with 10% prior strain. (b) First principal component of PCA, shown in sample coordinates [Figure 2.1(b)]. Colormap is unitless. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

The method was implemented (Papanikolaou et al. 2019) in an explicit model of 2D-DDD, where two slip systems are used, for 50 random initializations of sources and obstacles and 0.1%, 1%, and 10% prior loading of the samples. In this way, statistically reliable initial conditions are produced at various initial dislocation densities. The prior-deformed samples are subjected to a small compressive but noninvasive load of 0.1% testing strain, and the final strain images consist of the tests. The applied strain is small so that it does not introduce significant further plastic deformation on the samples. After removing the strain information present at the unload stage, corresponding to well-annealed samples, strain

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

41

correlation signatures are examined on strain profiles [see Figure 2.1(b)] created by the small load mechanical testing and are collected in a matrix D:  C [1] [r1 |hi hj ] · · · C [1] [rm |hi hj ]   .. D=  . 

C [n] [r

1 |hi hj

] · · · C [n] [r

[2.1]

m |hi hj ]

where each row of D contains the vector dij : dij = (C [k] [r1 |hi hj ], C [k] [r2 |hi hj ], · · · , C [k] [rm |hi hj ])

[2.2]

where the correlation function C [k] [rv |hi hj ], k = 1, n (n = number of samples) is modeled after the Materials Knowledge System in Python [PyMKS (Wheeler et al. 2014)] scheme. Due to the inherent 2D nature of the problem presented by Papanikolaou et al. (2019), shear band formation upon small reloading may be picked up in large samples using spatial correlations, for all prior deformation levels. Similar ML schemes may be used in a 2D cross-section of 3D problems, for example, on surface deformation fields of nano-indented samples. However, in 3D settings, a similar ML scheme would require the study of multiple volume cross-sections, in order to characterize prior deformation, or other mechanical properties. The validity of the ML workflow is quantified through the investigation of accuracy and Fβ -scores (Baeza-Yates and Ribeiro-Neto 2011). Accuracy is defined as the fraction of correct predictions of the classifier. The Fβ -scores are used to quantify the performance in each cluster: Fβ = (1 + β 2 ) ·

(β 2

p·r · p) + r

[2.3]

where precision p is the number of correctly classified samples in a cluster divided by the number of all classified samples in the cluster, and recall, r, is the number of correctly classified samples in a cluster divided by the number of samples that should have been in that cluster.

42

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 2.10. Measures of success for classification of samples (see Papanikolaou et al. 2019): (a) Accuracy score for the samples. (b) F1 -score of the three clusters that are formed. (c) F2 -score of the three clusters that have formed. (d) F0.5 -score of the three clusters. Red • : ϵprior = 0.1%, blue N : ϵprior = 1%, and green  : ϵprior = 10%. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Results for larger systems are shown in Figure 2.9, where we observe the results of unsupervised ML for systems sizes of w = 1 µm [Figure 2.9(a)] and the corresponding smooth correlations [Figure 2.9(b)]. The unsupervised ML results for all system sizes can be summarized in Figure 2.10, where the accuracy and Fβ -scores are shown. Maximum value 1 means that all samples have been correctly classified. For the F2 -score, the weight of r is increased, and the 0.7 maximum value is expected for the “square” cluster of smaller system sizes. For the F0.5 -score, the weight of r is decreased. A correspondence between strain correlations and prior deformation history is found with 100% success for large systems.

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

43

2.6. Predicting the mechanical response of crystalline materials using supervised machine learning While unsupervised ML is necessary when the number of distinct data classes is unknown, supervised ML can perform much improved classification tasks. In this section, we discuss the application of supervised ML approaches on the data set of Papanikolaou et al. (2019), assuming known prior deformation histories of 80% of the samples. The aim is to identify relationships that fully describe the connections between strain correlations and prior processing history. In addition, the understanding and classification of prior deformation are equivalent to knowing the deformation State. In that case, one should be able to perform predictions of future mechanical response, albeit at average levels. We show that we can statistically predict mechanical responses for test data (20% of the samples), which can be thought of as average future mechanical responses of classified specimens.

Figure 2.11. Schematic for obtaining the 1% strain mechanical response of unknown samples. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

44

Mechanics and Physics of Solids at Micro- and Nano-Scales

w 0.125 µm 0.25 µm 0.5 µm 1 µm 2 µm

Training set accuracy 90.38% 91.35% 100% 100% 100%

Test set accuracy 83.3% 100% 100% 100% 100%

Table 2.3. Neural networks. Accuracy scores on supervised machine learning for identification of prior deformation histories via spatial strain correlations

In supervised ML approaches, the data set consists of samples with known outputs/features (in our case, prior deformation history is known for each sample), and the goal is to create robust algorithms that recognize sample/feature correspondences with high accuracy. In a typical supervised ML workflow, collected data sets are split into training and testing data sets. The algorithm is trained on the training data sets, and then, it is tested as to the validity and accuracy of the testing data sets. In the absence of big data collections, it is common to perform an 80%–20% split for the training and testing data sets. We train two types of supervised ML algorithms on the training set: neural networks (Bishop et al. 1995) and decision trees (Quinlan 1986). Neural networks are a set of algorithms, modeled loosely after the human brain, that are designed to recognize patterns in data sets and consist of “neurons” from which the data set passes through and activates various input functions. Decision trees are a set of decisions for the features of the input matrix, modeled after trees. The algorithm finds patterns in the features and creates leaves of a tree. When all possible patterns have been found, we have multiple leaves in a tree, hence the name decision tree. The most accurate neural networks and decision trees may be found through a parameter search using an algorithm for parameter optimization [GridSearchCV (Bergstra et al. 2011)]. The GridSearchCV algorithm gives the input of multiple parameters of a given classifier and the output of the set of parameters, which will provide the highest accuracy for the problem. We employed the use of the GridSearchCV (Bergstra et al. 2011) algorithm for neural networks (Bishop et al. 1995) and decision trees (Quinlan 1986), in order to identify the parameters that produce the highest accuracy in the

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

45

supervised problem. In the case of neural networks, the parameter search included adaptive or constant learning rate ranging from 10−5 to 103 . For decision trees, the input parameters on the GridSearchCV algorithm were gini or entropy criteria with the maximum depth of the tree ranging from 12 to 16. With these parameters, the highest accuracy was provided for adaptive learning rate of 10−5 for neural networks, while for decision trees, the best criterion was gini (Quinlan 1986) with the maximum depth (for w = 0.125 µm) set at 14 leaves. w 0.125 µm 0.25 µm 0.5 µm 1 µm 2 µm

Training set accuracy 100% 100% 100% 100% 100%

Test set accuracy 83.3% 100% 100% 100% 100%

Table 2.4. Decision trees. Accuracy scores on supervised machine learning for identification of prior deformation histories via spatial strain correlations

The scores for the supervised problem (see Tables 2.3 and 2.4) exceed the scores of the unsupervised problem reported by Papanikolaou et al. (2019) (also see Figure 2.10). This result was expected since the deformation histories are now known for the training set and it is easier to establish connections between known input–outputs. With the application of supervised algorithms on the data set, we are able to find a relationship between the known prior deformation histories (three classes of uniaxial compressive strain) and spatial strain correlations in training samples and use it for the classification of testing samples with high accuracy. We assume that samples that belong in each class are “similar” in terms of their mechanical properties. We use classified samples as norms for the prediction of the mechanical response upon further compression. Figure 2.11 shows a schematic for the prediction of the mechanical response, and we discuss the detailed process of calculating the average response based on prior deformation. We create three separate data sets, one for each deformation class (irrespective of the accuracy of the algorithm). For samples in each class, we assume future deformation features (1% testing deformation) as known

46

Mechanics and Physics of Solids at Micro- and Nano-Scales

since there is a one-to-one correspondence between testing deformation levels1. While the classification of data sets with 1% reloading strain has not taken place (also see Figure 2.12), this is irrelevant to promoting predictions, since samples share the same initial dislocation ensemble, which may be found for small reload strain.

Figure 2.12. w = 0.5µm – Prior deformation history of samples, large reload strain (1%) (see Papanikolaou et al. 2019). (a) Colors follow Figure 2.9. The failure of the classifier is evident. (b) First principal component of PCA, shown in sample coordinates (Figure 2.1). The anisotropy of the component is largely due to the high localization effects upon reloading to higher strain. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Figure 2.13. Prediction of the mechanical response of samples with a known prior deformation history: (a) w = 2 µm. (b) w = 0.25 µm. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

1 The same sample that is loaded to 0.1% testing deformation to capture the strain correlation patterns is also loaded to 1% testing deformation (see Papanikolaou et al. 2019).

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

47

Figure 2.14. Size effects in thin films: the maximum predicted stress (see Figure 2.13) is plotted against sample widths. The relationships derived correspond to a power law with an exponent that changes depending on the degree of prior deformation history of specimens. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

For each data set, we collect the average reload response (1% strain) per width. The results can be seen in Figures 2.13 and 2.14. In Figure 2.13, we observe, for decreasing width, whether the prior deformation history controls the mechanical response and the hardening behavior of the material. Red lines are for ϵprior = 0.1%, blue lines for ϵprior = 1%, and green lines for ϵprior = 10%. We see a transition at w ≃ 1 µm, where the maximum stress response in further deformation changes from being high deformation history dominated [prior deformation history = 10%, Figure 2.13(a)] to low deformation history dominated [prior deformation history = 0.1%, Figure 2.13(b)]. Prior deformation history of samples directly connects to the relaxed dislocation configuration in the volume upon unloading, with prior strain levels indicating the corresponding dislocation density levels in the crystal. In Figure 2.14, the sample yield stress is plotted against the thin film width for different dislocation density levels (acquired through prior deformation), which demonstrate an evolving size effect σ ≈ w−a , where a is shown

48

Mechanics and Physics of Solids at Micro- and Nano-Scales

in the legend, with a → 0 as the dislocation density increases. This is consistent with the basic phenomenological expectation in crystalline size effects’ literature (El-Awady 2015; Papanikolaou et al. 2017a; Song et al. 2019a). It is worth noting that the discussed model, in this work, is the first discrete dislocation model demonstration of this well-suggested transition (since Taylor) as a function of pre-existing dislocation density. The origin and further consequences of these findings will be discussed elsewhere. Overall, the suggested approach for the prediction of mechanical responses implies that there is an accurate method to describe and predict far-fromequilibrium mechanical-response phenomena: given a sample of unknown origin, and a known database of prior deformation histories, one only needs to apply a small load mechanical test, capture spatial strain correlation features, use them as part of the test set in the supervised ML problem, and obtain a prediction of future mechanical response and the prior deformation history/dislocation density of the crystal.

2.7. Summary In this chapter, we presented recent advances in the multi-scale modeling of material science to understand how and when crystal plasticity of small finite volumes displays dependence on loading rate, specimen size, and pre-existing, load-induced, dislocation microstructures. We introduced and discussed an explicit model of discrete dislocations, which is both minimal (in model details) and rich (in results and conclusions). While we investigated only the simple example of uniaxial compression, the model is directly generalizable to any other geometry in mechanics. Intrinsic, plasticity-induced crackling noise allows for thorough, statistically reliable examination of event statistics in a finite-volume system. Through an extensive investigation of this model, there has been a thorough and deep understanding of the collective effects in nanocrystal plasticity. The ultimate results of these studies have been the development of predictions for further signatures of rate and size effects, especially the finding of a dislocation-density-dependent size effect that promotes a transition to Taylor work hardening for the very first time in discrete dislocation modeling efforts. In addition, a major result of these studies has been a precise machine learning method for mechanical predictions of deformation characteristics. Ultimately, beyond specific predictions in nanocrystal plasticity, these research efforts have led

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

49

to a unified investigation of finite-volume nanocrystal plasticity across rates, sizes, and prior deformation histories.

2.8. Acknowledgements This work was supported by the National Science Foundation, DMR-MPS, Award No. #1709568.

2.9. References Agnihotri, P. K. and der Giessen, E. V. (2015). On the rate sensitivity in discrete dislocation plasticity. Mechanics of Materials (Proceedings of the IUTAM Symposium on Micromechanics of Defects in Solids), 90, 37–46. Alava, M., Laurson, L., and Zapperi, S. (2014). Crackling noise in plasticity. The European Physical Journal Special Topics, 223(11), 2353–2367. Anderson, P. M., Hirth, J. P., and Lothe, J. (2017). Theory of Dislocations. Cambridge University Press, Cambridge, UK. Armstrong, R. W. and Walley, S. M. (2008). High strain rate properties of metals and alloys. International Materials Reviews, 53(3), 105–128. Asaro, R. and Lubarda, V. (2006). Mechanics of Solids and Materials. Cambridge University Press, Cambridge, UK. Baeza-Yates, R. and Ribeiro-Neto, B. (2011). Modern Information Retrieval: The Concepts and Technology Behind Search. ACM Press Books, Addison-Wesley, Boston, MA, USA. Bergstra, J. S., Bardenet, R., Bengio, Y., and K´egl, B. (2011). Algorithms for hyperparameter optimization. In Advances in Neural Information Processing Systems (NIPS Proceedings’ 2011), 2546–2554. Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press, Oxford, UK. Chaikin, P. M., Lubensky, T. C., and Witten, T. A. (1995). Principles of Condensed Matter Physics, vol. 1. Cambridge University Press, Cambridge, UK. Clifton, R. (1990). High strain rate behavior of metals. Applied Mechanics Reviews, 43, S9–S22. Clifton, R. (2000). Response of materials under dynamic loading. International Journal of Solids and Structures, 37, 105–113. Cui, Y., Po, G., and Ghoniem, N. (2016). Controlling strain bursts and avalanches at the nano-to micrometer scale. Physical Review Letters, 117(15), 155502.

50

Mechanics and Physics of Solids at Micro- and Nano-Scales

DeCost, B. L., Jain, H., Rollett, A. D., and Holm, E. A. (2017). Computer vision and machine learning for autonomous characterization of am powder feedstocks. The Journal of The Minerals, Metals & Materials Society, 69(3), 456–465. El-Awady, J. A. (2015). Unravelling the physics of size-dependent dislocationmediated plasticity. Nature Communications, 6, 5926. Fisher, D. (1998). Collective transport in random media: from superconductors to earthquakes. Physics Reports, 301(1), 113–150. Follansbee, P. and Kocks, U. (1988). A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Metallurgica, 36(1), 81–93. Greer, J. R. and De Hosson, J. T. M. (2011). Plasticity in small-sized metallic systems: intrinsic versus extrinsic size effect. Progress in Materials Science, 56(6), 654–724. Groma, I., Zaiser, M., and Isp´anovity, P. D. (2016). Dislocation patterning in a two-dimensional continuum theory of dislocations. Physical Review B, 93(21), 214110. Hu, J., Liu, Z., der Giessen, E. V., and Zhuang, Z. (2017). Strain rate effects on the plastic flow in submicron copper pillars: considering the influence of sample size and dislocation nucleation. Extreme Mechanics Letters, 17, 33–37. Huhn, S., Sonnenberg, H., Eggersgl¨uß, S., Clausen, B., and Drechsler, R. (2017). Revealing properties of structural materials by combining regression-based algorithms and nano indentation measurements. In 2017 IEEE Symposium Series on Computational Intelligence (SSCI), IEEE Staff (ed.). IEEE, Piscataway, NJ, USA, pp. 1–6. Hutchinson, J. W. (2000). Plasticity at the micron scale. International Journal of Solids and Structures, 37(1–2), 225–238. Iskakov, A., Yabansu, Y. C., Rajagopalan, S., Kapustina, A., and Kalidindi, S. R. (2018). Application of spherical indentation and the materials knowledge system framework to establishing microstructure-yield strength linkages from carbon steel scoops excised from high-temperature exposed components. Acta Materialia, 144, 758–767. Khosravani, A., Cecen, A., and Kalidindi, S. R. (2017). Development of high throughput assays for establishing process-structure-property linkages in multiphase polycrystalline metals: application to dual-phase steels. Acta Materialia, 123, 55–69. Kraft, O., Gruber, P. A., M¨onig, R., and Weygand, D. (2010). Plasticity in confined dimensions. Annual Review of Materials Research, 40, 293–317. Lebedkina, T. A., Zhemchuzhnikova, D. A., and Lebyodkin, M. A. (2018). Correlation versus randomization of jerky flow in an AlMgScZr alloy using acoustic emission. Physical Review E, 97, 013001.

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

51

Liu, R., Kumar, A., Chen, Z., Agrawal, A., Sundararaghavan, V., and Choudhary, A. (2015). A predictive machine learning approach for microstructure optimization and materials design. Scientific Reports, 5, 11551. Maass, R. and Derlet, P. (2017). Micro-plasticity and recent insights from intermittent and small-scale plasticity. arXiv preprint arXiv:1704.07297. Maass, R., Wraith, M., Uhl, J. T., Greer, J. R., and Dahmen, K. A. (2015). Slip statistics of dislocation avalanches under different loading modes. Physical Review E, 91, 042403. Meng, L., Breitkopf, P., Raghavan, B., Mauvoisin, G., Bartier, O., and Hernot, X. (2015). Identification of material properties using indentation test and shape manifold learning approach. Computer Methods in Applied Mechanics and Engineering, 297, 239–257. Meng, L., Raghavan, B., Bartier, O., Hernot, X., Mauvoisin, G., and Breitkopf, P. (2017). An objective meta-modeling approach for indentation-based material characterization. Mechanics of Materials, 107, 31–44. Miguel, M.-C., Vespignani, A., Zapperi, S., Weiss, J., and Grasso, J.-R. (2001a). Complexity in dislocation dynamics: model. Materials Science and Engineering: A, 309, 324–327. Miguel, M.-C., Vespignani, A., Zapperi, S., Weiss, J., and Grasso, J.-R. (2001b). Intermittent dislocation flow in viscoplastic deformation. Nature, 410(6829), 667–671. Mueller, T., Kusne, A. G., and Ramprasad, R. (2016). Machine learning in materials science: recent progress and emerging applications. Reviews in Computational Chemistry, 29, 186–273. Murphy, W., Higginbotham, A., Kimminau, G., Barbrel, B., Bringa, E., Hawreliak, J., Kodama, R., Koenig, M., McBarron, W., Meyers, M., Nagler, B., Ozaki, N., Park, N., Remington, B., Rothman, S., Vinko, S. M., Whitcher, T., and Wark, J. S. (2010). The strength of single crystal copper under uniaxial shock compression at 100 GPa. Journal of Physics: Condensed Matter, 22(6), 065404. Nicola, L., Xiang, Y., Vlassak, J., Van der Giessen, E., and Needleman, A. (2006). Plastic deformation of freestanding thin films: experiments and modeling. Journal of the Mechanics and Physics of Solids, 54(10), 2089–2110. ˇ ak, B., and Z´arubov´a, N. (1984). Plasticity of high purity iron Nov´ak, V., Sest´ single crystals (II) surface observations. Crystal Research and Technology, 19(6), 793–807. Oliver, W. C. and Pharr, G. M. (2010). Nanoindentation in materials research: past, present, and future. MRS Bulletin, 35(11), 897–907. Ovaska, M., Laurson, L., and Alava, M. J. (2015). Quenched pinning and collective dislocation dynamics. Scientific Reports, 5, 10580.

52

Mechanics and Physics of Solids at Micro- and Nano-Scales

Papanikolaou, S. (2016). Shearing a glass and the role of pinning delay in models of interface depinning. Physical Review E, 93, 032610. Papanikolaou, S., Bohn, F., Sommer, R., Durin, G., Zapperi, S., and Sethna, J. (2011). Universality beyond power laws and the average avalanche shape. Nature Physics, 7(4), 316–320. Papanikolaou, S., Cui, Y., and Ghoniem, N. (2017a). Avalanches and plastic flow in crystal plasticity: an overview. Modelling and Simulation in Materials Science and Engineering, 26(1), 013001. Papanikolaou, S., Dimiduk, D., Choi, W., Sethna, J., Uchic, M., Woodward, C., and Zapperi, S. (2012). Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator. Nature, 490(7421), 517–521. Papanikolaou, S., Luijten, E., and Fradkin, E. (2007). Quantum criticality, lines of fixed points, and phase separation in doped two-dimensional quantum dimer models. Physical Review B, 76(13), 134514. Papanikolaou, S., O’Hern, C. S., and Shattuck, M. D. (2013). Isostaticity at frictional jamming. Physical Review Letters, 110(19), 198002. Papanikolaou, S., Song, H., and Van der Giessen, E. (2017b). Obstacles and sources in dislocation dynamics: strengthening and statistics of abrupt plastic events in nanopillar compression. Journal of the Mechanics and Physics of Solids, 102, 17–29. Papanikolaou, S., Tzimas, M., Reid, A. C. E., and Langer, S. A. (2019). Spatial strain correlations, machine learning, and deformation history in crystal plasticity. Physical Review E, 99, 053003. Pilania, G., Wang, C., Jiang, X., Rajasekaran, S., and Ramprasad, R. (2013). Accelerating materials property predictions using machine learning. Scientific Reports, 3, 2810. Quinlan, J. R. (1986). Induction of decision trees. Machine Learning, 1(1), 81–106. Rabkin, E., Nam, H.-S., and Srolovitz, D. (2007). Atomistic simulation of the deformation of gold nanopillars. Acta Materialia, 55(6), 2085–2099. Raman, K. S., Fradkin, E., Moessner, R., Papanikolaou, S., and Sondhi, S. L. (2008). Quantum dimer models and exotic orders. In Quantum Magnetism, Barbara, B., Imry, Y., Sawatzky, G., and Stamp, P. C. E. (eds). Springer Netherlands, Heidelberg, Germany, 139–150. doi: 10.1007/978-1-4020-8512-3. Ramprasad, R., Batra, R., Pilania, G., Mannodi-Kanakkithodi, A., and Kim, C. (2017). Machine learning in materials informatics: recent applications and prospects. npj Computational Materials, 3(1), 54. Sahni, P. S., Srolovitz, D., Grest, G. S., P. Anderson, M., and Safran, S. A. (1983). Kinetics of ordering in two dimensions. II. Quenched systems. Physical Review B, 28, 2705.

Effects of Rate, Size, and Prior Deformation in Microcrystal Plasticity

53

Schreier, H., Orteu, J.-J., and Sutton, M. A. (2009). Image Correlation for Shape, Motion and Deformation Measurements. Springer US. Senger, J., Weygand, D., Motz, C., Gumbsch, P., and Kraft, O. (2011). Aspect ratio and stochastic effects in the plasticity of uniformly loaded micrometer-sized specimens. Acta Materialia, 59(8), 2937–2947. Shan, Z., Mishra, R. K., Asif, S. S., Warren, O. L., and Minor, A. M. (2008). Mechanical annealing and source-limited deformation in submicrometre-diameter ni crystals. Nature Materials, 7(2), 115. Song, H., Dimiduk, D., and Papanikolaou, S. (2019a). Universality class of nanocrystal plasticity: localization and self-organization in discrete dislocation dynamics. Physical Review Letters, 122, 178001. Song, H., Yavas, H., Van der Giessen, E., and Papanikolaou, S. (2019b). Discrete dislocation dynamics simulations of nanoindentation with pre-stress: hardness and statistics of abrupt plastic events. Journal of the Mechanics and Physics of Solids, 123, 332–347. Sparks, G. and Maass, R. (2018). Shapes and velocity relaxation of dislocation avalanches in Au and Nb microcrystals. Acta Materialia, 152, 86–95. Tong, W., Clifton, R. J., and Huang, S. (1992). Pressure-shear impact investigation of strain rate history effects in oxygen-free high-conductivity copper. Journal of the Mechanics and Physics of Solids, 40(6), 1251–1294. Uchic, M., Dimiduk, D., Florando, J., and Nix, W. (2002). Exploring specimen size effects in plastic deformation of Ni3 (Al, Ta). In MRS Proceedings, vol. 753, Cambridge University Press, Cambridge, pp. BB1–BB4. Uchic, M., Dimiduk, D., Florando, J., and Nix, W. (2003). Materials Research Society Symposium Proceedings, vol. 753. Materials Research Society, Pittsburgh, PA, pp. BB1.4.1–BB1.4.6. Uchic, M. D., Shade, P. A., and Dimiduk, D. M. (2009a). Plasticity of micrometerscale single crystals in compression. Annual Review of Materials Research, 39, 361–386. Uhl, J. T., Pathak, S., Schorlemmer, D., Liu, X., Swindeman, R., Brinkman, B. A., LeBlanc, M., Tsekenis, G., Friedman, N., Behringer, R., Denisov, D., Schall, P., Gu, X., Wright, W. J., Hufnagel, T., Jennings, A., Greer, J. R., Liaw, P. K., Becker, T., Dresen, G., and Dahmen, K. A. (2015). Universal quake statistics: from compressed nanocrystals to earthquakes. Scientific Reports, 5, 16493. Van der Giessen, E. and Needleman, A. (1995). Discrete dislocation plasticity: a simple planar model. Modelling and Simulation in Materials Science and Engineering, 3(5), 689. Weiss, J., Lahaie, F., and Grasso, J. (2000). Statistical analysis of dislocation dynamics during viscoplastic. Journal of Geophysical Research, 105(B1), 433–442.

54

Mechanics and Physics of Solids at Micro- and Nano-Scales

Weiss, J. and Marsan, D. (2003). Three-dimensional mapping of dislocation avalanches: clustering and space/time coupling. Science, 299(5603), 89–92. Wheeler, D., Brough, D., Fast, T., Kalidindi, S., and Reid, A. (2014). PyMKS: materials knowledge system in Python. figshare, available at: http://dx.doi.org/10.6084/m9.figshare.1015761. Xiang, Y. and Vlassak, J. (2006). Bauschinger and size effects in thin-film plasticity. Acta Materialia, 54(20), 5449–5460. Yamakov, V., Wolf, D., Phillpot, S., Mukherjee, A., and Gleiter, H. (2004). Deformation-mechanism map for nanocrystalline metals by molecular-dynamics simulation. Nature Materials, 3(1), 43. Yefimov, S., Groma, I., and Van der Giessen, E. (2004). A comparison of a statisticalmechanics based plasticity model with discrete dislocation plasticity calculations. Journal of the Mechanics and Physics of Solids, 52(2), 279–300. Zaiser, M. (2006). Scale invariance in plastic flow of crystalline solids. Advances in Physics, 55(1–2), 185–245. Zaiser, M. (2015). Local density approximation for the energy functional of threedimensional dislocation systems. Physical Review B, 92(17), 174120.

3 Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

3.1. Introduction Materials properties are strongly influenced by their microstructures at the mesoscale. Although dislocations – line defects present in crystalline materials – are key to control materials’ strain hardening during plastic deformation, the presence of defects other than dislocations, such as second phase precipitates and inclusions, can dramatically influence the strength of the materials. In general, the presence of a second phase can block the motion of dislocations in various manners, leading to improved materials’ mechanical properties. Since the resistance offered by precipitates generally increases with their size and volume fraction, aging is also a useful way to design alloy properties in the industry (Argon 2008). Dislocation–particle interactions are commonly classified as hard-contact interactions as in the case of incoherent particles (with negligible elastic mismatch) or coherent shearable particles that offer an additional resistance upon entry of the dislocation such as the formation of staking fault or anti-phase boundaries, or surface ledges. Soft-contact interactions may arise from size or elastic mismatch, leading to remote interactions with the dislocation stress field.

Chapter written by Sylvie AUBRY, Sylvain Q UEYREAU and Athanasios A RSENLIS.

Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

56

Mechanics and Physics of Solids at Micro- and Nano-Scales

Several metals, such as iron and tungsten, have been considered as part of structural components for future fission and fusion reactors. The main obstacle for their use is the production of helium due to the transmutation of metal atoms by irradiation. At low temperatures, helium atoms tend to group and form helium (He) bubbles. The presence of He bubbles in a metal leads to the alteration of its mechanical properties, and in particular, an increase in strength but a decrease in ductility, as explained by Arsenlis et al. (2004) and Haghighat and Schaublin (2010). These irradiation defects, along with voids (vacancy clusters) or stacking fault tetrahedrons, typically lead to dislocation interactions or macroscopic effects similar to those observed in the case of second-phase precipitates (Argon 2008). The plastic response of metals in the presence of He bubbles is a complex interplay between dislocations and He bubbles. Several mechanisms contributing to the resistance of dislocations to glide motion, and linked to the interactions between dislocations and bubbles, have been described in the literature [refer, for instance, to Arsenlis et al. (2004, 2005)]. Three major contributions can be expected. (1) A dislocation may shear a bubble and create a step which increases the surface area of the bubble. (2) The presence of bubbles creates a mismatch between elastic moduli inside the bubble and the surrounding solid. (3) The stress field of He bubbles due to the imbalance between the gas pressure of the He bubbles and its surface tension. Contributions 1 and 3 can be seen as hard-contact interactions and will be addressed in a different article. Contribution 2 can be seen as a remote soft-contact interaction and is a common dislocation-particle interaction. A number of analytical models have been proposed to describe the strengthening induced by a random distribution of punctual obstacles to dislocation motion [see reviews in Brown and Stobbs (1971), Nembach (1996), and Argon (2008)]. A key ingredient of these models is the statistical average of the resistance offered by the obstacles. In the case of the elastic interaction between a precipitate with elastic or lattice mismatch, the resulting force depends on the dislocation orientation, and particularly on the relative distance between the dislocation and the obstacle. To the authors’ knowledge, there is too few statistical sampling of these interactions based on DD. Formally, the stress field caused by inclusions is deduced from Eshelby’s stress expressions for spherical inclusions or precipitates in an isotropic medium. Eshelby derived the stress field of an inclusion in a matrix (Eshelby 1954, 1961) and

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

57

Cai et al. (2014). While in principle the inclusion can have any shape, only in the case of a spherical inclusion is the stress field defined fully analytically. While our ultimate goal is the quantification of the effects of He bubble microstructure on the plastic behavior of Fe and W, the present work is mostly a methodological paper on the more general interaction of dislocation with Eshelby inclusions. In this chapter, we propose a closed form integration of the Eshelby stress field along dislocation lines, which is essential to largescale DD simulations of the strengthening induced by Eshelby inclusions. These force routines exhibit high numerical fidelity and efficiency. First, in section 3.2, we review the existing DD approaches and investigations of particle strengthening. Then, in section 3.3, we present the derivation of the Eshelby forces and implementation in a DD code. Finally, in section 3.4, we demonstrate the efficiency of the proposed modeling through massive simulations involving fully 3-D microstructures and thousands of Eshelby inclusions. We conclude the chapter by a discussion. A complete modeling, including both hard-contact and remote interactions of dislocations with He bubbles and the quantification of the associated strengthening, is left for an upcoming paper.

3.2. Review of existing approaches 3.2.1. Modeling discrete precipitates with DD simulations A number of models have been proposed in the literature to predict precipitation hardening [see reviews in Brown and Stobbs (1971), Nembach (1996), and Argon (2008)]. Most of them rely on line tension approximations, and a random distribution of punctual obstacles to dislocation motion with constant force resistance. It has been understood early on that the statistical sampling in terms of effective diameters seen by dislocations and obstacle spacing was key in predicting precipitation strengthening of finite size precipitates. For spherical precipitates, the average radius and area sampled by dislocations are (Argon 2008): √ ( )2 ∫ r0 dz π z = r0 , [3.1] = r0 1 − r0 r0 4 0 < a > = π < r2 >=

32 < r >2 3π

[3.2]

58

Mechanics and Physics of Solids at Micro- and Nano-Scales

where z is the height of the cutting plane, r0 is the actual particle radius, and f the volume fraction. This leads to an average distance l between precipitate centers of: √ l =< r >

32 . 3πf

[3.3]

In the case of weak punctual defects, the Friedel–Flescher sampling relates the critical stress to the maximum force Fm offered to dislocations as follows (Argon 2008; Nabarro 1972): √ 3c Fm τF F ≈ , [3.4] 2b2 T where T is the line tension of the dislocation segment and c as the concentration per unit area. When considering strong obstacles, the Mott– Labusch sampling provides a different scaling for the critical stress as (Labusch 1970; Nabarro 1972): √ 4 2 3 Fm c w τM L ≈ , [3.5] 4b3 T where w is the interaction range. An experimental investigation of particle strengthening is possible but appears difficult as several mechanisms are occurring simultaneously, ranging from dislocation–dislocation interactions, solid solution and particle strengthening. As the combination of these various mechanisms is usually poorly known a priori and as it is difficult to modulate one of these mechanisms – by thermal or mechanical treatments – without affecting the others, a quantitative determination solely based on experiments seems challenging. In this context, Dislocation Dynamics (DD) simulations are particularly well-adapted to investigate the interaction of large ensembles of dislocations with thousands of precipitates. Various modeling approaches of the dislocation–particle interaction have been proposed in the literature depending on the type of precipitates under consideration. Obviously, in a real system, several effects may be operating simultaneously. Overall, the modeling of defects other than dislocations in the frame of DD simulations is rather natural, as they constitute additional stress sources, and these effects can be included in the collision detection and in the force balance on dislocation segments. For details on the DD method, see Arsenlis et al. (2007) and Devincre et al. (2011).

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

59

One of the first and most simple modeling of precipitates is certainly the case of impenetrable particles developed by Bacon et al. (1973) and Mohles and Nembach (2001). This model is particularly well-adapted to the case of incoherent particles with the hosting matrix, where the absence of gliding plane continuity across the particle interface prevents the propagation of matrix dislocations. A dislocation must by-pass the precipitate by bowing around it until the dislocation forms a dipole in front of the precipitate. The attractive dipolar interaction between the dipole leads to a collinear annihilation and the release of the dislocation from the obstacle. This mechanism – known as the Orowan by-passing – leads to a stored dislocation loop around the precipitate. The Orowan mechanism plays a central role in precipitation hardening theory, as it represents an upper bound to the resistance that a precipitate of a given size may represent (in the absence of long-range elastic interactions). From a practical standpoint, the modeling of an impenetrable particle in DD simulations can be made through collision detection (Mohles and Nembach 2001; Monnet 2006; Queyreau et al. 2009) or through a large and fictitious repulsive force (Bak´o and Hoffelner 2007); however, the latter procedure may induce numerical issues if not handled with care. Coherent and semi-coherent precipitates or other defects similar to precipitates such as voids and bubbles (Argon 2008), can be traversed by dislocations shearing them in the process. A simple way to account for this phenomenon in DD simulations is by means of a constant shear resistance associated with the dislocation shearing of particles. For this, the force balance of dislocation segments that are entirely or partially inside a precipitate is modified to include the shear resistance offered by the precipitate along the dislocation line. The physical origin of the shear resistance can be well-defined as in the case of a creation of a stacking fault of anti-phase boundary (Mohles 2005), or can average several elementary mechanisms when its value is provided from atomic scale simulations (Monnet et al. 2010; Queyreau et al. 2011a). A constant shear resistance for precipitates of the same nature may well reproduce most of the strength dependence upon the precipitate size (Monnet 2018; Monnet et al. 2010), if remote interactions are weak. Long-range interactions associated with elastic or size mismatch can also be included in DD simulations. Assuming that the superposition principle applies, dislocation segments experience an additional stress field than can be expressed in the form of the conventional stress field of an Eshelby inclusion (Eshelby 1954, 1961). The resulting dislocation–particle interaction

60

Mechanics and Physics of Solids at Micro- and Nano-Scales

can be attractive or repulsive depending on the shearing height. For more general boundary conditions at the precipitate interface or when the volume fraction is large enough so that inclusion interactions cannot be neglected anymore, DD simulations must be coupled with the boundary elements method (BEM) or the finite element method (FEM) (Giessen and Needleman 1995; Jamond et al. 2016; Lemarchand et al. 2001; Martinez and Ghoniem 2002; O’Day and Curtin 2004; Tang et al. 2006; Vattr´e et al. 2014; Weinberger et al. 2009; Zbib et al. 2001). Indeed, regular DD owes its numerical efficiency to the use of closed form analytical expressions when interactions of dislocations with various type of defects can be characterized analytically. However, these analytical expressions exist only for a handful of configurations, mostly in the case of dilute limit and an infinite domain. When associating DD to BEM or FEM, the classical infinite domain solution is corrected by an additional solution that enforces the prescribed conditions at the boundaries of the domain. However, care must be taken to variables being exchanged between DD and FEM and, in particular, the unbalanced tractions induced in the infinite domain stress field of dislocations at the boundary domain (Crone et al. 2014; Queyreau et al. 2014, 2019). These hybrid approaches have been employed, for example, to investigate the plastic behavior of Nickel (Ni)-based super alloys (Jamond et al. 2016; Vattr´e et al. 2014) and strengthening induced by θ′ particles in the Fe-Cu system (Santos-Guemes et al. 2018). In the case of semi-coherent interfaces of large precipitates with respect to the matrix, misfit dislocations may accommodate the large lattice and elastic mismatch and need to be identified. For planar interfaces, possible arrays of misfit dislocations are expected to comply with the so-called quantized −1 Frank–Bilby equation: B = (F−1 M − FP ) · k that relates the displacement gradient FM and FP for the matrix M and particle P with respect to a reference interface state, to the sum of interface Burgers vectors B crossed by an interface probe vector k. In theory, the number of possible arrays of dislocations capable of accommodating is not unique, and requires to find the reference state of the interface (Vattr´e and Demkowicz 2013, 2015). Vattr´e et al. (2016) devised a strategy in a number of papers to define a unique solution for semi-coherent interfaces in a bi-crystal, by testing different reference states, and using continuous mechanics and the Stroh anisotropic elasticity formulation. The dislocation array solution is then identified as the one canceling far field stress induced by the interface and associated with the lowest energy.

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

61

Finally, with the growing interest in the strengthening induced by irradiation defects or in oxide dispersed systems, a number of multiscale approaches have emerged. For instance, atomistic simulations in Bacon et al. (2009) have shown that conventional models based on line tension approximations fail for the smallest nanometric and periodic distributions of irradiation defects as voids or Copper (Cu) precipitates, suggesting complex atomistic and dislocation core effects. In the works of Lehtinen et al. (2016) and Bai and Fan (2018), a spatially varying force [derived from a Gauss potential in Lehtinen et al. (2016)] is employed to model remote and contact interaction with dislocation and is fitted using MD data in different systems. This model can mimic shearing and Orowan by-passing of precipitates, but the physical meaning of this approach is unclear. A very convincing approach has been proposed by Monnet in a series of papers (Monnet 2015; Monnet et al. 2010) and accounts for the assisting effects of temperature to overcome nanometric obstacles. A single shear resistance is determined from zero Kelvin atomistic simulations for various obstacle spacings and sizes. The change of potential energy observed in atomistic simulations is rationalized into different contributions as the elastic energy, curvature energy (in agreement with line tension models), and finally the interaction with the obstacle. Then, a thermodynamical model relying on Boltzmann statistics is proposed to define the activation energy function. A linear dependence of the activation energy is found with temperature and a “classical” decreasing power function with effective stress is recovered.

3.2.2. Investigation of precipitation strengthening and some related effects In this section, we review the findings of a number of DD investigations of the precipitation strengthening and some related effects, with special attention to massive simulations and full three-dimensional treatment when possible. In a series of papers published in early 2000s, Mohles and co-authors (Mohles 2005; Mohles and Nembach 2001) have proposed a number of models to describe the strengthening associated with impenetrable particles, chemical strengthening, and lattice mismatch in FCC alloys such as CopperCobalt (CuCo)-alloys and a commercial Ni-based superalloy. Most of the simulations have been conducted with periodic boundary conditions (PBCs), infinite screw and edge dislocations. The simulated critical shear stress was found to be dependent upon the dislocation character of the infinite dislocation

62

Mechanics and Physics of Solids at Micro- and Nano-Scales

which was not the case for larger scale simulations (Monnet 2006) with free (unpinned) initial loops. The dissociation of a dislocation into a pair of Shockley partials was also investigated, and different behavior was observed depending of the size of the shearable particle. When the Orowan mechanism is controlling the plastic flow as in the case of impenetrable precipitates or coherent particles that are too large to be sheared by dislocations, the observed strengthening agrees well with analytical model of dispersion strengthening by Bacon, Kocks, Scattergood (BKS) (Bacon et al. 1973; Sobie et al. 2015). The model proposed by BKS for impenetrable finite size particles relies on the Friedel statistical model for weak punctual obstacles, and stated for the first time that the dipolar interactions occurring behind precipitates are greatly helping the Orowan process (Bacon et al. 1973):

τBKS

[ ( ′) ] 3 [ ( )] 1 2 D L −2 µb log + 0.7 log = 2πL b b

[3.6]

A simple tweak of the strengthening dependence upon volume fraction was, however, required to match DD results, and was attributed to the interactions with the Orowan loops and islands left behind for large volume fraction of precipitate. The predictive capability of the BKS model was reinforced latter by other fully 3D simulations of precipitates in HCP Zirconium (Zr) alloys (Monnet 2006) and carbides in BCC pressure vessel steels (Queyreau et al. 2009). In Monnet (2015), the critical stress associated with a random population of shearable obstacles is found to increase with the shear resistance (defined from atomistics) offered by the precipitates until saturating to a value close to the BKS limit. Before saturation, the Friedel–Fleischer and the Mott– Labusch failed to predict the simulated precipitation hardening, but provided the correct tendency and a rather good upper and lower bounds, respectively. Particle strengthening is rarely the only strengthening mechanism at play during plastic deformation of alloys, and the question of combination of hardening effects has seldom been addressed in the literature. Most of the mixture laws adopt the form [see review (Brown and Stobbs 1971)]: ( )1/k τtotal = Σi τik

[3.7]

where τi is the critical stress induced by population of obstacles i and assumes that there is no synergy between the strengthening effects. The exponent k can

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

63

be rigorously derived only for simple cases like obstacles of same strength (k = 2) or same surface density (k = 3/2) but does not lead to very different predictions (Queyreau et al. 2009). The linear mixture law corresponding to k = 1 is considered to match only the case of the superposition of few strong obstacles among large densities of weak obstacles e.g. solid solution strengthening with other stronger interaction. Monnet investigated the combination of two strenghening effects associated with two different population of incoherent precipitates in Zr Alloy and concluded that only a mixture law similar to the one of Brown was capable of reproducing the DD results: ) ( ) ( ρ2 ρ1 τ1 + τ2 , [3.8] τtotal = ρ1 + ρ2 ρ1 + ρ2 where ρi denotes surface densities of the two types of obstacles. Queyreau et al. investigated the combination of incoherent spherical precipitates with forest hardening associated with junction formation. Interestingly, the quadratic law of mixture k = 2 was found to provide the best match with DD simulations. It must be recalled that the quadratic law of mixture is commonly employed to described the various resistance offered by dislocations reactions.

3.3. Dislocation dynamics modeling of dislocation interactions with Eshelby inclusions Our main goal is to investigate large-scale 3D dislocation microstructures evolving in interaction with thousands of He bubbles. Leaving aside hardcontact interactions for now, capturing the remote interactions between dislocations and He bubbles can be done through an additional Eshelby stress field for small volume fraction or using a hybrid DD-FEM modeling. However, the latter modeling does not seem particularly adapted to large number of obstacles of nanometric size. In this section, a closed-form analytical integration of the stress induced by an Eshelby inclusion which complies with nodal DD framework is proposed.

3.3.1. Stress field and forces at dislocation lines We define the interaction force induced by an Eshelby inclusion onto a straight segment of dislocation as the integral of the Eshelby stress field caused by the inclusion. Eshelby provides mathematical expressions for the stress field

64

Mechanics and Physics of Solids at Micro- and Nano-Scales

coming from (1) several inclusions, when the inclusions are composed of the same material as the surrounding matrix, and (2) for a single inclusion in a matrix, when the inclusion is made of a different material than the matrix. In this chapter, we use Eshelby’s stress field formula for this latter case, but for a few thousand He bubbles, in a metal. In particular, we make the assumption that the interaction force between the inclusions is negligible. Assuming the inclusions have a spherical shape, and that the strain is a volumetric strain only, an analytical formula for the force can be derived.

3.3.2. Stress at a point induced by an inclusion In a linear elastic domain, the elastic stress field at a point x due to a misfitting, spherical inclusion of center C and radius R, in an infinite medium is given by Eshelby’s analytical solution (Cai et al. 2014; Eshelby 1954, 1961) as σ(r) = −2ϵV Λin I, when |r| ≤ R ( ) I r⊗r 3 = ϵV Λout R −3 , when |r| > R |r|3 |r|5

[3.9]

(1+νout ) (1+νin ) and Λout = µout (1−2ν are two constants where Λint = µin (1−2ν out ) in ) respectively depending on the elastic properties of the inclusion and the matrix surrounding it. ϵV is a volumetric misfit, (µout , νout ) are the elastic constant of the material or matrix surrounding the inclusion, and (µin , νin ) are the elastic constant inside the inclusion. In equation [3.9], we make use of the following substitution r = x − C, where r corresponds to the vector from the center of the inclusion C to the point x where the stress is to be evaluated. I is the identity matrix in three dimensions. In this definition, the stress field inside the particle does not depend on its radius.

3.3.3. Force on a dislocation coming from an inclusion Eshelby’s stress field σ defined in equation [3.9] produces a Peach–Koehler force F on a closed dislocation loop of the form I F = C

N (y) σ(y) · b × t dy

[3.10]

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

65

where b is the Burgers vector of the dislocation loop C, N is a shape function, 1 and t = x2 −x is the dislocation line direction. L Within the DD framework, dislocation lines are typically discretized into straight segments, ended by nodes. The force at dislocation nodes are given by the sum of the forces associated with the dislocation segments attached to nodes under investigation. Shape functions N are thus required to transform the distributed force along the segment into a nodal force. If [x1 , x2 ] is a dislocation segment, then the force F2 at the point x2 coming from this dislocation segment can be defined from equation [3.10] as: ∫

x2

F2 = x1

(x − x1 ) σ(x) · b × t dx L

[3.11]

where b and t are the Burgers vector and the line direction of the dislocation segment, respectively, and L = |x2 − x1 | is the length of the dislocation segment. According to equation [3.9], the force on a dislocation line exhibits two definitions and depends on the relative position of the dislocation with respect to the inclusion. Care must be paid to the integration bounds and the intersection geometry of the dislocation segment with the inclusion’s sphere. Indeed, the dislocation segment [x1 , x2 ] may intersect the inclusion in one or two points, or may be entirely inside or outside the precipitate. The different geometries of the intersection between a straight segment with a spherical precipitate are illustrated in Figure 3.1. If there is no intersection between the dislocation segment and the inclusion, the Peach–Koehler force at x2 coming from the stress field, equation [3.9], is F2 =

) Λout ϵV R3 ( ˜ F (s2 ) − F˜ (s1 ) L

[3.12]

where s1 = ((x1 − C) − p) · t, s2 = ((x2 − C) − p) · t, p = (x1 − C) − ((x1 − C) · t)t,

[3.13]

66

Mechanics and Physics of Solids at Micro- and Nano-Scales

where q + s1 s q 2 − 3qs1 s − 2s1 s3 F˜ (s) = −(b × t) + (b · p)(p × t) ρq ρ3 q 2 s3 + qs1 [3.14] ρ3 q √ where we have defined ρ = s2 + q and q = p · p. More detail about this derivation is given in section 3.7. − (b · t)(p × t)

When the dislocation line intersects the inclusion, several scenarios can occur. The dislocation line (t, b) can intersect the inclusion in one or two points, but the dislocation segment can have zero, one, or two intersection points with the inclusion. In the case of an existing intersection between the dislocation line and the inclusion, the intersection points are denoted as xa and xb , respectively. It may happen that one or two intersection points belong to the dislocation line but is/are outside the dislocation segment. From these points, we can define √ √ sa = − R2 − q and sb = R2 − q. If the dislocation line intersects the inclusion in two points, q ≤ R2 . The definition of the force depends on the relative position of the dislocation segment with respect to the inclusion. Several definitions follow: – The dislocation line intersects the inclusion but only the point xa is located within the segment [x1 , x2 ], see Figure 3.1(a). This corresponds to the case: s1 < sa ≤ s2 ≤ sb , where s1 and s2 are defined in equation [3.13]. In this case, the force is F2 =

) Λout ϵV R3 ( ˜ F (sa ) − F˜ (s1 ) L 2Λin ϵV − (b × t) (s2 (s2 − 2s1 ) − sa (sa − 2s1 )) L

where F˜ (s) is defined in equation [3.14].

[3.15]

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

67

– The dislocation segment is completely inside the inclusion, see Figure 3.1(b): sa < s1 < s2 ≤ sb , and the force is 2Λin ϵV F2 = −(b × t) [3.16] L – The dislocation line intersects the inclusion but only the point xb is located within the segment [x1 , x2 ], see Figure 3.1(c) : sa ≤ s1 ≤ sb ≤ s2 , and the force is F2 =

) Λout ϵV R3 ( ˜ 2Λin ϵV F (s2 ) − F˜ (sb ) − (b × t) (sb − s1 )2 L L [3.17]

– The dislocation segment intersects the inclusion in two points xa and xb , see Figure 3.1(d): s1 < sa ≤ sb ≤ s2 , and the force is ) Λout ϵV R3 ( ˜ F (s2 ) − F˜ (s1 ) + F˜ (sa ) − F˜ (sb ) L 2LΛin ϵV − (b × t) (sb (sb − 2s1 ) − sa (sa − 2s1 )) L

F2 =

[3.18]

Figure 3.1. Different geometries of the intersection between a straight dislocation segment [x1 , x2 ] and a spherical inclusion. The dislocation line intersects the inclusion but in (a) only xa is inside, or in (b) xa and xb are outside the segment, or (c) xb is inside, or finally, (d) both intersection points are inside the dislocation segment. For a color version of this figure, see www.iste.co.uk/ionescu/ mechatronics.zip

68

Mechanics and Physics of Solids at Micro- and Nano-Scales

3.3.4. Far field interactions induced by an Eshelby inclusion In the absence of a truncation distance, the numerical cost of dislocation– inclusion interactions scales as Nd × Ni , where Nd and Ni are the number of dislocation segments and inclusions, respectively. While the Eshelby stress field is commonly categorized as short range because of the 1/|r|3 and 1/|r|5 spatial dependences (while weighted by the lattice or elastic mismatch), large-scale multithreading for simulations containing massive dislocation ensembles, often renders the fully analytical calculations of dislocations too expensive, refer to Arsenlis et al. (2007). We, therefore, employ a “classical” fast multipole method (FMM) for long-range interactions to reduce the computational expense of the Eshelby interaction calculations, refer to Greengard and Rokhlin (1987) and Chen et al. (2018). In practice, the simulation domain is separated into sub-domains and forces at a point are calculated directly only for a sub-domain surrounding this point. Outside of that cut-off region, a lumped source approximations enabled by FMM is used to determine forces. An FMM method is used to determine Eshelby forces for inclusions located in domains far-away from the dislocation line they interact with. Also, since these interactions are not as strong as the interaction force between dislocations, the Taylor series used in the FMM is truncated to a smaller number of computed coefficients. In this work, the sum of the Taylor series expansion order and the multipole order is limited to be less than 6. This value is chosen to be a good compromise between relative stress error sufficiently small for accurate calculations of remote forces due to Eshelby particles and a reduced cost. The Eshelby force can be integrated into a dislocation dynamics code using direct calculations given by equations [3.11]–[3.18], for close-range interactions between dislocations and particles, and using the fast multipole method for the farther ones. It is an added force acting on the dislocation nodes in the domain coming from the stress field of all the Eshelby inclusions present in the simulations.

3.3.5. Parallel implementation For similar efficiency requirement, the practical implementation of the Eshelby force was designed to comply with a parallel implementation to simulate the interactions and evolution of large ensembles of dislocations with

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

69

thousands of inclusions. In the dislocation dynamics code ParaDiS (Arsenlis et al. 2007), parallel algorithms have been proposed for the calculations of the elastic interactions among dislocation segments and the evaluation of the velocity and positions of dislocation nodes. When a few thousand Eshelby inclusions are added in the simulation domain, the parallel algorithm is slightly modified to account for the force from the inclusions on the dislocation lines in the network. In the ParaDiS code, the CPU owns the inclusions whose center’s positions are contained within its subdomain. The CPU maintains and updates the Eshelby force for the dislocation nodes it owns and sends the data to other CPUs required to perform computations involving its nodes. This subdivision of tasks allows for hundred of thousand of dislocation segments to interact with a few thousand He bubbles in an efficient and accurate manner.

3.4. DD simulations of the interaction with Eshelby inclusions In this section, we provide an illustration of the proposed modeling for Eshelby inclusions. A first example on a single segment interacting with a single inclusion demonstrates that the proposed nodal force captures the expected dislocation/particle interaction correctly. A second set of simulations shows that the proposed framework complies with the large-scale simulation requirements.

3.4.1. Eshelby force for a single dislocation and a single inclusion An example of the Eshelby force along the edge and screw components of a single dislocation crossing a single particle is shown in Figure 3.2(a) for a screw dislocation, and for an edge dislocation in Figure 3.2(b). The Eshelby force derived in section 3.3.1 varies with the volumetric strain and the elastic constants of the particles and the surrounding metal. In Figure 3.2, different values of the volumetric strain, set in the range [0.01, 0.05], are also plotted. The considered material is tungsten, and the particle has a shear modulus five order of magnitude smaller than the one chosen for tungsten, µ = 160 GPa. The inclusion has a radius set to R = 1|b|. In these examples, the dislocation does not intersect the middle of the particle but intersects slightly above it, leading to non-zero Eshelby force contributions in the glide plane. When the dislocation intersects the particle exactly in its middle, only the component of the Eshelby force projected along the glide plane normal does not vanish.

70

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 3.2. Profile of the Eshelby force projected along the the line direction t of the dislocation, and along be = b − (b · t)t, for (a) a screw dislocation and (b) an edge dislocation when the dislocation intersects an Eshelby particle of radius 1|b|. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Interestingly, the interaction for the screw dislocation is symmetrical and exhibits a maximum at the center of the inclusion while the force is asymmetrical and vanishes at the inclusion center for the edge dislocation. This force component also changes sign when the dislocation is now placed slightly below the hemispheric plane. Figure 3.2 shows that as the volumetric strain increases, the Eshelby force increases proportionally. The Eshelby force vanishes quickly when the dislocation segment leaves the bubble. These basic examples highlight some of the interactions that are possible between a single dislocation segment and an inclusion. As it will be discussed later, this constitutes one of the key challenges in the associated strengthening. After validation of the nodal force implementation, we illustrate large-scale dislocation dynamics simulations.

3.4.2. Simulations of bulk crystal plasticity The collective behavior of large ensembles of dislocations in interaction with inclusions is now investigated. Several key physical quantities that are expected to control precipitation hardening, namely, the particle density, the associated strain misfit, and strain rate, are varied. Several DD simulations are executed and involve two BCC metals of interest: iron and tungsten. These materials exhibit different elastic parameters as follows: a shear modulus of µ = 86 GPa, Poisson’s ratio of ν ≈ 0.29, and a lattice constant of a = 2.86A˙

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

71

for iron; and a shear modulus of µ = 160 GPa, a Poisson’s ratio of ν ≈ 0.28, and a lattice constant of a = 3.16A˙ for tungsten, known to be almost perfectly isotropic elastically. Simulations account for the twelve 1/2 110 slip systems of the BCC crystal structure. The initial dislocation configuration is made of infinitely long screw dislocations randomly distributed in the simulation box and on all slip systems, with an initial density of 1014 m−2 . In BCC materials, screw dislocations are known to exhibit a thermally activated character at low stress and low temperature (Monnet et al. 2004; Gilbert 2011).

Figure 3.3. Dislocation dynamics results in iron taking Eshelby pressures into account. (a) Stress versus total strain relationship, (b) stress versus plastic strain relationship, and (c) dislocation density evolution. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Focusing on the athermal regime, dislocation mobility simply exhibits a viscous behavior that corresponds to the phonon-drag phenomenon. Screw dislocations have the ability to easily cross-slip to a different gliding plane

72

Mechanics and Physics of Solids at Micro- and Nano-Scales

in BCC materials, and we use a pencil glide mobility law for BCC crystals as described in Arsenlis et al. (2007) to represent this behavior.

Figure 3.4. Dislocation dynamics results in tungsten taking Eshelby pressures into account (a) stress versus total strain relationship, and (b) dislocation density evolution. For a color version of this figure, see www.iste.co.uk/ionescu/ mechatronics.zip

The loading conditions of all simulations of bulk single crystal reported here is tension along [001] axis. The loading is applied under a constant strain rate ranging from 104 , 105 , and 106 s−1 , which are acceptable values with regard to the small simulation box size and large dislocation density explored in these simulations. These simulation conditions are also in agreement with irradiation conditions. We will also check that the strain rate has little impact on the simulated precipitation strengthening. Periodic Boundary Conditions are applied in all dimensions of the simulations domain. The inclusion microstructure is made of a distribution of 10, 000 to 30, 000 inclusions, which are introduced in the simulation domain at random without overlapping inclusions. Their diameter D is set to a small value of 5|b|. These particle number and diameter values correspond to an inclusion number density ρV ranging from 8 × 1022 to 2.4 × 1023 m−3 corresponding to typical densities of irradiation defects. This corresponds to volume fraction ranging √ from 8 × 10−5 to 2.4 × 10−4 , and an average spacing of the order 1/ ρV D ranging from 80 to 140 nm. The elastic constants for the Eshelby inclusions are chosen so that the shear modulus representing the inclusion is five order of magnitude smaller than one of the metals. The volumetric strain inside the Eshelby inclusion ranges from ϵV = 0.01 to ϵV = 0.05 depending on

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

73

the simulation and corresponds to strains observed in simulations for He bubbles (Caro et al. 2011).

Figure 3.5. Stress (a) and dislocation density (b) as function of the total strain obtained from dislocation dynamics in iron when the volume fraction is varied from 8 × 10−5 to 2.4 × 10−4 . The strain rate is 104 s−1 . For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Since our proposed modeling of Eshelby particle relies essentially on the mechanical balance existing on dislocation segments, we investigate the impact of the imposed strain rate in a first set of simulations. The corresponding stress–strain response and dislocation density evolution are presented in Figures 3.3 and 3.4 for Fe and W, respectively. In these simulations, the particle number is 20,000 and the volumetric strain inside the bubbles is set to ϵV = 0.04 for iron, and to ϵV = 0.03 for tungsten. Figure 3.3(a) shows the mechanical response in terms of stress as function of total or plastic strain for Fe with and without a distribution of Eshelby inclusions for different strain rates. Considering the minor contribution of elastic strain to the overall response, these curves will not be shown afterward. The stress–strain curves exhibit multiplication peaks in agreement with the small dimensions of the initial dislocation loops and growing with the imposed strain rate. When dislocation density [see Figure 3.3(b)] is sufficient to produce the required strain rate, a linear regime is observed on the dτ hardening rate. The linear hardening rate θ = dγ ≈ µ/120 − µ/150, which is slightly superior to the initial hardening rate observed in experiments for BCC metals when dynamic recovery is still low. Even in the absence of Eshelby particles, increasing the strain rate increases the dislocation density by the

74

Mechanics and Physics of Solids at Micro- and Nano-Scales

activation of dislocation loops and leads to a considerable density storage. However, most importantly, when comparing with the curves obtained with the Eshelby particle distribution, the mechanical response is simply shifted by a stress value over the entire deformation range, and corresponds to the strengthening induced by the random distribution of particles. The increase in terms of dislocation density is limited when comparing curves with and without bubbles. This small increase is explained by the dislocation length growth due to the additional curvature of dislocations and the larger stress induced by particles. The strengthening at the end of the multiplication peak is of the order of 250–300 MPa and is weakly affected by the strain rate. The difference between the curves with and without Eshelby inclusion tends to reduce with strain and dislocation density, suggesting that the law of mixture of the precipitate strengthening and forest interaction is not linear, and could be quadratic as in Queyreau et al. (2009). This allows us to validate our implementation of the Eshelby interaction and choice of deformation strain rates. To simplify the presentation of the results, we defined the additional stress induced ∆σ(ϵ) by the presence of Eshelby inclusion as the difference ∆σ(ϵ) = σtotal (ϵ) − σf orest (ϵ) between the stress with σtotal and without precipitates σf orest . ∆σ will, in particular, be evaluated just after the multiplication peak. The second law of Orowan can provide an estimate of < v > the average dislocation velocity: γ˙ = ρmob b < v >, where γ˙ is the plastic strain rate and ρmob the density of mobile dislocations. This latter quantity is typically taken as ρmob ≈ ρtot /10. In simulations, < v > decreases with the amount of plastic deformation as the dislocation density increases. The average velocity is, therefore, in the range of 2.4–7 m/s for a strain rate of 104 s−1 , for dislocation densities evaluated after the multiplication peak, and at the end of the simulation. < v > is in the range 12–60 m/s for ϵ˙ = 105 s−1 and 36–240 m/s for ϵ˙ = 106 s−1 . These values are certainly larger than the average values obtained in quasi-static experiments performed at the macro-scale, but are reasonable values for DD simulations (at least for ϵ˙ = 104 and 105 s−1 ) and are in line with the weak influence of the strain rate upon the precipitate strengthening. The mechanical response obtained for tungsten in the presence of Eshelby inclusions is slightly different, with a larger hardening rate observed in presence of inclusions. This is more noticeable at 104 s−1 and 105 s−1 strain rates. The dislocation density curves also show a larger density in the presence

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

75

of Eshelby inclusions. The extra storage of dislocations is due to the formation of Orowan loops and islands left around the denser regions of precipitates and larger diameters sheared by the dislocations. This is confirmed by the observation of microstructures with many loops and collinear loops found in the microstructure. The increased storage rate due to the Orowan loop left around impenetrable particles was found to be a linear function of the plastic strain in Queyreau et al. 2009, which seems to be the case here. Another interesting feature is the existence of stress fluctuations existing on all stress curves obtained for W, and are only visible for the lowest strain rate curves in the case of iron. The absence of serrations is certainly due the number of critical events occurring simultaneously and providing sufficient plastic strain to satisfy continuously the imposed loading. Metal Iron Iron Iron Tungsten

Vol. fraction 8 × 10−5 1.6 × 10−4 2.4 × 10−4 1.6 × 10−4

ϵ˙ = 104 s−1 121 168 236 252

ϵ˙ = 105 s−1 133 175 210 280

ϵ˙ = 106 s−1 159 166 214 372

Table 3.1. Approximate added stress contribution ∆σ in MPa coming from the presence of Eshelby inclusions in iron for a constant volumetric strain of 4% for iron, and 3% for tungsten as the strain rate is increased. This value is determined as the stress value after the peak

Next, we discuss the statistical representativeness of our simulations. The number n of particles intersected by dislocations in the course of a swept area A is n = DρV A. According to Orowan’s law, this swept area corresponds to a plastic increment ∆γ = Ab V in the simulated volume V . The number of intersected particles is thus n = DρV b∆γV (Queyreau et al. 2009). When considering the increment of plastic strain in all the simulations performed here, the number of intersected particles is in the range of 10,000–180,000 which is much larger that the criterion for statistical representativeness defined by Monnet (2006). Next, we vary the volume fraction of Eshelby inclusions contained in single crystal iron simulation from 8 × 10−5 to 2.4 × 10−4 , while preserving their diameter to 5|b|. Results are shown in Figure 3.5, and the corresponding critical shear stress values past the multiplication peak are given in Table 3.1. The stress increase ∆σ increases with the volume fraction as expected, and

76

Mechanics and Physics of Solids at Micro- and Nano-Scales

√ is proportional to 1/ DρV . Finally, we varied the mismatch elastic strain inside the inclusion in the Fe matrix, while preserving the volume fraction to 1.6 × 10−4 and the same diameter. Figure 3.6 shows these effects of increasing the pressure inside the inclusions by an order of magnitude from 0.5% to 4% on the stress/total strain, Figure 3.6(a); the stress/plastic strain, Figure 3.6(b); and the dislocation density in iron. As expected, the additional stress increases with the pressure in the inclusion but it is a clearly smaller effect than the particle density with the values under consideration here. For the last two sets of simulations, there is a minor change in the dislocation density which is related to increased dislocation curvature with stress and obstacle density, but no increased storage in the form of Orowan loops.

Figure 3.6. Dislocation dynamics results of iron when the pressure inside the particles is increased. (a) Stress versus total strain relationship and (b) dislocation density. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Volumetric strain 0.005 0.02 0.03 0.04

ϵ˙ = 105 /s 102 131 150 175

ϵ˙ = 106 /s 106 133 146 166

Table 3.2. Added stress contribution ∆σ in MPa coming from the presence of Eshelby inclusions in iron for an increasing volumetric strain and a constant volume fraction, set to about 1.6 × 10−4

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

77

3.5. Conclusion and discussion In the discussion, we propose an interpretation of the large-scale simulations and offer a summary of effects that have been rarely addressed in the literature and that we believe are important for the understanding of precipitation strengthening and possibly improving alloy design. For the range of volume fractions and Eshelby pressures investigated here for Fe, the large increase of flow stress induced by particle was found to be solely controlled by particle shearing in presence of an Eshelby interaction with dislocation, and little Orowan by-passing. As mentioned in section 3.2, a number of analytical models have been proposed to predict the strengthening induced by a random distribution of particles. These models, however, require the averaging of the strength due to an obstacle onto a single dislocation. In the preparation of the present manuscript, we used a similar procedure as Bacon, Kocks, and Scattergood’s approach (BKS) to average the strength of impenetrable precipitates. Interestingly, this procedure only partially worked to define an average obstacle strength but did not cover the entire range of simulations in terms of remote dislocation–particle interaction, dipolar interaction for smaller particle, and Orowan looping for stronger obstacle. Thus, providing a unique average associated with Eshelby inclusion is left for future work. We expect that lower and upper bounds can be derived as function of the average strength of obstacles (Argon 2008) in the form of the Friedel–Fleicher model for weak average obstacles and Mott–Labusch model for stronger obstacles. Second, as explained in section 3.2.2, in BKS (Bacon et al. 1973; Mohles and Nembach 2001; Sobie et al. 2015), the stress contribution coming from impenetrable particles distributed randomly in the domain is given by

τBKS

[ ( ′) ] 3 [ ( )] 1 2 D µb L −2 = log + 0.7 log 2πL b b

where L is the mean spacing between particles, L = ′

√ 1 ; N is the number ND DL = D+L is an effective

density; D is the diameter of the bubbles; and D particle diameter. It is expected that this should provide an upper bound for Fe where particles are sheared and could capture a part of the strengthening for W where stronger particle/dislocation interactions favor the Orowan mechanism.

78

Mechanics and Physics of Solids at Micro- and Nano-Scales

When using the particle size and spacings corresponding to our simulations, the BKS model τBKS leads to 54, 79, and 99 MPa in iron for a volume fraction of 8 × 10−5 , 1.6 × 10−4 , and 2.4 × 10−4 , respectively, and 147 MPa for tungsten and a volume fraction 1.6 × 10−4 . Assuming a quadratic law of mixture as found by Queyreau et al. (2009), the total shear stress can be expressed as a function of the critical stress associated with the Orowan bypassing τBKS and the one associated with dislocation interactions – or forest mechanism – τf orest : √ 2 + τf2orest τtotal = τBKS where the forest contribution is defined by √ τf orest = α ¯ µb ρtotal where α ¯ measures the average interaction strength of all dislocationdislocation interactions for a given dislocation density. Finally, the stress increment ∆σBKS is recovered by employing a Schmid factor of 0.408 for [001] loading direction and removing the stress without inclusions. Focusing on the smallest strain rates and values right after the multiplication peak, the calculated additional stress equals 130, 170, and 207 MPa for iron at inclusion volume fractions of 8 × 10−5 , 1.6 × 10−4 , and 2.4 × 10−4 , respectively, and equals 247 MPa in the case of tungsten. It is interesting to note that the calculated values of ∆σBKS , obtained by assuming Orowan strengthening and a quadratic law of mixture for precipitate and forest hardenings, provides a relatively close upper bound for the simulated total stresses obtained under the considered conditions, see Tables 3.1 and 3.2. This highlights the importance of accounting for Eshelby’s effects, when the strain mismatch is seemingly as small as few percents with respect to the matrix values. Finally, going back to the review of the literature in section 3.2, we highlight few research directions related to precipitate strengthening and that we intend to follow in future works. First, when comparing with Orowan strengthening, the case of shearable precipitates has been relatively less investigated in the literature and very few models to predict the corresponding strength exist. Coherent or semi-coherent precipitates in alloys are of industrial interest, as they lead to a smaller dislocation storage and thus smaller stress concentrations that may ultimately promote precipitate decohesion and fracture. Other defects as voids or He bubbles can be sheared by dislocations

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

79

and lead to similar dislocation/defects interactions, which constitutes another motivation for obtaining predictive models. DD simulations are ideally suited for studying the separate effects (shearable, unshearable, etc.) of voids or He bubbles in metals. MD simulations contain all these effects simultaneously, and the importance of each effect is usually difficult to analyze. However, it would be of interest to be able to precisely separate the observed atomistic interaction with dislocations into well-defined elementary mechanisms. This could help better definitions of the stain rate and temperature dependence of penetrable precipitates, for example. Finally, with the improvement of experimental setups to characterize materials microstructures or mesoscopic simulations of microstructure evolution like kinetic Monte Carlo or phase field approaches, more investigations should address the question of realistic precipitate distributions in terms of size and spacing, and correlation between these two parameters. Finally, precipitation strengthening is rarely the only mechanism at play in complex alloys, and the difficult question of the combination effects and possible synergy existing among various effects such as solid solution hardening, precipitation and forest interactions should be at reach. In this chapter, we have analyzed the effects of taking into account the Eshelby force on dislocation networks, due to the presence of He bubbles iron and tungsten. A detailed exploration of the variation of the mechanical properties of iron and tungsten such as stress–strain relationship and dislocation density as a function of the number of particles present in the network, the volumetric strain corresponding to a given pressure inside the He bubbles, and as a function of strain rate was shown. A constant stress contribution is observed as a function of increasing strain rate.

3.6. Acknowledgments This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DEAC52-07NA27344. Also, research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2-0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and

80

Mechanics and Physics of Solids at Micro- and Nano-Scales

distribute reprints for Government purposes notwithstanding any copyright notation herein. Release number: LLNL-JRNL-780040.

3.7. Appendix: derivation of the Eshelby force The Eshelby force F2 evaluated at point x2 , on the dislocation segment [x1 , x2 ], coming from the stress of an inclusion of center C and radius r, can be defined using Peach–Koelher’s relationship as ∫

x2

F2 = x1

(x − x1 ) σ(x) · b × t dx L

[3.19]

where the Eshelby stress is ( σ(r) = ϵV Λout R3

I r⊗r −3 |r|3 |r|5

) [3.20]

if the inclusion and the dislocation line do not intersect, as shown in section 3.3.1. Substituting the stress into the force equation gives ϵV Λout R3 F2 = L

∫

r2

r1

[( (r − r1 )

r⊗r I −3 |r|3 |r|5

)] · b × t dr [3.21]

where we have used the change of variables r = x − C, r2 = x2 − C, and r1 = x1 − C. Using the variables s1 = (r1 − p) · t,

s2 = (r2 − p) · t,

p = r1 − (r1 · t)t,

defined in section 3.3.1, another change of variables leads to [( )] ∫ ϵV Λout R3 s2 I (p + s t) ⊗ (p + s t) F2 = (s−s1 ) −3 ·b×t ds L ρ3 ρ5 s1 √ where ρ = s2 + q, and q = p · p.

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

81

Rearranging the terms, the force becomes ∫ s2 ϵV Λout R3 (s − s1 ) ds F2 = b×t L ρ3 s1 ∫ s2 ϵV Λout R3 (s − s1 ) −3 (b · p)(p × t) ds L ρ5 s1 ∫ s2 ϵV Λout R3 s(s − s1 ) ds −3 (b · t)(p × t) L ρ5 s1 since (b × t) p ⊗ p = [(b × t) · p] · p = (b · p)(p × t) (b × t) p ⊗ t = [(b × t) · p] · t = (b · t)(p × t) (b × t) t ⊗ t = [(b × t) · t] · t = 0 (b × t) t ⊗ p = [(b × t) · t] · p = 0 The solutions of the integrals are ∫ s2 s s2 ds = 3 qρ s1 s1 ρ ∫ s2 1 s2 sds = − 3 ρ s1 s1 ρ ( ) ∫ s2 2 1 s2 s ds + =− 5 3pq p q 2 s1 s1 ρ ∫ s2 s s2 sds = − 5 3ρ3 s1 s1 ρ ( ) ∫ s2 2 s 1 1 s2 s ds = − ρ5 3ρ p ρ2 s1 s1 Finally, the force is ) Λout ϵV R3 ( ˜ ˜ F2 = F (s2 ) − F (s1 ) L

[3.22]

82

Mechanics and Physics of Solids at Micro- and Nano-Scales

where q + s1 s q 2 − 3qs1 s − 2s1 s3 F˜ (s) = −(b × t) + (b · p)(p × t) ρq ρ3 q 2 − (b · t)(p × t)

s3 + qs1 ρ3 q

[3.23]

When the dislocation line intersects the particle in one or two points, the definition of the stress field differs for the parts of the dislocation segment located inside and outside the particle. The force is found by integrating the piecewise stress field on intervals of the type s1 < sa ≤ s2 ≤ sb , sa < s1 < s2 ≤ sb , sa ≤ s1 ≤ sb ≤ s2 , and s1 < sa ≤ sb ≤ s2 depending on the relative position of the intersection points xa and xb with the dislocation segment [x1 , x2 ].

3.8. References Argon, A. S. (2007). Strengthening Mechanisms in Crystal Plasticity. Oxford Series on Materials Modelling, Oxford University Press. Arsenlis, A., Cai, W., Tang, M., Rhee, M., Oppelstrup, T., Hommes, G., Pierce, T., and Bulatov, V. V. (2007). Enabling strain hardening simulations with dislocation dynamics. Modelling and Simulation in Materials Science and Engineering, 15(6), 553–595. Arsenlis, A., Wirth, B. D., and Rhee, M. (2004). Dislocation density-based constitutive model for the mechanical behaviour of irradiated Cu. Philosophical Magazine, 84(34), 3617–3635. Arsenlis, A., Wolfer, W., and Schwartz, A. (2005). Change in flow stress and ductility of δ-phase Pu–Ga alloys due to self-irradiation damage. Journal of Nuclear Materials, 336(1), 31–39. Bacon, D. J., Kocks, U. F., and Scattergood, R. O. (1973). The effect of dislocation self-interaction on the orowan stress. The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics, 28(6), 1241–1263. Bacon, D. J., Osetsky, Y., and Rodney, D. (2009). Dislocation-obstacle interactions at the atomic level. In Dislocations in Solids, vol. 15, Hirth, J. P. and Kubin L. (eds). Elsevier, pp. 1–90. Bai, Z. and Fan, Y. (2018). Abnormal strain rate sensitivity driven by a unit dislocation-obstacle interaction in bcc Fe. Physical Review Letters, 120, 125504. Bak´o, B. and Hoffelner, W. (2007). Cellular dislocation patterning during plastic deformation. Physical Review B, 76, 214108.

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

83

Brown, L. M. and Stobbs, W. M. (1971). The work-hardening of copper-silica. The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics, 23(185), 1185–1199. Cai, W., Sills, R., Barnett, D., and Nix, W. (2014). Modeling a distribution of point defects as misfitting inclusions in stressed solids. Journal of the Mechanics and Physics of Solids, 66(Supplement C), 154–171. Caro, A., Hetherly, J., Stukowski, A., Caro, M., Martinez, E., Srivilliputhur, S., Zepeda-Ruiz, L., and Nastasi, M. (2011). Properties of helium bubbles in Fe and FeCr alloys. Journal of Nuclear Materials, 418(1), 261–268. Chen, C., Aubry, S., Oppelstrup, T., Arsenlis, A., and Darve, E. (2018). Fast algorithms for evaluating the stress field of dislocation lines in anisotropic elastic media. Modelling and Simulation in Materials Science and Engineering, 4, 045007. Crone, J. C., Chung, P. W., Leiter, K. W., Knap, J., Aubry, S., Hommes, G., and Arsenlis, A. (2014). A multiply parallel implementation of finite element-based discrete dislocation dynamics for arbitrary geometries. Modelling and Simulation in Materials Science and Engineering, 22(3), 035014. Devincre, B., Madec, R., Monnet, G., Queyreau, S., Gatti, R., and Kubin, L. (2011). Modeling crystal plasticity with dislocation dynamics simulations: the micromegas code. In Mechanics of Nano-Objects, Forest, S., Ponchet, A., and Thomas, A. (eds). Presses des Mines, Paris. Eshelby, J. D. (1954). Distortion of a crystal by point imperfections. Journal of Applied Physics, 25, 255. Eshelby, J. D. (1961). Elastic inclusion and inhomogeneities. In Progress in Solid Mechanics, vol. 2. Amsterdam, pp. 89–140. Giessen, E. V. D. and Needleman, A. (1995). Discrete dislocation plasticity: a simple planar model. Modelling and Simulation in Materials Science and Engineering, 3(5), 689–735. Greengard, L. and Rokhlin, V. (1987). A fast algorithm for particle simulations. Journal of Computational Physics, 73(2), 325–348. Haghighat, S. M. H. and Schaublin, R. (2010). Influence of the stress field due to pressurized nanometric He bubbles on the mobility of an edge dislocation in iron. Philosophical Magazine, 90(7–8), 1075–1100. Jamond, O., Gatti, R., Roos, A., and Devincre, B. (2016). Consistent formulation for the Discrete-Continuous Model: improving complex dislocation dynamics simulations. International Journal of Plasticity, 80, 19–37. Lehtinen, A., Granberg, F., Laurson, L., Nordlund, K., and Alava, M. J. (2016). Multiscale modeling of dislocation-precipitate interactions in Fe: from molecular dynamics to discrete dislocations. Physical Review E, 93, 013309. Lemarchand, C., Devincre, B., and Kubin, L. P. (2001). Homogenization method for a discrete-continuum simulation of dislocation dynamics. Journal of the Mechanics and Physics of Solids, 49(9), 1969–1982.

84

Mechanics and Physics of Solids at Micro- and Nano-Scales

Martinez, R. and Ghoniem, N. M. (2002). The influence of crystal surfaces on dislocation interactions in mesoscopic plasticity: a combined dislocation dynamicsfinite element approach. Computer Modeling in Engineering & Sciences, 3, 229–243. Mohles, V. (2005). Dislocation Dynamics Simulations of Particle Strengthening. John Wiley and Sons, Ltd., London, pp. 375–395. Mohles, V. and Nembach, E. (2001). The peak- and overaged states of particle strengthened materials: computer simulations. Acta Materialia, 13, 2405–2417. Monnet, G. (2006). Investigation of precipitation hardening by dislocation dynamics simulations. Philosophical Magazine, 86(36), 5927–5941. Monnet, G. (2015). Multiscale modeling of precipitation hardening: application to the Fe–Cr alloys. Acta Materialia, 95, 302–311. Monnet, G. (2018). Multiscale modeling of irradiation hardening: application to important nuclear materials. Journal of Nuclear Materials, 508, 609–627. Monnet, G., Devincre, B., and Kubin, L. (2004). Dislocation study of prismatic slip systems and their interactions in hexagonal close packed metals: application to zirconium. Acta Materialia, 52(14), 4317–4328. Monnet, G., Osetsky, Y. N., and Bacon, D. J. (2010). Mesoscale thermodynamic analysis of atomic-scale dislocation-obstacle interactions simulated by molecular dynamics. Philosophical Magazine, 90(7–8), 1001–1018. Nabarro, F. (1972). The statistical problem of hardening. Journal of the Less Common Metals, 28(2), 257–276, available at: https://linkinghub.elsevier.com/ retrieve/pii/0022508872901294. Nembach, E. H. (1996). Particle Strengthening of Metals and Alloys. John Wiley & Sons, Inc, New York. O’Day, M. P. and Curtin, W. A. (2004). A superposition framework for discrete dislocation plasticity. Journal of Applied Mechanics, 71(6), 805. Queyreau, S., Monnet, G., and Devincre, B. (2009). Slip systems interactions in α-iron determined by dislocation dynamics simulations. International Journal of Plasticity, 25(2), 361–377. Queyreau, S., Monnet, G., Wirth, B. D., and Marian, J. (2011a). Modeling the dislocation-void interaction in a dislocation dynamics simulation. MRS Online Proceedings Library, 1297, 10. Queyreau, S., Marian, J., Gilbert, M. R., and Wirth, B. D. (2011b). Edge dislocation mobilities in bcc Fe obtained by molecular dynamics. Physical Review B, 84, 064106. Queyreau, S., Marian, J., Wirth, B. D., and Arsenlis, A. (2014). Analytical integration of the forces induced by dislocations on a surface element. Modelling and Simulation in Materials Science and Engineering, 22(3), 035004.

Dislocation Dynamics Modeling of the Interaction of Dislocations with Eshelby Inclusions

85

Queyreau, S., Hoang, K., Shi, X., Aubry, S., and Arsenlis, A. (2019). Analytical integration of the tractions induced by non-singular dislocations on an arbitrary shaped triangular quadratic element, available at: https://doi.org/10.7910/ DVN/FSVNB4. Santos-Guemes, R., Manzanares, G. E., Papadimitriou, I., Segurado, J., Capolungo, L., and Lorca, J. (2018). Discrete dislocation dynamics simulations of dislocationθ′ -precipitate interaction in Al-Cu alloys. Journal of the Mechanics and Physics of Solids, 118, 228–244. Sobie, C., Bertin, N., and Capolungo, L. (2015). Analysis of obstacle hardening models using dislocation dynamics: application to irradiation-induced defects. Metallurgical and Materials Transactions A, 46, 3761–3772. Tang, M., Cai, W., Xu, G., and Bulatov, V. V. (2006). A hybrid method for computing forces on curved dislocations intersecting free surfaces in threedimensional dislocation dynamics. Modelling and Simulation in Materials Science and Engineering, 14(7), 1139–1151. Vattr´e, A. and Demkowicz, M. (2013). Determining the Burgers vectors and elastic strain energies of interface dislocation arrays using anisotropic elasticity theory. Acta Materialia, 61(14), 5172–5187, available at: http://linkinghub. elsevier.com/retrieve/pii/S1359645413003741. Vattr´e, A. and Demkowicz, M. (2015). Partitioning of elastic distortions at a semicoherent heterophase interface between anisotropic crystals. Acta Materialia, 82, 234–243, available at: http://linkinghub.elsevier.com/retrieve/pii/ S1359645414006910. Vattr´e, A., Devincre, B., Feyel, F., Gatti, R., Groh, S., Jamond, O., and Roos, A. (2014). Modelling crystal plasticity by 3D dislocation dynamics and the finite element method. Journal of the Mechanics and Physics of Solids, 63, 491–505. Vattr´e, A., Jourdan, T., Ding, H., Marinica, M.-C., and Demkowicz, M. J. (2016). Non-random walk diffusion enhances the sink strength of semicoherent interfaces. Nature Communications, 7, 10424, available at: http://www.nature.com/doifinder/10.1038/ncomms10424. Weinberger, C. R., Aubry, S., Lee, S.-W., Nix, W. D., and Cai, W. (2009). Modelling dislocations in a free-standing thin film. Modelling and Simulation in Materials Science and Engineering, 17(7), 075007. Zbib, H. M., Diaz de la Rubia, T., and Bulatov, V. (2001). A multiscale model of plasticity based on discrete dislocation dynamics. Journal of Engineering Materials and Technology, 124(1), 78–87.

 

4 Scale Transition in Finite Element Simulations of Hydrogen–Plasticity Interactions

4.1. Introduction The development of hydrogen-based energetic supply chain leads to increased interest in the phenomenon of hydrogen–material interaction, especially for metallic materials [see (Djukic et al. 2016) and (Traidia et al. 2018) for a review of hydrogen-assisted cracking models; (Ghosh et al. 2018) for a global description of these interactions for pipeline and pressure vessel; (Djukic et al. 2019) for a comprehensive review of the embrittlement mechanisms interactions in steels and iron; and (Martin et al. 2019) for a review of hydrogen-enhanced localized plasticity experimental results and the recent warning by Lynch (2019) on these mechanisms]. Aside from the experimental aspect, driven by the progress of observation and characterization devices [such as nano-scale experiments (Alvaro et al. 2015; Barnoush et al. 2010; Deng and Barnoush 2018; Müller et al. 2019); 3D hydrogen localization by atom probe (Cheng et al. 2013; Koyama et al. 2017) or neutron tomography (Griesche et al. 2014; Pfretzschner et al. 2019); or 2D mapping by Kelvin Probe Force Microscopy (Evers et al. 2013; Melitz et al. 2011)], modeling tools have made notable progress. This progress was helped, on the one hand, by the increase in computational capacities and, on the other hand, by the development or the adaptation of specific approaches at several scales [see (Barrera et al. 2018) for a review].                                         Chapter written by Yann CHARLES, Hung Tuan NGUYEN, Kevin ARDON and Monique GASPERINI.

Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

88

Mechanics and Physics of Solids at Micro- and Nano-Scales

Extensive investigations on several materials or systems have been made from very local up to meso-scale using one of the following numerical tools [see (Aubert et al. 2019) for a global picture of the multi-scale modeling approach]: – Density Functional Theory (DFT): Lu et al. (2002) investigated the effect of H atoms on pure aluminum mechanical properties, while Jiang and Carter (2004a) focused on hydrogen diffusion in Iron; Metsue et al. (2018) investigated the interactions between vacancies and hydrogen in nickel. Traction–separation laws have been derived from DFT results (Van der Ven and Ceder 2003) for various materials, such as aluminum (Ehlers et al. 2016, 2017), iron (Jiang and Carter 2004b), or nickel (Alvaro et al. 2015). – Molecular Dynamic (MD): For instance, investigations on the hydrogenassisted failure have been focused on pre-cracked nickel single crystal (Wen et al. 2004) or bicrystals (Song and Curtin 2011), or aluminum single crystal coated by alumina (Verners et al. 2015). Hydrogen–material interactions have also been studied, as in pure iron nano-pillars (Xu et al. 2017), as well as hydrogen-vacancy interplay in a tungsten material (Fu et al. 2018). A complete picture is given by Tehranchi and Curtin (2019). – Kinetic Monte Carlo (KMC): Diffusion-related material parameters might be extracted from computations, as in zircon (Zhang et al. 2017). The influence of defects on diffusion might also be accounted for in the work of Ramasubramaniam et al. (2008) for dislocation in iron or in the work of Oda et al. (2015) for vacancies in tungsten. At the component scale, a finite element (FE) provides the most popular framework for performing numerical simulation on structure in the presence of hydrogen [to reproduce hydrogen-sensitivity characterization test, as in the work of Olden et al. (2009) for SENT, Charles et al. (2012) for Disk Pressure Test, Charles et al. (2017a) for U-Bend, or Ayadi et al. (2017) for shear and tensile test coupled with blistering]. In Abaqus software, the hydrogen diffusion and trapping are usually sequentially solved, as in the work of Moriconi et al. (2014) for instance, while classical elastoplasticity is considered [however, in few recent works, gradient plasticity models have also been proposed (Martínez-Pañeda et al. 2016)]. Fully-coupled resolution scheme, like the one used in the work of Charles et al. (2019), is not that numerous in Abaqus. Recently, investigations have been made at an intermediate scale: the single-crystal one. This scale is strategically pertinent because it fills the gap

Scale Transition in Finite Element Simulations of Hydrogen–Plasticity Interactions

89

between low scales and high ones, allowing one to obtain a better picture of hydrogen effects on materials at the component scale, improving their reliability. Two main tools have been used in such an approach: dislocation dynamics [e.g. to get the influence of Hydrogen on dislocation motion in steel or iron in 2D configurations (Taketomi et al. 2013) or 3D (Gu and El-Awady 2018; Yu et al. 2019) or to investigate the hydrogen dragging by mobile dislocations (Sills and Cai 2016)], and crystal plasticity finiteelement method (CPFEM). At the polycrystal scale, few studies use crystal plasticity, most of them only deal with hydrogen transport through heterogeneous mediums. This chapter is mainly dedicated to the scale transition between hydrogen transport and plasticity-induced trapping modeling from the polycrystal scale (using CPFEM), up to the component one. At the crystal scale, mechanical properties are anisotropic, the medium being heterogeneous, while at the macro-scale, every property is isotropic and the model is homogeneous. Hydrogen transport and trapping is classically described by the diffusion formulation proposed by Sofronis and McMeeking (1989) and later improved by Krom et al. (1999) and Sofronis and McMeeking (1989) (see below section 4.2.2), applied on small-scale yielding configurations. This approach has been implemented in commercial or home-made finite element software, coupled (or not) with mechanical fields, and used in numerous studies to investigate specific features of hydrogen–material interactions in homogeneous structures, including embrittlement or interactions with thermal fields [e.g. to study the plastic strain localization during a tensile test on steel (Miresmaeili et al. 2010), to analyze the permeation test on steel samples (Legrand et al. 2012), to model or analyze crack propagation (Takayama et al. 2011) or hydrogen repartition after welding in steel pipes (Yan et al. 2014), after welding in steel pipes, or to analyze the effect of a thermomechanical field in a tungsten plasma-facing component (Benannoune et al. 2019a)]. This transport and trapping equation is also used in all of the fewer studies conducted on polycrystalline configurations; among them, three main objectives might be found – the determination of polycrystalline aggregate effective diffusion properties; – an investigation of the hydrogen-assisted grain boundary failure;

90

Mechanics and Physics of Solids at Micro- and Nano-Scales

– the determination of the plastic activities in specific grains in conjunction with experimental results. These studies account nor for hydrogen diffusion neither for hydrogen–plasticity interactions. This chapter is dedicated to the average diffusion processes in polycrystals, accounting for both mechanical fields (stress and plastic strain) and non-Fickian effects. After quickly describing the simulations that can be found in the literature, and related to hydrogen-assisted failure or plastic strain repartition in a polycrystal, a survey of the finite-element works dedicated to the determination of effective diffusion properties through polycrystals will be presented. CPFEM computations have been used to study the influence of stress and hydrogen concentration heterogeneities on failure. Rimoli and Ortiz (2010) have used a 3D pre-cracked regular polycrystal to model an AISI 4340 steel, using trap-free grain boundary diffusion and hydrogen-sensitive cohesive elements to model a transgranular crack propagation. Benedetti et al. (2018) performed similar computations on steel, without any cracks, and for a given load, Benabou (2019) for copper alloys. Yu et al. (2017) focused on the effect of grain boundary disorientation on its hydrogen-assisted failure, considering a bi-crystal made of steel with elastic anisotropy and cohesive zones for the grain boundary. Finally, Wu and Zikry (2015) conducted 2D computations, considering a bulk hydrogen trapping and transport process and intergranular embrittlement modeled by an overlapping element method. Few diffusion-free CPFEM models have been used to correlate experimental data and numerical plastic localization [e.g. in steel (Aubert et al. 2016)] or failure [e.g. in aluminum alloys (Pouillier et al. 2012)]. While claiming to work at the polycrystal scale, numerous investigations are dedicated to the determination of the effective diffusion process through stress-free heterogeneous structures; as a consequence, only a classical Fick law is used with heterogeneous diffusion coefficients. In these studies, mainly in 2D, the grain morphology might be regular or based on a Voronoi tessellation and the diffusion properties depending on the studied materials. For nickel material, grain boundaries act as diffusion shortcut, following the pioneering work of Swiler et al. (1997) and latter (Zhu et al. 2001); the grain boundary type (Legrand et al. 2013), their density (Jothi et al. 2015a), and their connectivity (Osman Hoch et al. 2015) are the main features for extracting an effective diffusion coefficient. Trapping at grain boundaries has been introduced in pure nickel (Ilin et al. 2016; Jothi et al. 2015c) or

Scale Transition in Finite Element Simulations of Hydrogen–Plasticity Interactions

91

nickel alloys (Turk et al. 2018), without considering a coupling with mechanical fields. For two-phase steel polycrystals, a diffusion shortcut mechanism has also been introduced, not coupled with mechanical fields (Yazdipour et al. 2012); such work focuses mainly on the phase morphology influence on hydrogen transport, as in the work of Sezgin et al. (2019). Similar studies have also been performed on 3D aluminum polycrystals (Lacaille et al. 2014). The impacts of mechanical fields (pressure and trapping by dislocation) on the average diffusivity are not well-studied, and most of the works do not include crystal plasticity. For instance, Olden et al. (2014) consider J2 plasticity in a two-phased steel and focused on the influence of grain morphology on diffusion and trapping kinetic, while Shibamoto et al. (2017) used a pre-computed pressure field and a regular grain shape, for the same aim. Finally, Jothi et al. (2014) only consider elastic anisotropy in nickel polycrystals. CPFEM is used in a 3D regular iron polycrystal in the work of Charles et al. (2017b) and in a 2D Voronoi steel one in the work of Hassan et al. (2018): in these two works, the influence of a predeformation on hydrogen diffusion is investigated, accounting for both pressure stress and dislocationinduced traps. Finally, Ilin et al. (2014) focused on hydrogen redistribution in a 2D steel polycrystal due to an applied load, and at the crystal scale, FE investigations have also been performed to investigate the experimentally measured diffusion anisotropy in nickel single crystal (Li et al. 2017), considering elastic anisotropy and vacancy-induced strain. Very few works have investigated the reformulation of crystal plasticity laws to account for the hydrogen effect on slips (Bal et al. 2017; Birnbaum and Sofronis 1994; Cailletaud 2009; Kumar et al. 2019; Vasios 2015), or the correspondence between the diffusion process through a polycrystal and the equivalent homogeneous medium (Charles et al. 2018; Jothi et al. 2015b). From the previous picture, it appears that CPFEM is not commonly used in computations, especially for the effective diffusion coefficient evaluation. The modeling assumptions used in this chapter are first presented, and the tools developed are required to perform multi-scale finite element computations (section 4.2). Afterward, two specific points are focused on to illustrate the consequences of scale transition in finite-element modeling while dealing with hydrogen transport and trapping:

92

Mechanics and Physics of Solids at Micro- and Nano-Scales

– In section 4.3, a reformulation of the dislocation-related trap density function is proposed for iron material to obtain the same transport kinetic through a polycrystal and the corresponding homogenous sample. – Section 4.4 is dedicated to the presentation of the model proposed by Dadfarnia et al. (2015), aiming at including hydrogen dragging by dislocation in a homogeneous medium. An adaptation of this approach at the crystal scale is proposed. The consequences at the polycrystal scale of such formalism, accounting for textures, have been investigated. 4.2. Modeling assumptions The modeling assumptions at the polycrystal scale are directly adapted from the one used at the macroscopic scale, considering few adaptations linked to the specific way plasticity occurs in crystals. 4.2.1. Crystal plasticity mechanical behavior The anisotropic elasticity is defined through Cij elastic constants. The crystal plasticity is described by a classical viscous formulation (Asaro 1983) for numerical purpose only. The slip rate

on the th slip

system, defined by its normal vector n and its slip direction m , is related to the resolved shear stress

by a power-law relationship

[4.1]

is a reference strain rate, where c is the critical resolved shear stress, and n is the strain-rate sensitivity, chosen high enough to avoid viscous effects. From slip rates on each system, the global plastic strain rate tensor is [4.2]

 

Scale Transition in Finite Element Simulations of Hydrogen–Plasticity Interactions

93

is obtained by [4.3] where is the local stress field, denotes the tensorial product, and “:” is the tensorial contraction. The hardening law is described by Peirce et al. (1982) [4.4] where h

represent the self-hardening and h

,

qh

represents the

latent one, with h0sech 2

h

h0 s

[4.5] 0

is the cumulated shear strain. h0,

s ,

and 0 are the

material parameters. No influence of hydrogen on mechanical behavior is taken into account in the present work. 4.2.2. Hydrogen transport equation The hydrogen transport equation is based on the local balance between the hydrogen concentration in normal interstitial lattice sites (NILS) and the hydrogen concentration in trapping sites (Oriani 1970) KT

L

T

1

[4.6] T

where KT represents the equilibrium constant with KT = exp( WB/RT) (WB being the trap binding energy). NILS hydrogen concentration CL relates to NILS density NL by CL = NL L, whereas trapped hydrogen density CT relates

94

Mechanics and Physics of Solids at Micro- and Nano-Scales

to trap density NT by CT = NT T; L and T are, respectively, NILS and trapping sites occupancy, assuming that L 0 for compressive stresses and ε > 0 for compressive strains. Platelets, which are live cells responsible for clot contraction, cause a pre-tension in the clots in the rarefied phase (with straight fibers). As a result, the stress–strain law in the rarefied phase of fibrin clots is modified to ΓL (σ) =

σ − σpre , EL

[5.2]

where σpre is a pre-stress and it is negative (tensile) in the experiments described earlier. In the densified phase, fibers are bent and buckled with many contact points. The number of contact points Nc per unit volume in a dense isotropic network has been studied by Toll (1999), and shown to be a function of current fiber volume fraction ϕ as: Nc =

4 2 ϕ = Cϕ2 , πd3

[5.3]

where C = πd4 3 , and d is the diameter of a fiber. An approximate expression for the increase of volume fraction ϕ with compression of a fiber network is in the form ϕ = ϕ0 /(1 − ε) (Toll 1999) , where ε is the uniaxial compressive strain. The stress–strain relation for such a dense network was proposed by Van Wyk (1946) and Toll (1999) as: ( ) σ = kEs ϕ3 − ϕ30 ,

[5.4]

where the coefficient k is determined by the material and loading conditions. Furthermore, fibrin fibers making contact and adhering to each other cause a

Compression of Fiber Networks Modeled as a Phase Transition

143

reduction in the free energy of the network. We treat each formed contact point as a bond that releases free energy Ubond , resulting in a total energy per unit deformed volume which is ∫ E = Nc Ubond + σdε. [5.5] Plugging the expression of Nc from equation [5.3] and for σ from equation [5.4] into the above equation, and then differentiating this total energy, we get a new stress–strain law in the densified phase as: ( ) 2Cϕ20 Ubond K − ∆G σnew = kEs ϕ3 − ϕ30 − − K, 3 = (1 − ε) (1 − ε)3

[5.6]

where K = kEs ϕ30 is a constant with units of stress, and ∆G = 2CUbond ϕ20 is a normalized bonding energy density. From the measurements on densified fibrin networks by Kim et al. (2014, 2015), we estimate K to be around 10 kPa, and we find values in a similar range by our fits to the experiments described above. By inverting the aforementioned stress–strain relation for the densified phase, we get √ K − ∆G . [5.7] ε = ΓH (σ) = 1 − 3 σ+K A non-dimensional parameter γ that captures the competition between adhesion of fibers and the bending elasticity of fibers may be constructed as L γ = Ubond Es I , where L is the typical length of fiber between contact points and 4

I = πd 32 is the moment of inertia of the cross-section of a typical fiber of 2 diameter d. Remembering that the fiber volume fraction scales as ϕ0 ≈ Ld 2 and the low strain Young’s modulus of the network (dominated by bending) scales as E ≈ Es ϕ20 , one can show after a short calculation that Ubond L E∆G γ≈ ≈ Es I K2

(

E Es

)3/4 .

[5.8]

5.3.2. Phase transition theory Next, we give a short summary of the Abeyaratne–Knowles theory of phase transitions in continua (Abeyaratne and Knowles 2006) for one-dimensional

144

Mechanics and Physics of Solids at Micro- and Nano-Scales

quasistatic processes. During loading, as the stress σ increases from zero, the network starts in the rarefied phase. The transition to the densified phase nucleates at a critical stress σLH < σM , where σM is the maximum stress at which the rarefied phase can exist. Similarly, during unloading, the transition to the rarefied phase nucleates at critical stress σHL > σm , where σm is the minimum stress at which the densified phase can exist and σHL < σLH . For stresses between σHL and σLH , the two phases can co-exist at a stress σ. We define the strain difference between the two phases as a transformation strain: √ K − ∆G σ − σpre γT (σ) = ΓH (σ) − ΓL (σ) = 1 − 3 − , σ+K E σm ≤ σ ≤ σM .

[5.9]

In the absence of dynamic effects due to small sample size and the very low loading rate, the assumption of quasistatic process holds and the stress is constant all over the sample. Let 0 ≤ z ≤ h be the reference coordinate along the direction of loading, and w (z, t) be the local displacement. The bottom at z = 0 is fixed, hence w (0, t) = 0 for all t. At the top, the displacement is given as w (h, t) = δ (t). Suppose for σHL ≤ σ ≤ σLH , there is a separation at z = s (t) between two parts of our continuum representing the rarefied (for z < s(t)) and densified phases (for z > s(t)). Then, from two springs in series, the displacement at the top is given by w (h, t) = δ (t) = ΓL [σ (t)] s (t) + ΓH [σ (t)] [h0 − s (t)] .

[5.10]

We have denoted the position of the phase boundary by s(t). A kinetic law is needed to describe the motion of this phase boundary s (t) (Abeyaratne and Knowles 2006)   MLH (f − fLH ), s˙ = Φ(f ) = 0,  MHL (f − fHL ),

if f > fLH , if fHL ≤ f ≤ fLH , if f < fHL ,

[5.11]

where Φ is a material property (Abeyaratne and Knowles 2006), and ∫

σ

f (σ) = σ0

γT (σ ′ ) dσ ′

[5.12]

Compression of Fiber Networks Modeled as a Phase Transition

145

is the driving force on the phase boundary (Abeyaratne and Knowles 2006). f is assumed here to be a unique function of stress, and σ0 is a Maxwell stress, at which the Helmholtz free energy density of the two phases is equal. The parameters MLH > 0 and MHL > 0 in equation [5.11] are mobilities that could be fitted to the experimental data. Φ(f ) must satisfy the dissipation inequality which requires that f s˙ ≥ 0. Also, fLH and fHL correspond, respectively, to stresses σLH and σHL which may be determined using equation [5.12]. In order to complete the formulation of the problem, we need a nucleation criterion. For loading, when the specimen is being compressed, the densified phase nucleates in the rarefied phase at stress σLH where σLH can be assumed to be where driving force f is just greater than fLH . Similarly, for unloading, the rarefied phase nucleates in the densified phase at stress σHL where driving force f is just smaller than fHL . Differentiating equation [5.10] with respect to time and eliminating s˙ (t) using equation [5.11], we get: ] [ δ ′ ′ ′ σ˙ ΓL (σ)ΓH (σ) − ΓL (σ)ΓH (σ) − γT (σ) h0 + γT (σ)

γ2 δ˙ = T Φ (σ) . h0 h0

[5.13]

This governing equation gives the response of the network undergoing phasetransition during loading and unloading. The motion of the phase boundary during loading and unloading could be different due to the difference in the mobilities MLH , MHL and nucleation values fLH , fHL .

5.3.3. Effect of liquid pumping One important feature of compressing and decompressing a fibrin clot is that it contains serum that is pumped out and then back into the network. The pumping introduces a rate-dependence into the mechanical response of the network through the well-known mechanism of poroviscoelasticity. An analytic formula for the compressive stress for a poroviscoelastic foam under compression is given as (Gibson and Ashby 1999): ( ) Cw µε˙ D 2 σli = , 1−ε l

[5.14]

where µ is the dynamic viscosity of the liquid, ε and ε˙ are the compressive strain and its rate, D is the horizontal dimension of the foam sample, and l

146

Mechanics and Physics of Solids at Micro- and Nano-Scales

is the cell edge-length of the foam. The coefficient Cw is about unity. The expression above assumes that ε is uniform through the sample, hence we will take ε = w (h0 , t) /h0 where w(h0 , t) is given by equation [5.10]. This formula gives a non-linear rate-dependent stress, which must be added to the stress due to the deformation of the fibrin fibers. If we imagine this strain-ratedependent stress as one due to a dashpot, then the dashpot is in parallel with a spring-dashpot arrangement characterizing the response of the fiber network (plus platelets). The total stress, σtot , in the network is then σtot (ε, ε) ˙ = σ(ε, ε) ˙ + σli (ε, ε). ˙

[5.15]

5.3.4. Application of phase transition model to PPP and PRP clots The stress–strain curves of PPP and PRP clots under compression can be fitted using these ideas as shown in Figure 5.4. The fitting parameters are given in Table 5.1. We use the following procedure to fit the curves. First, we fit the low strain linear response using equation [5.2] and obtain σpre and EL . Next, we consider the initial steep part of the unloading curve (line which does not include the lower plateau) and remember that this curve is given by ε = x1 ΓL (σ) + (1 − x1 )ΓH (σ),

[5.16]

where x1 is a fixed fraction of the rarefied phase with 0 < x1 < 1 and ΓH (σ) is given by equation [5.7]. Since ΓL (σ) is already known the fitting parameters are x1 , K, and ∆G. With ΓL (σ) and ΓH (σ) known, we now want to fit the plateau region of the stress strain curves using the differential equation equation [5.13]. For the upper plateau, the fitting parameters are MLH and σLH , and for the lower plateau, the fitting parameters are MHL and σHL . We perform our fits for the experiments with maximum strain ε = 0.33 on both PPP and PRP clots. Each of the plots in Figure 5.4 contain data from three compression-decompression cycles on the clots. The first cycle gives a different response from the subsequent cycles, which is captured in our model through a difference in σLH during loading, and σHL during unloading. Typically, for both PPP and PRP clots, σLH becomes smaller from the first cycle to the second. This suggests that there is rearrangement in the clot structure at network and fiber levels, perhaps due to adhesion and breakage of fibers, which makes the buckling stress for fibers smaller after the first cycle. The theoretical curves match the experimental data very well for both PPP (Figure 5.4–left) and PRP clots (Figure 5.4–right) for each cycle.

Compression of Fiber Networks Modeled as a Phase Transition

147

Figure 5.4. Fits (lines) of stress–strain curves of (a) PPP and (b) PRP clots to ε = 0.33. Top-most and bottom-most series of dots are experimental data for first cycle. The dots for the second and third cycles are so close as to not be distinguishable. The hysteresis loop of the second and third cycles falls inside the hysteresis loop of the first cycle. Reprinted from Liang et al. (2017a). For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Fitting parameters

PPP clot

PRP clot

Pre-stress σpre (kPa) Young’s modulus in the rarefied phase EL (kPa) Stress constant in densified phase K (kPa) Normalized bonding energy density ∆G (kPa) Nucleation stress σLH (kPa) in 1st cycle Nucleation stress σLH (kPa) in 2nd cycle Nucleation stress σLH (kPa) in 3rd cycle Nucleation stress σHL (kPa) in 1st cycle Nucleation stress σHL (kPa) in 2nd cycle Nucleation stress σHL (kPa) in 3rd cycle Phase boundary mobility Mlh (kPa−1 s−1 ) Phase boundary mobility Mhl (kPa−1 s−1 ) Liquid pumping coefficient Cw

−2.2 10 10.25 10 1.5 1.4 1.6 −0.5 −0.6 −0.4 0.06 0.04 1

−2.2 10 10.25 10 2.1 1.3 1.3 −0.7 −0.7 −0.7 0.06 0.04 1

Table 5.1. Fitting parameters for compression experiments of fibrin clots

148

Mechanics and Physics of Solids at Micro- and Nano-Scales

5.3.5. Predictive capability of our model Liang et al. used the parameters extracted from these fits to predict the stress–strain curve of PPP and PRP clots compressed to ε = 0.5. The results are shown for PPP clots in Figure 5.5 (left) and PRP clots in Figure 5.5 (right). The agreement between the experimental data (dots) and the theoretical predictions (lines) is excellent. This shows that for low strain rates the assumption of a sharp transition front (as in the Abeyaratne–Knowles theory) is reasonable for fibrin clots. Note also that the upper plateau in each stress– strain curve is upward sloping in Figures 5.4 and 5.5. This is due to the effects of liquid pumping; if there was no fluid drainage from these clots, then the plateaus would have been relatively flat.

Figure 5.5. Prediction (solid lines) for (a) PPP and (b) PRP clots compressed to ε = 0.5. Similar to Figure 5.4, the hysteresis loops of the second and third cycle are almost indistinguishable and fall inside the hysteresis loop of the first cycle. Reprinted from Liang et al. (2017a). For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

5.3.6. Application of phase transition model to CNT networks For the CNT networks, the experiment is performed at a constant rate δ˙ < 0 during loading. Initially, the entire continuum is in the rarefied phase. As the compressive strain ε increases, the stress σ increases linearly and reaches the critical value σLH ; this is when a phase boundary nucleates at the top of the CNT pillar (x = h0 ), with the densified phase above and the rarefied phase below. The stress in the continuum is now governed by equation [5.13] with initial condition σ = σLH given by the nucleation criterion. The phase

Compression of Fiber Networks Modeled as a Phase Transition

149

boundary moves through the continuum and converts all the material into the densified phase. Once the phase boundary has reached x = 0, equation [5.13], the nucleation criterion and the kinetic relation are no longer required. The stress is the determined by the constitutive law ΓH (σ) since all the material is in the densified phase. When we unload δ˙ > 0, the stress declines along the curve ΓH (σ) until a critical value σHL is reached at which a phase boundary nucleates at x = 0. Then, σ(t) is again governed by equation [5.13] with initial condition σ = σHL . This differential equation remains relevant until the phase boundary has traversed the full length of the specimen reaching x = h0 . After this, the stress follows the curve ΓL (σ) in the rarefied phase. We have fitted the experimental data in Figure 5.3 from the measured loading and unloading response of three different samples using the methods described earlier. In Figure 5.3, the experimental data are represented by discrete markers and the model fit is shown as a continuous line. There are separate lines corresponding to the stress–strain curve of the rarefied phase during loading, mixture of rarefied and densified phases during loading (top plateau), densified phase during unloading, and mixture of rarefied and densified phase during unloading (bottom plateau). Note that around a strain of 0.8 there is a difference between the loading and unloading curves (for ˙ 0 for the densified phase) obtained from the experiments. The strain rate δ/h these experiments was 0.001 s−1 . The parameters obtained from the fits are summarized in Table 5.2. Our phase transitions model captures the main features of the stress–strain curve quite well. The fitting parameters are E of the rarefied phase, two constants K and ∆G of the densified phase (recall that ∆G captures the release of free energy per adhesive bond and K is a modulus with units of stress for the densified phase), σLH and MLH for the upper plateau, and σHL and MHL for the lower plateau. There is no need for a pre-stress σpre or a fluid pumping parameter Cw for fitting the CNT experiments. The theoretical lines fall on top of the experimental data for the linear elastic response in the rarefied phase and the non-linearly elastic response in the densified phase. The plateaus in loading/unloading are also captured except for the stress-jumps seen in the loading plateau which represent nucleation events [see Liang et al. (2017b) for analysis of these nucleation events]. We note that Young’s modulus in the rarefied phase of the bare CNT pillars is the smallest. Young’s modulus in the rarefied phase increases as the thickness of the ALD alumina coating increases. Similarly, the stress at which the densified phase nucleates

150

Mechanics and Physics of Solids at Micro- and Nano-Scales

is also lowest for the bare CNTs, and it increases as the thickness of the ALD layers increases. This is expected because increasing the thickness of the CNTs increases their stiffness, causing an increase in Young’s modulus in the rarefied phase. The densified phase is nucleated when the fibers buckle. The critical stress σLH is determined for foams from a knowledge of the buckling load of single fibers and the density of the network [see, for example, Cao et al. 2005]. If the single fiber bending stiffness is higher due to increased thickness of the ALD layers, then σLH also must increase. This is exactly what we see in fitting our model to the experimental data. Fitting parameters

Bare CNT ALD 5 cycle ALD 10 cycle

Pre-stress σpre (kPa) Young’s modulus in the rarefied phase EL (MPa) Stress constant in densified phase K (MPa) Normalized bonding energy density ∆G (MPa) Nucleation stress σLH (MPa) Nucleation stress σHL (MPa) Phase boundary mobility Mlh (MPa−1 s−1 ) Phase boundary mobility Mhl (MPa−1 s−1 ) Liquid pumping coefficient Cw

0 6

0 8

0 20

0.67

0.61

1.37

0.66

0.60

1.36

1 0 0.35

2 0.2 0.5

3.7 0.5 0.09

0.3

0.3

0.12

0

0

0

Table 5.2. Fitting parameters for compression experiments on CNT forests

The mobility parameters MLH and MHL have not been measured previously in the experiments. However, we can predict the stress–strain ˙ response in loading/unloading of the same samples at different strain rates δ. −1 We have done this exercise for two other strain rates (0.01 and 0.1 s ) with the parameters shown in Table 5.2. The corresponding experimental stress– strain curves for these strain rates are plotted together with the theoretical predictions in Figure 5.6. Remarkably, the agreement between theory and experiment indicates that the parameters we have obtained from fitting one set of experimental data are useful in describing the constitutive response of

Compression of Fiber Networks Modeled as a Phase Transition

151

the material at other strain rates as well. In particular, the nucleation stresses and our choice of the kinetic law gives a small strain-rate dependence of the hysteresis which is consistent with earlier experiments (Pathak et al. 2012; Raney et al. 2013).

Figure 5.6. Experimental data for compression stress–strain curves for three different strain rates 0.1, 0.01, and 0.001 s−1 (solid lines) for (a) Bare CNTs, (b) CNTs with 5 ALD cycles, and (c) CNTs with 10 ALD cycles, respectively. Dashed lines are curves predicted by our phase transition model with parameters given in Table 5.2. Reprinted from Liang et al. (2017b). For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

5.4. Conclusion In this chapter, we have summarized joint experimental and theoretical analyses of two types of fiber networks, namely, fibrin networks derived from blood clots and alumina-coated carbon nanotube forests. Although the origins of these networks are very different, there are many similarities in their compression behavior. First, the stress–strain response of both types of network shows hysteresis under repeated cycles of compression and decompression. The first cycle for both networks is different from the subsequent cycles because changes occur in the networks during the first cycle (Liang et al. 2017a; Pathak et al. 2012). These changes include damage to fibers, bundling of fibers due to adhesion, and an increase in pore size due to fiber bundling. These irreversible changes occur in the early cycles after which the compression response of the networks becomes repeatable. Second, there is coexistence of rarefied and densified phases in both types of fiber network over a range of stresses. In the rarefied phase, the fibers are mostly straight and they deform by bending and rotation in response to compressive (and tensile) loads. The stress–strain response of the rarefied phase is linear and Young’s modulus of the network can be written in terms of the volume fraction of fibers

152

Mechanics and Physics of Solids at Micro- and Nano-Scales

and the fiber Young’s modulus. In the densified phase, the fibers have buckled and a large number of contacts have been formed. The stress rises steeply for small changes in strain in this phase. The stress–strain curve for this phase can be modeled after the work of Van Wyk (1946) and Toll (1999) and it takes a simple form which is a function of the current fiber volume fraction (which is larger than the fiber volume fraction at zero stress), Young’s modulus of the fibers, and the energy of adhesion between the fibers. These phases coexist in the plateau regions of the cyclic stress–strain curves; as compressive strain increases the volume fraction of fibers in the densified phase increases and that in the rarefied phase decreases. Third, both types of fiber network exhibit propagating interfaces separating the rarefied and densified phases which move as compressive strain is increased. The fibrin clots showed one interface that nucleated at the top plate and moved downward as compression proceeded. The fiber network above the interface was in the densified phase and that below it was in the rarefied phase. In carbon nanotube forests, an interface nucleated at the top plate and moved downward as compressive strain increased, but there were nucleation events at other positions which caused sudden drops in the measured stress at the upper plateau of the stress–strain curve. In fibrin clots, the width of the interface was higher at higher strain rates. The interface was relatively sharp in fibrin clots compressed at low strain rates of 10 µm/min for a several hundred µm thick clot. In the carbon nanotube forests discussed in this chapter, the interfaces were shown to be sharp using digital image correlation techniques (Liang et al. 2017b). There is a competition between fiber-to-fiber adhesion and fiber bending elasticity in both networks which can be captured using the non-dimensional parameter γ. Using Es = 15 MPa for fibrin (Brown et al. 2009), we find that γ ≈ 4 × 10−3 for the fibrin networks in the work of Liang et al. (2017a); using Es ≈ 1000 GPa for carbon nanotubes (Cao et al. 2005), we find that γ ≈ 2 × 10−3 for the CNT forests in the work of Liang et al. (2017b). Thus, even though the materials and fiber geometries in these two types of networks are different, their overall qualitative response is similar because γ is the same in both types of networks. The plateaus in the theoretical stress–strain curves are flat in the case of CNT forests which contain air and upward sloping in the case of fibrin clots because poroelastic effects due to fluid drainage make a significant contribution in fibrin clots which contain water (whose viscosity is much larger than that of air). A hysteretic stress–strain response to cyclic loading and the propagation of an interface separating rarefied and densified phases of the fiber networks are

Compression of Fiber Networks Modeled as a Phase Transition

153

features that are typical of phase changing materials. For this reason, we have used a continuum theory for the evolution of phase transitions to describe the responses to cyclic compression of fibrin clots and carbon nanotube forests. The main inputs to this theory are the stress–strain responses of the rarefied and densified phases which can be measured in experiments and fitted with known constitutive relations. Another important input to the theory is a kinetic law governing the motion of the interface, or phase boundary, separating the rarefied and densified phases of the network. This kinetic relation is difficult to measure, but a hysteretic stress–strain response even at low strain rates suggests that the kinetic law should be of the “stick-slip” type (Abeyaratne and Knowles 2006). This is why we assume a simple stick-slip type kinetic law and fit the parameters in it to the plateau regions of the stress–strain curves of the two types of fiber network. We show that the fitted parameters can predict the hysteretic stress–strain curves at other strain rates. Although we concern ourselves in this chapter with fibrin networks and carbon nanotube forests, both at the micrometer scale in height, the characteristics described here are present in the compression response of other biological and non-biological materials. In particular, compressed collagen networks (Novak et al. 2016) seem to show a propagating interface similar to that seen in fibrin clots. Glass fiber networks in compression seem to follow closely the stress–strain law used for the densified phase in this chapter (Toll and Manson 1995). Cellulose networks under compression show a hysteretic stress–strain response (Paunonen et al. 2018). Thus, the features of the mechanics of fiber networks discussed in this chapter may be general, and the continuum phase transition model used here may be applied to other network materials too, at least in the quasistatic one-dimensional setting illustrated here. In the future, the phase transition model described here could be used to understand structure property relations of fiber networks, and it could help design fiber network materials (both biological and non-biological) with tailored mechanical properties.

5.5. References Abeyaratne, R. and Knowles, J. K. (2006). Evolution of Phase Transitions: A Continuum Theory. Cambridge University Press, Cambridge, UK. Boal D. (2002). Mechanics of the Cell. Cambridge University Press, Cambridge, UK. Brieland-Shoultz, A., Tawfick, S., Park, S. J., Bedewy, M., Maschmann, M. R., Baur, J. W., and Hart, A. J. (2014). Scaling the stiffness, strength, and toughness of

154

Mechanics and Physics of Solids at Micro- and Nano-Scales

ceramic coated nanotube foams into the structural regime. Advanced Functional Materials, 24(36), 5728–5735. Brown A. E. X., Litvinov, R. I., Discher, D. E., Purohit, P. K., and Weisel, J. W. (2009). Multiscale mechanics of fibrin polymer: gel stretching with protein unfolding and loss of water. Science, 325(5941), 741–744. Cao, A., Dickrell, P. L., Sawyer, W. G., Ghasemi-Nejhad, M. N., and Ajayan, P. M. (2005). Super-compressible foamlike carbon nanotube films. Science, 310(5752), 1307–1310. Ding, Y., Hou, H., Zhao, Y., Zhu, Z., and Fong, H. (2016). Electrospun polyimide nanofibers and their applications. Progress in Polymer Science, 61, 67–103. Gibson, L. J. and Ashby, M. F. (1999). Cellular Solids: Structure and Properties. Cambridge University Press, New York, USA. Hutchens, S. B., Needleman, A., and Greer, J. R. (2011). Analysis of uniaxial compression of vertically aligned carbon nanotubes. Journal of the Mechanics and Physics of Solids, 59(10), 2227–2237. Islam, M. R., Tudryn, G., Bucinell, R., Schadler, L., and Picu, R. C. (2017). Morphology and mechanics of fungal mycelium. Scientific Reports, 7, Article 13070. Janmey, P. A., Winer, J. P., and Weisel, J. W. (2009). Fibrin gels and their clinical and bioengineering applications. Journal of the Royal Society Interface, 6(30), 1–10. Kim, O. V., Liang, X., Litvinov, R. I., Weisel, J. W., Alber, M. S., and Purohit, P. K. (2016). Foam-like compression behavior of fibrin networks. Biomechanics and Modeling in Mechanobiology, 15, 213–228. Kim, O. V., Litvinov, R. I., Weisel, J. W., and Alber, M. S. (2014). Structural basis for the nonlinear mechanics of fibrin networks under compression. Biomaterials, 35(25), 6739–6749. Liang, X., Chernysh, I., Purohit, P. K., and Weisel, J. W. (2017a). Phase transitions during compression and decompression of clots from platelet-poor plasma, plateletrich plasma and whole blood. Acta Bioamterialia, 60, 275–290. Liang, X., Shin, J., Magagnosc, D., Jiang, Y., Park, S. J., Hart, A. J., Turner, K., Gianola, D. S., and Purohit, P. K. (2017b). Compression and recovery of carbon nanotube forests described as a phase transition. International Journal of Solids and Structures, 122–123, 196–209. Magagnosc, D. J., Ehrbar, R., Kumar, G., He, M. R., Schroers, J., and Gianola, D. S. (2013). Tunable tensile ductility in metallic glasses. Scientific Reports, 3, Article 1096. Mauseth, J. D. (2016). Botany: An Introduction to Plant Biology. Jones & Bartlett Learning, Burlington, MA, USA.

Compression of Fiber Networks Modeled as a Phase Transition

155

Novak, T., Seelbinder, B., Twitchell, C. M., van Donkellar, C. C., Voytik-Harbin, S. L., and Neu, C. P. (2016). Mechanisms and microenvironment investigation of cellularized high density gradient collagen matrices via densification. Advanced Functional Materials, 26, 2617–2628. Park, S. J., Schmidt, A. J., Bedewy, M., and Hart, A. J. (2013). Measurement of carbon nanotube microstructure relative density by optical attenuation and observation of size-dependent variations. Physical Chemistry Chemical Physics, 15(27), 11511–11519. Pathak, S., Lim, E. J., Pour Shahid Saeed Abadi, P., Graham, S., Cola, B. A., and Greer, J. R. (2012). Higher recovery and better energy dissipation at faster strain rates in carbon nanotube bundles: an in-situ study. ACS Nano, 6(3), 2189–2197. Paunonen, S., Timofeev, O., Torvinen, K., Turpeinen, T., and Ketoja, J. A. (2018). Improving compression recovery of foam-formed fiber materials. BioResources, 13(2), 4058–4074. Picu, R. C. (2012). Mechanical behavior of non-bonded fiber networks in compression. Procedia IUTAM, 3, 91–99. Purohit, P. K., Litvinov, R. I., Brown, A. E. X., Discher, D. E., and Weisel, J. W. (2011). Protein unfolding accounts for the unusual mechanical behavior of fibrin networks. Acta Biomaterialia, 7(6), 2374–2383. Raney, J. R., Fraternali, F., and Daraio, C. (2013). Rate-independent dissipation and loading direction effects in compressed carbon nanotube arrays. Nanotechnology, 24(25), 255707. Tawfick, S., O’brien, K., and Hart, A. J. (2009). Flexible high conductivity carbon nanotube interconnects made by rolling and printing. Small, 5(21), 2467–2473. Toll, S. (1998). Packing mechanics of fiber reinforcements. Polymer Engineering Science, 38(8), 1337–1350. Toll, S. and Manson, J.-A. E. (1995). Elastic compression of a fiber network. Journal of Applied Mechanics, 62, 223–226. Van Wyk, C. M. (1946). Note on the compressibility of wool. Journal of the Textile Institute Transactions, 37(12), T285–T292. Weisel, J. W. (2004). The mechanical properties of fibrin for basic scientists and clinicians. Biophysical Chemistry, 112(2), 267–276. Zbib, A. A., Mesarovic, S. D., Lilleodden, E. T., McClain, D., Jiao, J., and Bahr, D. F. (2008). The coordinated buckling of carbon nanotube turfs under uniform compression. Nanotechnology, 19(17), 175704. Zhao, X., Strickland, D. J., Derlet, P. M., He, M. R., Cheng, Y. J., Pu, J., and Gianola, D. S. (2015). In situ measurements of a homogeneous to heterogeneous transition in the plastic response of ion-irradiated < 111 > Ni microspecimens. Acta Materialia, 88, 121–135.

 

6 Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

6.1. Introduction Many materials contain fibers of various types and with various organizations. The most obvious examples are fiber composites, in which fibers are added to a matrix to increase its stiffness, strength, and toughness. In such cases, fibers may be short, randomly oriented and distributed in the matrix, or may be long and woven. Most high-performance composites are of the second type, with the fibers forming a fabric which is then impregnated with matrix material (usually epoxy). In other applications, such as in the case of materials used for insulation and filtration, fibers are not crosslinked, are not embedded in a matrix, and form a random network in 3D. Yet, a third category is formed by cases in which fibers are connected in a network that spans the entire domain, and which may or may not be embedded in a matrix. Paper, gels, and rubber are examples of such materials. Most biological materials are of a fibrous type. This likely represents an evolutionary adaptation since large volumes can be spanned using fibers at very low-volume fractions of actual material. In addition, fibrous structures have low modulus, but high strength and toughness. Connective tissue is composed of collagen fibrils organized in complex random networks (Parry and Craig 1984). Blood vessels are reinforced with collagen (Ottani et al. 2001) and elastin fibers (Green et al. 2014). Various membranes in the body, such as the diaphragm, the liver capsule, and the amnion, have a quasi-2D network (a fiber mat) of collagen fibers as their main structural component.                                         Chapter written by Catalin R. PICU and Vineet NEGI.

Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

158

Mechanics and Physics of Solids at Micro- and Nano-Scales

The component of the skin that provides mechanical strength – the dermis – is also a complex collagen network (Ottani et al. 2001). Fibers assemble in networks due to either crosslinking or surface interactions between fibers. Crosslinks are bonds that bind two fibers at specific sites. In biological networks, crosslinking is produced by specialized molecules such as glycosaminoglycans (Robins 2007). In the case of artificial fibers having a larger diameter, crosslinking can be introduced by literally merging the volumes of two fibers forcing them to interpenetrate. An example of this type is provided by the process of thermal bonding of polymeric fibers in non-wovens (Dharmadhikary et al. 1995). From a mechanistic perspective, crosslinking introduces kinematic constraints that force the two bonded fibers to move and/or rotate together at the site of the bond. From this point of view, crosslinks can be classified as “welds” that force kinematic compatibility both with respect to the displacements and rotations, “pin-joints” which enforce only displacement compatibility, and “rotating joints” that form between two continuous fibers in contact and allow their relative rotation, while restricting their relative displacements at the contact point. Surface interactions between fibers are of various types. Surface adhesion is due to short-range interactions which may be of van der Waals type or may be produced by surface groups forming transient bonding such as hydrogen bonding (Benítez and Walther 2017). Adhesion is generally too short-ranged to be able to bring together fibers separated by distances larger than few nanometers. However, if fibers are brought in contact by other means, adhesion may effectively stabilize the resulting network. Capillarity may also produce fiber bundling and organization. Capillarity in the context of the interaction of a fluid with a fibrous structure may efficiently organize the fibers due to the long-range action of capillary forces (De Volder and Hart 2013). Capillary forces produce a variety of effects in a soft matter, some of these being useful in various applications, as reviewed by Style et al. (2016). Flocculation takes place in fiber suspensions due to other types of interactions, among which osmotic and entropic forces are important (Tempel et al. 1996; Yunoki et al. 2015). Hydrodynamic forces may also produce organization of the solid phase in suspensions. Surface interactions are usually most effective in organizing the fibrous structure when the fibers have a small diameter. In this case, the relatively weak surface forces can overcome the bending energy penalty associated with bundling and organization of a set of initially randomly-oriented and

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

159

distributed straight fibers. The self-organization process is pronounced in the case of molecular filaments. Most biological materials are actually composed of bundles of proto-filaments which are assembled by surface interactions. Tropocollagen molecules of well-defined length self-organizes in fibrils which are stabilized by enzymatic bonds (Reznikov et al. 2016; Snedeker and Gautieri 2014). The resulting fibrils organize into collagen fibers which form random networks, as in the membranes and cartilage, or preferentially oriented structures, as in tendon (Franchi et al. 2007; Ottani et al. 2001). Fibrin, another type of filament with plaque formation and wound healing properties (Piechocka et al. 2010), and spider silk (van Beek et al. 2002) have a similar multiscale structure, containing at the small scales proto-filaments of the respective biomacromolecule, which organize into larger-scale fibrils and fibers. While the mechanical behavior of fiber networks with and without crosslinks has been studied extensively (Broedersz and Mackintosh 2014; Durville 2005; Kulachenko and Uesaka 2012; Liu and Dzenis 2016; Picu 2011; Rodney et al. 2005), relatively little attention was devoted to the adhesion-driven self-organization of networks. Li and Kröger (2012a, 2012b) numerically studied the structure and mechanical response of buckypaper. They observe intense bundling and conclude that, for CNTs with weaker adhesion, the structure is controlled by entanglements, while CNTs with stronger adhesion bundle intensely. The pore size could be controlled from 7 to 50 nm by increasing the bending stiffness of the filaments. Volkov and Zhigilei (2010) also simulated assemblies of CNTs and concluded that the resulting structures can be stabilized, provided the CNT length is larger than a threshold and CNT bending-buckling is considered in the model. A demonstration of the effect of adhesion between filaments on the overall mechanical behavior of the network was provided by Xu et al. (2010) using random networks of long non-crosslinked CNTs. They observe strong energy dissipation under cyclic loading in such structures due to the bundling/unbundling of filaments. Since the system is athermal, the measured system-scale storage and loss moduli are temperature independent in a broad range of temperatures. Simulations reproducing this result were presented by Li and Kröger (2012c). The present review is based on these works and on the recent articles (Negi and Picu 2019; Picu and Sengab 2018; Sengab and Picu 2018) which address the problem of networks self-organization in both crosslinked (Negi and Picu 2019) and non-crosslinked networks (Picu and Sengab 2018; Sengab and Picu 2018). It was observed that in the absence of crosslinks,

160

Mechanics and Physics of Solids at Micro- and Nano-Scales

adhesive interactions lead to the formation of a new network structure in which the connectors are fiber bundles and the crosslinks are specific triangular constructs of fiber bundles which stabilize the network of bundles. This is discussed in sections 6.3 and 6.4 of this chapter. The case with crosslinks is discussed in sections 6.5 and 6.6. An isostatic crosslinked network of fibers shrinks under the action of adhesive forces. The degree of shrinkage depends on the strength of adhesion. If the network is sub-isostatic, the shrinkage is more pronounced. The mechanical behavior of such networks is controlled in part by the adhesive interactions and by the underlying crosslinked network structure. These aspects are details in section 6.6. The results presented in Negi and Picu (2019), Picu and Sengab (2018), and Sengab and Picu (2018) are obtained with 2D (in the case of crosslinked networks) and quasi-2D (in the case of non-crosslinked networks) models and are expressed in terms of non-dimensional parameters to render them generally applicable. The resulting phenomenology is identical to that observed in 3D (Li and Kröger 2012a, 2012b, 2012c). Experimental work to date documented extensively bundling due to adhesion and other surface forces (De Volder and Hart 2013; Lieleg et al. 2007; Linares et al. 2009; Liu et al. 1998; Streichfuss et al. 2011), and the formation of networks of bundles. However, a quantitative comparison with theoretical results is not possible at this time due to the lack of dedicated experiments and direct measurements of the relevant system parameters. 6.2. Mechanics in the presence of adhesion In order to clarify the key physical processes and identify the essential parameters of the problem, it is useful to consider first simple structures constructed from several fibers. To this end, we discuss in this section two cases: two pin-jointed fibers interacting adhesively, and a self-equilibrated triangular structure of fiber bundles which emerges upon the self-organization of non-crosslinked networks under the action of inter-fiber adhesion. 6.2.1. The adhesive interaction of two fibers Consider fibers of identical diameter, , of identical length, L, and made from the same linear elastic material characterized by Young’s modulus, . The fibers are large enough to be considered athermal and hence behave and , mechanically as beams, with axial and bending rigidities

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

161

where and are the area and the moment of inertia of the fiber crosssection. Adhesion is defined by parameter , which represents the energy gain per unit length of contact when two fibers are brought together. Note that has units of N and is related to the specific work of adhesion, , and to the elasticity of the two contacting fibers. The Johnson–Kendall–Roberts / / , such that a softer (JKR) (Johnson et al. 1971) theory predicts ~ fiber material leads to stronger adhesive forces (larger ) for given . Figure 6.1 shows two initial configurations [Figures 6.1(a) and 6.1(c)] and their corresponding final states after the fibers are allowed to stick. In the case of Figure 6.1(a), the crosslink at B is a pin-joint which allows free rotation, and the motion of ends A and C is constrained in the horizontal direction. The fibers form a common segment of length s. The length of this segment can be evaluated analytically and is given by: s L

cos

2

9

Ef If L2

1/ 4

sin 2

2

[6.1]

The solution depends uniquely on a non-dimensional material parameter: L2 , Ef I f

[6.2]

which represents the interplay of adhesion and bending resistance. This ⁄ parameter can be re-arranged as Ψ , where the characteristic ⁄ has been defined in the literature as the length elastocapillarity length (Bico et al. 2004). Hence, Ψ represents the ratio of a characteristic geometric length scale of the problem and the material parameter . It is important to observe that ⁄ (equation [6.1]) decreases rapidly as the fiber diameter increases at constant since ~ . Hence, for a given , fibers of nanoscale diameter are more susceptible to form bundles than fibers of larger diameter. The total energy, computed as the difference between the bending and adhesion energies, decreases continuously from 0 in the state in Figure 6.1(a) to a minimum in the state of Figure 6.1(b). Hence, no activation is needed to initiate the bundling process.

162

Mechanics and Physics of Solids at Micro- and Nano-Scales

The case shown in Figure 6.1(c) is different since joint B is a weld that constrains angle α. Adhesion stabilizes the structure in Figure 6.1(d). However, a large energy barrier associated with bending of the two fibers must be overcome in order to bring points A and C together such that adhesion may become active. Therefore, the effect of adhesion is expected to be weak in crosslinked networks in which crosslinks are rigid welds. Filaments have to be free to rotate so as to enable the adhesion mechanism to operate. Even in such cases, the kinematics of filaments is important and may introduce energy barriers that require mechanical activation.

Figure 6.1. Evolution of two configurations under the action of adhesion. (a) and (b) show a pin-jointed structure, while in (c) and (d), the fibers are welded at B and are free at A and C. (a) and (c) show the initial state, and (b) and (d) show the respective final states

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

163

6.2.2. Triangle of fiber bundles Consider now the triangular structure of fiber bundles shown in Figure 6.2. As discussed in section 6.3, this structure emerges upon the relaxation of a non-crosslinked network of fibers with inter-fiber adhesion. All nodes of the resulting network of bundles are of this type. Here, we show that the structure is self-equilibrated and define some of its geometric characteristics. Each branch of this structure is a bundle of parallel fibers much longer than any dimension shown in Figure 6.2. The number of fibers in each bundle segment is defined by . External bundles AA’, BB’, and CC’ are of size , , and and form angles , , and . The sub-bundles , , and , and the obvious connecting nodes A, B, and C are of size , , and conservation conditions hold.

Figure 6.2. Parameters defining a triangle of fiber bundles

Bundles AB, BC, and AC forming the triangle are loaded in pure bending and hence are arcs of the circles of radii , , and . Since these circles must be tangent to each other at A, B, and C, segments OA, OB, and OC are also of equal length, . If the incoming bundles AA’, BB’, and CC’ are straight, i.e. the entire bending energy is concentrated in the triangle, the

164

Mechanics and Physics of Solids at Micro- and Nano-Scales

bending moments loading each of the three edges of the triangle are equal, . The equilibrium configuration of the structure results by minimizing the total energy, for the given bundle sizes and set of angles, relative to the size of the triangle. The bending moment in each edge of the triangle results: M tr

2 E f I f L2

B

[6.3]

where B depends only on the bundle sizes and angles B

n n (n ) tan k 1,3 ik c k

k

2

nik nc (nik )(

k)

: [6.4]

Figure 6.3. Probability distribution function for angles and defining the geometry of a minimum energy triangle (Figure 6.2) for a broad range of bundle sizes, , , and . The figure shows that triangles with very different values for the close to 120°, three bundle sizes relax into the symmetric state with , , and i.e. the network of bundles tends to become a honeycomb network

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

165

It is of interest to enquire how such a triangle evolves if it is free to adjust and , 1,3, the three angles while keeping the bundle sizes, (with and being the total constant. The total energy bending and adhesive energies of the triangle) is minimized relative to the 1,3 while imposing the condition 2 . three angles , This procedure is applied to a large number of configurations with bundle sampled from a discrete, binomial distribution of set sizes , , and mean and broad and adjustable variance. Each configuration minimizing the triangle energy for each set of is characterized by angles . Figure 6.3 and . It shows the resulting probability distribution function for angles is seen that although the distribution of bundle sizes is broad and the three bundles forming a triangle are not of the same size ( are different), the triangle configuration that minimizes the total energy is close to being equilateral, with 120°. This analysis implies that the honeycomb structure is the attractor for the evolution of networks of bundles. However, realistic networks of bundles are topologically trapped in one of the energy minima of the configurational phase space, possibly in the vicinity of the honeycomb structure. 6.3. Structure of non-crosslinked networks with inter-fiber adhesion In this section, we consider assemblies of athermal fibers which are not crosslinked but interact adhesively. This problem has been studied in the context of carbon nanotube (CNT) structures and buckypaper by Picu and Sengab (2018), Sengab and Picu (2018), Volkov and Zhigilei (2010), and Xie et al. (2011). CNTs have very strong adhesion and, during sufficient proximity, self-assemble into bundles (Li and Kröger 2012b; Liu et al. 1998; Sengab and Picu 2018; Volkov and Zhigilei 2010). CNTs are grown in a furnace bundle while floating, before being deposited in the mat. Once deposited, further adhesion-driven self-organization of the mat is possible, provided the level of friction with the background is not too high. The resulting structure is a network of CNT bundles. This section (and Picu and Sengab 2018; Sengab and Picu 2018) discusses the geometric features of such a network. The strong bundling tendency of CNTs is also important for the formation of yarns and ropes of CNTs. Significant interest has been devoted over the last decade to the development of CNT yarns of strength

166

Mechanics and Physics of Solids at Micro- and Nano-Scales

comparable with that of carbon fibers, for use in structural applications (Barber et al. 2005; Jeon et al. 2017; Mirzaeifar et al. 2015; Zhang et al. 2004). Several processes allowing the production of yarns of this type have been developed. Some of these start with a mat of CNT bundles and require stretching the buckypaper uniaxially to produce bundle alignment. This usually requires using a plasticizing embedding, which may be, for example, polymeric. In the absence of a plasticizer, the buckypaper fractures before sufficient alignment results due to frictional forces between CNTs. However, if bundles can be sufficiently drawn and aligned, an adhesion-stabilized yarn forms despite the fact that its internal structure is not quite perfect and includes folded and/or mispacked CNTs or CNT sub-bundles (Jeon et al. 2017; Lu et al. 2013; Miao et al. 2010).

Figure 6.4. Self-organization of a mat of fibers interacting adhesively

The concept of self-organization of a fibrous structure under the action of surface interactions is shown in Figure 6.4. Consider a network of fibers deposited on a substrate and forming a quasi-2D mat. The structure is threedimensional, but fibers are placed on top of each other and are preferentially oriented in the plane of the mat. It is not particularly important if the fibers are straight or crimped in the initial state. If friction is low (as is the case with CNTs), and if some degree of activation is provided, either by mechanical means or by capillarity, the fibers begin to assemble under the action of adhesion. The cellular structure, shown in the right panel of Figure 6.4, is a network of fiber bundles. The crosslinks of this network are triangles of bundles such as that shown in Figure 6.2. As discussed in the following, these triangles stabilize the network of bundles.

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

167

It is of interest to enquire how the parameters of the system influence the structure of the network of bundles in Figure 6.4. The system parameters are the fiber length, , the initial network density, defined as the total length of and , and fiber per unit area of the mat, , fiber properties, such as the strength of adhesion represented by . A dimensional analysis (Sengab and Picu 2018) indicates that there are only two important non-dimensional groups in this problem: and . The analysis presented by Picu and Sengab (2018) and Sengab and Picu (2018) leads to a phase diagram in the space of these two system parameters indicating the three possible states of the network, Figure 6.5.

Figure 6.5. Phase diagram in the field of non-dimensional network parameters and indicating the stable states of the system

The figure indicates that for small , 5.71, the initial set of fibers do not form a network. Such initial structures are too sparse to percolate [the limit value of 5.71 represents the transport percolation threshold of 2D fiber networks derived by Stauffer and Bunde (1994) and Wilhelm and Frey (2003)] and hence the problem addressed here is not defined. For larger , initial fiber networks form and three possible final states exist, function of the non-dimensional parameter Ψ. Small values of Ψ correspond to weak adhesion. Specifically, small Ψ results either when is small or when the fibers are stiff in bending, i.e.

168

Mechanics and Physics of Solids at Micro- and Nano-Scales

is large. Under such conditions, the structure does not evolve and remains locked in the initial state. For large values of Ψ, the structure of fibers evolves. The boundary between the domains of locked and evolving , which structures (AC in Figure 6.5) is described by the equation Ψ~ is derived by Sengab and Picu (2018). This boundary also depends on interfiber friction (Sengab and Picu 2018). With nanofibers, friction is usually not Coulombic (friction force proportional to the normal force on the contact), rather a situation in which the resistant force is constant or scales weakly with the relative velocity of the two surfaces (viscous) is more probable. If friction takes place at the contacts between fibers, boundary AC moves up, increasing the range of locked structures. We conclude that, in order for the structure to evolve, the magnitude of adhesion should be large enough, fibers should be thin and flexible such that is low, and friction should be relatively weak. The density and the associated mean segment length between the contacts should also be low. These conditions are expected to be fulfilled in the case of nanofilaments, such as CNTs, collagen, and fibrin. Network structures evolving under the action of adhesion may end up disintegrating or forming stable cellular structures similar to that shown in Figure 6.4. To understand the cause of network disintegration, consider that two fibers in contact, both of length , are driven by adhesion to fully align, i.e. to form a bundle of length . Hence, the lowest energy state of the system of fibers is a single bundle of length , containing all fibers in the system. This lowest energy state cannot be reached due to topological reasons. However, the network may separate in a multitude of isolated bundles, each of length . This is shown schematically in the left upper . Note that, panel of Figure 6.5. This phenomenon is expected at low according to the Kallmes–Corte relation, the mean segment length in a 2D ⁄2 (Kallmes and Corte 1960). Hence, ~ ⁄ , which network is represents how much longer is the fiber relative to the mean segment length of the network. ~ ⁄ is sufficiently large, the topology of the evolving When network of bundles prevents disintegration. This argument is discussed in detail by Sengab and Picu (2018). Under such conditions, cellular structures such as that shown in Figure 6.4 (and in the right upper panel of Figure 6.5) form and become stable. It is of interest to observe that the evolution from the initial to the cellular state of the network requires a small amount of in-

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

169

plane tension in the initial regime. Once the cellular network develops, the in-plane tension vanishes, and this restriction is not necessary. Cellular networks of bundles are stabilized by triangular features such as that shown in Figure 6.2. These are self-equilibrated and store strain energy. The entire structure stores adhesion energy. Bundle segments between triangles (i.e. the cell walls) tend to be straight and do not store strain energy. These structures are stable as long as the cell size remains small relative to . The results presented here define a new type of network: cellular networks of fiber bundles. Such structures are observed in various nanofibrous materials, such as buckypaper (Stallard et al. 2018) and collagen constructs (Holder et al. 2018). However, their structure and mechanics have not been studied to date. The following section provides preliminary data on the mechanical behavior of the cellular structure. It indicates that networks of bundles stabilized by triangles of bundles are particularly stable under load and, although not physically crosslinked, exhibit elastic behavior similar to that of crosslinked networks. 6.4. Tensile behavior of non-crosslinked networks with interfiber adhesion A specific characteristic of cellular networks, such as those discussed in section 6.3, is the fact that bundles may split into sub-bundles which, in turn, may regroup and re-bundle. A node linking three bundles of the cellular network may evolve in multiple ways via this non-bundling and re-bundling mechanism. Such processes are termed here, collectively, as relaxation processes. The movement of the sub-bundle branching points is expected to cause the transformation of strain energy into adhesion energy or vice versa. This mechanism is absent in fiber networks without adhesive interactions. Another mechanism specific to cellular networks is the sliding of fibers within a bundle. In the presence of inter-fiber friction, the sliding of fibers is a dissipative process. Consequently, it contributes to load transfer within the bundle.

170

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 6.6. Relaxed structure of a cellular network in the strain-free state. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Figure 6.7. Stress-strain response of the cellular network in Figure 6.6 loaded in the two conditions described in the text

 

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

171

We construct here a cellular network with a structure similar to that of a 2D Voronoi network. The segments defined by the network nodes are replaced with fiber bundles of identical size (number of fibers) and identical length. Figure 6.6 shows the resulting network after relaxation. This network is in equilibrium with respect to peeling and zipping of bundle branches. We consider that sliding of fibers within bundles does not take place, which is a reasonable assumption in the presence of some degree of inter-fiber friction and for sufficiently long fibers. The network is loaded up to 10% strain while allowing the relaxation process (non-bundling and re-bundling) to take place at bundle branching points. In a separate simulation, we prevent relaxation processes from taking place in order to quantify their importance in the mechanics of the network. Figure 6.7 shows stress–strain curves for these two loading cases of the network in Figure 6.6. It is seen that the relaxation processes lead to a more compliant network behavior. This is attributed to the transfer of the strain energy into adhesion energy which is stored in the bundles. 6.5. Structure crosslinks

of

networks

with

inter-fiber

adhesion

and

In the presence of crosslinks, fibers are not entirely free to re-arrange under the action of surface forces. Therefore, the degree of structural re-organization of the network is expected to be much smaller than in the case of non-crosslinked networks (section 6.3). This issue has been studied in detail by Negi and Picu (2019) using a novel finite element-based technology applied to 2D models of fiber networks. The model and the simulation method are described in the respective reference. Here, it suffices to mention that fibers are represented as Timoshenko beams and the crosslinks are considered pin-joints that transmit forces but not moments. Since surface interactions between fibers engage the bending mode of fibers, the resulting structure is not simply a truss network. We discuss two types of networks: isostatic structures, which have nonzero stiffness in the initial configuration without surface interactions activated; and structures sub-isostatic in the initial state and without surface forces acting. In the second case, the network has zero stiffness at small strains, but acquires stiffness as it deforms.  

172

Mechanics and Physics of Solids at Micro- and Nano-Scales

In this section, we discuss the configuration of the network without external loads applied and under the action of inter-fiber adhesion. It is observed that in both cases, adhesive interactions lead to a reduction of the network volume (or area, in 2D). In the isostatic case, this contraction is opposed by the intrinsic elasticity of the network. In the sub-isostatic case, the contraction is much more pronounced since the structure acquires stiffness after some degree of deformation (volume reduction). Figure 6.8 shows the variation of the model area upon the activation of adhesive interactions. Figure 6.8(a) refers to an isostatic network, while Figure 6.8(b) represents the case of a sub-isostatic network. The vertical axis shows the engineering volumetric strain. It is observed that, as the strength of adhesion, Ψ, increases, the network shrinks more. This trend is well-defined in the isostatic case but is weak in the sub-isostatic case. This difference can be understood based on the degree of shrinkage. Sub-isostatic networks are much floppier and can shrink by more than 80% even when adhesion is weak. In these cases, a further increase in the strength of adhesion does not lead to a substantial increase in shrinkage. This is important in biological cases where fibers engaged in the process are thin and the network connectivity parameter, z (the mean number of fiber segments emerging from a crosslink), is between 3 and 4. Networks with z smaller than 4 and fibers of zero bending stiffness are sub-isostatic even in 2D. The isostaticity threshold z in 3D is 6. This indicates that realistic networks, which have much smaller z than the threshold, should compact when subjected to inter-fiber adhesion, even at small Ψ. A similar situation is encountered in molecular networks of flexible polymers, such as in polyisoprene rubber. The common (compacted) state of the rubber is maintained by adhesive (van der Waals) interactions between strands. In turn, this enables excluded volume interactions of the strands. The interplay between adhesion and excluded volume enforces the constant-volume deformation restriction. The situation is different in hydrated gels (nondraining case) and other swollen networks, in which the fluid embedded in the network prevents the network from collapsing and prevents the molecular strands from coming in close contact.

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

173

Figure 6.8. Reduction of area as a function of for (a) an isostatic network and (b) a sub-isostatic network. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

6.6. Tensile behavior of crosslinked networks with inter-fiber adhesion Networks with adhesive interactions and crosslinks exhibit a characteristic behavior under external loading. Consider a structure which is collapsed under the action of adhesive forces, as described in section 6.5. The degree of bundling in such state is much less extensive than that observed in non-crosslinked networks (section 6.3), simply because the kinematic constraints imposed by the crosslinks prevent a large number of fibers to come together to form a bundle. Nevertheless, even bundling of fibers emerging from a common crosslink is expected to lead to stress–strain curves quite different from those typically observed in the absence of adhesion. This problem was studied by Negi and Picu (2019), and a brief account of the main results is presented here. It is instructive to discuss first the behavior in uniaxial tension of a regular network with no inter-fiber surface interactions. Two typical responses are shown in Figures 6.9 and 6.10. The work conjugate stress– strain pair reported here is the second Piola–Kirchhoff stress, , and the Green–Lagrange strain, E. Furthermore, the stress and the tangent stiffness ( ) are normalized with / and denoted by and , respectively.

174

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 6.9. (a) Normalized stress–strain and (b) normalized tangent stiffness–stress curves for Voronoi-type sub-isostatic ( 3.5) network. A realization of the network is shown in the inset to (a). The Voronoi-type structure is a network of trusses with freely rotating pin-joints as crosslinks

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

175

Figure 6.10. (a) Normalized stress–strain and (b) normalized tangent stiffness–stress curves for a 2D Delaunay network. A realization of the network is shown in the inset to (a). The Delaunay structure is a network of trusses with freely rotating pin-joints as crosslinks

Figure 6.9 corresponds to a Voronoi type network of trusses of low z ( 3.5) in 2D. This network is sub-isostatic ( 4 in 2D is the limit of isostaticity) and, therefore, has zero stiffness at small strains if the crosslinks are pin-jointed. Such networks can pick up stiffness when sufficiently strained (Arzash et al. 2019; Broedersz et al. 2011; Licup et al. 2016). The typical response of a structure of this type to uniaxial loading has three regimes (Arzash et al. 2019; Licup et al. 2016), Figure 6.9(a). The small strain response has zero stiffness. This regime extends to only a few % strain, depending on the deviation from isostaticity. In the second regime,

176

Mechanics and Physics of Solids at Micro- and Nano-Scales

the network acquires non-zero stiffness and further strain stiffens. The exact type of stiffening depends on the network structure (Islam and Picu 2018; Žagar et al. 2015), but, in most cases, biological networks strain-stiffen exponentially, ~ exp . Other types of networks exhibit power-law stiffening. Finally, the third regime is linear and occurs at large strains. The second regime appears better defined in the representation of Figure 6.9(b), where the tangent stiffness is plotted versus the stress. In this log-log plot, regime I is not shown since the stiffness goes to zero, and regime II shows a constant slope of ~0.8 (which corresponds to power-law stiffening). The transition from regime II to regime III can also be seen in Figure 6.9(b) as the curve starts to plateau. The second case, shown in Figure 6.10, is that of a Delaunay network of trusses. In this case, z is large (z = 6) and the network is unconditionally stable in the unloaded state. This response of the network shows two regimes. In the first regime defined at small strains, the stiffness is high and the network deformation is affine. The regime ends when compressive stress in the fibers running roughly perpendicular to the loading direction exceeds their buckling stress. This instability leads to softening [see Figure 6.10(b)] and allows for the gradual re-orientation of the fibers in the loading direction. Due to the realignment of the fibers in the longitudinal direction, the network may strain to stiffen again (the second regime) as can be seen in the tangent stiffness versus stress representation shown in Figure 6.10(b). Consider now the response of networks in which inter-fiber adhesion is enabled. To this end, a 2D Delaunay network similar to that leading to the constitutive response shown in Figure 6.10(a) is considered, however, now the trusses are replaced with beams. Figure 6.11(a) shows a set of stress– strain curves for the same type of network and with increasing Ψ. The representation of the curves as tangent stiffness versus stress is shown in Figure 6.11(b). The key observations from Figure 6.11 are: – The small strain modulus decreases significantly as the strength of adhesion increases. The network becomes very soft and the strain range of this linear regime I increase considerably. While regime I barely extends to 1% strain in the reference case without adhesion, it extends to almost 20% strain when Ψ = 11.11, the largest value considered in the respective study. – At larger strains, the network with adhesion strain stiffens (regime II) just like the reference network.

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

177

The behavior is controlled by adhesion at small strains and by the underlying crosslinked network at large strains. The observation that adhesion drastically transforms the network response is of importance in applications. It implies that the stiffness of some “soft materials” is controlled primarily by the inter-fiber surface interactions of adhesive or other type, and depends less on the network architecture. It is of great interest that adhesion may lead to a drastic increase of the range of the linear regime I, even when working with an underlying network with a very nonlinear response in the absence of adhesion.

Figure 6.11. (a) Normalized second Piola–Kirchhoff stress versus Green–Lagrange strain for Delaunay networks with various subjected to uniaxial tension. The side panels represent deformed configurations with = 1.23 at 5% strain and with = 11.11 at 25% strain. The data in (a) are re-plotted in (b) as normalized tangent stiffness, , versus stress. For a color version of this figure, see www.iste.co.uk/ ionescu/mechatronics.zip

178

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 6.12. (a) Normalized Second Piola–Kirchhoff stress versus Green–Lagrange strain for Voronoi networks with various subjected to uniaxial tension. (b) Extension of the curve in (a) corresponding to 2 to large strains. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Sub-isostatic networks with adhesion exhibit similarly interesting behavior. Figure 6.12 shows stress-strain curves for Voronoi networks with various values of Ψ. In the reference state (without adhesion), these networks have zero stiffness at infinitesimal strains. The stress–strain and tangent stiffness–stress curves for the same network with adhesion turned off is shown in Figures 6.9(a) and 6.9(b). Figure 6.12(a) shows that, in the

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

179

presence of adhesion, the small strain stiffness is finite and increases with increasing Ψ. This is expected since at small strains, the structure unbundles as it deforms, and this requires external work to be performed. The stress– strain curve softens as unbundling progresses. This type of stress–strain curve is not observed in networks without inter-fiber surface interactions. As in the case of the Delaunay networks (Figure 6.11), the response becomes controlled by the crosslinked structure of the network at larger strains. This can be seen in Figure 6.12(b) where one of the curves in Figure 6.12(a) is extended to large strains. Strain stiffening is observed in this regime, as expected. Hence, once again, adhesion introduces a bi-modal response, controlled by adhesion at small strains and more akin to that of the reference network without adhesion at large strains. These observations indicate that interesting behavior can be obtained in applications by combining the effect of inter-fiber surface interactions with that of the underlying network. 6.7. Conclusion Surface interactions between fibers impart interesting properties to the respective network. In assemblies of non-crosslinked fibers, adhesive interactions re-organize the structure in complex ways. If adhesion is weak or/and fibers are prevented from moving relative to each other by friction, the network remains in the initial configuration and no evolution is expected. If adhesion is stronger, the network either disintegrates (at low ) or reorganizes into a network of fiber bundles. This structure is stabilized by the formation of triangular features, similar to the Plateau triangles in foams, at each node connecting three converging fiber bundles. Rules are developed to predict the stable structure of the network function of structural and mechanical parameters such as the network density, fiber length, fiber bending stiffness, and strength of adhesion. Networks of fiber bundles stabilized by triangles are exceptionally stable during loading, despite the fact that they are not crosslinked. Their mechanical behavior is similar to that of a network of similar configuration in which all triangles are replaced by crosslinks. In the presence of inter-fiber adhesion and without external loads, crosslinked networks collapse to a pre-stressed state of much smaller volume. When loaded, these structures exhibit a behavior influenced at small

180

Mechanics and Physics of Solids at Micro- and Nano-Scales

strains by adhesion and controlled by the crosslinked network structure at large strains. If the corresponding network without adhesion is isostatic in the unloaded state, adhesion makes the mechanical behavior softer. However, if the network without adhesion is sub-isostatic in the unloaded state, adhesion makes the small strain response stiffer. This brief overview demonstrates that surface interactions, and adhesion, in particular, can be used to engineer the mechanical behavior of the network, leading to interesting, new properties. On the other hand, these observations help understand the features of the mechanics of biological materials and artificial networks in which adhesion is important, such as buckypaper. We expect that other unusual properties will be identified by future investigations of the effect of surface interactions on the physical behavior of random fibrous materials. 6.8. References Arzash, S., Shivers, J. L., Licup, A. J., Sharma, A., and Mackintosh, F. C. (2019). Stress-stabilized subisostatic fiber networks in a ropelike limit. Physical Review E, 99(4), 042412, available at: https://doi.org/10.1103/PhysRevE.99.042412. Barber, A. H., Andrews, R., and Schadler, L. S. (2005). On the tensile strength distribution of multiwalled carbon nanotubes. Citation: Applied Physics Letters, 87, 1–3, available at: https://doi.org/10.1063/1.2130713. Benítez, A. J. and Walther, A. (2017). Cellulose nanofibril nanopapers and bioinspired nanocomposites: a review to understand the mechanical property space. Journal of Materials Chemistry A, available at: https://doi.org/10.1039/c7ta02006f. Bico, J., Roman, B., Moulin, L., and Boudaoud, A. (2004). Elastocapillary coalescence in wet hair. Nature, 432(7018), 690, available at: https://doi.org/10.1038/432690a. Broedersz, C. P. and Mackintosh, F. C. (2014). Modeling semiflexible polymer networks. Reviews of Modern Physics, 86(3), 995–1036, available at: https://doi.org/10.1103/RevModPhys.86.995. Broedersz, C. P., Mao, X., Lubensky, T. C., and MacKintosh, F. C. (2011). Criticality and isostaticity in fibre networks. Nature Physics, 7(12), 983–988, available at: https://doi.org/10.1038/nphys2127. De Volder, M. and Hart, A. J. (2013, February 25). Engineering hierarchical nanostructures by elastocapillary self-assembly. Angewandte Chemie – International Edition, 52, 2412–2425, available at: https://doi.org/10.1002/anie.201205944.

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

181

Dharmadhikary, R. K., Gilmore, T. F., Davis, H. A., and Batra, S. K. (1995). Thermal Bonding of nonwoven fabrics. Textile Progress, 26(2), 1–37, available at: https://doi.org/10.1080/00405169508688865. Durville, D. (2005). Numerical simulation of entangled materials mechanical properties. Journal of Materials Science, 40(22), 5941–5948, available at: https://doi.org/10.1007/s10853-005-5061-2. Franchi, M., Trirè, A., Quaranta, M., Orsini, E., and Ottani, V. (2007). Collagen structure of tendon relates to function. The Scientific World Journal, 7, 404–420, available at: https://doi.org/10.1100/tsw.2007.92. Green, E. M., Mansfield, J. C., Bell, J. S., and Winlove, C. P. (2014, April 6). The structure and micromechanics of elastic tissue. Interface Focus, 4(2):20130058, available at: https://doi.org/10.1098/rsfs.2013.0058. Holder, A. J., Badiei, N., Hawkins, K., Wright, C., Williams, P. R., and Curtis, D. J. (2018). Control of collagen gel mechanical properties through manipulation of gelation conditions near the sol-gel transition. Soft Matter, 14(4), 574–580, available at: https://doi.org/10.1039/c7sm01933e. Islam, M. R. and Picu, R. C. (2018). Effect of network architecture on the mechanical behavior of random fiber networks. Journal of Applied Mechanics, 85(8), 081011, available at: https://doi.org/10.1115/1.4040245. Jeon, S. Y., Jang, J., Koo, B. W., Kim, Y. W., and Yu, W. R. (2017). A predictive model of the tensile strength of twisted carbon nanotube yarns. Nanotechnology, 28(1), available at: https://doi.org/10.1088/0957-4484/28/1/015703. Johnson, K. L., Kendall, K., and Roberts, A. D. (1971). Surface energy and the contact of elastic solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 324(1558), 301–313, available at: https://doi.org/10.1098/rspa.1971.0141. Kallmes, O. J. and Corte, H. (1960). The structure of paper – the statistical geometry of an ideal two dimensional fiber network. Tappi Journal, 43(9), 737–752. Kulachenko, A. and Uesaka, T. (2012). Direct simulations of fiber network deformation and failure. Mechanics of Materials, 51, 1–14, available at: https://doi.org/10.1016/j.mechmat.2012.03.010. Li, Y. and Kröger, M. (2012a). A theoretical evaluation of the effects of carbon nanotube entanglement and bundling on the structural and mechanical properties of buckypaper. Carbon, 50(5), 1793–1806, available at: https://doi.org/10.1016/ j.carbon.2011.12.027. Li, Y. and Kröger, M. (2012b). Computational study on entanglement length and pore size of carbon nanotube buckypaper. Applied Physics Letters, 100(2), available at: https://doi.org/10.1063/1.3675912.

182

Mechanics and Physics of Solids at Micro- and Nano-Scales

Li, Y. and Kröger, M. (2012c). Viscoelasticity of carbon nanotube buckypaper: zipping-unzipping mechanism and entanglement effects. Soft Matter, 8(30), 7822–7830, available at: https://doi.org/10.1039/c2sm25561h. Licup, A. J., Sharma, A., and Mackintosh, F. C. (2016). Elastic regimes of subisostatic athermal fiber networks. Physical Review E, 93(1), 012407, available at: https://doi.org/10.1103/PhysRevE.93.012407. Lieleg, O., Claessens, M. M. A. E., Heussinger, C., Frey, E., and Bausch, A. R. (2007). Mechanics of bundled semiflexible polymer networks. Physical Review Letters, 99(8), available at: https://doi.org/10.1103/PhysRevLett.99.088102. Linares, A. V., Vandevelde, F., Pantigny, J., Falcimaigne-Cordin, A., and Haupt, K. (2009). Polymer films composed of surface-bound nanofilaments with a high aspect ratio, molecularly imprinted with small molecules and proteins. Advanced Functional Materials, 19(8), 1299–1303, available at: https://doi.org/10.1002/ adfm.200801222. Liu, J., Rinzler, A. G., Dai, H., Hafner, J. H., Kelley Bradley, R., Boul, P. J., Lu, A., Iverson, T., Shelimov, K., Huffman, C. B., Rodriguez-Macias, F., Shon, Y.-S., Randall Lee, T., Colbert, D. T., and Smalley, R. E. (1998). Fullerene pipes. Science, 280(5367), 1253–1256, available at: https://doi.org/10.1126/science.280.5367.1253. Liu, Y. and Dzenis, Y. (2016). Explicit 3D finite-element model of continuous nanofibre networks. Micro & Nano Letters, 11(11), 727–730, available at: https://doi.org/10.1049/mnl.2016.0147. Lu, W., Liu, X., Li, Q., Byun, J.-H., and Chou, T.-W. (2013). Mechanical behavior and structural evolution of carbon nanotube films and fibers under tension: a coarsegrained molecular dynamics study. Journal of Applied Mechanics, 80(5), 051015, available at: https://doi.org/10.1115/1.4023684. Miao, M., Mcdonnell, J., Vuckovic, L., and Hawkins, S. C. (2010). Poisson’s ratio and porosity of carbon nanotube dry-spun yarns. Carbon, 48, 2802–2811, available at: https://doi.org/10.1016/j.carbon.2010.04.009. Mirzaeifar, R., Qin, Z., and Buehler, M. J. (2015). Mesoscale mechanics of twisting carbon nanotube yarns. Nanoscale, 7(12), 5435–5445, available at: https://doi.org/10.1039/c4nr06669c. Negi, V. and Picu, R. C. (2019). Mechanical behavior of cross-linked random fiber networks with inter-fiber adhesion. Journal of the Mechanics and Physics of Solids, 122, 418–434, available at: https://doi.org/10.1016/j.jmps.2018.09.027. Ottani, V., Raspanti, M., and Ruggeri, A. (2001, April 1). Collagen structure and functional implications. Micron, 32, 251–260, available at: https://doi.org/10.1016/ S0968-4328(00)00042-1. Parry, D. A. D. and Craig, A. S. (1984). Growth and development of collagen fibrils in connective tissue. In Ultrastructure of the Connective Tissue Matrix. Springer US, Boston, MA, pp. 34–64, available at: https://doi.org/10.1007/978-1-4613-2831-5_2.

Mechanics of Random Networks of Nanofibers with Inter-Fiber Adhesion

183

Picu, R. C. (2011, July 19). Mechanics of random fiber networks – a review. Soft Matter, 7, 6768–6785, available at: https://doi.org/10.1039/c1sm05022b. Picu, R. C. and Sengab, A. (2018). Structural evolution and stability of noncrosslinked fiber networks with inter-fiber adhesion. Soft Matter, 14(12), 2254–2266, available at: https://doi.org/10.1039/C7SM02555F. Piechocka, I. K., Bacabac, R. G., Potters, M., MacKintosh, F. C., and Koenderink, G. H. (2010). Structural hierarchy governs fibrin gel mechanics. Biophysical Journal, 98(10), 2273–2280, available at: https://doi.org/10.1016/j.bpj.2010.01.040. Reznikov, N., Steele, J. A. M., Fratzl, P., and Stevens, M. M. (2016, August 14). A materials science vision of extracellular matrix mineralization. Nature Reviews Materials,1, 16041, available at: https://doi.org/10.1038/natrevmats.2016.41. Robins, S. P. (2007). Biochemistry and functional significance of collagen crosslinking. Biochemical Society Transactions, 35(5), 849–852, available at: https://doi.org/10.1042/bst0350849. Rodney, D., Fivel, M., and Dendievel, R. (2005). Discrete modeling of the mechanics of entangled materials. Physical Review Letters, 95(10), 2–5, available at: https://doi.org/10.1103/PhysRevLett.95.108004. Sengab, A. and Picu, R. C. (2018). Filamentary structures that self-organize due to adhesion. Physical Review E, 97(3), 032506, available at: https://doi.org/10.1103/PhysRevE.97.032506. Snedeker, J. G. and Gautieri, A. (2014, July). The role of collagen crosslinks in ageing and diabetes – the good, the bad, and the ugly. Muscles, Ligaments and Tendons Journal, 4, 303–308, available at: https://doi.org/10.11138/mltj/ 2014.4.3.303. Stallard, J. C., Tan, W., Smail, F. R., Gspann, T. S., Boies, A. M., and Fleck, N. A. (2018). The mechanical and electrical properties of direct-spun carbon nanotube mats. Extreme Mechanics Letters, 21, 65–75, available at: https://doi.org/10.1016/ j.eml.2018.03.003. Stauffer, D. and Bunde, A. (1994). Introduction to Percolation Theory. Physics Today (Vol. 40). CRC Press, available at: https://doi.org/10.1063/1.2820231. Streichfuss, M., Erbs, F., Uhrig, K., Kurre, R., Clemen, A. E. M., Böhm, C. H. J., Haraszti, T, and Spatz, J. P. (2011). Measuring forces between two single actin filaments during bundle formation. Nano Letters, 11(9), 3676–3680, available at: https://doi.org/10.1021/nl201630y. Style, R. W., Jagota, A., Hui, C.-Y., and Dufresne, E. R. (2016). Elastocapillarity: surface tension and the mechanics of soft solids. Annual Review of Condensed Matter Physics, 8(1), 99–118, available at: https://doi.org/10.1146/annurevconmatphys-031016-025326.

184

Mechanics and Physics of Solids at Micro- and Nano-Scales

Tempel, M., Isenberg, G., and Sackmann, E. (1996). Temperature-induced sol-gel transition and microgel formation in α-actinin cross-linked actin networks: a rheological study. Physical Review E – Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 54(2), 1802–1810, available at: https://doi.org/10.1103/PhysRevE.54.1802. van Beek, J. D., Hess, S., Vollrath, F., and Meier, B. H. (2002). The molecular structure of spider dragline silk: folding and orientation of the protein backbone. Proceedings of the National Academy of Sciences, 99(16), 10266–10271, available at: https://doi.org/10.1073/pnas.152162299. Volkov, A. N. and Zhigilei, L. V. (2010). Mesoscopic interaction potential for carbon nanotubes of arbitrary length and orientation. Journal of Physical Chemistry C, 114(12), 5513–5531, available at: https://doi.org/10.1021/jp906142h. Wilhelm, J. and Frey, E. (2003). Elasticity of stiff polymer networks. Physical Review Letters, 91(10), 108103, available at: https://doi.org/10.1103/ PhysRevLett.91.108103. Xie, B., Liu, Y., Ding, Y., Zheng, Q., and Xu, Z. (2011). Mechanics of carbon nanotube networks: microstructural evolution and optimal design. Soft Matter, 7(21), 10039, available at: https://doi.org/10.1039/c1sm06034a. Xu, M., Futaba, D. N., Yamada, T., Yumura, M., and Hata, K. (2010). Carbon nanotubes with temperature-invariant viscoelasticity from -196° to 1000°C. Science, 330(6009), 1364–1368, available at: https://doi.org/10.1126/science.1194865. Yunoki, S., Hatayama, H., Ebisawa, M., Kondo, E., and Yasuda, K. (2015). A novel fabrication method to create a thick collagen bundle composed of uniaxially aligned fibrils: an essential technology for the development of artificial tendon/ligament matrices. Journal of Biomedical Materials Research – Part A, 103(9), 3054–3065, available at: https://doi.org/10.1002/jbm.a.35440. Žagar, G., Onck, P. R., and Van Der Giessen, E. (2015). Two fundamental mechanisms govern the stiffening of cross-linked networks. Biophysical Journal, 108(6), 1470–1479, available at: https://doi.org/10.1016/j.bpj.2015.02.015. Zhang, M., Atkinson, K. R., and Baughman, R. H. (2004). Multifunctional carbon nanotube yarns by downsizing an ancient technology. Science, 306(5700), 1358–1361, available at: https://doi.org/10.1126/science.1104276.

7 Surface Effects on Elastic Structures

7.1. Introduction Classical continuum elasticity is scale invariant: as long as proportions are kept constant, the size of an object does not matter in regard to elastic deformations. For example, buckling of a structure of a given shape would occur for the same imposed strains, independently of its size. We would, therefore, expect the same phenomena observable at the macroscopic scale to occur at the micro-scale (and maybe even at the nano-scale so long as continuum approximation holds). However, the presence of a surface energy γ modifies this picture by introducing a length scale to the problem. In particular, the coupling between surface interactions and elasticity is characterized by a length scale that compares the force per unit length exerted by the surface effect to the rigidity of the solid: ℓec = γ/E, where E is Young’s modulus of the material. In this situation, the size L of the system, therefore, matters, and a structure bigger than ℓec is insensitive to surface effect, while a smaller structure might be strongly deformed. In fact, we show that for slender structures, even when L ≫ ℓec , large deflections may still occur because slenderness implies a weak stiffness in bending. In this chapter, we are interested in how two different surface effects can deform elastic structures. We start with the interaction between capillary surface tension and slender mechanics, and how a droplet can deform a thin

Chapter written by Hadrien B ENSE, Benoit ROMAN and Jos´e BICO.

Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

186

Mechanics and Physics of Solids at Micro- and Nano-Scales

sheet. We then focus on electrostatic surface effects and their use to actuate electroactive polymer membranes.

7.2. Liquid surface energy In this first part, we show how the surface tension of a liquid interface in contact with an elastic solid may deform it. Recent years have seen a large body of research effort in this field. Here we only give scaling arguments and present the consequences of a surface tension force at the micro-scale. Thorough reviews are available (see, e.g. Andreotti and Snoeijer in press; Bico et al. 2018).

7.2.1. Can a liquid deform a solid? Surface tension is the surface energy cost, γ, associated with the creation of an interface between two materials (be they liquid or solid). γ is positive: surface tension tends to minimize the area of an interface. These capillary interactions are responsible for a large number of phenomena in liquids, such as imbibition, the motion of insects at the surface of water, and the spherical shape of small drops and bubbles (de Gennes et al. 2002).

Figure 7.1. Deformation of a solid by a liquid. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

In this section, we are interested in how surface tension can deform a solid that is not slender. Consider, for example, the effect of a droplet deposited on a solid: it exerts a torque on the latter. Indeed, the air/liquid surface tension pulls up the solid (γ13 in Figure 7.1), while the Laplace pressure pushes down (note that Laplace pressure ensures the vertical equilibrium of forces in the drop,

Surface Effects on Elastic Structures

187

and the horizontal projection leads to the classical Young–Dupr´e’s relation) so that the solid may deform (Lester 1961). However, taking a standard value of γ ≈ 70 mN/m for liquid water, and Young’s modulus on the order of 70 GPa for the solid (typically glass), one finds that the fluid can deform the solid on a length scale on the order of γ/E ≈ 1 pm. At such a small scale (smaller than intermolecular distances), the use of continuum mechanics is dubious, and such elasto-capillary effects are irrelevant. However, the recent development of microfabrication methods, ultra soft gels, and observation techniques have led to the study of the coupling between capillary forces and elasticity. For example, the deformation near the contact line of a droplet lying on a very deformable substrate (E ≈ 3 kPa) has been observed with confocal microscopy. It has been shown that the substrate adopts a shape that does not depend on its thickness nor on the droplet size, but only on the liquid composition (Style et al. 2013). Another example is the rounding off of extremely soft gels (shear modulus between 35 and 350 Pa). These gels are casted with sharp angles, but the surface tension of the solid softens the corners so that they exhibit a curvature on the order of E/γ (see Mora et al. 2013). This experimental observation of the action of a solid’s surface tension has led to many numerical and theoretical developments. In conclusion, we have seen that a liquid can deform a solid in its bulk, but only at a very small scale, and for very soft solids. However, we will show that on slender structures, surface or capillary effects may produce macroscopic deformations, even in materials with a high Young’s modulus.

7.2.2. Slender structures A slender structure is a structure with at least one dimension that is small when compared to the others. A plate, for example, has a very small thickness h compared to its length L and width w. Due to their slenderness, these structures can be considered with two modes of deformation. On the one hand, the structures can undergo stretching (or equivalently compression), with an elastic energy Estretch ≃ EhLwϵ2 for an in-plane strain ϵ, where E is Young’s modulus of the plate (left scheme in Figure 7.2).

188

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 7.2. The two modes of deformation of a slender structure: stretching (left) and bending (right)

On the other hand, the plate can be given a radius of curvature R. The bending energy associated with this deformation is Ebend ≃

B LW, R2

where B ≃ Eh3 is the bending stiffness of the plate (right scheme of Figure 7.2). Stretching energy is, therefore, proportional to the thickness h, whereas bending energy is proportional to h3 . Consequently, the bending energy of a thin structure (small thickness h) vanishes very quickly when the thickness vanishes. What are the practical consequences of this very compliant mode of deformation?

7.2.3. Wrapping a cylinder A simple way to understand how surface forces may bend a slender structure is to consider a solid cylinder of radius R, covered with a liquid of surface tension γ, and a sheet of thickness h. We assume the liquid to perfectly wet both solids. What are the conditions for the sheet to spontaneously wrap the cylinder (Figure 7.3)? Wrapping occurs if surface tension is strong enough to bend the sheet with a radius of curvature R. This is a pure bending problem, and the energetic cost associated with this deformation is proportional to Eh3 Lw . However, covering the cylinder with the sheet reduces the interface R2 between the liquid and the air, and thus is energetically favorable. The associated energetic gain is 2γLw. These two energies are in competition, and wrapping will be possible when the energetic gain due the reduction of the

Surface Effects on Elastic Structures

189

air/liquid interface is higher than the energetic cost of the elastic deformation of the sheet, i.e. when E

h3 Lw < 2γLw. R2

This condition can be expressed in terms of a cylinder critical size: √ Eh3 R > ℓB = , γ where ℓB is the capillary-bending length, the relevant length scale when dealing with bending deformations of slender structures by capillary effects. It compares the bending stiffness of the elastic structure to the surface tension of the liquid. If a cylinder has a radius smaller than this length, surface tension will not be able to sufficiently deform the sheet for wrapping to occur.

Figure 7.3. Can a thin sheet wrap around a wet cylinder?

Elasto-capillary deformation is not simply an interesting academic problem, it also has practical consequences in the field of microfabrication. Indeed, the main technique used to manufacture microelectromechanical systems (MEMS), or micro-electronic elements, is photolithography. After an insulation step, a photo-sensitive resin is put into a solvent solution. During the drying process, capillary bridges may form inside these objects, which can cause deformations, stictions, or even fractures. These irreversible events are strong limiting factors in the elaboration of slender microstructures, such as nanolines, microcantilevers, or microstamps (Hui et al. 2002; Namatsu et al. 1995; Tas et al. 1997).

190

Mechanics and Physics of Solids at Micro- and Nano-Scales

7.2.4. Capillary origamis

Figure 7.4. a) Cubic boxes obtained after the fusion of the welding deposited on the hinge of the initial pattern (taken from Cho et al. 2010); b) Capillary origami Py et al. (2007)

Manufacturing 3D structures at a micro-scale with classical microfabrication techniques is a very complicated task, partly because of the limiting role played by surface tension. This force can also be harnessed. The top row of Figure 7.4(a) presents the examples of micrometer cubes obtained, thanks to capillary interactions. A liquid metal droplet is deposited at the hinge of an initially flat metallic sheet, and as it tends to minimize its interface with the air, it folds the cube (see scheme 7.4(a) and Cho et al. 2010). We will focus on the technologically simpler case where a macroscopic sheet is deformed by a droplet (Figure 7.4(b)). What happens when a water droplet is deposited on a thin polymer sheet? Does the drop spread, or does the sheet wrap the droplet? If, for example, the droplet is deposited on a 50-µm-thin silicone elastomer square sheet, we first observe that the corners of the sheet wrap the droplet. As the liquid evaporates, the sheet bends more and more until its complete closing (see Figure 7.4). Once the droplet has completely evaporated, the sheet may return to its flat state, or remain curved, depending on the intensity of Van der Waals interactions. How do the capillary forces act on the sheet? On the one hand, liquid/air surface tension exerts a traction that pulls the sheet up; on the other hand, Laplace pressure exerts a pressure that pushes the sheet down (the drop being curved, the pressure inside the liquid is higher than outside). The drop is, therefore, exerting a moment on the sheet (scheme in Figure 7.4(b)). The capillary torque scales as γL2 , while the typical torque

Surface Effects on Elastic Structures

191

to bend the sheet on its own size scales as B ≃ Eh3 . Therefore, we expect wrapping if the sheet exceeds the size limit: √ B Lcrit ≃ ≃ ℓB . γ Experiments have been carried out with sheets of different thicknesses and different shapes (triangles or squares). In each case, the critical length has been determined. Lcrit is indeed found to be proportionnal to ℓB , with a prefactor that depends on the geometry of the sheet: Lcrit ≈ 12ℓB for triangles and Lcrit ≈ 7ℓB for squares (see Py et al. 2007). Interestingly, this law holds up to nanometric scales as shown with simulations on graphene sheets (see Patra et al. 2009). At a higher scale, gravity starts to play a role, and the maximal size is fixed by the capillary length. Finally, the initial shape of the sheet can be tuned to obtain different 3D shapes, as illustrated in Figure 7.5.

Figure 7.5. Different patterns lead to different 3D shapes. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

In this first part, we studied how a liquid can deform a solid through surface tension interactions. We focused on the interplay between the peculiar mechanics of slender structure and surface tension. The slenderness of

192

Mechanics and Physics of Solids at Micro- and Nano-Scales

the structure is what allows the surface tension to produce remarkable deformation, even though γ/E remains very small. As we explained earlier, the addition of surface effects brings new length scales to problems that would otherwise be scale invariant. More precisely, we have seen that the relevant length scale when dealing with slender structure interacting with capillary forces is the capillary bending length ℓB . This length captures the slender elasticity by comparing the bending rigidity to the surface tension. In the following, we examine the deformation of thin plates mediated via an electrostatic surface energy.

7.3. Dielectric elastomers: a surface effect? We now focus on the description of dielectric elastomers, with an emphasis on electrostatic interactions seen as a surface effect, which play an important role.

7.3.1. Introduction: surface energy

electrostatic energy of a capacitor as a

These last years have seen the rapid development of a novel class of robotics made of a compliant material called “soft robotics”. Electroactive polymers are, among the possible technologies to build such robots, a particularly cost-effective and easy-to-manufacture technology. The large deformation electroactive materials can achieve up to 500% in area strain (Huang et al. 2012), and their harmless contact make them suitable for a wide range of potential applications: from bioinspired actuators (Carpi et al. 2005; Carpi and Rossi 2007) to soft grippers (Araromi et al. 2015; Shintake et al. 2016), or tunable lenses (Carpi et al. 2011; Maffli et al. 2015; Son et al. 2012), and even energy harvesting systems (Foo et al. 2012; Kaltseis et al. 2011; McKay et al. 2011). Although their rediscovery is certainly recent, the principle of dielectric actuation can be traced back to R¨ontgen (Keplinger et al. 2010) at the end of the 19th Century. The basic idea is indeed fairly simple: a dielectric elastomer is a soft capacitor, in which both electrodes and the insulating material can deform under the action of Coulombian interactions (see Figure 7.6(a)). Charges from the opposite faces of the membrane attract each other, leading to a reduction of its thickness and, therefore, to in-plane extension. However, a surface effect is also at play in this system. Indeed, the electrostatic energy

Surface Effects on Elastic Structures

193

Eels of a planar capacitor whose electrodes are separated by H and have a surface S reads: 1 V2 Eels = − ε S 2 H

[7.1]

where ε is the permittivity and V is the potential difference. For a fixed gap H, this energy is proportional to the area of the electrode and can, therefore, be seen as a (negative) surface energy, which would tend to increase the surface S.

Figure 7.6. a) An electroactive polymer setup is made of two compliant electrodes separated by a dielectric polymer. When a voltage is applied, the area of the electrodes increases. b) The electrostatic interactions can be decomposed in two effects: an electrostatic pressure, acting through the thickness of the membrane, and a negative surface tension, acting along the membrane

To gain intuition on this surface term, we can evoke electrowetting. Lipmann, during his PhD (Lippmann 1875), imposed a potential difference V between a conductive drop and a metallic surface separated by a dielectric layer (see Figure 7.7) with thickness H. The drop is observed to spread, and we can interpret this as a result of the repulsion of charges with the same sign at its surface. It is as if the electric field was modifying the drop surface 2 tension γ into: γ ′ = γ − 12 ε VH . More precisely, the electric field reduces the apparent surface tension compared with the initial one, and therefore, it seems that its effect can be represented as forces localized at the liquid interface.

194

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 7.7. Electrowetting: under the application of a voltage, the drop spreads

In the case of an electroactive polymer (Figure 7.6), the material expanding is a solid unlike in the electrowetting experiment, but similar line forces are expected to take place at the boundary of the electrode. We wish to clarify the consequences of such a negative surface tension (with an electrostatic origin) in this elastic solid. In this chapter, we intend to sketch the formal derivation of the equilibrium equations of an electroactive polymer, with an emphasis on the case where the electrodes are not entirely covering the membrane. In doing so, we will shed light on the practical consequences of the tensile stress existing in the system, and its physical origin, by drawing analogies with capillary surface tension. The rationale developed in this first part will then be applied to an experimental study of a buckling instability triggered by inhomogeneous actuation of the system.

7.3.2. Mechanics of dielectric elastomers Our approach derives the equations for the deformation of the polymer under an electric field through a variational approach. We compute both the elastic and electrostatic energies and minimize the total energy to obtain the equations coupling electrostatics and elasticity. We will then give several interpretations of these equations. For simplicity, we will consider the planar case of a dielectric strip of thickness H, width W , and length L at rest. In this reference state, the system is parametrized with the curvilinear abscissa S and the coordinates X, Y , and Z running, respectively, along the length, width, and thickness of the strip. Upper case letters refer to the reference, undeformed state, and lower case

Surface Effects on Elastic Structures

195

letters refer to the deformed state. We assume small strains e ≪ 1 and a plane strain configuration: ey = 0 (a plane stress configuration with σyy = 0 would not qualitatively change the results). A more detailed derivation and discussion is the subject of a future article by Bense et al.

Figure 7.8. Scheme of a partially activated dielectric elastomer. The grey part represents the electrode. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

In the deformed state, the curvilinear coordinate is denoted as s and θ is the local slope (see Figure 7.8). We introduce the tangential strain averaged over the thickness of the membrane et , which determines the stretching: ds = (1 + et )dS Similarly, we define the average normal strain en as: h = (1 + en )H And finally, the curvature: κ=

dθ 1 dθ dθ = ≈ ds 1 + et dS dS

With the small strains and slender-body approximations, the latter indeed means that the curvature is small: (Hκ)2 ≪ 1. We can, therefore, neglect any nonlinear term involving products of both κ and et or en .

7.3.2.1. Elastic energy In the case of plane strain assumed here (where there is no strain along direction Y ), Hooke’s law leads to the in-plane stretching energy per unit surface of the strip, depends on the tangential and normal strains (et , en ) and reads ( ) 1 2ν 2 2 Estretch = Y et + en et + en , 2 1−ν

196

Mechanics and Physics of Solids at Micro- and Nano-Scales

where we have introduced the effective stretching modulus Y=

(1 − ν)EH (1 − 2ν)(1 + ν)

[7.2]

This expression takes an unusual form because the variation of thickness en appears here as an unknown. It is indeed customary in thin-plate mechanics to assume that stresses on the faces of the plate vanish, thus setting the value of en = −ν/(1 − ν)et (for this case of plane strain) and 2 EH Estretch = 21 1−ν 2 (en ) . Following this, the in-plane force per unit distance EH (or tension) T = 1−ν 2 et would be proportional to stretching strain, as expected. We find that electrostatic pressure on the faces is not negligible, and they play an important role, modifying the relation between tension T and deformation et . The bending energy per unit surface takes, however, the usual form EH 3 1 . [7.3] Ebend = Bθ′2 , where the bending stiffness is B = 2 12(1 − ν 2 )

7.3.2.2. Electrostatic energy The electrostatic energy of the system comprising the electroactive polymer and the generator imposing a fixed voltage V writes Eels = − 12 CV 2 . In order to take the curvature of the capacitor into account, we consider the capacity of a capacitor formed by two coaxial cylinders: 2πεw ( ),

C= ln

1+κh/2 1−κh/2

[7.4]

where ε is the dielectric constant of the polymer. We expand [7.4] for small curvatures κh ≪ 1 to find the expression for the electrostatic energy (per unit width) in the slender-body approximation: ∫ Eels = −

[ ] εV2 1 2 ds 1− (hκ) . 2 h 12

[7.5]

This energy is a generalization of equation [7.1] and is still proportional to the surface (remember Eels above is written per unit width), making it

Surface Effects on Elastic Structures

197

analogous to a surface tension with a negative sign. Moreover, we notice a term proportional to the curvature κ2 , analogous to a elastic bending rigidity in equation [7.3]. The associated electrostatic bending modulus is: BV =

εV 2 h εV 2 H ≃ , 12 12

[7.6]

at leading order in strain. Finally, to simplify the following calculations, we express the electrostatic energy in the reference coordinate and as a function of en (S), et (S), and θ(S): Eels

εV2 =− 2H

∫

1 + et dS + 1 + en

∫

1 dS BV θ′2 . 2

[7.7]

7.3.2.3. Variations We assume here, for the sake of simplicity, that both end positions of the strip are fixed. The details of the boundary conditions may not affect the main results of our discussion. The end point position measured from the origin, therefore, reads: ∫ rend =

dS (1 + et )t,

[7.8]

where t is a unit vector tangent to the sheet (having, therefore, an angle θ with the x axis). This constraint will be imposed with a Lagrange multiplier f, which corresponds to an externally applied force. Finally, the total energy to be minimized is ∫ F = dS F(et , en , θ), [7.9] with the surface density of energy given by the sum of all contributions ( ) 1 2ν 1 2 2 F(et , en , θ) = Y et + en et + en + (B + BV )θ′2 2 1−ν 2 −

ε V 2 1 + et − (1 + et )f · t, 2 H 1 + en

[7.10]

198

Mechanics and Physics of Solids at Micro- and Nano-Scales

where this expression must be used with BV = 0 and V = 0 in the portion of the strip that is not covered by the electrode. We now minimize this functional with respect to different parameters to obtain the set of equilibrium equations: δF = −(B + BV )θ′′ − (1 + et )f · n = 0, δθ(S) ( ) δF ν ε V 2 1 + et = 0, = Y en + et + δen (S) 1−ν 2 H (1 + en )2 ( ) δF ν εV2 = Y et + en − − f · t = 0. δet (S) 1−ν 2H

[7.11] [7.12] [7.13]

These equations are geometrically nonlinear and, as such, are valid for large displacements. In the remainder of this section, we will consider small strains and, therefore, replace 1 + en ≈ 1 and 1 + et ≈ 1. Equations [7.11] and [7.13] can be interpreted as force balance equations projected, respectively, along the normal and tangential directions. Defining T = f · t as the membrane tension (which may be compressive if negative) and P as the electrostatic pressure, as the usual attractive pressure between two parallel conducting plates. P=

ϵ V2 , 2 H2

[7.14]

Combining equations [7.11] and [7.13] to eliminate et , we can obtain the following equation governing the tangential strain et : T =

EH 1 HP. et − 2 1−ν 1−ν

[7.15]

The total tension may be decomposed into the usual elastic tension of the membrane (first term, following the usual Hooke’s law) and a compressive (negative) tension induced by electrostatics (second term). Similarly, we can compute the normal strain en , which reads: EH 1 ν en + HP = − T. 2 1−ν 1−ν 1−ν

[7.16]

Note that equations [7.15] and [7.16] are equally valid in the regions without electrode by taking P = 0.

Surface Effects on Elastic Structures

199

These two equations unveil the first effect of the electro-actuation: at the boundary between the covered and non-covered membranes as the electrostatic pressure P drops to zero, the strains et and en are discontinuous. This discontinuity is also naturally observed in the stresses, if computed from these strains using Hooke’s law, which we will discuss in section 7.3.2.4. In equation [7.11], we also recognize the standard Elastica equation (Love 2011) for a strip submitted to a force f at its end: (B + BV )θ′′ + f · n = 0,

[7.17]

with a difference in the added electrostatic bending rigidity BV (valid only in the electroactive region). The energy minimization that we performed, therefore, shows us that the peculiar mode of actuation of this plate does not modify the global buckling equation (apart from an added bending rigidity). Nonetheless, it introduces a discontinuity in the stress and strain fields at the boundary between the electrode and the membrane (although the total force transmitted through the membrane f is continuous). In the following, we propose several interpretations of these equations.

7.3.2.4. Electrostatic surface tension and pressure Qualitatively, the electrostatic interactions can be thought of as giving rise to two effects. On the one hand, the attraction of opposite sign charges between the faces of the dielectric generates an electrostatic pressure that compresses the membrane along its thickness. This effect can be made apparent by manipulating the force balance along the z axis (equation [7.16]). Indeed, let us assume, for example, that no external force is applied to the membrane, i.e. T = 0, and then, using Hooke’s relation, equation [7.16] can be rewritten as: 1 εV 2 . 2 h2 This compressive stress of electrostatic origin is the classical Maxwell pressure that acts on a rigid plate capacitor. Through the Poisson effect, this pressure contributes to the extension of the membrane. σzz = −P = −

On the other hand, repulsion of the same sign charges on each electrode creates a tensile stress. Similarly, equation [7.15], equilibrium of longitudinal

200

Mechanics and Physics of Solids at Micro- and Nano-Scales

forces, can be rewritten in the absence of externally applied membrane tension T = 0, 1 εV 2 . 2 h2 An electrostatic positive (tensile) stress, of equal magnitude as the electrostatic pressure, acts along the electrode and helps its expansion. As a result, the system undergoes a compressive stress across its thickness and a tensile stress along its electrode (see Figure 7.6), leading to the strain derived in equations [7.15] and [7.16]. Interestingly, both effects have the same magnitude. σxx = P =

We may now comment on the discontinuities in the elastic stresses at the electrode boundary for a strip submitted to a total membrane force T . The discontinuity in σzz simply results from the fact that an electrostatic pressure σzz = P is applied under the electrode, whereas σzz = 0 outside, where the electrostatic pressure vanishes. The situation is more subtle for the in-plane stresses σxx . Applying [7.15] inside and outside the electrode and using Hooke’s law, we can deduce the value of the elastic stresses jump V2 ∆σxx = P = 2ϵ H 2 . This jump results from a localized force at the boundary ϵ V2 of the electrode 2 H , consistently with the negative surface tension deduced from [7.1]. We conclude that elastic stress in the material results from (i) an electrostatic pressure on its faces and (ii) a surface tension force acting on the boundaries of the electrode. We propose different interpretations of these stress distributions in the next two sections.

7.3.2.5. Doubled electrostatic pressure A very common approach to describe electroactive polymers was introduced by Ronald Pelrine in his seminal work (Pelrine et al. 1998). To better understand, let us consider a strip subjected to a simple mechanical pressure σzz = −P0 on one specific region. In that case, Hooke’s relations lead to: ( ) 1 ν T = Hw Eet − P [7.18] 1 − ν2 1−ν If we now compare equation [7.15] (electrostatic actuation) and equation [7.18] (mechanical actuation), we note that both equations are equivalent (i.e.

Surface Effects on Elastic Structures

201

electrostatic and mechanical actuations produce the same strain) if P0 = P/ν. In the common case where the dielectric polymer is an elastomer, we have ν = 0.5. For an electroactive elastomer, the state of strain under actuation can, therefore, be derived by considering that a doubled electrostatic pressure acts on the system, without any negative surface tension effect. Our variational approach agrees that this point of view is still valid in the case where the electrodes do not entirely cover the membrane. This approach is simpler when one is interested in the strain field in the membrane. However, the state of stress and the added electrostatic bending rigidity are missed.

7.3.2.6. Equivalent growth Equations [7.15] and [7.16] can be rewritten as: EH (et − e0t ) 1 − ν2 EH ν T, (en − e0n ) = − 1 − ν2 1−ν

T =

[7.19] [7.20]

if we define e0t = (1 + ν)

P ; E

e0n = −(1 + ν)

P . E

[7.21]

Written in this way, these equations are exactly the ones describing an elastic strip whose reference state differs by a strain (e0t , e0n ) from the initial stress-free reference sate. By reference state, we mean that the state was obtained without external mechanical loading T = 0. This rewriting, therefore, provides another interpretation of the electrostatic loading: applying a voltage is equivalent to defining a new stress-free reference state for the membrane. This situation is similar to the cases of inelastic strain, such as those produced by growth or plastic deformation, which effectively redefines the reference state. Hence, the electrostatic loading can be seen as a modification of the rest length of the strip, together with providing a new bending rigidity B + BV . This interpretation differs from other common approaches such as those described earlier. It presents the advantage of completely capturing the electrostatic effects and provides a useful framework when dealing with non-homogeneous actuation.

202

Mechanics and Physics of Solids at Micro- and Nano-Scales

7.3.2.7. Conclusion and order of magnitude In this section, we demonstrated how the electrostatic loading can be decomposed in two effects, first, a compressive pressure P=

1 εV 2 2 H2

on the faces of the electrode, and second, a (negative) surface tension γ=−

1 εV 2 . 2 H

[7.22]

on the electrode. Both effects tend to expand the membrane, with the same order of magnitude (in fact, for ν = 1/2, both effects produce the exact same strain). Referring to the introduction of this chapter, we may estimate the effect of, say, the surface tension term by computing the electrostatic equivalent to the bulk elasto-capillary length γ/E, in which γ designates the electrostatic 2 surface tension defined in [7.22], γ = Eε VH . In typical experiments, we find γ/E = HP/E ≈ 10−6 m. As for the capillary case, this length scale is too small to play an important role in the problem. In fact, P/E can be interpreted as the typical strains in the material, which remain modest and do not induce large shape change. However, we get into the detailed study of a specific example that mechanical instabilities can be harnessed in an electro-actuated thin membrane to obtain interesting shape changes.

7.3.3. Buckling experiments We experimentally investigate how a non-uniform spatial distribution of voltage can trigger out-of-plane buckling patterns in electro-activated polymers. This work is inspired by non-uniform growth of plant leaves or material swelling that leads to complex 3D shapes (Dervaux and Ben Amar 2008; Klein et al. 2007; Kim et al. 2012; Wu et al. 2013). We focus on a model axisymmetric configuration where a circular membrane floats freely on a bath of water. A compliant circular electrode

Surface Effects on Elastic Structures

203

is deposited in its center, surrounded by a passive material (see Figure 7.9). When submitted to a voltage, the area of this region tends to increase up to a point where a buckling instability occurs (see Figure 7.9 (bottom)). Actuation here is non-homogeneous since one region is subjected to the voltage, while the rest of the membrane is not. We use linear elasticity and weak nonlinear equations of thin plates within the framework developed earlier to investigate the buckled morphologies.

Figure 7.9. Sketch and picture of the setup. A membrane of PVS floats freely on a bath of soapy water. A circular electrode (black circle) of radius a is connected to a high-voltage amplifier, and water is connected to the ground. When a threshold voltage has been reached, a buckling instability occurs, as can be seen from the deflected laser line on the picture. For a color version of this figure, see www.iste.co.uk/ ionescu/mechatronics.zip

7.3.3.1. Description of the experiment Dielectric membranes We use dielectric membranes made of polyvinil siloxane elastomers of Young’s modulus E = 250 ± 15 kPa and dielectric permittivity εr = 2.5 ± 0.6. The polymer is spin-coated before its curing is over to obtain a circular membrane, with radius b and thickness H ranging from 100 to 300 µm. The compliant electrode consists of carbon black powder manually brushed through a circular stencil of radius a on the surface of the cured polymer. We refer to the part covered with the electrode as the active part, whereas the uncovered membrane is designated as the passive part.

204

Mechanics and Physics of Solids at Micro- and Nano-Scales

This powder strongly adheres to the polymer and thus provides an electrical conductivity even when the membrane is strained at 40% (the strains we achieve experimentally are on the order of 10%).

Experimental setup Once prepared, the membrane is gently deposited at the surface of soapy water, where it floats freely. Surfactants allow us to impose a controlled value of the surface tension γ ≈ 30 mN.m−1 and enhance the electric conductivity of water. The voltage is imposed through a thin metallic wire (of radius 10 µm) in light contact with the circular electrode. The wire is connected to a high-voltage amplifier, driven by a signal generator. Water is connected to the ground and plays the role of a second compliant electrode. Voltages applied to the system typically range from 200 V to 5 kV. Out-of-plane deformations are measured with a laser sheet with a grazing incidence projected on the active part of the membrane. The deflection of the laser is recorded using a camera above the set-up and is directly proportional to the vertical displacement of the membrane. The voltage is increased by 100 V every 30 s so that the experiment can be considered as quasistatic, and any viscoelastic effect of the polymer can be neglected.

7.3.3.2. Membrane stresses below the buckling threshold What is the state of stress in the actuated membrane prior to buckling? We start with accounting for the effect of water surface tension. It induces a tensile strain on the order of γ/EH ≈ 10−2 for a 200-µm-thick membrane with free edges. We have previously shown how the electrostatic actuation could be modeled by an equivalent growth. Adapting equation [7.21] to this axisymmetric situation, the electrostatic actuation imposes a new rest length to the electrode, with an extensional strain 1ε e0 = 2E

(

V H

)2 .

[7.23]

If we now express the strain imposed by the water surface tension, we find an equivalent voltage of 2γH/ε = 700 V . This contribution is not negligible. However, as we are here interested in the regime below buckling, mechanics remain linear. Surface tension effects and electro-actuation are simply additive,

Surface Effects on Elastic Structures

205

and therefore, we consider the pre-strained state as the reference state. Water surface tension will be taken into account when we focus on buckling and post-buckling. If the problem is axisymmetric, the mechanical equilibrium reads: ∂rσr − σθ = 0, ∂r

[7.24]

where (r, θ) are the cylindrical coordinates, with origin the center of the membrane. σr and σθ are the radial and azimuthal stresses. In our equivalent growth approach, Hooke’s relations can be written as: σr = E ∗ ((er − e0 ) + ν(eθ − e0 ))

[7.25]

σθ = E ∗ ((eθ − e0 ) + ν(er − e0 )) ,

[7.26]

with E ∗ = E/(1 − ν 2 ), and e0 = 0 in the passive region. Finally, noting that radial and azimuthal strains are related to the radial displacement u(r) through er = du/dr and eθ = r/r, equation [7.24] can be rewritten as: r2 u′′ + ru′ − u = 0,

[7.27]

where .′ means the derivation with respect to r. The symmetry of the problem imposes u(0) = 0 and σr (b) = 0, while the equilibrium conditions imply the continuity of σr and u at the interface between the active and passive regions. With these conditions, equation [7.27] can be solved analytically to obtain the following expressions for the stress: σθA

−P = 2

σrA

=

σθP

P a2 = 2 b2

( ) a2 1− 2 0, r

[7.28]

[7.29]

( V )2 with P = 21 ε H . In the active region, both radial and azimuthal stresses are compressive, whereas in the passive region, only radial stresses are compressive and the azimuthal stresses are tensile. Both stresses decaying away from the active region (note also the strong discontinuity in orthoradial stress at r = a).

206

Mechanics and Physics of Solids at Micro- and Nano-Scales

7.3.3.3. Out-of-plane buckling As the voltage is increased, the active region expands in conflict with the passive region, and we have shown that radial compressive stresses build up along the whole membrane. When this stress reaches a high enough value, the membrane undergoes axisymmetric buckling. In contrast to Euler buckling, the out-of-plane deformation is strongly localized in the active region. We also note that the global mode of the instability depends on the size of the active zone (see Figures 7.10(a) and 7.10(b)). This mode, however, does not depend on the magnitude of the actuation. The superposition of the deflected laser line at different voltages in Figure 7.10 clearly shows that increasing the voltage beyond the buckling threshold only increases the amplitude of the deflection.

Figure 7.10. a) Different buckling modes at V = 5 kV and H = 210 µm for different radii of the active zone (from left to right: a = 0.5 cm, a = 1 cm, and a = 3 cm. The upper row is a picture of the membrane taken from above, and the lower picture highlights the profile of the membrane. The scale bar is the elastogravity length scale (ℓeg ≈ 1.4 cm). b) Superposition of laser profiles obtained for increasing applied voltage (from 0 to 5 kV , H = 210 µm, a = 1.5 cm). The buckling mode does not change, only its amplitude increases. For a color version of this figure, see www.iste.co.uk/ ionescu/mechatronics.zip

The water foundation underneath the membrane introduces a new length scale that comes from the competition between gravity and bending stiffness of the plate. Balancing both effects leads to an elastrogravity length scale ( )1/4 EH 3 (Pi˜neirua et al. 2013; Pocivavsek et al. 2008): ℓeg = 2π 12(1−ν , 2 )ρg where ρ is the volumetric mass of the water, and g is the acceleration of gravity. For a typical membrane of H = 200 µm, we obtain ℓeg ≈ 1.4 cm.

Surface Effects on Elastic Structures

207

In Figure 7.10(a), we use ℓeg as a scale bar. When a ≫ ℓeg , we observe that the buckling is localized near the edges of the electrode, and the width of the corresponding annulus is comparable with this length scale (see Figure 7.10(b) (middle and right)). Conversely, when a ≤ ℓeg , the whole active region is deformed (see Figure 7.10(a), left). We assess the onset of the buckling threshold and its evolution by monitoring the profile as a function of the applied voltage (Figure 7.10). We have seen that electrostatic loading produces an actuation strain e0 in [7.21], which we compare with the critical compressive strain for the buckling of a 1D strip lying on water 2 ec1D = 2π3 (h/ℓeg )2 . Results are presented in Figure 7.11. We observe a clear increase in the amplitude once a certain threshold voltage has been reached. This evolution is, however, not as sharp as in the case of a classical pitchfork bifurcation. The smoothness of the experimental transition is probably a consequence of imperfections. Among them, we can mention the slight deformation caused by the contact with the wire and the possible migration of charges outside the active region. To describe the buckling and the post-buckling behavior, we use axisymmetric weakly nonlinear plate equations. We assume that the plate keeps a uniform thickness (strains remain low in our experiments). We use the F¨oppl–Von K´arm´an framework. We are, therefore, left with the resolution of a system of two differential equations: the in-plane equilibrium and the torque equilibrium. We do not take into account the added electrostatic bending rigidity BV that we evidenced previously. Indeed, in this particular case, the total bending rigidity is only modified by 10% for the highest voltage accessible in our experiments. In the axisymmetric regime, the equations can be written as: 1 − ν ′2 rw + r2 w′ w′′ = 0. 2 w′ B∆2 w = Nr w′′ + Nθ +q r r2 u′′ + ru′ − u +

[7.30] [7.31]

with Nr = Hσr and Nθ = Hσθ and boundary conditions w′ (0) = w′′ (0) = 0, Nr (b) = γ, w′′ (0) = 0’. Experiments and numerics are in fairly good agreement, showing the relevance of our modeling to capture this buckling instability.

208

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 7.11. Maximal amplitude A of the deflection normalized by ℓeg as a function of eo /e1D , where eo is the expansion strain triggered in the active zone by electrostatic loading, and e1D is the typical compressive strain leading to buckling of a 1D strip floating on water. Experimental data (circles) are compared to the numerical integration of equations [7.30] and [7.31] (continuous line). No fitting parameter. Inset: superposition of experimental (red) and numerical (profiles) at the point indicated by the arrow. a) a = 6 mm. b) a = 30 mm. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Surface Effects on Elastic Structures

209

7.3.3.4. Conclusion on dielectric elastomers We introduced a framework for dealing with electrostatic surface effect. In particular, we drew an analogy with biological growth. Building on this analogy, we have shown how a non-uniform voltage distribution can trigger buckling instability in a free-floating electroactive elastomer sheet. We have seen that the buckling mode depends on the size of the electrode and is set by an interplay between hydrostatics and the bending stiffness of the membrane. We also demonstrated how the framework developed in the first part, coupled with classical weakly nonlinear plate equations, allows us to capture well the behavior of the system. This tool can then be used to study the influence of the size of the electrode on the buckling threshold of the system (see Bense et al. 2017). This study represents a simple demonstration of how inhomogeneous growth can trigger 3D shapes in electroactive polymers. Building on this idea, more complicated electrode geometries have been considered, which lead to different buckling modes (Hajiesmaili and Clarke 2019; Li et al. 2017). In parallel, numerical tools are developed to study these types of problems (see Langham et al. (2018)).

7.4. Conclusion In this chapter, we discussed the effects of two different interfacial energies: one of capillary nature, and the other of electrostatic nature. On the one hand, capillary surface tension stems from the energetic cost of creating an interface (due to molecular interaction at the interface). On the other hand, an “electrostatic surface tension” can be associated with electrostatic repulsion along the surface: charges on the edge of the electrode are pushed further away by their neighbors. This analogy also helps us understand that the effects of these surface forces are contradictory: capillary surface tension tends to limit the size of the interface, while electrostatic effects favor the expansion of the electrode. The existence of a surface energy γ brings a new length scale in elasticity ℓec = γ/E. In particular, we introduced both the elasto-capillary (with γ the liquid–vapor surface energy) and elasto-electro (with γ = 12 εV 2 /H) lengths that demonstrate how small the deformation produced by these surface effects are in general, even for relatively soft elastomers (a few hundreds

210

Mechanics and Physics of Solids at Micro- and Nano-Scales

of kPa. Indeed, γ/E typically ranges from 10−7 m to 10−5 m. It is clear that elastomer structures smaller or comparable to this length are prone to large deformations, either by capillary or by electrostatic forces. For structures made of stiff materials, we do not expect capillary nor electrostatic surface energies to generate large strains, even at the micro-scale, because γ/E is on the order of a picometer, smaller than the inter-atomic distance. Nevertheless, we have shown that, in slender structures, the effect of small deformations may be amplified greatly. In such problems, a bending elasto√ capillary length ℓB = B/γ, where B is the bending stiffness, is more relevant. When ℓb becomes on the order of the size of the system, surface forces are expected to generate strong bending: we have shown how a droplet may fold a thin plate and form 3D origamis through capillary forces tending to reduce the droplet surface. By contrast, electrostatic surface tension generates an expansion of the electrode. We presented a case of inhomogeneous actuation that triggers a buckling instability in a macroscopic thin plate. In both the capillary (ℓB ∼ h3/2 ) and the electrostatic cases (ℓB ∼ h2 ), the typical radius of curvature induced by surface forces lB vanishes faster than the thickness h when a structure is scaled down: eventually, this radius of curvature becomes of the order of the structure size, leading to significant bending. We conclude that both types of surface effects become all the more important when microor nanostructures are considered. Such effects could be used to manipulate or efficiently actuate microstructures.

7.5. References Andreotti, B. and Snoeijer, J. H. (in press). Statics & dynamics of soft wetting. Annual Review of Fluid Mechanics, 52. Araromi, O. A., Gavrilovich, I., Shintake, J., Rosset, S., Richard, M., Gass, V., and Shea, H. R. (2015). Rollable multisegment dielectric elastomer minimum energy structures for a deployable microsatellite gripper. ASME Transactions on Mechatronics, 20(1), 438–446. Bense, H., Trejo, M., Reyssat, E., Bico, J., and Roman, B. (2017). Buckling of elastomer sheets under non-uniform electro-actuation. Soft Matter, 13, 2876–2885, available at: http://dx.doi.org/10.1039/C7SM00131B. Bico, J., Reyssat, E., and Roman, B. (2018). Elastocapillarity: when surface tension deforms elastic solids. Annual Review of Fluid Mechanics, 50(1), 629–659, available at: https://doi.org/10.1146/annurev-fluid-122316-050130.

Surface Effects on Elastic Structures

211

Carpi, F., Migliore, A., Serra, G., and Rossi, D. D. (2005). Helical dielectric elastomer actuators. Smart Materials and Structures, 14(6), 1210, available at: http://stacks.iop.org/0964-1726/14/i=6/a=014. Carpi, F. and Rossi, D. D. (2007). Bioinspired actuation of the eyeballs of an android robotic face: Concept and preliminary investigations. Bioinspiration & Biomimetics, 2(2), S50–S63, available at: http://stacks.iop.org/1748-3190/2/i=2/a= S06?key=crossref.b7fe86d7b650c6fd1d3b23dd6fc97ed8. Carpi, F., Frediani, G., Turco, S., and De Rossi, D. (2011). Bioinspired tunable lens with muscle-like electroactive elastomers. Advanced Functional Materials, 21(21), 4152–4158, available at: http://doi.wiley.com/10.1002/adfm.201101253. Cho, J.-H., Datta, D., Park, S.-Y., Shenoy, V. B., and Gracias, D. H. (2010). Plastic deformation drives wrinkling, saddling, and wedging of annular bilayer nanostructures. Nano Letters, 10(12), 5098–5102, available at: http://pubs.acs. org/doi/abs/10.1021/nl1035447. de Gennes, P., Brochard-Wyart, F., and Qu´er´e, D. (2002). Gouttes, bulles, perles et ´ ondes. Collection Echelles, Belin, available at: https://books.google.nl/books?id= Y6KOAAAACAAJ. Dervaux, J. and Ben Amar, M. (2008). Morphogenesis of growing soft tissues. Physical Review Letters, 101, 068101, available at: https://link.aps.org/doi/ 10.1103/PhysRevLett.101.068101. Foo, C., Koh, S. J. A., Keplinger, C., Kaltseis, R., Bauer, S., and Suo, Z. (2012). Performance of dissipative dielectric elastomer generators. Journal of Applied Physics, 111(9), 094107. Hajiesmaili, E. and Clarke, D. R. (2019). Reconfigurable shape-morphing dielectric elastomers using spatially varying electric fields. Nature Communications, 10(1), 183, available at: https://doi.org/10.1038/s41467-018-08094-w. Huang, J., Li, T., Chiang Foo, C., Zhu, J., Clarke, D. R., and Suo, Z. (2012). Giant, voltage-actuated deformation of a dielectric elastomer under dead load. Applied Physics Letters, 100(4), 041911, available at: https://doi.org/10.1063/1.3680591. Hui, C. Y., Jagota, A., Lin, Y. Y., and Kramer, E. J. (2002). Constraints on microcontact printing imposed by stamp deformation. Langmuir, 18(4), 1394–1407, available at: http://pubs.acs.org/doi/abs/10.1021/la0113567. Kaltseis, R., Keplinger, C., Baumgartner, R., Kaltenbrunner, M., Li, T., M¨achler, P., Schw¨adiauer, R., Suo, Z., and Bauer, S. (2011). Method for measuring energy generation and efficiency of dielectric elastomer generators. Applied Physics Letters, 99(16), 162904, available at: https://doi.org/10.1063/1.3653239. Keplinger, C., Kaltenbrunner, M., Arnold, N., and Bauer, S. (2010). R¨ontgen’s electrode-free elastomer actuators without electromechanical pull-in instability. Proceedings of the National Academy of Sciences, 107(10), 4505–4510, available at: https://www.pnas.org/content/107/10/4505.

212

Mechanics and Physics of Solids at Micro- and Nano-Scales

Kim, J., Hanna, J. A., Byun, M., Santangelo, C. D., and Hayward, R. C. (2012). Designing responsive buckled surfaces by halftone gel lithography. Science, 335(6073), 1201–1205, available at: https://science.sciencemag.org/content/335/ 6073/1201. Klein, Y., Efrati, E., and Sharon, E. (2007). Shaping of elastic sheets by prescription of non-euclidean metrics. Science, 315(5815), 1116–1120, available at: https://science.sciencemag.org/content/315/5815/1116. Langham, J., Bense, H., and Barkley, D. (2018). Modeling shape selection of buckled dielectric elastomers. Journal of Applied Physics, 123(6), 065102, available at: https://doi.org/10.1063/1.5012848. Lester, G. (1961). Contact angles of liquids at deformable solid surfaces. Journal of Colloid Science, 16(4), 315–326, available at: http://www.sciencedirect.com/ science/article/pii/0095852261900320. Li, K., Wu, W., Jiang, Z., and Cai, S. (2017). Voltage-induced wrinkling in a constrained annular dielectric elastomer film. Journal of Applied Mechanics, 85(1), 011007–011017, available at: http://dx.doi.org/10.1115/1.4038427. Lippmann, G. (1875). Relations entre les ph´enom`enes e´ lectriques et capillaires. PhD thesis, Facult´e des sciences de Paris, Paris. Love, A. E. H. (2011). A Treatise on the Mathematical Theory of Elasticity. Dover editions, USA. Maffli, L., Rosset, S., Ghilardi, M., Carpi, F., and Shea, H. (2015). Ultrafast all-polymer electrically tunable silicone lenses. Advanced Functional Materials, 25(11), 1656–1665, available at: https://onlinelibrary.wiley.com/doi/abs/10.1002/ adfm.201403942. McKay, T. G., O’Brien, B. M., Calius, E. P., and Anderson, I. A. (2011). Soft generators using dielectric elastomers. Applied Physics Letters, 98(14), 142903, available at: https://doi.org/10.1063/1.3572338. Mora, S., Maurini, C., Phou, T., Fromental, J.-M., Audoly, B., and Pomeau, Y. (2013). Solid drops: Large capillary deformations of immersed elastic rods. Physical Review Letters, 111(11), 114301, available at: http://link.aps.org/doi/10.1103/ PhysRevLett.111.114301. Namatsu, H., Kurihara, K., Nagase, M., Iwadate, K., and Murase, K. (1995). Dimensional limitations of silicon nanolines resulting from pattern distortion due to surface tension of rinse water. Applied Physics Letters, 66(20), 2655–2657, available at: https://doi.org/10.1063/1.113115. Patra, N., Wang, B., and Kr´al, P. (2009). Nanodroplet activated and guided folding of graphene nanostructures. Nano Letters, 9(11), 3766–3771, available at: https://doi.org/10.1021/nl9019616. Pelrine, R., Kornbluh, R., and Joseph, J. (1998). Electrosctriction of polymer dielectrics with compliant electrodes as a means of actuation. Sensors and Actuators A, 64, 77–85.

Surface Effects on Elastic Structures

213

Pi˜neirua, M., Tanaka, N., Roman, B., and Bico, J. (2013). Capillary buckling of a floating annulus. Soft Matter, 9, 10985–10992, available at: http://dx.doi.org/10. 1039/C3SM51825F. Pocivavsek, L., Dellsy, R., Kern, A., Johnson, S., Lin, B., Lee, K. Y. C., and Cerda, E. (2008). Stress and fold localization in thin elastic membranes. Science, 320(5878), 912–916, available at: https://science.sciencemag.org/content/320/ 5878/912. Py, C., Reverdy, P., Doppler, L., Bico, J., Roman, B., and Baroud, C. N. (2007). Capillary origami: Spontaneous wrapping of a droplet with an elastic sheet. Physical Review Letters, 98, 156103, available at: https://link.aps.org/doi/10.1103/ PhysRevLett.98.156103. Shintake, J., Rosset, S., Schubert, B., Floreano, D., and Shea, H. (2016). Versatile soft grippers with intrinsic electroadhesion based on multifunctional polymer actuators. Advanced Materials, 28(2), 231–238. Son, S.-i., Pugal, D., Hwang, T., Choi, H. R., Koo, J. C., Lee, Y., Kim, K., and Nam, J.-D. (2012). Electromechanically driven variable-focus lens based on transparent dielectric elastomer. Applied Optics, 51(15), 2987–2996, available at: http://ao.osa.org/abstract.cfm?URI=ao-51-15-2987. Style, R. W., Boltyanskiy, R., Che, Y., Wettlaufer, J. S., Wilen, L. A., and Dufresne, E. R. (2013). Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses. Physical Review Letters, 110, 066103, available at: https://link.aps.org/doi/10.1103/PhysRevLett.110.066103. Tas, N., Vogelzang, B., Elwenspoek, M., and Legtenberg, R. (1997). Adhesion and Friction in MEMS. Springer Netherlands, Dordrecht, pp. 621–628. Wu, Z. L., Moshe, M., Greener, J., Therien-Aubin, H., Nie, Z., Sharon, E., and Kumacheva, E. (2013). Three-dimensional shape transformations of hydrogel sheets induced by small-scale modulation of internal stresses. Nature Communications, 4, 1586, available at: https://doi.org/10.1038/ncomms2549.

8 Stress-driven Kirigami: From Planar Shapes to 3D Objects

8.1. Introduction At length scales lower than 1 µm, the planar technology (molecular beam epitaxy, metal organic chemical vapor deposition, ALD, spin coating, etc.) is dominant and most of the fabrication methods allow very fine tuning of both composition ( a2 then m > 0 2 (the upper layer is in tension).

218

Mechanics and Physics of Solids at Micro- and Nano-Scales

In the particular case of cubic materials and spherical pre-strain, a E straightforward computation shows that Cˆ11 + Cˆ12 = 1−ν so that, the real number ) /( ) ( E2 E1 C= , [8.6] 1 − ν1 1 − ν2 defines the contrast between the reduced elastic moduli. Then, equation [8.2.1] is reduced to (Ch1 + h2 )ε +

h1 h2 (1 − C)κ = −h2 m, 2

[8.7]

while [8.2.2] reads h1 h2 1 h1 h2 (1 − C)ε + (C(h31 + 3h1 h22 ) + h32 + 3h2 h21 )κ = −m . 2 12 2 [8.8] Multiplying [8.7] by 1/h1 , and [8.8] by h12h2 and denoting by ξ = h2 /h1 and κ ˆ = κh2 , the system of equations [8.2] becomes  1   κ = −mξ,  (C + ξ)ε + 2 (1 − C)ˆ ( ( ) ) 1 1 3   C +3 +ξ+ κ ˆ = −m,  (1 − C)ε + 6 ξ2 ξ

[8.9]

whose general solution is ε=

−mξ(ξ 3 + 3Cξ 2 + 3Cξ + C) , C 2 + 2Cξ(2ξ 2 + 3ξ + 2) + ξ 4

[8.10]

κ ˆ=

−6mCξ 2 (1 + ξ) . C 2 + 2Cξ(2ξ 2 + 3ξ + 2) + ξ 4

[8.11]

For most of the applications, we have low-contrast materials for which C ≃ 1 so that, we obtain, the classical result ε=

−mξ , 1+ξ

κ=

−6mξ 2 . h2 (1 + ξ)3

[8.12]

Stress-driven Kirigami: From Planar Shapes to 3D Objects

219

2 Note that ε = −m h1h+h is exactly the thickness weighted average of the 2 pre-strain. Moreover, since the curvature κ ˆ > 0, the bilayer is concave, in agreement with the expected relaxed shape. We observe that in the limit h1 ≫ h2 , we have ξ → 0 and, as a consequence, as expected ε → 0 and κ ˆ → 0, too. On the contrary, if h1 ≪ h2 then ξ → ∞ and as expected ξ → −m and κ ˆ → 0. The curvature has a maximum value (at fixed total thickness) at ξ = 1 (both layers have same thickness). Obviously, the thinner the films the higher the curvature.

8.3. Constant curvature ribbons and geodesic curvature It is well-known that straight thin strips relax into constant radius circles. To generalize this result, a first natural question is the following: assume we have a two-dimensional planar strip with isotropic pre-stress. What can be said about the relaxed object obtained? A quite general answer to this question is given by Danescu et al. (2013). Note that while for planar curves the knowledge of the curvature κ2 determines, up to an isometry, the shape of the curve, and the shape of a curve in R3 is determined, up to an isometry, by the knowledge of both the curvature κ3 and torsion τ. However, since the result of the previous section shows that (locally) a flat surface relaxes into an object with spherical curvature (i.e. a sphere of radius 1/κ), the image of the relaxed planar strip will be a spherical curve. On the contrary, a well-known result in differential geometry relates the curvature κ3 and torsion τ of a spherical curve through d ds

(

1 dκ3 τ κ23 ds

) =

τ . κ3

[8.13]

This result also shows that spherical curves have constant curvature iff τ = 0, i.e if they are planar curves. Danescu et al. (2013) have shown that κ22 = κ23 − κ2 ,

[8.14]

so that in particular, straight strips (κ2 = 0) relax to geodesics κ3 = κ. The right-hand side in [8.14] is known as the geodesic curvature of a threedimensional curve. A numerical illustration of a wireframe designed to relax into equal latitude circles on a sphere is shown in Figure 8.1.

220

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 8.1. The planar wireframe (left) with local planar curvature equal to the geodesic curvature of equal latitude circles on a sphere and the relaxed object (right). For a color version of this figure, see www.iste.co.uk/ionescu/ mechatronics.zip

8.3.1. Experimental evidence

Figure 8.2. Overview of a quarter-circle ribbon with planar radius 75 µm and close view of the fixed end. Note that the chamfer geometry is needed in order to avoid fracture of the ultra-thin layer during the underetching process

In order to illustrate the role of the geodesic curvature, we have grown a 130-nm-thick InP layer on a sacrificial layer followed by a 77-nm-thick indium gallium phosphide (InGaP) layer. The fraction of the Ga material in the InGaP alloy was fixed so as to obtain a pre-strain of 0.08%. From relation [8.12.2] with ξ = 2, m = 0.008, and h2 = 75 nm, we estimate κ = 18.5 µm. We

Stress-driven Kirigami: From Planar Shapes to 3D Objects

221

designed four series of constant curvature quarter circles (4 µm width) with radii and lengths given (see Figure 8.2) in the first two columns of Table 8.1 and, subsequently, print them on the bilayer structure. Planar radius R (µm)

Length = R π2 (µm)

Relaxed angle (◦ ) = √ 90◦ 1 + (κR)2

75

118

360◦ + 15◦

100

157

360◦ + 134◦

125

196

360◦ + 254◦

150

236

360◦ + 375◦

Table 8.1. Values of the planar radii, lengths of quarter-circle ribbons and computed values of the angular coordinate of the free-standing ribbons after stress relaxation

The final underetching process release the pre-strains, so that the structures evolve toward their relaxed configurations whose theoretical curvature is 18.5 µm. The last column in Table 8.1 gives the position of the free-end of the quarter circles as predicted by the theory, and it can be compared to those observed by SEM and illustrated in Figure 8.3. Experimental results show very good agreement with the theoretical predictions: excepting several particular situations (see Figure 8.3, right picture on the second line), the relaxed structures are planar as can be noticed on both the first and the last lines in Figure 8.3. From a quantitative point of view and in agreement with predicted angles in Table 8.1, the free-end of the quarter circles is found at the correct position, which is a consequence of the curvature estimate given by the geodesic curvature in formula [8.14]. Moreover, on the second line (left picture), the SEM-measured diameter of the relaxed structure is also in good agreement with the theoretical estimate, i.e. 18.5 µm. Note that, in order to avoid fracture during relaxation, the chamfer geometry at the connection between the planar design and the support of the structure is mandatory.

222

Mechanics and Physics of Solids at Micro- and Nano-Scales

8.3.2. Geodesic objects

Figure 8.3. From top to bottom: large field view of series of relaxed quarter-circle ribbons shows a reproducible planar shape (first line). Second line: relaxed shapes for 75- and 100-µm quarter-circle ribbons. Third line: relaxed shapes for 125- and 150-µm quarter-circle ribbons

Stress-driven Kirigami: From Planar Shapes to 3D Objects

223

In the particular situation in which the planar design contains only straight lines, the geodesic curvature of the relaxed shape vanishes, which is exactly the definition of the geodesic on the sphere. As a consequence, our result provides a receipt to fabricate any 3D object formed exclusively by pieces of geodesics on the sphere of radius 1/κ, further called geodesic objects. In particular, one can fabricate all spherical versions of the regular polyhedra using only planar designs with equal straight lines. However, for all geodesic objects, and in particular for regular polyhedra, the planar design should respect two important restrictions: (i) On the planar design, the angles between any two straight lines intersecting at a vertex should be those measured in the tangent plane to the circumscribed sphere. This is obviously the case for arbitrary geodesic objects but require special attention for spherical versions of regular polyhedra. For instance (see Danescu et al. 2018), in the case of the truncated icosahedron, the angles of the regular polyhedron are θ5 = 108◦ (regular pentagon) and θ6 = 120◦ (regular hexagon), respectively, while for the spherical version they become θˆ5 = 111.4◦ and θˆ6 = 124.3. (ii) The planar design should not contain any closed polygon. This requirement is the consequence of the spherical geometry result: for any spherical triangle, the sum of the angles is greater than 180◦ (and less than 270◦ ). It follows that the sum of the angles of any closed spherical n-vertex polygon is greater than 180◦ ·(n−2). Figure 8.4 illustrates this requirement: the bending stress due to spherical angle incompatibility between a planar regular polygon design (angles = 108◦ ) and the relaxed shape leads to fracture along one of the polygon sides. In the work of Danescu et al. (2018), we provide the details for the fabrication of the spherical truncated icosahedron, a five-fold symmetry object, starting from a planar bilayer with isotropic pre-strain and cubic symmetry. We note that, in general, the planar design, even restricted by (i) and (ii) above, is not unique. The fabrication method proves robust and able to preserve a constant curvature over significant length (≃ 250 µm).

8.4. Directional bending of large surfaces In order to fully exploit the potential of pre-strained layers, it is essential to understand not only the shape changes of small width ribbons but also that of large surfaces. This case is particularly interesting since large surfaces cannot

224

Mechanics and Physics of Solids at Micro- and Nano-Scales

accommodate simultaneous bending along two perpendicular directions. From this perspective, the situation is somehow similar to that encountered in buckling under dead loads of the three-dimensional bodies (Ball and Schaeffer 1983; Danescu 1991). In that case, low symmetry solutions appear (platelike and/or beam-like) and they are energetically preferred for sufficiently large values of the dead loads. As a result, of this analogy, we expect that for large surfaces one dominant direction will prevail so that, as a consequence, the bending in the normal direction will be inhibited. As a result, large enough square surfaces will relax rather to cylindrical shapes than to spherical shapes.

Figure 8.4. Fracture induced by the incompatibility between the angles of a planar regular pentagon (θ5 = 108◦ ) and the relaxed spherical regular pentagon (θˆ5 = 111.4◦ ).

8.4.1. Photonic crystals tubes As an illustration of the directional bending, we report in Danescu et al. (2018) the fabrication of photonic crystal (PC) tubes, which represent the analogs of the single wall carbon nanotubes (SWCNT) in the class of classical crystals. Although the design of planar PC is straightforward, threedimensional PCs are much more difficult to fabricate, as they need selective (periodic) refraction index changes. As an illustration, we design and fabricate a planar pre-strained PC build on a triangular pattern pre-stressed bilayer with PC parameter 1.2 µm and

Stress-driven Kirigami: From Planar Shapes to 3D Objects

225

cylindrical cavities with radius 475 nm. The orientation of the design on the sacrificial layer was designed so that the maximum etching direction to be aligned normal to the rolling axis. Figure 8.5 illustrates the details of the design (first line left picture), close-up SEM image of the PC structure (second image on the first line, and first image on the second line) before and after underetching, respectively, and a close-up image of the relaxed structure.

Figure 8.5. First line: details of the design of the planar PC. The blue color indicates the material to be removed, the white-filled zone is the anchorage of the cylindrical structure (left). A close-up SEM image of the PC before underetching (right). Second line: close-up SEM image of the PC after underetching (left) and close-up SEM image of the free cylinder (right)

8.4.2. Control the directional bending For practical purposes, if some structures can be fabricated by choosing the bending direction and inhibiting the second one, flexibility of the method goes far behind this curvature selection. For instance, once the PC cylinder is fabricated, its spatial orientation may be important in order to measure its physical characteristics. A natural question is then: can we select different

226

Mechanics and Physics of Solids at Micro- and Nano-Scales

bending directions for different parts of the same structure? The answer to this question is positive and determined by both the geometry of various parts and the geometry of the transition area between these. To illustrate this issue, we have designed a contact between the object and the substrate such that during underetching the contact bends around the direction which is inhibited by the relaxation of the cylinder. This kind of design needs a smooth geometric transition, typically oriented along the lowvelocity underetching direction. Figure 8.6 shows the details of the contact design, as well as the relaxed PC at the top of the contact but actually normal to the substrate. It is obvious that other various combinations of successive bending directions can be designed illustrating the range of the proposed fabrication method.

Figure 8.6. First line: large view of a contact that bends along the same axis as the PC tube, thus prohibiting the orientation of the tube with respect to the substrate (left). A close-up view of a contact with edges parallel to the low-velocity underetching. This geometry allows initiation of the bending along the direction normal to the cylinder axis. Second line: SEM images of free-standing PC tubes positioned normal to the substrate using the contact shape in the first line

Stress-driven Kirigami: From Planar Shapes to 3D Objects

227

8.5. Conclusion Stress engineering for the fabrication of 3D structures requires fine control of both material composition and thickness and can be obtained for ultrathin films (several nanometers thick by MBE). Understanding the interplay between mechanical aspects and the geometric features involved in the prestress and/or pre-strain relaxation of (small width and large areas), planar structures provide a flexible tool for the fabrication of 3D objects. While the classical estimates of the linear theory of multi-layered plates correctly predict the curvature of the resulting objects (at least for small width ribbons), the role of non-linearities (multiple solutions, path selection, curvature inhibition, etc.) inherent to the problem is not completely understood. Although recent developments toward a nonlinear effective theory of plates and rods with pre-strains (Kohn and O’Brien 2018; Lewicka et al. 2017) are under way, an intermediate two-dimensional low-cost largerotation–small strain theory including pre-strains is still missing. From a different perspective, while approximate solutions of elastic equilibrium (local minima) can be numerically obtained, the role of the underetching process, i.e. the path selecting one particular solution of interest is still an open problem.

8.6. References Ball, J. M. and Schaeffer, D. G. (1983). Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions. Mathematical Proceedings of the Cambridge Philosophical Society, 94(2), 315–339. Chen, Z., Huang, G., Trase, I., Han, X., and Mei, Y. (2016). Mechanical self-assembly of a strain-engineered flexible layer: wrinkling, rolling, and twisting. Physical Review Applied, 5(1), 017001. Danescu, A. (1991). Bifurcation in the traction problem for a transversely isotropic material. Mathematical Proceedings of the Cambridge Philosophical Society, 110(2), 385–394. Danescu, A., Chevalier, C., Grenet, G., Regreny, P., Letartre, X., and Leclercq, J.-L. (2013). Spherical curves design for micro-origami using intrinsic stress relaxation. Applied Physics Letters, 102(12), 123111.

228

Mechanics and Physics of Solids at Micro- and Nano-Scales

Danescu, A., Regreny, P., Cremillieu, P., and Leclercq, J. (2018). Fabrication of self-rolling geodesic objects and photonic crystal tubes. Nanotechnology, 29(28), 285301. Fritzler, K. B. and Prinz, V. Y. (2019). 3D printing methods for micro-and nanostructures. Uspekhi Fizicheskikh Nauk, 189(1), 55–71. Golod, S., Prinz, V. Y., Mashanov, V., and Gutakovsky, A. (2001). Fabrication of conducting GeSi/Si micro-and nanotubes and helical microcoils. Semiconductor Science and Technology, 16(3), 181. Kohn, R. V. and O’Brien, E. (2018). On the bending and twisting of rods with misfit. Journal of Elasticity, 130(1), 115–143. Lewicka, M., Raoult, A., and Ricciotti, D. (2017). Plates with incompatible prestrain of high order. Annales de l’Institut Henri Poincar´e, 34, 1883–1912. Li, J. and Liu, Z. (2018). Focused-ion-beam-based nano-kirigami: from art to photonics. Nanophotonics, 7(10), 1637–1650. Li, X. (2011). Self-rolled-up microtube ring resonators: a review of geometrical and resonant properties. Advances in Optics and Photonics, 3(4), 366–387. Prinz, V. Y., Chekhovskiy, A., Preobrazhenskii, V., Semyagin, B., and Gutakovsky, A. (2002). A technique for fabricating ingaas/gaas nanotubes of precisely controlled lengths. Nanotechnology, 13(2), 231. Prinz, V. Y., Gr¨utzmacher, D., Beyer, A., David, C., Ketterer, B., and Deckardt, E. (2001). A new technique for fabricating three-dimensional micro-and nanostructures of various shapes. Nanotechnology, 12(4), 399. Prinz, V. Y., Naumova, E. V., Golod, S. V., Seleznev, V. A., Bocharov, A. A., and Kubarev, V. V. (2017). Terahertz metamaterials and systems based on rolled-up 3D elements: designs, technological approaches, and properties. Scientific Reports, 7, 43334. Prinz, V. Y., Seleznev, V., Gutakovsky, A., Chehovskiy, A., Preobrazhenskii, V., Putyato, M., and Gavrilova, T. (2000). Free-standing and overgrown InGaAs/GaAs nanotubes, nanohelices and their arrays. Physica E: Low-Dimensional Systems and Nanostructures, 6(1–4), 828–831. Ren, Z. and Gao, P.-X. (2014). A review of helical nanostructures: growth theories, synthesis strategies and properties. Nanoscale, 6(16), 9366–9400. Shaltout, A. M., Kildishev, A. V., and Shalaev, V. M. (2016). Evolution of photonic metasurfaces: from static to dynamic. JOSA B, 33(3), 501–510. Vorob’ev, A. and Prinz, V. Y. (2002). Directional rolling of strained heterofilms. Semiconductor Science and Technology, 17(6), 614. Yang, S., Choi, I.-S., and Kamien, R. D. (2016). Design of super-conformable, foldable materials via fractal cuts and lattice kirigami. MRS Bulletin, 41(2), 130–138.

Stress-driven Kirigami: From Planar Shapes to 3D Objects

229

Yu, X., Zhang, L., Hu, N., Grover, H., Huang, S., Wang, D., and Chen, Z. (2017). Shape formation of helical ribbons induced by material anisotropy. Applied Physics Letters, 110(9), 091901. Zhang, Y., Yan, Z., Nan, K., Xiao, D., Liu, Y., Luan, H., Fu, H., Wang, X., Yang, Q., Wang, J., Ren, W., Si, H., Liu, F., Yang, L., Li, H., Wang, J, Guo, X., Luo, H., Wang, L., Huang, Y., and Rogers, J. A. (2015). A mechanically driven form of kirigami as a route to 3D mesostructures in micro/nanomembranes. Proceedings of the National Academy of Sciences, 112(38), 11757–11764. Zheludev, N. I. and Plum, E. (2016). Reconfigurable nanomechanical photonic metamaterials. Nature Nanotechnology, 11(1), 16.

9 Modeling the Mechanics of Amorphous Polymer in the Glass Transition

9.1. Introduction The mechanical behaviors of polymer melts and elastomers at temperatures well above their glass transition temperature are well understood. At present, sophisticated models are able to describe the flow properties of molecules with various architectures [Pom-Pom (Bishko et al. 1997), Ring polymers (Ge et al. 2016)], as well as the nonlinear mechanics of elastomers (Rubinstein and Panyukov 2002). By contrast, understanding of the mechanics of amorphous polymers near or below their glass transition remains poor. Close to glass transition, amorphous polymers exhibit slow relaxations – often described by a stretched exponential function (or KWW relaxation function). It is only 20 years ago that a clear picture of polymer dynamics close to glass transition has emerged (Ediger 2000; Sillescu 1999). In its glass transition domain, a polymer can be considered as a constituent of nanometric domains, the sizes of which weakly vary with temperature. Each domain can change its local arrangement by thermally activated hops. Each arrangement has a specific lifetime. The probability density of lifetime is extremely broad – typically 4 decades for a homopolymer, and more for a copolymer or a polymer mixture. The specific dynamical behavior of domains hoping has been called “dynamical heterogeneities” (Berthier 2011). The macroscopic signature of the huge broadness of the time distribution is a response function that is close to a stretched exponential.                                         Chapter written by Hélène MONTES, Aude BELGUISE, Sabine CANTOURNET and François LEQUEUX.

Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

232

Mechanics and Physics of Solids at Micro- and Nano-Scales

In this chapter, we introduce a mechanical model that is built on the picture of dynamical heterogeneities. As explained earlier, a single domain can hop from a local arrangement to another one. An applied stress results in biasing hops of the dynamical heterogeneities along the stress field direction: under stress, hopping is associated with stress relaxations, as described by Eyring’s model. The aim of our model is to describe such a mechanical process in the presence of high disorder of the picture of dynamical heterogeneities and to discuss its macroscopic consequence. Let us focus on the consequences of the dynamical heterogeneities on stress relaxation. A step strain is applied to a glassy polymer initially at rest. Just after step strain, the stress field is initially homogenous. It further evolves as a result of stress relaxation by various domains. As the domains have very different relaxation times, the stress field may become very heterogeneous. This random stress field may itself modify the relaxation of the domains and is thus of importance to describe the macroscopic relaxation. We will show, in this chapter, that macroscopic relaxation is different from the simple average of the local ones. Indeed, experimental results are usually well captured using phenomenologically generalized Maxwell models (Dooling et al. 2004; Dreistadt et al. 2009; Tervoort et al. 1996) or stretched exponential function to describe the linear viscoelastic response. However, the link between such a macroscopic description and the physics of dynamical heterogeneities is still an open question. We will explain how our approach can give a quantitative description of the linear viscoelastic response. Moreover, we will show that the effect of dynamical disorder on the mechanical properties is modified in confined geometry where the sample size is of the order of the length scale of a few heterogeneities, as observed experimentally (Vogt 2018). Finally, we will show that dynamical heterogeneities also influence the nonlinear mechanical response of glassy polymers. In order to analyze the effect of the dynamical disorder on the mechanical properties of polymer near their glass transition, we have developed a model where the heterogeneities are explicitly taken into account and combined with continuum mechanics (Masurel et al. 2015). In order to avoid the complexity of reptation and other related processes, we will discuss the peculiar situation of an elastomer and we will show that the Rouse mode – conformation relaxation at the scale of the chains strands – is negligible. We will thus relate the macroscopic mechanical response to the local stress and

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

233

local strain fields. The model provides a description not only of linear relaxation but also of the effect of confinement (Masurel et al. 2017a) and nonlinearities (Masurel et al. 2017b). 9.2. Modeling the mechanics of amorphous 9.2.1. Input physics

Figure 9.1. Schematic representation of the model. A sample is a set of domains that are cubic in 3D and hexagonal or square in 2D. One domain is composed of a given d number of elements – defined by r with d = 2 in 2D and d = 3 in 3D – that have the same mechanical response. We assume a Zener response for each element characterized by a glassy GG, a rubber GR modulus, and a relaxation time i . i depends on the intrinsic relaxation time i that describes the intrinsic dynamical fluctuations existing in a glass at rest. i is randomly distributed on domains without any spatial correlation assuming a log-normal time distribution of width s. Thus the spatial correlation of length of i field is equal to the size of one domain, typically of 5 nm. 0 is the center of the distribution. Periodic conditions are applied to the boundaries. For a color version of this figure, see www.iste.co.uk/ionescu/ mechatronics.zip

234

Mechanics and Physics of Solids at Micro- and Nano-Scales

In order to describe the picture of dynamical heterogeneities, the bulk is divided into domains, each differing by its own mechanical response or more precisely by its own lifetime i (as shown in Figure 9.1). This intrinsic time aims to mimic the thermally activated configurational hops occurring at rest. The hop relaxation leads to a nearly complete and instantaneous relaxation of the local stress. However, it is difficult to implement these types of behaviors in continuum mechanics formalism or in finite element models. However, for a large number of domains, a random hop with a given lifetime is equivalent to an exponential decay. We approximate the relaxation of each domain by an exponential decay of the stress, with a relaxation time i. Indeed, replacing the hopping by an exponential decay, we only lose the stepwise stress relaxation that may be observed in very small systems that are out of the scope of this work. We thus assign a generalized Maxwell model to each domain (see Figure 9.1). In this frame, the response of each domain is glassy at short times with a shear – respectively, Young’s modulus – GGi or EGi. The modulus originates from the Van der Waal interactions and is typically of the order of 1 GPa. At a long time, each domain behaves as polymer chains in their rubber state characterized by a modulus – GRi or ERi – originating from the entropy of the strands and typically of 1 MPa. As explained earlier, we limit ourselves to the situation of an elastomer. In the picture of dynamical heterogeneities, each domain differs by its own intrinsic relaxation time i. We choose the latter to be randomly drawn from a log-normal distribution P(ln( i)) that is characterized by a width s and a mean time 0.

√









[9.1]

Thus the spatial correlation of the relaxation time is equal to the size of one domain. The latter distribution not only can be deduced from recent models (Long and Merabia 2002) but can also be inferred from assuming Gaussian distributions of the energy barriers of each domain, which is consistent with domains containing a large number of monomers. As time i is proportional to the exponential of the energy barrier, a log-normal distribution is expected for its distribution.  

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

235

For simplicity, we assume the following in our model. – The intrinsic lifetime time.

i

of a domain remains constant with the elapsed

– The local glassy (GGi or EGi ) and rubber (GRi or ERi) moduli are constant for all unit domains of a given system. We will call them GG and GR in the following. – The system is almost incompressible. Thus, the KGi and KRi moduli are significantly larger than the glassy and rubber moduli GG and GR. KGi and KRi are set to a unique value K equal to 107 MPa. – We have also neglected the contribution of the Rouse dynamics. We show, in section 9.7, that the Rouse modes have no significant contribution to the stress decay. In a 3D model, the response of one domain is thus given by the following stress–strain relation ∶ ̿



[9.2]

where and are the fourth-order isotropic elasticity tensors 3 2 with and such as 1⁄2 (I is the fourth-order identity). The glassy and rubber shear moduli are equal to GG and GR, respectively, and compressibility moduli K >> GG. i is the relaxation time of the domain i. Moduli GG and GR are constant for all domains. The stress relaxation of an isolated domain undergoing a shear strain step is thus given by:









0

[9.3]

This can be easily extended to a nonlinear relaxation. The introduction of the nonlinear relaxation will be described in the following. 9.2.2. Temperature dependence of the intrinsic relaxation times Time–temperature superposition is a very useful tool for the observation of mechanical relaxation of polymers. For instance, in a step strain

236

Mechanics and Physics of Solids at Micro- and Nano-Scales

experiment performed within a linear regime, the macroscopic stress relaxation occurs at shorter times when the temperature increases. In a reasonable approximation, the stress relaxations at different temperatures correspond to the ones observed at different timescales. Two macroscopic stress relaxations measured at different temperatures can be superimposed, applying a shift factor ⁄ on the timescale (Ferry 1980). More precisely, ,



,

⁄

with

⁄

. This feature is the so-called

time–temperature superposition principle that is observed in the glass transition domain of polymer systems. The temperature dependence of the shift factor ⁄ can be described using the empirical WLF relation: ⁄

[9.4]

where C1 and C2 are the material constants whose values depend on the reference temperature T0 and can be found in the literature (Ferry 1980). The macroscopic time–temperature rescaling law is related to the increase in hop’s rate with temperature. The time–temperature superposition is known to fail because of a variation of the intrinsic time distribution with temperature, as measured by dielectric relaxation methods (Ding and Sokolov 2006). However, this effect is marginal in mechanics – because at a given temperature, the experimental range of times is at most of 4 decades. Therefore, in the present model, we assume that the width s of the intrinsic time distribution does not depend on the temperature. In addition, we assume that the variations of 0 with temperature follow a WLF’s law. In the frame of the latter assumptions, quantitative comparisons of our numerical prediction model and experiments are made possible. 9.2.3. Length scales in the model The model is implemented in a finite element code. The behavior of each domain is described by the constitutive equation [9.2] and with an intrinsic relaxation time randomly drawn from a time distribution function given by equation [9.1]. Thus, the spatial correlation length of the intrinsic relaxation time field is equal to the size of one domain. The last point to be set is the length scale of the domains. The size of the domains is the objective of extended literature in numerical (Berthier 2011) and theoretical (Long and Merabia 2002) works, but only a few experimental observations have been reported (Ediger 2000; Sillescu 1999; Tracht et al. 1998). The domain size is

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

237

generally considered to be of ca. 5 nm and to weakly increase for decreasing temperatures. As a result, we assume a size of around 5 nm in our model. We recall that, in the thermodynamic limit, the macroscopic response of bulk systems does not depend on the domain size. Consistently, the length scale of 5 nm is relevant only for the simulation of confined systems. Moreover, simulations of bulk systems require that the number of domains is large enough for predicted mechanical responses independent of system size. In order to provide an estimate for the ideal size of our virtual bulk system composed by N domains, the macroscopic modulus was calculated with varying N. The standard deviation of the macroscopic modulus becomes negligible for √ 20. In that frame, 2D simulations were performed using at least 1,600 domains for bulk systems. 3D computations were done with bulk samples constituted of 4,096 cubic domains. For confined samples (sandwiched or free-standing thin films), 3D computations were performed using at least 512 domains. However, a question remains on the typical length scale for the entropic contribution of the strands, i.e. the GR modulus. The typical length scale of the strands may be larger – by a factor two – than the one of the glassy domains, as observed by Casas et al. (2008). However, as demonstrated in section 9.7, the contribution of the Rouse modes to the rubber modulus is in practice negligible. In practice, the only relevant length scale in our problem is one of the dynamical heterogeneities. Thus, we have introduced a physical model that can be simply implemented in a finite element code both for linear and nonlinear conditions. Before discussing the results, we will first detail the technical points required for the computation of the model. 9.2.4. Numerical implementation The finite-element method is a procedure for obtaining numerical approximations to the mechanical equilibrium solution of problems resulting from boundary values. In the finite element code, the sample is divided into subdomains called finite elements or elements in a shortened form. The response of each element is defined by the value of degrees of freedom (DOF) at a set of nodal points. The number of DOF depends on the dimension of the sample. The stresses in each element are related to the strains by use of the time integral equation of [9.2]. Because the problem is nonlinear in time step, an iterative procedure should be followed in order to

238

Mechanics and Physics of Solids at Micro- and Nano-Scales

ensure equilibrium in each time increment. Here the Newton−Raphson method was used for that purpose. The sample of dynamical heterogeneities is modeled by a set of domains. Each domain exhibits its own mechanical response characterized by a time i. We attribute the same intrinsic time to each element of one domain. The number of elements per domain, Nel, was set such that there are enough degrees of freedom to ensure the calculation of the mechanical response of the sample with enough precision. We tested different ratios r between the number of elements and the number of domains. r is defined as

,

where d is the dimension of the sample (d = 2 for 2D and d = 3 for 3D geometry). We adapted the value of r and the number of nodal points per element depending on the experiment geometry, on the width of the intrinsic time distribution P(ln( i ) and on the shape chosen for the domains. 2D computations were performed using either square domains or hexagonal ones. 3D simulations were done using cubic domains. Characteristics of the samples in each geometry are summarized in Table 9.1. Dimension Geometry D 2

Bulk

2

Bulk

3

Bulk

2

Confined systems

3

Confined systems

Domain (Shape, Number) Square 41 41 Hexagon 40 41 Cubic 16 16 16 Square 1,024 Cubic ≥512

Characteristics of Nodal Points √4

C2D8 2 DOF*; 8 nodes/element

√24

C2D6 2 DOF*, 6 nodes/element

√8

C3D20 3 DOF*, 20 nodes/element

r = 1 or 2

C2D8 2 DOF*; 8 nodes/element C3D20 3 DOF*, 20 nodes/element

r=2

Table 9.1. Characteristics chosen for 2D and 3D samples for FE calculations: the domain number that sets the size of the sample are chosen above the minimal number of domains required to get a representative elementary volume (REV), the ratio r that gives the number of element per domain, and the nature of nodal points characterized by the number of degree of freedom per nodal point (DOF) and the number of nodal point per element

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

239

We work in the 2D plane strain condition for all bulk systems and 2D plane stress condition for free-standing films. We use periodic boundary conditions, and we solve it in time, using logarithmic time step, in order to cover about 12 decades of time. We use the ZéBuLon FE software (Ryckelynck et al. 2012; Z-set). 9.3. Linear regime in bulk geometry 9.3.1. Stress relaxation Figure 9.2 shows the macroscopic modulus relaxation predicted by our 2D model in a shear strain step for hexagonal domains. Calculations were performed taking s = 4.6 and 0 = 0.1. A similar shear modulus relaxation is obtained with square domains taking the same time distribution function (Masurel et al. 2015). As shown in Figure 9.2, the relaxation is significantly broadened when compared to the Maxwell relaxation of a homogeneous system (s = 0).

Figure 9.2. The shear modulus relaxation predicted by FE 2D model is shown as a function of the time normalized by the center of the intrinsic time distribution. Calculations have been performed using hexagonal domains and applying the following conditions: 0 = 0.1, s = 4.6, GG = 1 GPa, and GR = 1 MPa. The response of a homogenous sample (s = 0) is plotted for comparison. Stress field ( i12) is shown at different stages of the macroscopic relaxation. The corresponding values of t/ 0 are equal to: 10 5, 10 4, 10 3, 0.5, 1, 5, and 50. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

240

Mechanics and Physics of Solids at Micro- and Nano-Scales

9.3.2. Numerical predictions versus experiments in the linear regime We first check whether the model gives a good description of stress relaxation experiments performed in a linear regime during a uniaxial strain step. Figure 9.3 shows the comparison of the modulus relaxation measured on a crosslinked Poly(methyl methacrylate) (PMMA) sample at 373 K and the numerical macroscopic response predicted by the model. In that case, calculations were performed with a width s of 4.4 for the lognormal intrinsic time distribution and an average time 0 equal to 103.6 s. The values of the glassy and rubber Young’s moduli were taken equal, respectively, to 1 GPa and 2 MPa.

Figure 9.3. The macroscopic Young’s modulus relaxation measured on crosslinked PMMA chains (square) and 50% in weight mixture of 50% of crosslinked SBR and PB chains (circle) are plotted versus time. Experimental results are compared to the predictions given by our 3D finite element model (in solid line). The best fit obtained by applying a KWW function is plotted in dashed lines. For PMMA crosslinked chains, FE calculations were performed by considering s = 4.4 and 0 = 4000 s at Tref = 100°C, ER = 2 MPa, and EG = 1 GPa. Parameters of the KWW fit are: = 0.235 and KWW = 3600 s. For the SBR/PB mixture at Tref = −80 °C, FE simulations were calculated taking s = 10.4, 0 = 1 s at T = 193K, ER = 1.9 MPa, and EG = 2.5 GPa. Parameters of the KWW fit are: = 0.127 and KWW = 1.88 s. The inset shows the relation between the width of the intrinsic time distribution function s and the exponent of the KWW function . For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

241

A good description was achieved for various polymer systems, including miscible polymer mixtures that show very broad stress relaxation (Shi et al. 2013). The mechanical response measured at T = 193K on an SBR/PB mixture composed of 50% in weight of each homopolymer is shown in Figure 9.3. In that case, a good description was obtained, taking values for the width s and the center of the time distribution 0 as, respectively, 10.4 and 1 s. The value of s increases with the broadness of the modulus relaxation. Generally, experimental results are fitted to a stretched exponential KWW function exp

[9.5]

where KWW is a characteristic relaxation time and is a parameter ranging between 0 and 1, which is related to the width of the dynamical disorder. The correlation between the width of the intrinsic time distribution s and the exponent involved in the KWW function is plotted in the inset of Figure 9.3. s decreases with increasing However, as shown in Figure 9.3, the agreement with the KWW fit is poorer than the one from the present model. The KWW fit does not describe the full relaxation of the modulus and, in particular, fails significantly at the end of the relaxation. 9.3.3. Role of elastic coupling between domains Our model gives a good description of the macroscopic mechanical response of real polymer systems measured for several experimental conditions in the glass transition regime. We recall that our model describes the whole mechanics of a system made of domains, each one exhibiting a simple relaxation, but with its own relaxation time, all the domains being mechanically coupled. Consequently, the stress and strain fields can provide evidence for the role of the mechanical coupling between domains. In the following, we focus on the stress fields observed during the relaxation. Figure 9.2 presents the stress maps associated with different stages of the macroscopic stress relaxation after shear strain step. At short times, the response of each domain is glassy. The local stresses equal to its glassy

242

Mechanics and Physics of Solids at Micro- and Nano-Scales

value (in dark red). As time increases, some domains that have short intrinsic relaxation times relax and their local stresses reach its rubber value. They appear in sky blue on the maps. The number of relaxed domains increases with time such that lines formed by high-stress domains appear (in redorange). The non-relaxed domains are organized into lines and form a rigid network. They are oriented parallel to the eigenvectors of the strain tensor. Thus, the relaxation time disorder leads to the structuring of the stress field. When the rigid network disappears as a result of additional relaxations of domains belonging to these high-stress lines, there are large and steep decreases of the macroscopic stress. After that stage, rigid domains are embedded in a soft matrix. Similar high-stress lines are observed if we chose either square (Masurel et al. 2015) or hexagonal domains. In order to characterize the effect of the mechanical coupling between all the domains, it is important to focus on the correlation function of stress. As explained by Masurel et al. (2015), the correlation function of stress decays slower than the one expected from the Eshelby estimation and thus exhibits long-range elastic coupling between the domains, reminiscent from percolation. To discuss the effect of elastic coupling, we will define an effective relaxation time that we will compare to the intrinsic relaxation time of the domain. The effective relaxation time of one domain is chosen to be the first time at which the stress undergone by the domain during its relaxation is equal to 1/10 of its initial value, i.e. GGe0. The choice of 1/10 is arbitrary, but the following discussion is general. More precisely, the two effective times and are defined as the first time the stress component in the shear direction (12 directions) reaches the values: 0 ⁄10

[9.6]

and ⁄10 with

0



[9.7]

We can now compare the stress relaxation of one domain in the heterogeneous sample to the one it would have if the domain was alone, without interactions with neighbors with the mean of the effective time. The response of such isolated domains is controlled by its intrinsic relation time – randomly drawn – i. Figures 9.4a and 9.4b present two typical local stress relaxations observed for domains having a short or a long intrinsic relaxation time. Figure 9.4c presents the correlation of the effective relaxation time

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

243

of each domain with its intrinsic relaxation time. At short times, the effective time of the local stress relaxation of the fastest domains is nearly equal to the initial relaxation time. However, the effective time reaches at a value around 1 for large intrinsic time. In other words, the stress relaxation of the slowest domains occurs much faster than expected from their initial time. We deduce that mechanical coupling deeply modifies the stress relaxation of the slow domains. This can be understood as follows. Once a domain is surrounded by low stress domains, it cannot sustain a high level of stress any more. It thus relaxes its stress. Therefore, because the stress is organized in stress paths, as soon as, in a stress path, a domain relaxes its stress, the stress path disappears. As a consequence, the macroscopic stress relaxation is faster – once stress paths become predominant – than the sum of the local stress relaxations. The slowest domains thus have a very weak contribution to stress relaxation because they relax after the disappearance of the stress paths.

Figure 9.4. The local stress relaxation and the corresponding strain are plotted versus time. Calculations were performed with hexagonal domains. The relaxation of the local modulus is shown for comparison. (a) and (b) present the data of one domain having an intrinsic time equal to 0.0034 and 1,200, respectively. The effective time values were determined such that the local stress (modulus) reaches a value equal to the tenth of its glassy value according to equation [9.6]. (c) and (d) present the correlation of eff ( Meff) with the intrinsic time i. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

244

Mechanics and Physics of Solids at Micro- and Nano-Scales

However, the local apparent elastic modulus relaxation – or the ratio between stress and strain – is not quantitatively affected by the elastic coupling as is highly correlated with I,  as shown in Figure 9.4d. The mechanical contribution originating from the relaxation of the other domains thus somehow average: it does not significantly shift the decay of the local elastic modulus compared to the one for isolated domains. In the conclusion of that discussion on the relaxation of bulk samples, we can say that the macroscopic relaxation is governed by the relaxation of stress paths. Indeed, the stress paths cease to sustain the macroscopic strain as soon as one of their domains relaxes. 9.4. Linear regime in confined geometries It is known that, since 25 years, the dynamical and mechanical properties of the glassy polymer have been depending on the geometry of the samples. When a sample is shaped in a suspended thin film, its apparent modulus appears to be smaller than its bulk modulus (Vogt 2018). Similarly, when a sample is confined between two solid surfaces, its apparent modulus appears to be larger than the bulk modulus. We will show that our model accounts for these confinement effects. 9.4.1. Apparent linear viscoelasticity in various geometries First, we have considered a sandwiched thin film sheared between two rigid plates (see Figure 9.5). We have computed its viscoelastic response when one plate moves relatively parallel to the other. The lateral boundary conditions are periodic in strain. We have varied the film thickness from one to eight domains. Figure 9.6 shows the variation of the apparent viscoelastic modulus as a function of frequency in that geometry, using cubic domains. The calculation has been done with GG = 1 GPa, GR = 1 MPa, and s = 4.6. As confinement increases, the relaxation of the modulus of sandwiched thin films is shifted to lower frequencies. A broadening of the modulus frequency dependency is also observed as film thickness decreases. Indeed, the shift toward slower frequency is more pronounced for decreasing value of the modulus.

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

245

Figure 9.5. A schematic representation of the model for confined geometry is presented. Each domain behaves according to a Zener model with its own intrinsic relaxation time, as presented in Figure 9.1. Calculations were performed applying a pure shear strain for sandwich thin films and a uniaxial stretching along the y direction for free-standing films. The confinement is set by the thickness h of the thin film that is the product of the number of domain layer n with the size of each domain . Confinement is varied by changing the number of domain layers. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Figure 9.6. Log–log plot of the viscoelastic shear modulus of a sandwiched thin film as a function of frequency for varying confinement degrees, h. 3D calculations were performed taking s = 4.6, 0 = 1s, GR = 1 MPa, and GG = 1 GPa. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

246

Mechanics and Physics of Solids at Micro- and Nano-Scales

For a one-domain-thick film, each domain is submitted to a simple shear with the same deformation as the macroscopic one. As a result, the viscoelastic modulus for one domain thickness G1 is simply the parallel average of the domains viscoelastic modulus – Voigt approximation – and can thus be written as: [9.8] The real and imaginary parts of G1 are shown in Figure 9.6 in solid lines, confirming the quality of the numerical calculation. The second and third situations we have considered are a one-domainthick suspended film and a one-domain-thick thread (Figure 9.7) in tensile test conditions. In both cases, the relaxation of the modulus is shifted to larger frequencies for large values of the modulus, as compared to one of the bulk systems. Finally, note that the one-domain-thick thread corresponds exactly to the average of the compliance of the domains and thus to the Reuss limit. Therefore, the effect of confinement is to vary the elastic response from Voigt to Reuss limits. Because relaxation times span more than four decades, the evolution of the modulus with the shape of the sample (thread or sandwiched film) is very large. We will now compare these results with experimental results, focusing on polymers confined between solid surfaces.

Figure 9.7. The frequency dependence of the viscoelastic Young’s modulus on a onedomain-thick suspended thin film (blue square) is compared to the bulk one (red circle). 3D calculations were performed by taking s = 4.6, 0 = 1 s, ER = 3 MPa, and EG = 3 GPa. The response of a thread (Reuss limit) is shown in a green solid line. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

247

9.4.2. Comparison of the results of our model with the observation of Tg shift in filled elastomers It has been known for decades that confinement modifies the dynamics of polymer chains. Generally, this influence is reported through measurements of varying glass transition temperature of confined polymer chains compared . Measurements have shown that the glass to its value in the bulk transition of confined chains depends on film thickness and on the confinement geometry (Fryer et al. 2001). For free-standing thin films, a Tg decrease as a function of the film thickness h was reported. By contrast, a Tg increase can be observed for thin films that have strong interactions with a solid substrate (Berriot et al. 2002; Fryer et al. 2001). This dynamical change can be well-described assuming that the temperature of glass transition Tg varies as the inverse of the distance from the particle surface, z, according to the equation: 1

[9.9]

where is a characteristic length of the order of the nanometer, which sets the amplitude of the Tg change as a function of film thickness. The sign is set by the interaction conditions at the polymer/substrate interface (Fryer et al. 2001). It is negative for repulsive or weak interactions or free surfaces and positive for strong and attractive interactions. Such chain dynamic changes occur in filled elastomers in which a part of the polymer chain is confined between the interfaces of neighboring particles. For instance, NMR measurements performed on filled elastomers have shown that there is a fraction of polymer chains whose dynamics can be significantly slowed down as compared to the one measured on the pure matrix (Kaufman et al. 1971; Litvinov and Steeman 1999; Papon et al. 2012). It is also well known that the macroscopic viscoelastic response of an elastomer matrix is significantly changed when solid nanoparticles are embedded. In most experiments performed on filled elastomers, it is reported that the glass transition domain –measured by viscoelasticity – is broadened as compared to the pure elastomer: the low-frequency part of the modulus relaxation is significantly broadened while only very small variations of the macroscopic glass transition temperature are observed (Wang 1998). It reveals the existence of a network of solid particles that are mechanically connected by confined polymers (Heinrich et al. 2002). The addition of fillers to an elastomer matrix results in a macroscopic mechanical response that is

248

Mechanics and Physics of Solids at Micro- and Nano-Scales

qualitatively similar to the one predicted by our model for confined sandwiched thin films, as shown in Figure 9.8.

Figure 9.8. (a) presents the damping factor tan( ) predicted by our 3D model for sandwiched and free-standing thin films. Values of the complex modulus G*(T, 1 Hz) were deduced from data plotted in Figures 9.6 and 9.7. Frequencies are converted into temperature applying a WLF time–temperature law such that log( ) = aT/Tref (C1 = 10 and C2 = 55). (b) shows the data measured on Duradene matrix filled with N234 carbon black particles from Wang (1998). The temperature dependence of tan( ) is mainly controlled by the response of polymer confined between fillers. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

However, additional effects could also be involved in the modifications of the viscoelastic response of polymer chains. For instance, strong polymer–substrate interactions lead to structural modifications of polymer chains in the very vicinity of the surface (conformational changes and free volume fluctuations) (Barrat et al. 2010). Our model provides an estimate of the contribution of dynamical disorder to slowing down of confined polymer chains that are in strong interaction with a solid substrate. As a result, the viscoelastic response expected in a confined geometry can be predicted from its bulk response. The changes in macroscopic properties induced by confinement can then be expressed as a shift of Tg. This frequency-dependent shift can further be compared to the one measured in model-filled systems. We have considered model silica-filled-crosslinked polyethyl acrylate (PEA) elastomers in which there are no covalent bonds between silica particles and polymer chains. In such samples, NMR measurements have shown that the crosslink density of the polymer chains is not modified by the silica particles (Papon et al. 2011). Consequently, the viscoelastic response

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

249

in the bulk geometry of the crosslinked PEA chains can be deduced from rheological measurements performed on the pure crosslinked matrix. NMR analysis has revealed that a fraction of the crosslinked PEA chains moves slower in the filled elastomer than in the pure matrix. We have shown that magnetization relaxation curves can be well-described, assuming a Tg gradient for the PEA polymer chains in filled elastomers (see equation [9.9]). The value of the characteristic length was found to be 0.1 nm. We have thus compared these experimental results to numerical predictions given by our model for sandwiched crosslinked PEA thin films. Figure 9.9a shows the viscoelastic response measured for crosslinked PEA chains submitted to an oscillating strain. A good description is obtained by the model with s = 4.8, GR = 0.3 MPA, and GG = 0.95 GPa. Figure 9.9a shows the response of sandwiched crosslinked PEA thin films, which has been computed by keeping the same intrinsic time distribution function and for different confinement degrees. For a given modulus value, the frequency shift as a function of the confinement can be found. For instance, the frequency at which the macroscopic modulus G’ is equal to GG/10 is shifted by about 1 decade between the bulk PEA sample and the most confined geometry (n = 1). The frequency shift results in a glass temperature shift ∆ → ∞ using the bulk WLF coefficients inferred from rheological curves: C1 = 18 and C2 = 77 K at Tref = Tg = 253 K. We compare the Tg shift measured by NMR (in line) with the ones deduced from our numerical simulations, taking a modulus threshold of GG/10 (in circle), as shown in Figure 9.9b. Similar results are obtained by taking a modulus threshold of GG/100 (in filled square). Using the classical value 5 nm, the model predicts that ∆ varies as the inverse of the film thickness. The 1/h scaling is observed irrespective of the reference value of G’ chosen to characterize the frequency shift of the dynamics in glass transition of the confined system. The amplitude of the apparent Tg shift increases for decreasing modulus values, but this effect is rather small in this case. Thus, we can conclude that, in the case of weak interactions between polymer chains and particles surfaces, a comparison between experimental data and simulations shows that the dynamical heterogeneities contribute up to 80% to the slowing down of the PEA chains close to the particle surfaces.

250

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 9.9. The frequency dependence of the real part of the shear modulus measured on a crosslinked PEA chains is presented in (a). Experiments are compared to the FE prediction for bulk conditions given by our 3D model with: s = 2.1, GG = 0.95 GPa, GR = 0.3 MPa, and 0 = 0.1 s. The mechanical responses of equivalent PEA sandwiched thin films have been computed applying our model. Figure 9.9b compares a log–log plot of the normalized shift in Tg versus h, as obtained by the model with the experimental law deduced from NMR. The thickness of one FE domain was taken equal to 5 nm. For color versions of the figures in this book, see www.iste.co.uk/ionescu/mechatronics.zip

9.4.3. Role of mechanical coupling in confined geometry 9.4.3.1. Confined thin films In order to understand the effect of confinement, we can have a look at the correlation of the stress effective time, as defined in equation [9.6], and the intrinsic time. Figure 9.10 presents the correlation of the effective with the intrinsic time of the domain relaxation time of the local stress for large i. We observe that, for increasing confinement, the plateau of values of intrinsic time tends to disappear. Obviously, for a complete confinement (n = 1), the effective time is equal to the intrinsic time. Thus, the plateau of the effective time, which is ruled by the propagation of stress in paths, disappears for increasing confinement. It is the loss of efficiency of the elastic interactions between domains that are at the origin of the effect of confinement on the average relaxation modulus.

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

251

Figure 9.10. Correlation of the effective stress relaxation times to the intrinsic times for sandwiched thin films. 2D FE calculations were performed using square domains and applying two confinement degrees [h = 1, h = 2, and h = 32 (bulk)]. Values of were determined using equation [9.6] . For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

9.4.3.2. Free-standing thin films The situation is symmetric for free-standing films. Figure 9.11a shows the comparison of the stress relaxation for a freestanding film of one-domain thickness (64 64 1) and the one deduced from a bulk sample (16 16 16 domains). Calculations for the suspended film were made with 2D plane stress conditions on a 64 64 sample. Stress lines clearly appear in the stress maps (see Figure 9.11c). They progressively disappear with time. The effective times of the initially slowest domains are in that case smaller to the ones of the bulk sample. In fact, in suspended thin films, the stress paths must remain in two dimensions, while they can expand in three dimensions in bulk samples, which is at the origin of their smaller lifetimes. As a result, a small number of relaxations are enough for the stress lines to relax and the effective time plateau of the slowest domains (see Figure 9.11b) is smaller for 2D free-standing films.

252

Mechanics and Physics of Solids at Micro- and Nano-Scales

Figure 9.11. The macroscopic relaxation of a one-domain-thick free-standing film and a bulk sample uniaxially stretched are shown as a function of time. Simulations were performed with log( 0) = 1, s = 4.6, EG = 3 GPa, ER = 3 MPa, and = 0.01. Local stress maps are shown at different stages of the macroscopic relaxation. The correlation of eff with the intrinsic times is plotted. The values of eff were determined, applying i11( eff)=Eg* /e. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

9.4.4. Conclusion on the effects of confinement In conclusion, confinement results in a change of the elastic interactions between domains. The stress of slow domains consequently decreases, provided they are surrounded by fast domains. The stress that is organized in 3D paths in bulk systems is modified by confinement. In films confined between solid surfaces, the stress lines only extend between the two solid surfaces; thus, they are shorter than without confinement. They contain less domains, the probability they relax is consequently smaller, which results in a slowing down of the long-time part of the relaxation modulus. By contrast, for free-standing films, lines are confined in 2D, and relaxation is more probable, which results in a faster relaxation. We have shown that at least in confined films, our model accurately describes the mechanical behaviors

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

253

reported in filled elastomers, which are controlled by the polymer confined between the solid particles. We now focus on the effect of nonlinearities. 9.5. Nonlinear mechanics The nonlinear mechanics of the glassy polymer in the glass transition regime have been poorly investigated. Except for a few ones (for instance Nanzai 1993), many studies address nonlinearities either above glass transition – in the rubber state, or below glass transition – in the glassy state. In the rubber state, the origin of mechanical nonlinearities is geometric. The network strands that are responsible for the mechanical properties exhibit a complex response under large deformation due to the complex topology of the polymer network. The response is well-described by sophisticated models that include the sliding of entanglements (Rubinstein and Panyukov 2002). The latter effect becomes significant for deformations close to unity and thus requires the sophisticated nonlinear formalism of mechanics. By contrast, in the glassy state, nonlinearities are governed by local rearrangements of the monomers, often described as hops from one arrangement to the other, the arrangement being called Eyring’s cage. The nonlinearities are significant for strains of a few per cents, and their description does not require the nonlinear formalism. However, the models remain very phenomenological (Boyce et al. 1992; Dreistadt et al. 2009; Hasan and Boyce 1995; Klompen et al. 2005; Mulliken and Boyce 2006; Tervoort et al. 1996). Recent models that include dynamical heterogeneities have been developed (Dequidt et al. 2016; Dequidt et al. 2012; Medvedev and Caruthers 2013). Nevertheless, except in the work of Dequidt et al. (2016), the mechanical coupling between domains is not properly described and, in addition, has not been implemented in finite element code. With our model, we can deal with the onset of these glassy nonlinearities near the glass transition. Unfortunately, a quantitative comparison is possible with only a few experiments (G’Sell and Souahi 1997; Roetling 1965a, 1965b). We will first explain how to simply take into account glassy nonlinearities in our model, and we will limit ourselves to the discussion of the effect of nonlinearities on the elastic coupling between domains.

254

Mechanics and Physics of Solids at Micro- and Nano-Scales

9.5.1. Input of nonlinearities In the literature, the most popular microscopic model for glassy mechanical nonlinearities is Eyring’s model (Ree and Eyring 1955). In that frame, the energy barriers between the two configurations are linearly changed by the local stress. Hops in the stress direction are favored compared to the ones in the opposite direction. In 1D, the stress-dependent relaxation time of a dynamical heterogeneity is given by: [9.10] where i is the intrinsic relaxation time of the cage, v is an activation volume, k is the Boltzmann constant, T is the temperature, and is the local stress. The subscript is used to emphasize the stress dependence of the relaxation time. In order to extend the scalar version of the Eyring model to a 2D or 3D model, we first recall that pressure is known to modify the glass transition temperature (Ferry 1980). A typical value of the derivative of Tg with pressure is around 0.2 K/MPa. Therefore, the relaxation time is modified by pressure, typically a factor of 10 for a pressure change of 20 MPa. That result can be generalized to any stress field, using the Drucker Prager or modified von Mises criterion that is known to describe yielding of a “glassy polymer” (Quinson et al. 1997; Rottler and Robins 2005). Thus, we can write that

where



with



∶

–

[9.11]

is the deviatoric part of the stress tensor and

. Y corresponds, for a given domain, P is the pressure defined as to the yield stress. Y is related to temperature and to the activation volume associated with the configurational rearrangement. That flow criterion accounts both for an acceleration of the dynamics with increasing von Mises stress and for its slowing down with increasing pressure. The shift of relaxation time with pressure is controlled by the ratio /Y. The parameter ranges typically between 0.05 and 0.4 in a great variety of polymeric systems (Ward 1983), and /Y is of the order of 2 MPa.

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

255

The nonlinear version of our model just consists in replacing i in according to equation [9.11]. As for the linear case, the equation [9.2] by spatial correlation of the i field is equal to the size of one domain. However, due to the spatial correlation of the stress field, we expect that the stressdependent relaxation times are correlated. 9.5.2. Results of the model

Figure 9.12. True stress as a function of true strain, for a constant engineering strain rate applied on crosslinked PMMA samples whose linear modulus relaxation is shown in Figure 9.3. Experiments were performed at different strain rates and temperatures T. For each experiment, we deduced the equivalent strain rate equal to at the reference temperature Tref = 373 K, applying the WLF coefficient / / determined in the linear regime (C1 = 15.3 and C2 = 50 at T = 373K). Predictions given by our FE model in its nonlinear version are also shown. Calculations have been performed with the parameter values determined in the linear regime: s = 4.4 and 0 = 4000 s at T = 373K, ER = 2 MPa, and EG = 1 GPa. We assume a value for the critical stress Y equal to 5 MPa for all strain rates. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

We first examine the behavior for a constant strain rate. The responses are shown in Figure 9.12 together with the experimental ones. Experiments were performed at different strain rates ε and temperatures T. For each at the experiment, we deduced the equivalent strain rate equal to ε/a / reference temperature Tref = 373K, applying the WLF coefficient determined in the linear regime (C1 = 15.3 and C2 = 50 at T = 373 K). The chosen values

256

Mechanics and Physics of Solids at Micro- and Nano-Scales

of the parameters 0, s, EG, and ER result from a fit to the linear response, as explained earlier. The only adjusted parameter is Y – and we set to 0.3. We expect that has a minor role on the mechanical response in the case of simple shear or uniaxial loading. However, the effect of pressure may be large in more complex loading situations (Negi and Picu 2018). Y was set in order to adjust the model mechanical response to the experimental one at 4. 105 s 1 for the reference temperature T = 373K. The same value of Y is used to compute the mechanical responses at other strain rate values for T = 373K. The discrepancies between the model and experiments likely originate from the weak variations of the activation volume with temperature. See the work of Long et al. (2018) for a more precise discussion. However, the agreement is surprisingly good, considering nonlinear effects that are described earlier. 9.5.3. Role of elastic coupling in the nonlinear regime

Figure 9.13. Probability density functions of the effective stress relaxation times are deduced from FE 3D simulations performed, applying a shear strain step 0 in linear (triangles) and nonlinear conditions (squares). Values of were determined, / . FE calculations were performed using cubic domains applying 12 with: GG =1 GPa, GR = 1 MPa, 0 = 0.01, s = 4.6, and 0 = 0.04 and for nonlinear conditions Y = 10 MPa. The values of < > are equal to for linear condition and 13.5 for nonlinear condition. Solid and dash dotted lines are guide for the eye. The 10 intrinsic probability density function P(ln( i )) (circles) is shown as a reference. For a color version of this figure, see www.iste.co.uk/ionescu/mechatronics.zip

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

257

In order to discuss the role of mechanical coupling, we have performed nonlinear step strain computations. Figure 9.13 shows the distribution of the effective times for stress relaxation – defined by [9.6] – for both linear and nonlinear relaxations with an amplitude of 0.04. The effective times of the linear response are the ones shown in Figure 9.4c. Their distribution is truncated for large time values as a result of relaxation times of the slow domains smaller than their intrinsic time. By contrast, in the nonlinear case, the effective time distribution is symmetric as the stress paths relax faster in that case, as a result of the shortening of the intrinsic relaxation times under stress – introduced in [9.11]. Consequently, for a constant rate loading, we even observe that the stress field is not correlated any more (Masurel et al. 2017b), whereas it is highly correlated in the linear case. In consequence, nonlinearities mostly result in a faster relaxation of high-stress paths and their disappearance. As a result, the stress field disorder decreases when nonlinearities increase. 9.6. Conclusion In this chapter, we have described a model for the mechanics of amorphous polymer near their glass transition. The model is based on the dynamical heterogeneities of polymer glasses. It accounts for dynamical disorder and thus describes the random strain/stress field that is expected to take place in amorphous polymer under stress. It is implemented as a stochastic finite element code and thus yields – with a limited number of parameters, the glassy and rubber elastic modulus, the center and the width of the time distribution – a very good description of the linear viscoelastic properties of amorphous polymers at the glass transition. In addition, it predicts almost quantitatively the effect of confinement that is observed in filled elastomers and provides a steady basis for the description of weak nonlinearities in amorphous polymers mechanics. Finally, it highlights the crucial role of elastic coupling between the different domains: the macroscopic stress behavior is a complex average of the local ones, and this average depends on the sample geometry. Varying the confinement from free standing to sandwiched thin films, we show that the mechanical response greatly evolves, from the Reuss to the Voigt mechanical limits. Finally, the Eyring-like mechanical nonlinearities tend to decrease the stress disorder.

258

Mechanics and Physics of Solids at Micro- and Nano-Scales

9.7. Appendix We have checked that the contribution of the Rouse dynamics is negligible on the modulus relaxation predicted by our model. For that purpose, we have implemented the Rouse modes, i.e. the modes of relaxation of the network strands between crosslinks. Their contribution is well-known in the literature (Colby and Rubinstein 2003). For a step strain relaxation, the Rouse modes lead to a time decay of the rubber stress contribution. Thus, the rubber modulus GR has to be replaced by the sum of each Rouse contribution, thus

∑

1

⁄

with

and

⁄ , where bk is the Kuhn segment length and Nk is the number of Kuhn segments of the strands. The Maxwell response of an isolated domain under a shear strain step becomes: ⁄

∑

⁄

1

[9.12]

Figure 9.14 shows the comparison of the 3D modulus relaxations predicted with and without taking the Rouse dynamics into account. We choose s = 4.8 and GR = 0.3 MPa, which correspond to a number of Kühn segment in each domain of NK = 13 with bk = 1 nm and T = 300 K. Here we assume that the length scale associated with the distance between the extremities of polymer strands is the same as the size of a domain. The effect of the Rouse dynamics is thus negligible compared to the modulus variations over the whole frequency range we consider. Obviously, for increasing Nk, the discrepancy between the models with and without Rouse modes increases. Figure 9.14 shows the results obtained for an extremely weak network – clearly beyond what can be synthesized in practice – with GR = 0.04 MPa, i.e. Nk = 100 with bk = 1 nm and T = 300 K. The contribution of the Rouse modes, in that case, is significant only in the last decade of the stress relaxation and remains below about 10% of the total stress relaxation. Consequently, as long as elastomers are concerned, the contribution of the Rouse mode is completely negligible. However, for melt polymers, we expect that the model is no more valid about one decade above the rubber elastic plateau.

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

259

Figure 9.14. Time relaxation of the shear modulus predicted by the 3D numerical 3 approach taking s = 4.8, 0 = 1 s, and GG = 10 MPa. Calculations were performed using square domains for two values of GR: GR = 0.3 MPa and GR = 0.04 MPa. Predictions obtained taking the Rouse dynamics into account are compared to the ones neglecting the Rouse dynamics. Each value of GR corresponds to a number of Kuhn segments. Assuming a length of a Kuhn segment b = 1 nm and a temperature T =3 00K, the number of Kuhn segments Nk is equal to 13 for GR = 0.3 MPa. For GR = 0.04 MPa, Nk = 100. For a color version of this figure, see www.iste.co.uk/ ionescu/mechatronics.zip

9.8. References Barrat, J. B., Baschnagel, J., and Lyulin, A. (2010). Molecular dynamics simulations of glassy polymers. Soft Matter, 6, 3430–3446. Berriot, J., Montes, H., Lequeux, F., Long, D. R., and Sotta, P. (2002). Evidence for the shift of the glass transition near the particles in silica-filled elastomers. Macromolecules, 35(26), 9756–9762. Berthier, L. (2011). Trend: dynamic heterogeneity in amorphous materials. Physics, 4, 42. Bishko, G., McLeish, T. C. B., Harlen, O. G., and Larson, R. G. (1997). Theoretical molecular rheology of branched polymers in simple and complex flows: the pompom model. Physical Review Letters, 79, 2352. Boyce, M. C., Montagut, E. L., and Argon, A. S. (1992). The effects of thermomechanical coupling on the cold drawing process of glassy polymers. Polymer Engineering and Science, 38(16), 1073–1085.

260

Mechanics and Physics of Solids at Micro- and Nano-Scales

Casas, F., Alba-Simionesco, C., Montes, H., and Lequeux, F. (2008). Length-scale of glassy polymer plastic flow: a neutron scattering study. Macromolecules, 41(3), 860–865. Colby, R. H. and Rubinstein, M. (2003). Polymer Physics. Oxford University Press Inc., New York. Dequidt, A., Conca, L., Delannoy, J.-Y., Sotta, P., Lequeux, F., and Long, D. (2016). Heterogeneous dynamics and polymer plasticity. Macromolecules, 49(23), 9148–9162. Dequidt, A., Long, D. R., Sotta, P., and Sanseau, O. (2012). Mechanical properties of thin confined polymer films close to the glass transition in the linear regime of deformation: theory and simulations. European Physical Journal E, 35, 61. Ding, Y. and Sokolov, A. (2006). Breakdown of time−temperature superposition principle and universality of chain dynamics in polymers. Macromolecules, 39(9), 3322–3326. Dooling, P. J., Buckley, C. P., and Hinduja, S. (2004). The onset of nonlinear viscoelasticity in multiaxial creep of glassy polymers: a constitutive model and its application to PMMA. Polymer Engineering and Science, 38(6), 892–904. Dreistadt, C., Bonnet, A. S., Chevrier, P., and Lipinski, P. (2009). Experimental study of the polycarbonate behavior during complex loadings and comparison with the Boyce, parks and argon model predictions. Materials & Design, 30(8), 3126–3140. Ediger, M. (2000). Spatially heterogeneous dynamics in supercooled liquids. Annual Review of Physical Chemistry, 51, 99–128. Ferry, J. D. (1980). Viscoelastic Properties of Polymers, 3rd ed. Wiley, New York. Fryer, D. S., Peters, R. D., Kim, E. J., Tomaszewski, J. E., de Pablo, J. J., Nealey, P. F., White, C. C., and Wu, W. L. (2001). Dependence of the glass transition temperature of polymer films on interfacial energy and thickness. Macromolecules, 34(16), 5627–5634. Ge, T., Panyukov, S., and Rubinstein, M. (2016). Self-similar conformations and dynamics in entangled melts and solutions of nonconcatenated ring polymers. Macromolecules, 49(2), 708–722. G’Sell, C. and Souahi, A. (1997). Influence of crosslinking on the plastic behavior of amorphous polymers at large strains. Journal of Engineering Materials and Technology, 119, 223–227. Hasan, O. A. and Boyce, M. C. (1995). A constitutive model for the nonlinear viscoelastic viscoplastic behavior of glassy polymers. Polymer Engineering and Science, 35(4), 331–344. Heinrich, G., Klüppel, M., and Vilgis, T. A. (2002). Reinforcement of elastomers. Current Opinion in Solid State and Materials Science, 6(3), 195–203.

Modeling the Mechanics of Amorphous Polymer in the Glass Transition

261

Kaufman, S., Slichter, W. P., and Davis, D. D. (1971). Nuclear magnetic resonance study of rubber–carbon black interactions. Journal of Polymer Science Part A-2 Polymer Physics, 9(5), 829–839. Klompen, E. T. J. P., Engels, T. A., Govaert, L. E., and Meijer, H. E. H. (2005). Modeling of the postyield response of glassy polymers: influence of thermomechanical history. Macromolecules, 38, 6997–7008. Litvinov, V. M. and Steeman, P. A. M. (1999). EPDM−carbon black interactions and the reinforcement mechanisms, as studied by low-resolution 1H NMR. Macromolecules, 32(25), 8476–8490. Long, D. R., Conca, L., and Sotta, P. (2018). Dynamics in glassy polymers: the Eyring model revisited. Physical Review Materials, 2, 105601. Long, D. R. and Merabia, S. (2002). Heterogeneous dynamics at the glass transition in van der Waals liquids: determination of the characteristic scale. European Physical Journal E, 9, 195–206. Masurel, R. J., Cantournet, S., Dequidt, A., Long, D. R., Montes, H., and Lequeux, F. (2015). Role of dynamical heterogeneities on the viscoelastic spectrum of polymers: a stochastic continum mechanics model. Macromolecules, 48(18), 6690–6702. Masurel, R. J., Gelineau, P., Cantournet, S., Dequidt, A., Long, D. R., Lequeux, F., and Montes, H. (2017a). Role of dynamical heterogeneities on the mechanical response of confined polymer. Physical Review Letters, 118, 047801. Masurel, R. J., Gelineau, P., Lequeux, F., Cantournet, S., and Montes, H. (2017b). Dynamical heterogeneties and mechanical non linearities: modeling the onset of plasticity in polymer in the glass transition. European Physical Journal E, 40(12), 116. Medvedev, G. A. and Caruthers, J. M. (2013). Development of a stochastic constitutive model for prediction of postyield softening in glassy polymers. Journal of Rheology, 57(3), 949–1002. Mulliken, A. D. and Boyce, M. C. (2006). Mechanics of the rate-dependent elastic– plastic deformation of glassy polymers from low to high strain rates. International Journal of Solids and Structures, 43, 1331–1356. Nanzai, Y. (1993). Molecular kinetics of yield deformation and ductile fracture in polymer glasses. Progress in Polymer Science, 18, 437–479. Negi, V. and Picu, R. C. (2018). Elastic-plastic transition in stochastic heterogeneous materails: size effect and triaxiality. Mechanics of Materials, 120, 26–33. Papon, A., Montes, H., Hanafi, M., Lequeux, F., Guy, L., and Saalwächter, K. (2012). Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry. Physical Review Letters, 108, 065702.

262

Mechanics and Physics of Solids at Micro- and Nano-Scales

Papon, A., Saalwächter, K., Schäler, K., Guy, L., Lequeux, F., and Montes, H. (2011). Low-field NMR investigations of nanocomposites: polymer dynamics and network effects. Macromolecules, 44(4), 913–922. Quinson, R., Perez, J., Rink, M., and Pavan, A. (1997). Yield criteria for amorphous glassy polymers. Journal of Materials Science, 32(5), 1371–1379. Ree, T. and Eyring, H. (1955). Theory for non-Newtonian flow I. Solid plastic system. Journal of Applied Physics, 26(7), 793–800. Roetling, J. (1965a). Yield stress behavior of poly(ethyl methacrylate) in the glass transition region. Polymer, 6, 615–619. Roetling, J. (1965b). Yield stress behavior of polymethylmethacrylate. Polymer, 6, 311–317. Rottler, J. and Robbins, M. O. (2005). Macroscopic friction laws and shear yielding of glassy solids. Computer Physics Communications, 169(1–3), 177–182. Rubinstein, M. and Panyukov, S. (2002). Elasticity of polymer networks. Macromolecules, 35(17), 6670–6686. Ryckelynck, D., Vincent, F., and Cantournet, S. (2012). Multidimensional a priori hyper-reduction of mechanical models involving internal variables. Computer Methods in Applied Mechanics and Engineering, 225, 28–43. Shi, P., Schach, R., Munch, E., Montes, H., and Lequeux, F. (2013). Glass transition distribution in miscible polymer blends: from calorimetry to rheology. Macromolecules, 46(9), 3611–3620. Sillescu, H. (1999). Heterogeneity at the glass transition: a review. Journal of NonCrystalline Solids, 243(2–3), 81–108. Tervoort, T. A., Klompen, E. T. J., and Govaert, L. E. (1996). A multi-mode approach to finite, three-dimensional, nonlinear viscoelastic behavior of polymer glasses. Journal of Rheology, 40, 779–797. Tracht, U., Wilhelm, M., Heuer, A., Feng, H., Schmidt-Rohr, K. W. and Spiess, H. (1998). Length scale of dynamic heterogeneities at the glass transition determined by multidimensional nuclear magnetic resonance. Physical Review Letters, 81, 2727. Vogt, B. D. (2018). Mechanical and viscoelastic properties of confined amorphous polymers. Journal of Polymer Science Part B: Polymer Physics, 56, 9–30. Wang, M.-J. (1998). Effect of polymer-filler and filler-filler interactions on dynamic properties of filled vulcanizates. Rubber Chemistry and Technology, 71(3), 520–589. Ward, I. M. (1983). Mechanical Properties of Solid Polymers, 2nd ed. John Wiley, New York. Z-Set. Available at: http:// www.zset-software.com.

List of Authors

Kevin ARDON CNRS/University Paris 13 Nord LSPM Paris France

Hadrien BENSE AMOLF Amsterdam The Netherlands

Sylvie AUBRY Lawrence Livermore National Laboratory California USA

José BICO Physique et Mécanique des Milieux Hétérogènes CNRS UMR7636 Ecole Supérieure de Physique et Chimie Industrielles de Paris Paris Sciences et Lettres Research University Sorbonne Université Université de Paris France

Roberta BAGGIO CNRS/University Paris 13 Nord LSPM Paris France

Sabine CANTOURNET Centre des Matériaux MINES ParisTech Paris France

Aude BELGUISE Sciences et Ingénierie de la Matière Molle Paris France

Yann CHARLES CNRS/University Paris 13 Nord LSPM Paris France

Athanasios ARSENLIS Lawrence Livermore National Laboratory California USA

Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

264

Mechanics and Physics of Solids at Micro- and Nano-Scales

Pierre CRÉMILIEU Lyon Institute of Nanotechnology France Alexandre DANESCU Lyon Institute of Nanotechnology France Monique GASPERINI CNRS/University Paris 13 Nord LSPM Paris France Ioan R. IONESCU CNRS/University Paris 13 Nord LSPM Villetaneuse France Jean-Louis LECLERCQ Lyon Institute of Nanotechnology France François LEQUEUX Sciences et Ingénierie de la Matière Molle Paris France Hélène MONTES Sciences et Ingénierie de la Matière Molle Paris France Vineet NEGI Rensselaer Polytechnic Institute Troy USA

Hung Tuan NGUYEN CNRS/University Paris 13 Nord LSPM Paris France Stefanos PAPANIKOLAOU West Virginia University Morgantown USA Catalin R. PICU Rensselaer Polytechnic Institute Troy USA Prashant K. PUROHIT University of Pennsylvania Philadelphia USA Sylvain QUEYREAU University Paris 13 Nord LSPM Paris France Philippe REGRENY Lyon Institute of Nanotechnology France Benoit ROMAN Physique et Mécanique des Milieux Hétérogènes CNRS UMR7636 Ecole Supérieure de Physique et Chimie Industrielles de Paris Paris Sciences et Lettres Research University Sorbonne Université Université de Paris France

List of Authors

Oguz Umut SALMAN CNRS/University Paris 13 Nord LSPM Paris France  

Michail TZIMAS West Virginia University Morgantown USA

265

Index

3D structures, 190, 191, 202, 209, 210 A, B, C Abaqus, 88, 95, 98, 118 acoustic tensor, 10, 14, 16 adhesion, 143 ALD (atomic layer deposition), 138 avalanches, 26, 29, 34–38 bilayer plates, 216 BKS (Bacon, Kocks, and Scattergood), 62, 77, 78 blood clots, 133, 134 buckling, 185, 194, 199, 202–210 buckypaper, 159, 165, 166, 169, 180 capillary bending length, 189, 192 CNT (carbon nanotube networks), 133 collagen, 157–159, 168, 169 confinement, 233, 244–252, 257 confocal microscope, 136 CPFEM (crystal plasticity finite element method), see crystal plasticity, polycrystalline, 89, 90, 91

crystal plasticity, see CPFEM, 1, 26, 27, 29, 34–36, 40, 48, 49, 89, 91, 92, 95, 118 curvature, 217 D deformation, 3, 5–7, 10, 11, 12, 18 densified, 136, 137 diffusion, 88–91, 94–99, 103, 110, 114, 115, 117, 118 directional bending, 223 dislocation, 88, 89, 91, 92, 94, 97, 103–106, 114, 116, 118 annihilation, 13–15 density, 26, 27, 37, 39, 40, 47, 48 dynamics, 26, 30, 33, 55 nucleation, 1 dissipation inequality, 145 dynamical heterogeneities, 231, 232, 234, 237, 238, 249, 253, 254, 257 E, F elasticity, 185, 187, 192, 194, 203, 209

Mechanics and Physics of Solids at Micro- and Nano-Scales, First Edition. Edited by Ioan R. Ionescu, Sylvain Queyreau, Catalin R. Picu and Oguz Umut Salman. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

268

Mechanics and Physics of Solids at Micro- and Nano-Scales

elasto-capillary, 187, 189, 202, 209, 210 elastocapillarity, 161 electroactive polymer/dielectric elastomer, 186, 192–196, 200, 209 electrostatic energy, 192–194, 196, 197 Eshelby force, 68–70, 79, 80 inclusions, 55 stress, 57, 63, 68, 80 fiber networks, 133 finite-element method, 237 finite volume, 25, 26, 28–30, 34, 36, 48 Föppl–Von Kármán, 207 G, H, I geodesic curvature, 219 objects, 222 GL(2, Z), 6–8, 11 glass transition, 231, 232, 236, 241, 247, 249, 253, 254, 257 He bubbles, 56, 57, 63, 64, 69, 73, 79 hysteresis, 136 iron, 56, 70, 71, 73, 75, 76, 78, 79 irradiation, 56, 61, 72 K, L, M kinetic law, 144 Landau theory, 1 lattice square, 7–9, 13, 14 triangular, 7, 9 material science, 25, 48 membrane strain, 217 metal aluminum, 88, 90, 91 copper, 90

iron, 87–89, 91, 92, 96, 97 nickel, 88, 90, 91, 103 polycrystal, 89–92, 96–99, 101–104, 113–118 single crystal, 88, 91, 96, 106, 107, 110, 112, 113, 116, 118 steel, 87, 89–91 tungsten, 88, 89 ML (machine learning), 27, 38–45, 48, 49 multiscale, 26, 48, 88, 91 N, O, P netwoks cellular, 169–171 Delaunay, 175–177, 179 sub-isostatic, 160, 171–174 Voronoi, 171, 178 nucleation, 140, 145 obstacles, 56–59, 61–63, 76, 77 Orowan, 59, 61, 62, 74–78 phase boundary, 137, 144 transitions, 133 photonic crystals tubes, 224 Plateau triangles, 179 polycrystalline, see CPFEM, 89, 97, 98, 100–104, 113, 114 polymer glasses, 257 poro-viscoelasticity, 145 power-law stiffening, 176 pre-strain, 215 pre-stress, 215 precipitation, 55–63, 65, 70, 72, 74, 75, 77–79 pressure, 56, 62, 71, 72, 76, 77, 79 R, S rarefied, 136, 137 rate dependence/effect, 25–27, 29–32, 34, 36 saddle points, 18

Index

simulation, 26–30, 36, 37 size dependence/effect, 25–27, 31, 33, 36, 37, 47, 48 slender structure, 185, 187–189, 191, 192, 210 slip direction, 28 plane, 27–31, 34, 35 system, 27, 28, 31, 32, 40 statistics, 31–34, 36, 38, 39, 48 strain correlation, 39–46, 48 strengthening, 55–63, 70, 72, 74, 77–79 surface tension, 185–194, 197, 199–202, 204, 205, 209, 210

269

T, U, V Timoshenko beams, 171 transport, 89–95, 97, 99, 100–106, 108, 110, 111, 113–118 trapping, 88–91, 93–95, 97, 99–103, 108, 113, 117, 118 tungsten, 56, 69–75, 78, 79 ultra-thin films, 216 uniaxial compression, 25–27, 29, 30, 36, 37, 45, 48 viscoelasticity, 232, 244–249, 257

Other titles from

in Civil Engineering and Geomechanics

2019 KOTRONIS Panagiotis Risk Evaluation and Climate Change Adaptation of Civil Engineering Infrastructures and Buildings: Project RI-ADAPTCLIM LAMBERT David Edward, PASILIAO Crystal L., ERZAR Benjamin, REVILBAUDARD Benoit, CAZACU Oana Dynamic Damage and Fragmentation PERROT Arnaud 3D Printing of Concrete: State of the Art and Challenges of the Digital Construction Revolution SALENÇON Jean Viscoelastic Modeling for Structural Analysis ZEMBRI-MARY Geneviève Project Risks: Actions Around Uncertainty in Urban Planning and Infrastructure Development

2018 FROSSARD Etienne Granular Geomaterials Dissipative Mechanics: Theory and Applications in Civil Engineering KHALFALLAH Salah Structural Analysis 1: Statically Determinate Structures Structural Analysis 2: Statically Indeterminate Structures VERBRUGGE Jean-Claude, SCHROEDER Christian Geotechnical Correlations for Soils and Rocks

2017 LAUZIN Xavier Civil Engineering Structures According to the Eurocodes PUECH Alain, GARNIER Jacques Design of Piles Under Cyclic Loading: SOLCYP Recommendations SELLIER Alain, GRIMAL Étienne, MULTON Stéphane, BOURDAROT Éric Swelling Concrete in Dams and Hydraulic Structures: DSC 2017

2016 BARRE Francis, BISCH Philippe, CHAUVEL Danièle et al. Control of Cracking in Reinforced Concrete Structures: Research Project CEOS.fr PIJAUDIER-CABOT Gilles, LA BORDERIE Christian, REESS Thierry, CHEN Wen, MAUREL Olivier, REY-BETHBEDER Franck, DE FERRON Antoine Electrohydraulic Fracturing of Rocks TORRENTI Jean-Michel, LA TORRE Francesca Materials and Infrastructures 1 Materials and Infrastructures 2

2015 AÏT-MOKHTAR Abdelkarim, MILLET Olivier Structure Design and Degradation Mechanisms in Coastal Environments

MONNET Jacques In Situ Tests in Geotechnical Engineering

2014 DAÏAN Jean-François Equilibrium and Transfer in Porous Media – 3-volume series Equilibrium States – Volume 1 Transfer Laws – Volume 2 Applications, Isothermal Transport, Coupled Transfers – Volume 3

2013 AMZIANE Sofiane, ARNAUD Laurent Bio-aggregate-based Building Materials: Applications to Hemp Concretes BONELLI Stéphane Erosion in Geomechanics Applied to Dams and Levees CASANDJIAN Charles, CHALLAMEL Noël, LANOS Christophe, HELLESLAND Jostein Reinforced Concrete Beams, Columns and Frames: Mechanics and Design GUÉGUEN Philippe Seismic Vulnerability of Structures HELLESLAND Jostein, CHALLAMEL Noël, CASANDJIAN Charles, LANOS Christophe Reinforced Concrete Beams, Columns and Frames: Section and Slender Member Analysis LALOUI Lyesse, DI DONNA Alice Energy Geostructures: Innovation in Underground Engineering LEGCHENKO Anatoly Magnetic Resonance Imaging for Groundwater

2012 BONELLI Stéphane Erosion of Geomaterials

JACOB Bernard et al. ICWIM6  Proceedings of the International Conference on Weigh-In-Motion OLLIVIER Jean-Pierre, TORRENTI Jean-Marc, CARCASSES Myriam Physical Properties of Concrete and Concrete Constituents PIJAUDIER-CABOT Gilles, PEREIRA Jean-Michel Geomechanics in CO2 Storage Facilities

2011 BAROTH Julien, BREYSSE Denys, SCHOEFS Franck Construction Reliability: Safety, Variability and Sustainability CREMONA Christian Structural Performance: Probability-based Assessment HICHER Pierre-Yves Multiscales Geomechanics: From Soil to Engineering Projects IONESCU Ioan R. et al. Plasticity of Crystalline Materials: from Dislocations to Continuum LOUKILI Ahmed Self Compacting Concrete MOUTON Yves Organic Materials for Sustainable Construction NICOT François, LAMBERT Stéphane Rockfall Engineering PENSÉ-LHÉRITIER Anne-Marie Formulation PIJAUDIER-CABOT Gilles, DUFOUR Frédéric Damage Mechanics of Cementitious Materials and Structures RADJAI Farhang, DUBOIS Frédéric Discrete-element Modeling of Granular Materials RESPLENDINO Jacques, TOUTLEMONDE François Designing and Building with UHPFRC

2010 ALSHIBLI A. Khalid Advances in Computed Tomography for Geomechanics BUZAUD Eric, IONESCU Ioan R., VOYIADJIS Georges Materials under Extreme Loadings / Application to Penetration and Impact LALOUI Lyesse Mechanics of Unsaturated Geomechanics NOVA Roberto Soil Mechanics SCHREFLER Bernard, DELAGE Pierre Environmental Geomechanics TORRENTI Jean-Michel, REYNOUARD Jean-Marie, PIJAUDIER-CABOT Gilles Mechanical Behavior of Concrete

2009 AURIAULT Jean-Louis, BOUTIN Claude, GEINDREAU Christian Homogenization of Coupled Phenomena in Heterogenous Media CAMBOU Bernard, JEAN Michel, RADJAI Fahrang Micromechanics of Granular Materials MAZARS Jacky, MILLARD Alain Dynamic Behavior of Concrete and Seismic Engineering NICOT François, WAN Richard Micromechanics of Failure in Granular Geomechanics

2008 BETBEDER-MATIBET Jacques Seismic Engineering CAZACU Oana Multiscale Modeling of Heterogenous Materials

HICHER Pierre-Yves, SHAO Jian-Fu Soil and Rock Elastoplasticity JACOB Bernard et al. HVTT 10 JACOB Bernard et al. ICWIM 5 SHAO Jian-Fu, BURLION Nicolas GeoProc2008

2006 BALAGEAS Daniel, FRITZEN Claus-Peter, GÜEMES Alfredo Structural Health Monitoring DESRUES Jacques et al. Advances in X-ray Tomography for Geomaterials FSTT Microtunneling and Horizontal Drilling MOUTON Yves Organic Materials in Civil Engineering

2005 PIJAUDIER-CABOT Gilles, GÉRARD Bruno, ACKER Paul Creep Shrinkage and Durability of Concrete and Concrete Structures CONCREEP – 7

WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA.